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Bonferroni correction
In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem.
Background
The method is named for its use of the Bonferroni inequalities.[1] An extension of the method to confidence intervals was proposed by Olive Jean Dunn.[2]
Statistical hypothesis testing is based on rejecting the null hypothesis if the likelihood of the observed data under the null hypotheses is low. If multiple hypotheses are tested, the probability of observing a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., making a Type I error) increases.[3]
The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of $\alpha /m$, where $\alpha $ is the desired overall alpha level and $m$ is the number of hypotheses.[4] For example, if a trial is testing $m=20$ hypotheses with a desired $\alpha =0.05$, then the Bonferroni correction would test each individual hypothesis at $\alpha =0.05/20=0.0025$. Likewise, when constructing multiple confidence intervals the same phenomenon appears.
Definition
Let $H_{1},\ldots ,H_{m}$ be a family of hypotheses and $p_{1},\ldots ,p_{m}$ their corresponding p-values. Let $m$ be the total number of null hypotheses, and let $m_{0}$ be the number of true null hypotheses (which is presumably unknown to the researcher). The family-wise error rate (FWER) is the probability of rejecting at least one true $H_{i}$, that is, of making at least one type I error. The Bonferroni correction rejects the null hypothesis for each $p_{i}\leq {\frac {\alpha }{m}}$, thereby controlling the FWER at $\leq \alpha $. Proof of this control follows from Boole's inequality, as follows:
${\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}=m_{0}{\frac {\alpha }{m}}\leq \alpha .$
This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.[5]
Extensions
Generalization
Rather than testing each hypothesis at the $\alpha /m$ level, the hypotheses may be tested at any other combination of levels that add up to $\alpha $, provided that the level of each test is decided before looking at the data.[6] For example, for two hypothesis tests, an overall $\alpha $ of 0.05 could be maintained by conducting one test at 0.04 and the other at 0.01.
Confidence intervals
The procedure proposed by Dunn[2] can be used to adjust confidence intervals. If one establishes $m$ confidence intervals, and wishes to have an overall confidence level of $1-\alpha $, each individual confidence interval can be adjusted to the level of $1-{\frac {\alpha }{m}}$.[2]
Continuous problems
When searching for a signal in a continuous parameter space there can also be a problem of multiple comparisons, or look-elsewhere effect. For example, a physicist might be looking to discover a particle of unknown mass by considering a large range of masses; this was the case during the Nobel Prize winning detection of the Higgs boson. In such cases, one can apply a continuous generalization of the Bonferroni correction by employing Bayesian logic to relate the effective number of trials, $m$, to the prior-to-posterior volume ratio.[7]
Alternatives
Main article: Family-wise error rate § Controlling procedures
There are alternative ways to control the family-wise error rate. For example, the Holm–Bonferroni method and the Šidák correction are universally more powerful procedures than the Bonferroni correction, meaning that they are always at least as powerful. Unlike the Bonferroni procedure, these methods do not control the expected number of Type I errors per family (the per-family Type I error rate).[8]
Criticism
With respect to FWER control, the Bonferroni correction can be conservative if there are a large number of tests and/or the test statistics are positively correlated.[9]
The correction comes at the cost of increasing the probability of producing false negatives, i.e., reducing statistical power.[10][9] There is not a definitive consensus on how to define a family in all cases, and adjusted test results may vary depending on the number of tests included in the family of hypotheses. Such criticisms apply to FWER control in general, and are not specific to the Bonferroni correction.
References
1. Bonferroni, C. E., Teoria statistica delle classi e calcolo delle probabilità, Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze 1936
2. Dunn, Olive Jean (1961). "Multiple Comparisons Among Means" (PDF). Journal of the American Statistical Association. 56 (293): 52–64. CiteSeerX 10.1.1.309.1277. doi:10.1080/01621459.1961.10482090.
3. Mittelhammer, Ron C.; Judge, George G.; Miller, Douglas J. (2000). Econometric Foundations. Cambridge University Press. pp. 73–74. ISBN 978-0-521-62394-0.
4. Miller, Rupert G. (1966). Simultaneous Statistical Inference. Springer. ISBN 9781461381228.
5. Goeman, Jelle J.; Solari, Aldo (2014). "Multiple Hypothesis Testing in Genomics". Statistics in Medicine. 33 (11): 1946–1978. doi:10.1002/sim.6082. PMID 24399688. S2CID 22086583.
6. Neuwald, AF; Green, P (1994). "Detecting patterns in protein sequences". J. Mol. Biol. 239 (5): 698–712. doi:10.1006/jmbi.1994.1407. PMID 8014990.
7. Bayer, Adrian E.; Seljak, Uroš (2020). "The look-elsewhere effect from a unified Bayesian and frequentist perspective". Journal of Cosmology and Astroparticle Physics. 2020 (10): 009. arXiv:2007.13821. doi:10.1088/1475-7516/2020/10/009. S2CID 220830693.
8. Frane, Andrew (2015). "Are per-family Type I error rates relevant in social and behavioral science?". Journal of Modern Applied Statistical Methods. 14 (1): 12–23. doi:10.22237/jmasm/1430453040.
9. Moran, Matthew (2003). "Arguments for rejecting the sequential Bonferroni in ecological studies". Oikos. 100 (2): 403–405. doi:10.1034/j.1600-0706.2003.12010.x.
10. Nakagawa, Shinichi (2004). "A farewell to Bonferroni: the problems of low statistical power and publication bias". Behavioral Ecology. 15 (6): 1044–1045. doi:10.1093/beheco/arh107.
External links
• Bonferroni, Sidak online calculator
| Wikipedia |
Equilateral polygon
In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting.
Examples
All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square).
Convex equilateral pentagon
Concave equilateral pentagon
A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the shape of the pentagon.
A tangential polygon (one that has an incircle tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular.[1]
Measurement
Viviani's theorem generalizes to equilateral polygons:[2] The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point.
The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a, there exists a principal diagonal d1 such that[3]
${\frac {d_{1}}{a}}\leq 2$
and a principal diagonal d2 such that
${\frac {d_{2}}{a}}>{\sqrt {3}}$.
Optimality
Main article: Reinhardt polygon
When an equilateral polygon is inscribed in a Reuleaux polygon, it forms a Reinhardt polygon. Among all convex polygons with the same number of sides, these polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.[4]
References
1. De Villiers, Michael (March 2011), "Equi-angled cyclic and equilateral circumscribed polygons" (PDF), Mathematical Gazette, 95: 102–107, doi:10.1017/S0025557200002461.
2. De Villiers, Michael (2012), "An illustration of the explanatory and discovery functions of proof", Leonardo, 33 (3): 1–8, doi:10.4102/pythagoras.v33i3.193, explaining (proving) Viviani's theorem for an equilateral triangle by determining the area of the three triangles it is divided up into, and noticing the 'common factor' of the equal sides of these triangles as bases, may allow one to immediately see that the result generalises to any equilateral polygon.
3. Inequalities proposed in “Crux Mathematicorum”, , p.184,#286.3.
4. Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic", Geometriae Dedicata, 198: 1–18, arXiv:1405.5233, doi:10.1007/s10711-018-0326-5, MR 3933447, S2CID 119629098
External links
• Media related to Equilateral polygons at Wikimedia Commons
• Equilateral triangle With interactive animation
• A Property of Equiangular Polygons: What Is It About? a discussion of Viviani's theorem at Cut-the-knot.
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
| Wikipedia |
Triangle-free graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.
By Turán's theorem, the n-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible.
Triangle finding problem
The triangle finding or triangle detection problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph.
It is possible to test whether a graph with $m$ edges is triangle-free in time ${\tilde {O}}{\bigl (}m^{2\omega /(\omega +1)}{\bigr )}$ where the ${\tilde {O}}$ hides sub-polynomial factors and $\omega $ is the exponent of fast matrix multiplication.[1] As of 2023 it is known that $\omega <2.372$ from which it follows that triangle detection can be solved in time $O(m^{1.407})$. Another approach is to find the trace of A3, where A is the adjacency matrix of the graph. The trace is zero if and only if the graph is triangle-free. For dense graphs, it is more efficient to use this simple algorithm which again relies on matrix multiplication, since it gets the time complexity down to $O(n^{\omega })$, where $n$ is the number of vertices.
Even if matrix multiplication algorithms with time $O(n^{2})$ were discovered, the best time bounds that could be hoped for from these approaches are $O(m^{4/3})$ or $O(n^{2})$. In fine-grained complexity, the sparse triangle hypothesis is an unproven computational hardness assumption asserting that no time bound of the form $O(m^{4/3-\delta })$ is possible, for any $\delta >0$, regardness of what algorithmic techniques are used. It, and the corresponding dense triangle hypothesis that no time bound of the form $O(n^{2-\delta })$ is possible, imply lower bounds for several other computational problems in combinatorial optimization and computational geometry.[2]
As Imrich, Klavžar & Mulder (1999) showed, triangle-free graph recognition is equivalent in complexity to median graph recognition; however, the current best algorithms for median graph recognition use triangle detection as a subroutine rather than vice versa.
The decision tree complexity or query complexity of the problem, where the queries are to an oracle which stores the adjacency matrix of a graph, is Θ(n2). However, for quantum algorithms, the best known lower bound is Ω(n), but the best known algorithm is O(n5/4).[3]
Independence number and Ramsey theory
An independent set of $\lfloor {\sqrt {n}}\rfloor $ vertices (where $\lfloor \cdot \rfloor $ is the floor function) in an n-vertex triangle-free graph is easy to find: either there is a vertex with at least $\lfloor {\sqrt {n}}\rfloor $ neighbors (in which case those neighbors are an independent set) or all vertices have strictly less than $\lfloor {\sqrt {n}}\rfloor $ neighbors (in which case any maximal independent set must have at least $\lfloor {\sqrt {n}}\rfloor $ vertices).[4] This bound can be tightened slightly: in every triangle-free graph there exists an independent set of $\Omega ({\sqrt {n\log n}})$ vertices, and in some triangle-free graphs every independent set has $O({\sqrt {n\log n}})$ vertices.[5] One way to generate triangle-free graphs in which all independent sets are small is the triangle-free process[6] in which one generates a maximal triangle-free graph by repeatedly adding randomly chosen edges that do not complete a triangle. With high probability, this process produces a graph with independence number $O({\sqrt {n\log n}})$. It is also possible to find regular graphs with the same properties.[7]
These results may also be interpreted as giving asymptotic bounds on the Ramsey numbers R(3,t) of the form $\Theta ({\tfrac {t^{2}}{\log t}})$: if the edges of a complete graph on $\Omega ({\tfrac {t^{2}}{\log t}})$ vertices are colored red and blue, then either the red graph contains a triangle or, if it is triangle-free, then it must have an independent set of size t corresponding to a clique of the same size in the blue graph.
Coloring triangle-free graphs
Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored.[8] However, nonplanar triangle-free graphs may require many more than three colors.
The first construction of triangle free graphs with arbitrarily high chromatic number is due to Tutte (writing as Blanche Descartes[9]). This construction started from the graph with a single vertex say $G_{1}$ and inductively constructed $G_{k+1}$ from $G_{k}$ as follows: let $G_{k}$ have $n$ vertices, then take a set $Y$ of $k(n-1)+1$ vertices and for each subset $X$ of $Y$ of size $n$ add a disjoint copy of $G_{k}$ and join it to $X$ with a matching. From the pigeonhole principle it follows inductively that $G_{k+1}$ is not $k$ colourable, since at least one of the sets $X$ must be coloured monochromatically if we are only allowed to use k colours. Mycielski (1955) defined a construction, now called the Mycielskian, for forming a new triangle-free graph from another triangle-free graph. If a graph has chromatic number k, its Mycielskian has chromatic number k + 1, so this construction may be used to show that arbitrarily large numbers of colors may be needed to color nonplanar triangle-free graphs. In particular the Grötzsch graph, an 11-vertex graph formed by repeated application of Mycielski's construction, is a triangle-free graph that cannot be colored with fewer than four colors, and is the smallest graph with this property.[10] Gimbel & Thomassen (2000) and Nilli (2000) showed that the number of colors needed to color any m-edge triangle-free graph is
$O\left({\frac {m^{1/3}}{(\log m)^{2/3}}}\right)$
and that there exist triangle-free graphs that have chromatic numbers proportional to this bound.
There have also been several results relating coloring to minimum degree in triangle-free graphs. Andrásfai, Erdős & Sós (1974) proved that any n-vertex triangle-free graph in which each vertex has more than 2n/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2n/5 neighbors per vertex. Motivated by this result, Erdős & Simonovits (1973) conjectured that any n-vertex triangle-free graph in which each vertex has at least n/3 neighbors can be colored with only three colors; however, Häggkvist (1981) disproved this conjecture by finding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent set of a carefully chosen size. Jin (1995) showed that any n-vertex triangle-free graph in which each vertex has more than 10n/29 neighbors must be 3-colorable; this is the best possible result of this type, because Häggkvist's graph requires four colors and has exactly 10n/29 neighbors per vertex. Finally, Brandt & Thomassé (2006) proved that any n-vertex triangle-free graph in which each vertex has more than n/3 neighbors must be 4-colorable. Additional results of this type are not possible, as Hajnal[11] found examples of triangle-free graphs with arbitrarily large chromatic number and minimum degree (1/3 − ε)n for any ε > 0.
See also
• Andrásfai graph, a family of triangle-free circulant graphs with diameter two
• Henson graph, an infinite triangle-free graph that contains all finite triangle-free graphs as induced subgraphs
• Shift graph, a family of triangle-free graphs with arbitrarily high chromatic number
• The Kneser graph $KG_{3k-1,k}$ is triangle free and has chromatic number $k+1$
• Monochromatic triangle problem, the problem of partitioning the edges of a given graph into two triangle-free graphs
• Ruzsa–Szemerédi problem, on graphs in which every edge belongs to exactly one triangle
References
Notes
1. Alon, Yuster & Zwick (1994).
2. Abboud et al. (2022); Chan (2023); Jin & Xu (2023)
3. Le Gall (2014), improving previous algorithms by Lee, Magniez & Santha (2013) and Belovs (2012).
4. Boppana & Halldórsson (1992) p. 184, based on an idea from an earlier coloring approximation algorithm of Avi Wigderson.
5. Kim (1995).
6. Erdős, Suen & Winkler (1995); Bohman (2009).
7. Alon, Ben-Shimon & Krivelevich (2010).
8. Grötzsch (1959); Thomassen (1994)).
9. Descartes (1947); Descartes (1954)
10. Chvátal (1974).
11. see Erdős & Simonovits (1973).
Sources
• Abboud, Amir; Bringmann, Karl; Khoury, Seri; Zamir, Or (2022), "Hardness of approximation in P via short cycle removal: cycle detection, distance oracles, and beyond", in Leonardi, Stefano; Gupta, Anupam (eds.), STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, {ACM}, pp. 1487–1500, arXiv:2204.10465, doi:10.1145/3519935.3520066
• Alon, Noga; Ben-Shimon, Sonny; Krivelevich, Michael (2010), "A note on regular Ramsey graphs", Journal of Graph Theory, 64 (3): 244–249, arXiv:0812.2386, doi:10.1002/jgt.20453, MR 2674496, S2CID 1784886.
• Alon, N.; Yuster, R.; Zwick, U. (1994), "Finding and counting given length cycles", Proceedings of the 2nd European Symposium on Algorithms, Utrecht, The Netherlands, pp. 354–364.
• Andrásfai, B.; Erdős, P.; Sós, V. T. (1974), "On the connection between chromatic number, maximal clique and minimal degree of a graph" (PDF), Discrete Mathematics, 8 (3): 205–218, doi:10.1016/0012-365X(74)90133-2.
• Belovs, Aleksandrs (2012), "Span programs for functions with constant-sized 1-certificates", Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing (STOC '12), New York, NY, USA: ACM, pp. 77–84, arXiv:1105.4024, doi:10.1145/2213977.2213985, ISBN 978-1-4503-1245-5, S2CID 18771464.
• Bohman, Tom (2009), "The triangle-free process", Advances in Mathematics, 221 (5): 1653–1677, arXiv:0806.4375, doi:10.1016/j.aim.2009.02.018, MR 2522430, S2CID 17701040.
• Boppana, Ravi; Halldórsson, Magnús M. (1992), "Approximating maximum independent sets by excluding subgraphs", BIT, 32 (2): 180–196, doi:10.1007/BF01994876, MR 1172185, S2CID 123335474.
• Brandt, S.; Thomassé, S. (2006), Dense triangle-free graphs are four-colorable: a solution to the Erdős–Simonovits problem (PDF).
• Chan, Timothy M. (2023), "Finding triangles and other small subgraphs in geometric intersection graphs", in Bansal, Nikhil; Nagarajan, Viswanath (eds.), Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, {SIAM}, pp. 1777–1805, arXiv:2211.05345, doi:10.1137/1.9781611977554.ch68
• Chiba, N.; Nishizeki, T. (1985), "Arboricity and subgraph listing algorithms", SIAM Journal on Computing, 14 (1): 210–223, doi:10.1137/0214017, S2CID 207051803.
• Descartes, Blanche (April 1947), "A three colour problem", Eureka, 21.
• Descartes, Blanche (1954), "Solution to Advanced Problem no. 4526", Amer. Math. Monthly, 61: 352.
• Chvátal, Vašek (1974), "The minimality of the Mycielski graph", Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973), Lecture Notes in Mathematics, vol. 406, Springer-Verlag, pp. 243–246.
• Erdős, P.; Simonovits, M. (1973), "On a valence problem in extremal graph theory", Discrete Mathematics, 5 (4): 323–334, doi:10.1016/0012-365X(73)90126-X.
• Erdős, P.; Suen, S.; Winkler, P. (1995), "On the size of a random maximal graph", Random Structures and Algorithms, 6 (2–3): 309–318, doi:10.1002/rsa.3240060217.
• Gimbel, John; Thomassen, Carsten (2000), "Coloring triangle-free graphs with fixed size", Discrete Mathematics, 219 (1–3): 275–277, doi:10.1016/S0012-365X(00)00087-X.
• Grötzsch, H. (1959), "Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel", Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe, 8: 109–120.
• Häggkvist, R. (1981), "Odd cycles of specified length in nonbipartite graphs", Graph Theory (Cambridge, 1981), vol. 62, pp. 89–99, doi:10.1016/S0304-0208(08)73552-7.
• Imrich, Wilfried; Klavžar, Sandi; Mulder, Henry Martyn (1999), "Median graphs and triangle-free graphs", SIAM Journal on Discrete Mathematics, 12 (1): 111–118, doi:10.1137/S0895480197323494, MR 1666073, S2CID 14364050.
• Itai, A.; Rodeh, M. (1978), "Finding a minimum circuit in a graph", SIAM Journal on Computing, 7 (4): 413–423, doi:10.1137/0207033.
• Jin, Ce; Xu, Yinzhan (2023), "Removing additive structure in 3SUM-based reductions", in Saha, Barna; Servedio, Rocco A. (eds.), Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, {ACM}, pp. 405–418, arXiv:2211.07048, doi:10.1145/3564246.3585157
• Jin, G. (1995), "Triangle-free four-chromatic graphs", Discrete Mathematics, 145 (1–3): 151–170, doi:10.1016/0012-365X(94)00063-O.
• Kim, J. H. (1995), "The Ramsey number $R(3,t)$ has order of magnitude ${\tfrac {t^{2}}{\log t}}$", Random Structures and Algorithms, 7 (3): 173–207, doi:10.1002/rsa.3240070302, S2CID 16658980.
• Le Gall, François (October 2014), "Improved quantum algorithm for triangle finding via combinatorial arguments", Proceedings of the 55th Annual Symposium on Foundations of Computer Science (FOCS 2014), IEEE, pp. 216–225, arXiv:1407.0085, doi:10.1109/focs.2014.31, ISBN 978-1-4799-6517-5, S2CID 5760574.
• Lee, Troy; Magniez, Frédéric; Santha, Miklos (2013), "Improved quantum query algorithms for triangle finding and associativity testing", Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), New Orleans, Louisiana, pp. 1486–1502, ISBN 978-1-611972-51-1{{citation}}: CS1 maint: location missing publisher (link).
• Mycielski, J. (1955), "Sur le coloriage des graphes", Colloq. Math., 3 (2): 161–162, doi:10.4064/cm-3-2-161-162.
• Nilli, A. (2000), "Triangle-free graphs with large chromatic numbers", Discrete Mathematics, 211 (1–3): 261–262, doi:10.1016/S0012-365X(99)00109-0.
• Shearer, J. B. (1983), "Note on the independence number of triangle-free graphs", Discrete Mathematics, 46 (1): 83–87, doi:10.1016/0012-365X(83)90273-X.
• Thomassen, C. (1994), "Grötzsch's 3-color theorem", Journal of Combinatorial Theory, Series B, 62 (2): 268–279, doi:10.1006/jctb.1994.1069.
External links
• "Graphclass: triangle-free", Information System on Graph Classes and their Inclusions
| Wikipedia |
Cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation
$F(x,y,z)=0$
"Cubic curve" redirects here. For information on polynomial functions of degree 3, see Cubic function.
applied to homogeneous coordinates $(x:y:z)$ for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation. Here F is a non-zero linear combination of the third-degree monomials
$x^{3},y^{3},z^{3},x^{2}y,x^{2}z,y^{2}x,y^{2}z,z^{2}x,z^{2}y,xyz$
These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem.
A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve. The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points.
The real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two 'ovals'. One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, two points.
A non-singular plane cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic. This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form. There are many cubic curves that have no such point, for example when K is the rational number field.
The singular points of an irreducible plane cubic curve are quite limited: one double point, or one cusp. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or a tacnode (if a conic and a line), or up to three double points or a single triple point (concurrent lines) if three lines.
Cubic curves in the plane of a triangle
Suppose that △ABC is a triangle with sidelengths $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Relative to △ABC, many named cubics pass through well-known points. Examples shown below use two kinds of homogeneous coordinates: trilinear and barycentric.
To convert from trilinear to barycentric in a cubic equation, substitute as follows:
$x\to bcx,\quad y\to cay,\quad z\to abz;$
to convert from barycentric to trilinear, use
$x\to ax,\quad y\to by,\quad z\to cz.$
Many equations for cubics have the form
$f(a,b,c,x,y,z)+f(b,c,a,y,z,x)+f(c,a,b,z,x,y)=0.$
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:
$\sum _{\text{cyclic}}f(x,y,z,a,b,c)=0$.
The cubics listed below can be defined in terms of the isogonal conjugate, denoted by X*, of a point X not on a sideline of △ABC. A construction of X* follows. Let LA be the reflection of line XA about the internal angle bisector of angle A, and define LB and LC analogously. Then the three reflected lines concur in X*. In trilinear coordinates, if $X=x:y:z,$ then $X^{*}={\tfrac {1}{x}}:{\tfrac {1}{y}}:{\tfrac {1}{z}}.$
Neuberg cubic
Trilinear equation: $\sum _{\text{cyclic}}(\cos {A}-2\cos {B}\cos {C})x(y^{2}-z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}(a^{2}(b^{2}+c^{2})+(b^{2}-c^{2})^{2}-2a^{4})x(c^{2}y^{2}-b^{2}z^{2})=0$
The Neuberg cubic (named after Joseph Jean Baptiste Neuberg) is the locus of a point X such that X* is on the line EX, where E is the Euler infinity point (X(30) in the Encyclopedia of Triangle Centers). Also, this cubic is the locus of X such that the triangle △XAXBXC is perspective to △ABC, where △XAXBXC is the reflection of X in the lines BC, CA, AB, respectively
The Neuberg cubic passes through the following points: incenter, circumcenter, orthocenter, both Fermat points, both isodynamic points, the Euler infinity point, other triangle centers, the excenters, the reflections of A, B, C in the sidelines of △ABC, and the vertices of the six equilateral triangles erected on the sides of △ABC.
For a graphical representation and extensive list of properties of the Neuberg cubic, see K001 at Berhard Gibert's Cubics in the Triangle Plane.
Thomson cubic
Trilinear equation: $\sum _{\text{cyclic}}bcx(y^{2}-z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}x(c^{2}y^{2}-b^{2}z^{2})=0$
The Thomson cubic is the locus of a point X such that X* is on the line GX, where G is the centroid.
The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices A, B, C, the excenters, the midpoints of sides BC, CA, AB, and the midpoints of the altitudes of △ABC. For each point P on the cubic but not on a sideline of the cubic, the isogonal conjugate of P is also on the cubic.
For graphs and properties, see K002 at Cubics in the Triangle Plane.
Darboux cubic
Trilinear equation:$\sum _{\text{cyclic}}(\cos {A}-\cos {B}\cos {C})x(y^{2}-z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}(2a^{2}(b^{2}+c^{2})+(b^{2}-c^{2})^{2}-3a^{4})x(c^{2}y^{2}-b^{2}z^{2})=0$
The Darboux cubic is the locus of a point X such that X* is on the line LX, where L is the de Longchamps point. Also, this cubic is the locus of X such that the pedal triangle of X is the cevian triangle of some point (which lies on the Lucas cubic). Also, this cubic is the locus of a point X such that the pedal triangle of X and the anticevian triangle of X are perspective; the perspector lies on the Thomson cubic.
The Darboux cubic passes through the incenter, circumcenter, orthocenter, de Longchamps point, other triangle centers, the vertices A, B, C, the excenters, and the antipodes of A, B, C on the circumcircle. For each point P on the cubic but not on a sideline of the cubic, the isogonal conjugate of P is also on the cubic.
For graphics and properties, see K004 at Cubics in the Triangle Plane.
Napoleon–Feuerbach cubic
Trilinear equation: $\sum _{\text{cyclic}}\cos(B-C)x(y^{2}-z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}(a^{2}(b^{2}+c^{2})+(b^{2}-c^{2})^{2})x(c^{2}y^{2}-b^{2}z^{2})=0$
The Napoleon–Feuerbach cubic is the locus of a point X* is on the line NX, where N is the nine-point center, (N = X(5) in the Encyclopedia of Triangle Centers).
The Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices A, B, C, the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of △ABC.
For a graphics and properties, see K005 at Cubics in the Triangle Plane.
Lucas cubic
Trilinear equation: $\sum _{\text{cyclic}}\cos(A)x(b^{2}y^{2}-c^{2}z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}(b^{2}+c^{2}-a^{2})x(y^{2}-z^{2})=0$
The Lucas cubic is the locus of a point X such that the cevian triangle of X is the pedal triangle of some point; the point lies on the Darboux cubic.
The Lucas cubic passes through the centroid, orthocenter, Gergonne point, Nagel point, de Longchamps point, other triangle centers, the vertices of the anticomplementary triangle, and the foci of the Steiner circumellipse.
For graphics and properties, see K007 at Cubics in the Triangle Plane.
1st Brocard cubic
Trilinear equation:$\sum _{\text{cyclic}}bc(a^{4}-b^{2}c^{2})x(y^{2}+z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}(a^{4}-b^{2}c^{2})x(c^{2}y^{2}+b^{2}z^{2})=0$
Let △A'B'C' be the 1st Brocard triangle. For arbitrary point X, let XA, XB, XC be the intersections of the lines XA′, XB′, XC′ with the sidelines BC, CA, AB, respectively. The 1st Brocard cubic is the locus of X for which the points XA, XB, XC are collinear.
The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.
For graphics and properties, see K017 at Cubics in the Triangle Plane.
2nd Brocard cubic
Trilinear equation: $\sum _{\text{cyclic}}bc(b^{2}-c^{2})x(y^{2}+z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}(b^{2}-c^{2})x(c^{2}y^{2}+b^{2}z^{2})=0$
The 2nd Brocard cubic is the locus of a point X for which the pole of the line XX* in the circumconic through X and X* lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices of the 2nd and 4th Brocard triangles.
For a graphics and properties, see K018 at Cubics in the Triangle Plane.
1st equal areas cubic
Trilinear equation: $\sum _{\text{cyclic}}a(b^{2}-c^{2})x(y^{2}-z^{2})=0$
Barycentric equation: $\sum _{\text{cyclic}}a^{2}(b^{2}-c^{2})x(c^{2}y^{2}-b^{2}z^{2})=0$
The 1st equal areas cubic is the locus of a point X such that area of the cevian triangle of X equals the area of the cevian triangle of X*. Also, this cubic is the locus of X for which X* is on the line S*X, where S is the Steiner point. (S = X(99) in the Encyclopedia of Triangle Centers).
The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd Brocard points, and the excenters.
For a graphics and properties, see K021 at Cubics in the Triangle Plane.
2nd equal areas cubic
Trilinear equation: $(bz+cx)(cx+ay)(ay+bz)=(bx+cy)(cy+az)(az+bx)$
Barycentric equation:$\sum _{\text{cyclic}}a(a^{2}-bc)x(c^{3}y^{2}-b^{3}z^{2})=0$
For any point $X=x:y:z$ (trilinears), let $X_{Y}=y:z:x$ and $X_{Z}=z:x:y.$ The 2nd equal areas cubic is the locus of X such that the area of the cevian triangle of XY equals the area of the cevian triangle of XZ.
The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points in Encyclopedia of Triangle Centers indexed as X(31), X(105), X(238), X(292), X(365), X(672), X(1453), X(1931), X(2053), and others.
For a graphics and properties, see K155 at Cubics in the Triangle Plane.
See also
• Cayley–Bacharach theorem, on the intersection of two cubic plane curves
• Twisted cubic, a cubic space curve
• Elliptic curve
• Witch of Agnesi
• Catalogue of Triangle Cubics
References
• Bix, Robert (1998), Conics and Cubics: A Concrete Introduction to Algebraic Curves, New York: Springer, ISBN 0-387-98401-1.
• Cerin, Zvonko (1998), "Locus properties of the Neuberg cubic", Journal of Geometry, 63 (1–2): 39–56, doi:10.1007/BF01221237, S2CID 116778499.
• Cerin, Zvonko (1999), "On the cubic of Napoleon", Journal of Geometry, 66 (1–2): 55–71, doi:10.1007/BF01225672, S2CID 120174967.
• Cundy, H. M. & Parry, Cyril F. (1995), "Some cubic curves associated with a triangle", Journal of Geometry, 53 (1–2): 41–66, doi:10.1007/BF01224039, S2CID 122633134.
• Cundy, H. M. & Parry, Cyril F. (1999), "Geometrical properties of some Euler and circular cubics (part 1)", Journal of Geometry, 66 (1–2): 72–103, doi:10.1007/BF01225673, S2CID 119886462.
• Cundy, H. M. & Parry, Cyril F. (2000), "Geometrical properties of some Euler and circular cubics (part 2)", Journal of Geometry, 68 (1–2): 58–75, doi:10.1007/BF01221061, S2CID 126542269.
• Ehrmann, Jean-Pierre & Gibert, Bernard (2001), "A Morley configuration", Forum Geometricorum, 1: 51–58.
• Ehrmann, Jean-Pierre & Gibert, Bernard (2001), "The Simson cubic", Forum Geometricorum, 1: 107–114.
• Gibert, Bernard (2003), "Orthocorrespondence and orthopivotal cubics", Forum Geometricorum, 3: 1–27.
• Kimberling, Clark (1998), "Triangle Centers and Central Triangles", Congressus Numerantium, 129: 1–295. See Chapter 8 for cubics.
• Kimberling, Clark (2001), "Cubics associated with triangles of equal areas", Forum Geometricorum, 1: 161–171.
• Lang, Fred (2002), "Geometry and group structures of some cubics", Forum Geometricorum, 2: 135–146.
• Pinkernell, Guido M. (1996), "Cubic curves in the triangle plane", Journal of Geometry, 55 (1–2): 142–161, doi:10.1007/BF01223040, S2CID 123411561.
• Salmon, George (1879), Higher Plane Curves (3rd ed.), Dublin: Hodges, Foster, and Figgis.
External links
• A Catalog of Cubic Plane Curves (archived version)
• Points on Cubics
• Cubics in the Triangle Plane
• Special Isocubics in the Triangle Plane (pdf), by Jean-Pierre Ehrmann and Bernard Gibert
• "Real and Complex Cubic Curves - John Milnor, Stony Brook University [2016]". YouTube. Graduate Mathematics. June 27, 2018. lecture on July 2016, ICMS, Edinburgh at conference in honour of Dusa McDuff's 70th birthday
Topics in algebraic curves
Rational curves
• Five points determine a conic
• Projective line
• Rational normal curve
• Riemann sphere
• Twisted cubic
Elliptic curves
Analytic theory
• Elliptic function
• Elliptic integral
• Fundamental pair of periods
• Modular form
Arithmetic theory
• Counting points on elliptic curves
• Division polynomials
• Hasse's theorem on elliptic curves
• Mazur's torsion theorem
• Modular elliptic curve
• Modularity theorem
• Mordell–Weil theorem
• Nagell–Lutz theorem
• Supersingular elliptic curve
• Schoof's algorithm
• Schoof–Elkies–Atkin algorithm
Applications
• Elliptic curve cryptography
• Elliptic curve primality
Higher genus
• De Franchis theorem
• Faltings's theorem
• Hurwitz's automorphisms theorem
• Hurwitz surface
• Hyperelliptic curve
Plane curves
• AF+BG theorem
• Bézout's theorem
• Bitangent
• Cayley–Bacharach theorem
• Conic section
• Cramer's paradox
• Cubic plane curve
• Fermat curve
• Genus–degree formula
• Hilbert's sixteenth problem
• Nagata's conjecture on curves
• Plücker formula
• Quartic plane curve
• Real plane curve
Riemann surfaces
• Belyi's theorem
• Bring's curve
• Bolza surface
• Compact Riemann surface
• Dessin d'enfant
• Differential of the first kind
• Klein quartic
• Riemann's existence theorem
• Riemann–Roch theorem
• Teichmüller space
• Torelli theorem
Constructions
• Dual curve
• Polar curve
• Smooth completion
Structure of curves
Divisors on curves
• Abel–Jacobi map
• Brill–Noether theory
• Clifford's theorem on special divisors
• Gonality of an algebraic curve
• Jacobian variety
• Riemann–Roch theorem
• Weierstrass point
• Weil reciprocity law
Moduli
• ELSV formula
• Gromov–Witten invariant
• Hodge bundle
• Moduli of algebraic curves
• Stable curve
Morphisms
• Hasse–Witt matrix
• Riemann–Hurwitz formula
• Prym variety
• Weber's theorem (Algebraic curves)
Singularities
• Acnode
• Crunode
• Cusp
• Delta invariant
• Tacnode
Vector bundles
• Birkhoff–Grothendieck theorem
• Stable vector bundle
• Vector bundles on algebraic curves
| Wikipedia |
Ternary plot
A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant.[1] It graphically depicts the ratios of the three variables as positions in an equilateral triangle. It is used in physical chemistry, petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. In population genetics, a triangle plot of genotype frequencies is called a de Finetti diagram. In game theory, it is often called a simplex plot.[2] Ternary plots are tools for analyzing compositional data in the three-dimensional case.
In a ternary plot, the values of the three variables a, b, and c must sum to some constant, K. Usually, this constant is represented as 1.0 or 100%. Because a + b + c = K for all substances being graphed, any one variable is not independent of the others, so only two variables must be known to find a sample's point on the graph: for instance, c must be equal to K − a − b. Because the three numerical values cannot vary independently—there are only two degrees of freedom—it is possible to graph the combinations of all three variables in only two dimensions.
The advantage of using a ternary plot for depicting chemical compositions is that three variables can be conveniently plotted in a two-dimensional graph. Ternary plots can also be used to create phase diagrams by outlining the composition regions on the plot where different phases exist.
The values of a point on a ternary plot correspond (up to a constant) to its trilinear coordinates or barycentric coordinates.
Reading values on a ternary plot
There are three equivalent methods that can be used to determine the values of a point on the plot:
1. Parallel line or grid method. The first method is to use a diagram grid consisting of lines parallel to the triangle edges. A parallel to a side of the triangle is the locus of points constant in the component situated in the vertex opposed to the side. Each component is 100% in a corner of the triangle and 0% at the edge opposite it, decreasing linearly with increasing distance (perpendicular to the opposite edge) from this corner. By drawing parallel lines at regular intervals between the zero line and the corner, fine divisions can be established for easy estimation.
2. Perpendicular line or altitude method. For diagrams that do not possess grid lines, the easiest way to determine the values is to determine the shortest (i.e. perpendicular) distances from the point of interest to each of the three sides. By Viviani's theorem, the distances (or the ratios of the distances to the triangle height) give the value of each component.
3. Corner line or intersection method. The third method does not require the drawing of perpendicular or parallel lines. Straight lines are drawn from each corner, through the point of interest, to the opposite side of the triangle. The lengths of these lines, as well as the lengths of the segments between the point and the corresponding sides, are measured individually. The ratio of the measured lines then gives the component value as a fraction of 100%.
A displacement along a parallel line (grid line) preserves the sum of two values, while motion along a perpendicular line increases (or decreases) the two values an equal amount, each half of the decrease (increase) of the third value. Motion along a line through a corner preserves the ratio of the other two values.
• Figure 1. Altitude method
• Figure 2. Intersection method
• Figure 3. An example ternary diagram, without any points plotted.
• Figure 4. An example ternary diagram, showing increments along the first axis.
• Figure 5. An example ternary diagram, showing increments along the second axis.
• Figure 6. An example ternary diagram, showing increments along the third axis.
• Figure 7. Empty ternary plot
• Figure 8. Indication of how the three axes work.
• Unlabeled triangle plot with major grid lines
• Unlabeled triangle plot with major and minor grid lines
Derivation from Cartesian coordinates
Figure (1) shows an oblique projection of point P(a,b,c) in a 3-dimensional Cartesian space with axes a, b and c, respectively.
If a + b + c = K (a positive constant), P is restricted to a plane containing A(K,0,0), B(0,K,0) and C(0,0,K). If a, b and c each cannot be negative, P is restricted to the triangle bounded by A, B and C, as in (2).
In (3), the axes are rotated to give an isometric view. The triangle, viewed face-on, appears equilateral.
In (4), the distances of P from lines BC, AC and AB are denoted by a′, b′ and c′, respectively.
For any line l = s + t n̂ in vector form (n̂ is a unit vector) and a point p, the perpendicular distance from p to l is
$\left\|(\mathbf {s} -\mathbf {p} )-{\bigl (}(\mathbf {s} -\mathbf {p} )\cdot \mathbf {\hat {n}} {\bigr )}\mathbf {\hat {n}} \right\|\,.$
In this case, point P is at
$\mathbf {p} ={\begin{pmatrix}a\\b\\c\end{pmatrix}}\,.$
Line BC has
$\mathbf {s} ={\begin{pmatrix}0\\K\\0\end{pmatrix}}\quad {\text{and}}\quad \mathbf {\hat {n}} ={\frac {{\begin{pmatrix}0\\K\\0\end{pmatrix}}-{\begin{pmatrix}0\\0\\K\end{pmatrix}}}{\left\|{\begin{pmatrix}0\\K\\0\end{pmatrix}}-{\begin{pmatrix}0\\0\\K\end{pmatrix}}\right\|}}={\frac {\begin{pmatrix}0\\K\\-K\end{pmatrix}}{\sqrt {0^{2}+K^{2}+{(-K)}^{2}}}}={\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\,.$
Using the perpendicular distance formula,
${\begin{aligned}a'&=\left\|{\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}-\left({\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}\cdot {\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right){\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right\|\\[10px]&=\left\|{\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}-\left(0+{\frac {K-b}{\sqrt {2}}}+{\frac {c}{\sqrt {2}}}\right){\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right\|\\[10px]&=\left\|{\begin{pmatrix}-a\\K-b-{\frac {K-b+c}{2}}\\-c+{\frac {K-b+c}{2}}\end{pmatrix}}\right\|=\left\|{\begin{pmatrix}-a\\{\frac {K-b-c}{2}}\\{\frac {K-b-c}{2}}\end{pmatrix}}\right\|\\[10px]&={\sqrt {{(-a)}^{2}+{\left({\frac {K-b-c}{2}}\right)}^{2}+{\left({\frac {K-b-c}{2}}\right)}^{2}}}={\sqrt {a^{2}+{\frac {{(K-b-c)}^{2}}{2}}}}\,.\end{aligned}}$
Substituting K = a + b + c,
$a'={\sqrt {a^{2}+{\frac {{(a+b+c-b-c)}^{2}}{2}}}}={\sqrt {a^{2}+{\frac {a^{2}}{2}}}}=a{\sqrt {\frac {3}{2}}}\,.$
Similar calculation on lines AC and AB gives
$b'=b{\sqrt {\frac {3}{2}}}\quad {\text{and}}\quad c'=c{\sqrt {\frac {3}{2}}}\,.$
This shows that the distance of the point from the respective lines is linearly proportional to the original values a, b and c.[3]
Plotting a ternary plot
Cartesian coordinates are useful for plotting points in the triangle. Consider an equilateral ternary plot where a = 100% is placed at (x,y) = (0,0) and b = 100% at (1,0). Then c = 100% is $ ({\frac {1}{2}},{\frac {\sqrt {3}}{2}}),$ and the triple (a,b,c) is
$\left({\frac {1}{2}}\cdot {\frac {2b+c}{a+b+c}},{\frac {\sqrt {3}}{2}}\cdot {\frac {c}{a+b+c}}\right)\,.$
Example
This example shows how this works for a hypothetical set of three soil samples:
SampleClaySiltSandNotes
Sample 150%20%30%Because clay and silt together make up 70% of this sample, the proportion of sand must be 30% for the components to sum to 100%.
Sample 210%60%30%The proportion of sand is 30% as in Sample 1, but as the proportion of silt rises by 40%, the proportion of clay decreases correspondingly.
Sample 310%30%60%This sample has the same proportion of clay as Sample 2, but the proportions of silt and sand are swapped; the plot is reflected about its vertical axis.
Plotting the points
• Plotting Sample 1 (step 1):
Find the 50% clay line
• Plotting Sample 1 (step 2):
Find the 20% silt line
• Plotting Sample 1 (step 3):
Being dependent on the first two, the intersect is on the 30% sand line
• Plotting all the samples
• Ternary triangle plot of soil types sand clay and silt programmed with Mathematica
List of notable ternary diagrams
• Chromaticity diagram
• de Finetti diagram
• Dalitz plot
• Flammability diagram
• Jensen cation plot
• Piper diagram, used in hydrochemistry
• UIGS Classification diagram for Ultramafic rock
• USDA Soil texture diagram by particle sizes
See also
• Apparent molar property
• Viviani's theorem
• Barycentric coordinates (mathematics)
• Compositional data
• List of information graphics software
• Earth sciences graphics software
• IGOR Pro
• Origin (data analysis software)
• R has a dedicated package ternary maintained on the Comprehensive R Archive Network (CRAN)
• Sigmaplot
• Project triangle
• Trilemma
References
1. Weisstein, Eric W. "Ternary Diagram". mathworld.wolfram.com. Retrieved 2021-06-05.
2. Karl Tuyls, "An evolutionary game-theoretic analysis of poker strategies", Entertainment Computing January 2009 doi:10.1016/j.entcom.2009.09.002, p. 9
3. Vaughan, Will (September 5, 2010). "Ternary plots". Archived from the original on December 20, 2010. Retrieved September 7, 2010.
External links
• "Excel Template for Ternary Diagrams". serc.carleton.edu. Science Education Resource Center (SERC) Carleton College. Retrieved 14 May 2020.
• "Tri-plot: Ternary diagram plotting software". www.lboro.ac.uk. Loughborough University – Department of Geography / Resources Gateway home > Tri-plot. Retrieved 14 May 2020.
• "Ternary Plot Generator – Quickly create ternary diagrams on line". www.ternaryplot.com. Retrieved 14 May 2020.
• Holland, Steven (2016). "Data Analysis in the Geosciences – Ternary Diagrams developed in the R language". strata.uga.edu. University of Georgia. Retrieved 14 May 2020.
Wikimedia Commons has media related to Ternary plots.
Chemical solutions
Solution
• Ideal solution
• Aqueous solution
• Solid solution
• Buffer solution
• Flory–Huggins
• Mixture
• Suspension
• Colloid
• Phase diagram
• Phase separation
• Eutectic point
• Alloy
• Saturation
• Supersaturation
• Serial dilution
• Dilution (equation)
• Apparent molar property
• Miscibility gap
Concentration
and related quantities
• Molar concentration
• Mass concentration
• Number concentration
• Volume concentration
• Normality
• Molality
• Mole fraction
• Mass fraction
• Isotopic abundance
• Mixing ratio
• Ternary plot
Solubility
• Solubility equilibrium
• Total dissolved solids
• Solvation
• Solvation shell
• Enthalpy of solution
• Lattice energy
• Raoult's law
• Henry's law
• Solubility table (data)
• Solubility chart
• Miscibility
Solvent
• (Category)
• Acid dissociation constant
• Protic solvent
• Polar aprotic solvent
• Inorganic nonaqueous solvent
• Solvation
• List of boiling and freezing information of solvents
• Partition coefficient
• Polarity
• Hydrophobe
• Hydrophile
• Lipophilic
• Amphiphile
• Lyonium ion
• Lyate ion
| Wikipedia |
Triangle graph
In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle.[1]
Not to be confused with Triangular graph or Ternary plot.
Triangle graph
The triangle graph
Vertices3
Edges3
Radius1
Diameter1
Girth3
Automorphisms6 (D3)
Chromatic number3
Chromatic index3
Properties2-regular
Vertex-transitive
Edge-transitive
Unit distance
Hamiltonian
Eulerian
Notation$C_{3}$ or $K_{3}$
Table of graphs and parameters
The triangle graph is also known as the cycle graph $C_{3}$ and the complete graph $K_{3}$.
Properties
The triangle graph has chromatic number 3, chromatic index 3, radius 1, diameter 1 and girth 3. It is also a 2-vertex-connected graph and a 2-edge-connected graph.
Its chromatic polynomial is $(x-2)(x-1)x.$
See also
• Triangle-free graph
References
1. Weisstein, Eric W. "Triangle Graph". MathWorld.
| Wikipedia |
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.
Definition
Let l, m, n be integers greater than or equal to 2. A triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/l, 2π/m and 2π/n. Therefore, if the generating reflections are labeled a, b, c and the angles between them in the cyclic order are as given above, then the following relations hold:
1. $a^{2}=b^{2}=c^{2}=1$
2. $(ab)^{l}=(bc)^{n}=(ca)^{m}=1.$
It is a theorem that all other relations between a, b, c are consequences of these relations and that Δ(l,m,n) is a discrete group of motions of the corresponding space. Thus a triangle group is a reflection group that admits a group presentation
$\Delta (l,m,n)=\langle a,b,c\mid a^{2}=b^{2}=c^{2}=(ab)^{l}=(bc)^{n}=(ca)^{m}=1\rangle .$
An abstract group with this presentation is a Coxeter group with three generators.
Classification
Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles (π/l, π/m, π/n), and the space is tiled by reflections of the triangle. The sum of the angles of the triangle determines the type of the geometry by the Gauss–Bonnet theorem: it is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly smaller than π. Moreover, any two triangles with the given angles are congruent. Each triangle group determines a tiling, which is conventionally colored in two colors, so that any two adjacent tiles have opposite colors.
In terms of the numbers l, m, n > 1 there are the following possibilities.
The Euclidean case
${\frac {1}{l}}+{\frac {1}{m}}+{\frac {1}{n}}=1.$
The triangle group is the infinite symmetry group of a certain tessellation (or tiling) of the Euclidean plane by triangles whose angles add up to π (or 180°). Up to permutations, the triple (l, m, n) is one of the triples (2,3,6), (2,4,4), (3,3,3). The corresponding triangle groups are instances of wallpaper groups.
(2,3,6) (2,4,4) (3,3,3)
bisected hexagonal tiling tetrakis square tiling triangular tiling
More detailed diagrams, labeling the vertices and showing how reflection operates:
The spherical case
${\frac {1}{l}}+{\frac {1}{m}}+{\frac {1}{n}}>1.$
The triangle group is the finite symmetry group of a tiling of a unit sphere by spherical triangles, or Möbius triangles, whose angles add up to a number greater than π. Up to permutations, the triple (l,m,n) has the form (2,3,3), (2,3,4), (2,3,5), or (2,2,n), n > 1. Spherical triangle groups can be identified with the symmetry groups of regular polyhedra in the three-dimensional Euclidean space: Δ(2,3,3) corresponds to the tetrahedron, Δ(2,3,4) to both the cube and the octahedron (which have the same symmetry group), Δ(2,3,5) to both the dodecahedron and the icosahedron. The groups Δ(2,2,n), n > 1 of dihedral symmetry can be interpreted as the symmetry groups of the family of dihedra, which are degenerate solids formed by two identical regular n-gons joined together, or dually hosohedra, which are formed by joining n digons together at two vertices.
The spherical tiling corresponding to a regular polyhedron is obtained by forming the barycentric subdivision of the polyhedron and projecting the resulting points and lines onto the circumscribed sphere. In the case of the tetrahedron, there are four faces and each face is an equilateral triangle that is subdivided into 6 smaller pieces by the medians intersecting in the center. The resulting tesselation has 4 × 6=24 spherical triangles (it is the spherical disdyakis cube).
These groups are finite, which corresponds to the compactness of the sphere – areas of discs in the sphere initially grow in terms of radius, but eventually cover the entire sphere.
The triangular tilings are depicted below:
(2,2,2) (2,2,3) (2,2,4) (2,2,5) (2,2,6) (2,2,n)
(2,3,3) (2,3,4) (2,3,5)
Spherical tilings corresponding to the octahedron and the icosahedron and dihedral spherical tilings with even n are centrally symmetric. Hence each of them determines a tiling of the real projective plane, an elliptic tiling. Its symmetry group is the quotient of the spherical triangle group by the reflection through the origin (-I), which is a central element of order 2. Since the projective plane is a model of elliptic geometry, such groups are called elliptic triangle groups.[1]
The hyperbolic case
${\frac {1}{l}}+{\frac {1}{m}}+{\frac {1}{n}}<1.$
The triangle group is the infinite symmetry group of a tiling of the hyperbolic plane by hyperbolic triangles whose angles add up to a number less than π. All triples not already listed represent tilings of the hyperbolic plane. For example, the triple (2,3,7) produces the (2,3,7) triangle group. There are infinitely many such groups; the tilings associated with some small values:
Hyperbolic plane
Poincaré disk model of fundamental domain triangles
Example right triangles (2 p q)
(2 3 7)
(2 3 8)
(2 3 9)
(2 3 ∞)
(2 4 5)
(2 4 6)
(2 4 7)
(2 4 8)
(2 4 ∞)
(2 5 5)
(2 5 6)
(2 5 7)
(2 6 6)
(2 ∞ ∞)
Example general triangles (p q r)
(3 3 4)
(3 3 5)
(3 3 6)
(3 3 7)
(3 3 ∞)
(3 4 4)
(3 6 6)
(3 ∞ ∞)
(6 6 6)
(∞ ∞ ∞)
Hyperbolic triangle groups are examples of non-Euclidean crystallographic group and have been generalized in the theory of Gromov hyperbolic groups.
Von Dyck groups
Denote by D(l,m,n) the subgroup of index 2 in Δ(l,m,n) generated by words of even length in the generators. Such subgroups are sometimes referred to as "ordinary" triangle groups[2] or von Dyck groups, after Walther von Dyck. For spherical, Euclidean, and hyperbolic triangles, these correspond to the elements of the group that preserve the orientation of the triangle – the group of rotations. For projective (elliptic) triangles, they cannot be so interpreted, as the projective plane is non-orientable, so there is no notion of "orientation-preserving". The reflections are however locally orientation-reversing (and every manifold is locally orientable, because locally Euclidean): they fix a line and at each point in the line are a reflection across the line.[3]
The group D(l,m,n) is defined by the following presentation:
$D(l,m,n)=\langle x,y\mid x^{l},y^{m},(xy)^{n}\rangle .$
In terms of the generators above, these are x = ab, y = ca, yx = cb. Geometrically, the three elements x, y, xy correspond to rotations by 2π/l, 2π/m and 2π/n about the three vertices of the triangle.
Note that D(l,m,n) ≅ D(m,l,n) ≅ D(n,m,l), so D(l,m,n) is independent of the order of the l,m,n.
A hyperbolic von Dyck group is a Fuchsian group, a discrete group consisting of orientation-preserving isometries of the hyperbolic plane.
Overlapping tilings
Further information: Schwarz triangle
Triangle groups preserve a tiling by triangles, namely a fundamental domain for the action (the triangle defined by the lines of reflection), called a Möbius triangle, and are given by a triple of integers, (l,m,n), – integers correspond to (2l,2m,2n) triangles coming together at a vertex. There are also tilings by overlapping triangles, which correspond to Schwarz triangles with rational numbers (l/a,m/b,n/c), where the denominators are coprime to the numerators. This corresponds to edges meeting at angles of aπ/l (resp.), which corresponds to a rotation of 2aπ/l (resp.), which has order l and is thus identical as an abstract group element, but distinct when represented by a reflection.
For example, the Schwarz triangle (2 3 3) yields a density 1 tiling of the sphere, while the triangle (2 3/2 3) yields a density 3 tiling of the sphere, but with the same abstract group. These symmetries of overlapping tilings are not considered triangle groups.
History
Triangle groups date at least to the presentation of the icosahedral group as the (rotational) (2,3,5) triangle group by William Rowan Hamilton in 1856, in his paper on icosian calculus.[4]
Applications
External video
Warped modular tiling[5] – visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.
Triangle groups arise in arithmetic geometry. The modular group is generated by two elements, S and T, subject to the relations S² = (ST)³ = 1 (no relation on T), is the rotational triangle group (2,3,∞) and maps onto all triangle groups (2,3,n) by adding the relation Tn = 1. More generally, the Hecke group Hq is generated by two elements, S and T, subject to the relations S2 = (ST)q = 1 (no relation on T), is the rotational triangle group (2,q,∞), and maps onto all triangle groups (2,q,n) by adding the relation Tn = 1 the modular group is the Hecke group H3. In Grothendieck's theory of dessins d'enfants, a Belyi function gives rise to a tessellation of a Riemann surface by reflection domains of a triangle group.
All 26 sporadic groups are quotients of triangle groups,[6] of which 12 are Hurwitz groups (quotients of the (2,3,7) group).
See also
• Schwarz triangle
• The Schwarz triangle map is a map of triangles to the upper half-plane.
• Geometric group theory
References
1. (Magnus 1974)
2. (Gross & Tucker 2001)
3. (Magnus 1974, p. 65)
4. Sir William Rowan Hamilton (1856), "Memorandum respecting a new System of Roots of Unity" (PDF), Philosophical Magazine, 12: 446
5. Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp
6. (Wilson 2001, Table 2, p. 7)
• Magnus, Wilhelm (1974), "II. Discontinuous groups and triangle tessellations", Noneuclidean tesselations and their groups, Academic Press, pp. 52–106, ISBN 978-0-12-465450-1
• Gross, Jonathan L.; Tucker, Thomas W. (2001), "6.2.8 Triangle Groups", Topological graph theory, Courier Dover Publications, pp. 279–281, ISBN 978-0-486-41741-7
• Wilson, R. A. (2001), "The Monster is a Hurwitz group", Journal of Group Theory, 4 (4): 367–374, doi:10.1515/jgth.2001.027, MR 1859175
External links
• Elizabeth r chen triangle groups (2010) desktop background pictures
This article incorporates material from Triangle groups on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
| Wikipedia |
Triangle of partition numbers
In the number theory of integer partitions, the numbers $p_{k}(n)$ denote both the number of partitions of $n$ into exactly $k$ parts (that is, sums of $k$ positive integers that add to $n$), and the number of partitions of $n$ into parts of maximum size exactly $k$. These two types of partition are in bijection with each other, by a diagonal reflection of their Young diagrams. Their numbers can be arranged into a triangle, the triangle of partition numbers, in which the $n$th row gives the partition numbers $p_{1}(n),p_{2}(n),\dots ,p_{n}(n)$:[1]
k
n
1 2 3 4 5 6 7 8 9
1 1
2 11
3 111
4 1211
5 12211
6 133211
7 1343211
8 14553211
9 147653211
Recurrence relation
Analogously to Pascal's triangle, these numbers may be calculated using the recurrence relation[2]
$p_{k}(n)=p_{k-1}(n-1)+p_{k}(n-k).$
As base cases, $p_{1}(1)=1$, and any value on the right hand side of the recurrence that would be outside the triangle can be taken as zero. This equation can be explained by noting that each partition of $n$ into $k$ pieces, counted by $p_{k}(n)$, can be formed either by adding a piece of size one to a partition of $n-1$ into $k-1$ pieces, counted by $p_{k-1}(n-1)$, or by increasing by one each piece in a partition of $n-k$ into $k$ pieces, counted by $p_{k}(n-k)$.
Row sums and diagonals
In the triangle of partition numbers, the sum of the numbers in the $n$th row is the partition number $p(n)$. These numbers form the sequence
1, 2, 3, 5, 7, 11, 15, 22, ...,
omitting the initial value $p(0)=1$ of the partition numbers. Each diagonal from upper left to lower right is eventually constant, with the constant parts of these diagonals extending approximately from halfway across each row to its end. The values of these constants are the partition numbers 1, 1, 2, 3, 5, 7, ... again.[3]
References
1. Sloane, N. J. A. (ed.), "Sequence A008284 (Triangle of partition numbers)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
2. Arndt, Jörg (2011), "16.4.1: Unrestricted partitions and partitions into $m$ parts", Matters Computational: Ideas, Algorithms, Source Code (PDF), Springer, pp. 345–348
3. Hopkins, Brian (2009), "Column-to-row operations on partitions: the envelopes" (PDF), Integers, 9 (Supplement): A6:1–A6:11, MR 2521954
| Wikipedia |
Splitter (geometry)
In Euclidean geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle.[1][2] They are not to be confused with cleavers, which also bisect the perimeter but instead emanate from the midpoint of one of the triangle's sides.
Properties
The opposite endpoint of a splitter to the chosen triangle vertex lies at the point on the triangle's side where one of the excircles of the triangle is tangent to that side.[1][2] This point is also called a splitting point of the triangle.[2] It is additionally a vertex of the extouch triangle and one of the points where the Mandart inellipse is tangent to the triangle side.[3]
The three splitters concur at the Nagel point of the triangle,[1] which is also called its splitting center.[2]
Generalization
Some authors have used the term "splitter" in a more general sense, for any line segment that bisects the perimeter of the triangle. Other line segments of this type include the cleavers, which are perimeter-bisecting segments that pass through the midpoint of a triangle side, and the equalizers, segments that bisect both the area and perimeter of a triangle.[4]
References
1. Honsberger, Ross (1995), "Chapter 1: Cleavers and Splitters", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, vol. 37, Washington, DC: Mathematical Association of America, pp. 1–14, ISBN 0-88385-639-5, MR 1316889
2. Avishalom, Dov (1963), "The perimetric bisection of triangles", Mathematics Magazine, 36 (1): 60–62, JSTOR 2688140, MR 1571272
3. Juhász, Imre (2012), "Control point based representation of inellipses of triangles" (PDF), Annales Mathematicae et Informaticae, 40: 37–46, MR 3005114
4. Kodokostas, Dimitrios (2010), "Triangle equalizers", Mathematics Magazine, 83 (2): 141–146, doi:10.4169/002557010X482916
External links
• Weisstein, Eric W., "Splitter", MathWorld
| Wikipedia |
Trapezo-rhombic dodecahedron
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has D3h symmetry. A concave form can be constructed with an identical net, seen as excavating trigonal trapezohedra from the top and bottom. It is also called the trapezoidal dodecahedron.[1]
Trapezo-rhombic dodecahedron
TypePlesiohedron
Johnson solid dual
Faces6 rhombi
6 trapezoids
Edges24
Vertices14
Vertex configuration(2) 4.4.4
(6) 4.4.4.4
(6) 4.4.4
Symmetry groupD3h, [3,2], (*322), order 12
Rotation groupD3, [3,2]+, (322), order 6
Dual polyhedronTriangular orthobicupola
Propertiesconvex
Net
Concave configuration
Construction
This polyhedron could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts.
Space-filling tessellation
A space-filling tessellation, the trapezo-rhombic dodecahedral honeycomb, can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, or a rhombille tiling, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up.
:
In the special case that the long sides of the trapezoids equals twice the length of the short sides, the solid now represents the 3D Voronoi cell of a sphere in a hexagonal close packing, next to face-centered cubic an optimal way to stack spheres in a lattice. It is therefore related to the rhombic dodecahedron, which can be represented by turning the lower half of the picture at right over an angle of 60 degrees. The rhombic dodecahedron is a Voronoi cell of the other optimal way to stack spheres. The two shapes differ in their combinatorial structure as well as in their geometry: in the rhombic dodecahedron, every edge connects a degree-three vertex to a degree-four vertex, whereas the trapezo-rhombic dodecahedron has six edges that connect vertices of equal degrees.
As the Voronoi cell of a regular space pattern, it is a plesiohedron. It is the polyhedral dual of the triangular orthobicupola.
Variations
The trapezo-rhombic dodecahedron can be seen as an elongation of another dodecahedron, which can be called a rhombo-triangular dodecahedron, with 6 rhombi (or squares) and 6 triangles. It also has d3h symmetry and is space-filling. It has 21 edges and 11 vertices. With square faces it can be seen as a cube split across the 3-fold axis, separated with the two halves rotated 180 degrees, and filling the gaps with triangles. When used as a space-filler, connecting dodecahedra on their triangles leaves two cubical step surfaces on the top and bottom which can connect with complementary steps.
See also
• Elongated dodecahedron
• Hexagonal prismatic honeycomb
References
1. Lagarias, Jeffrey C. (2011). "The Kepler conjecture and its proof". The Kepler Conjecture: The Hales–Ferguson proof. Springer, New York. pp. 3–26. doi:10.1007/978-1-4614-1129-1_1. MR 3050907.; see especially p. 11
Further reading
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 170. ISBN 0-486-23729-X.
• Mathematical Recreations and Essays Walter William Rouse Ball, Harold Scott Macdonald Coxeter, p.151
• Structure in Nature Is a Strategy for Design, Peter Jon Pearce, p.48 Spacefilling systems based on rhombic dodecahedron
External links
• Weisstein, Eric W. "Space-filling polyhedron". MathWorld.
• VRML model
| Wikipedia |
Triangle wave
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Triangle wave
A bandlimited triangle wave[1] pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).
General information
General definition$x(t)=4\left\vert t-\left\lfloor t+3/4\right\rfloor +1/4\right\vert -1$
Fields of applicationElectronics, synthesizers
Domain, Codomain and Image
Domain$\mathbb {R} $
Codomain$\left[-1,1\right]$
Basic features
ParityOdd
Period1
Specific features
Root$\left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} $
DerivativeSquare wave
Fourier series$x(t)=-{\frac {8}{{\pi }^{2}}}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{k}}{\left(2k-1\right)^{2}}}\sin \left(2\pi \left(2k-1\right)t\right)$
Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).
Definitions
Definition
A triangle wave of period p that spans the range [0,1] is defined as:
$x(t)=2\left|{\frac {t}{p}}-\left\lfloor {\frac {t}{p}}+{\frac {1}{2}}\right\rfloor \right|$
where $\lfloor \,\ \rfloor $ is the floor function. This can be seen to be the absolute value of a shifted sawtooth wave.
For a triangle wave spanning the range [−1,1] the expression becomes:
$x(t)=2\left|2\left({\frac {t}{p}}-\left\lfloor {t \over p}+{1 \over 2}\right\rfloor \right)\right|-1.$
A more general equation for a triangle wave with amplitude $a$ and period $p$ using the modulo operation and absolute value is:
$y(x)={\frac {4a}{p}}\left|\left(\left(x-{\frac {p}{4}}\right){\bmod {p}}\right)-{\frac {p}{2}}\right|-a.$
For example, for a triangle wave with amplitude 5 and period 4:
$y(x)=5{\bigl |}\left((x-1){\bmod {4}}\right)-2{\bigr |}-5.$
A phase shift can be obtained by altering the value of the $-p/4$ term, and the vertical offset can be adjusted by altering the value of the $-a$ term.
As this only uses the modulo operation and absolute value, it can be used to simply implement a triangle wave on hardware electronics.
Note that in many programming languages, the % operator is a remainder operator (with result the same sign as the dividend), not a modulo operator; the modulo operation can be obtained by using ((x % p) + p) % p in place of x % p. In e.g. JavaScript, this results in an equation of the form 4*a/p * Math.abs((((x-p/4)%p)+p)%p - p/2) - a.
Relation to the square wave
The triangle wave can also be expressed as the integral of the square wave:
$x(t)=\int _{0}^{t}\operatorname {sgn} \left(\sin {\frac {u}{p}}\right)\,du.$
Expression in trigonometric functions
A triangle wave with period p and amplitude a can be expressed in terms of sine and arcsine (whose value ranges from −π/2 to π/2):
$y(x)={\frac {2a}{\pi }}\arcsin \left(\sin \left({\frac {2\pi }{p}}x\right)\right).$
The identity $ \cos {x}=\sin \left({\frac {p}{4}}-x\right)$ can be used to convert from a triangle "sine" wave to a triangular "cosine" wave. This phase-shifted triangle wave can also be expressed with cosine and arccosine:
$y(x)=a-{\frac {2a}{\pi }}\arccos \left(\cos \left({\frac {2\pi }{p}}x\right)\right).$
Expressed as alternating linear functions
Another definition of the triangle wave, with range from −1 to 1 and period p, is:
$x(t)={\frac {4}{p}}\left(t-{\frac {p}{2}}\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor }$
Harmonics
It is possible to approximate a triangle wave with additive synthesis by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by π) and multiplying the amplitude of the harmonics by one over the square of their mode number, n (which is equivalent to one over the square of their relative frequency to the fundamental).
The above can be summarised mathematically as follows:
${\begin{aligned}x_{\mathrm {triangle} }(t)&{}={\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin \left(2\pi f_{0}nt\right)\end{aligned}}$
where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), $f_{0}$ is the fundamental frequency, and i is the harmonic label which is related to its mode number by $n=2i+1$.
This infinite Fourier series converges quickly to the triangle wave as N tends to infinity, as shown in the animation.
Arc length
The arc length per period for a triangle wave, denoted by s, is given in terms of the amplitude a and period length p by
$s={\sqrt {(4a)^{2}+p^{2}}}.$
See also
• List of periodic functions
• Sine wave
• Square wave
• Sawtooth wave
• Pulse wave
• Sound
• Triangle function
• Wave
• Zigzag
References
1. Kraft, Sebastian; Zölzer, Udo (5 September 2017). "LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms". Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17). 20th International Conference on Digital Audio Effects (DAFx-17). Edinburgh. pp. 255–259.
• Weisstein, Eric W. "Fourier Series - Triangle Wave". MathWorld.
Waveforms
• Sine wave
• Non-sinusoidal
• Rectangular wave
• Sawtooth wave
• Square wave
• Triangle wave
| Wikipedia |
Triangular array
In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ith row contains only i elements.
Not to be confused with Triangular matrix.
Examples
Notable particular examples include these:
• The Bell triangle, whose numbers count the partitions of a set in which a given element is the largest singleton[1]
• Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched[2]
• Euler's triangle, which counts permutations with a given number of ascents[3]
• Floyd's triangle, whose entries are all of the integers in order[4]
• Hosoya's triangle, based on the Fibonacci numbers[5]
• Lozanić's triangle, used in the mathematics of chemical compounds[6]
• Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings[7]
• Pascal's triangle, whose entries are the binomial coefficients[8]
Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.[9]
Generalizations
Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial.[10]
Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.[11]
Applications
Apart from the representation of triangular matrices, triangular arrays are used in several algorithms. One example is the CYK algorithm for parsing context-free grammars, an example of dynamic programming.[12]
Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers.[13]
The Boustrophedon transform uses a triangular array to transform one integer sequence into another.[14]
See also
• Triangular number, the number of entries in such an array up to some particular row
References
1. Shallit, Jeffrey (1980), "A triangle for the Bell numbers", A collection of manuscripts related to the Fibonacci sequence (PDF), Santa Clara, Calif.: Fibonacci Association, pp. 69–71, MR 0624091.
2. Kitaev, Sergey; Liese, Jeffrey (2013), "Harmonic numbers, Catalan's triangle and mesh patterns", Discrete Mathematics, 313 (14): 1515–1531, arXiv:1209.6423, doi:10.1016/j.disc.2013.03.017, MR 3047390, S2CID 18248485.
3. Velleman, Daniel J.; Call, Gregory S. (1995), "Permutations and combination locks", Mathematics Magazine, 68 (4): 243–253, doi:10.2307/2690567, JSTOR 2690567, MR 1363707.
4. Miller, Philip L.; Miller, Lee W.; Jackson, Purvis M. (1987), Programming by design: a first course in structured programming, Wadsworth Pub. Co., pp. 211–212, ISBN 9780534082444.
5. Hosoya, Haruo (1976), "Fibonacci triangle", The Fibonacci Quarterly, 14 (2): 173–178.
6. Losanitsch, S. M. (1897), "Die Isomerie-Arten bei den Homologen der Paraffin-Reihe", Chem. Ber., 30 (2): 1917–1926, doi:10.1002/cber.189703002144.
7. Barry, Paul (2011), "On a generalization of the Narayana triangle", Journal of Integer Sequences, 14 (4): Article 11.4.5, 22, MR 2792161.
8. Edwards, A. W. F. (2002), Pascal's Arithmetical Triangle: The Story of a Mathematical Idea, JHU Press, ISBN 9780801869464.
9. Barry, P. (2006), "On integer-sequence-based constructions of generalized Pascal triangles" (PDF), Journal of Integer Sequences, 9 (6.2.4): 1–34, Bibcode:2006JIntS...9...24B.
10. Rota Bulò, Samuel; Hancock, Edwin R.; Aziz, Furqan; Pelillo, Marcello (2012), "Efficient computation of Ihara coefficients using the Bell polynomial recursion", Linear Algebra and Its Applications, 436 (5): 1436–1441, doi:10.1016/j.laa.2011.08.017, MR 2890929.
11. Fielder, Daniel C.; Alford, Cecil O. (1991), "Pascal's triangle: Top gun or just one of the gang?", in Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F. (eds.), Applications of Fibonacci Numbers (Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications, Wake Forest University, N.C., U.S.A., July 30–August 3, 1990), Springer, pp. 77–90, ISBN 9780792313090.
12. Indurkhya, Nitin; Damerau, Fred J., eds. (2010), Handbook of Natural Language Processing, Second Edition, CRC Press, p. 65, ISBN 9781420085938.
13. Thacher Jr., Henry C. (July 1964), "Remark on Algorithm 60: Romberg integration", Communications of the ACM, 7 (7): 420–421, doi:10.1145/364520.364542, S2CID 29898282.
14. Millar, Jessica; Sloane, N. J. A.; Young, Neal E. (1996), "A new operation on sequences: the Boustrouphedon transform", Journal of Combinatorial Theory, Series A, 76 (1): 44–54, arXiv:math.CO/0205218, doi:10.1006/jcta.1996.0087, S2CID 15637402.
External links
• Weisstein, Eric W., "Number Triangle", MathWorld
| Wikipedia |
Triangular bipyramid
In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
Triangular bipyramid
TypeBipyramid,
Johnson
J11 – J12 – J13
Faces6 triangles
Edges9
Vertices5
Schläfli symbol{ } + {3}
Coxeter diagram
Symmetry groupD3h, [3,2], (*223), order 12
Rotation groupD3, [3,2]+, (223), order 6
Dual polyhedronTriangular prism
Face configurationV3.4.4
PropertiesConvex, face-transitive
As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four.
The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, (J12). A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1] As a Johnson solid with all faces equilateral triangles, it is also a deltahedron.
Formulae
The following formulae for the height ($H$), surface area ($A$) and volume ($V$) can be used if all faces are regular, with edge length $L$:[2]
$H=L\cdot {\frac {2{\sqrt {6}}}{3}}\approx L\cdot 1.632993162$
$A=L^{2}\cdot {\frac {3{\sqrt {3}}}{2}}\approx L^{2}\cdot 2.598076211$
$V=L^{3}\cdot {\frac {\sqrt {2}}{6}}\approx L^{3}\cdot 0.235702260$
Dual polyhedron
The dual polyhedron of the triangular bipyramid is the triangular prism, with five faces: two parallel equilateral triangles linked by a chain of three rectangles. Although the triangular prism has a form that is a uniform polyhedron (with square faces), the dual of the Johnson solid form of the bipyramid has rectangular rather than square faces, and is not uniform.
Triangular prism Net
Related polyhedra and honeycombs
The triangular bipyramid, dt{2,3}, can be in sequence rectified, rdt{2,3}, truncated, trdt{2,3} and alternated (snubbed), srdt{2,3}:
The triangular bipyramid can be constructed by augmentation of smaller ones, specifically two stacked regular octahedra with 3 triangular bipyramids added around the sides, and 1 tetrahedron above and below. This polyhedron has 24 equilateral triangle faces, but it is not a Johnson solid because it has coplanar faces. It is a coplanar 24-triangle deltahedron. This polyhedron exists as the augmentation of cells in a gyrated alternated cubic honeycomb. Larger triangular polyhedra can be generated similarly, like 9, 16 or 25 triangles per larger triangle face, seen as a section of a triangular tiling.
The triangular bipyramid can form a tessellation of space with octahedra or with truncated tetrahedra.[3]
Layers of the uniform quarter cubic honeycomb can be shifted to pair up regular tetrahedral cells which combined into triangular bipyramids.
The gyrated tetrahedral-octahedral honeycomb has pairs of adjacent regular tetrahedra that can be seen as triangular bipyramids.
When projected onto a sphere, it resembles a compound of a trigonal hosohedron and trigonal dihedron. It is part of an infinite series of dual pair compounds of regular polyhedra projected onto spheres. The triangular bipyramid can be referred to as a deltoidal hexahedron for consistency with the other solids in the series, although the "deltoids" are triangles instead of kites in this case, as the angle from the dihedron is 180 degrees.
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4
See also
• Trigonal bipyramidal molecular geometry
• Boerdijk–Coxeter helix, an extension of the triangular bipyramid by adding more tetrahedrons
"Regular" right (symmetric) n-gonal bipyramids:
Bipyramid name Digonal bipyramid Triangular bipyramid
(See: J12)
Square bipyramid
(See: O)
Pentagonal bipyramid
(See: J13)
Hexagonal bipyramid Heptagonal bipyramid Octagonal bipyramid Enneagonal bipyramid Decagonal bipyramid ... Apeirogonal bipyramid
Polyhedron image ...
Spherical tiling image Plane tiling image
Face config. V2.4.4V3.4.4V4.4.4V5.4.4V6.4.4V7.4.4V8.4.4V9.4.4V10.4.4...V∞.4.4
Coxeter diagram ...
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Sapiña, R. "Area and volume of the Johnson solid J₁₂". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-01.
3. "J12 honeycomb".
External links
• Eric W. Weisstein, Triangular dipyramid (Johnson solid) at MathWorld.
• Conway Notation for Polyhedra Try: dP3
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
Triangular distribution
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b.
Triangular
Probability density function
Cumulative distribution function
Parameters $a:~a\in (-\infty ,\infty )$
$b:~a<b\,$
$c:~a\leq c\leq b\,$
Support $a\leq x\leq b\!$
PDF ${\begin{cases}0&{\text{for }}x<a,\\{\frac {2(x-a)}{(b-a)(c-a)}}&{\text{for }}a\leq x<c,\\[4pt]{\frac {2}{b-a}}&{\text{for }}x=c,\\[4pt]{\frac {2(b-x)}{(b-a)(b-c)}}&{\text{for }}c<x\leq b,\\[4pt]0&{\text{for }}b<x.\end{cases}}$
CDF ${\begin{cases}0&{\text{for }}x\leq a,\\[2pt]{\frac {(x-a)^{2}}{(b-a)(c-a)}}&{\text{for }}a<x\leq c,\\[4pt]1-{\frac {(b-x)^{2}}{(b-a)(b-c)}}&{\text{for }}c<x<b,\\[4pt]1&{\text{for }}b\leq x.\end{cases}}$
Mean ${\frac {a+b+c}{3}}$
Median ${\begin{cases}a+{\sqrt {\frac {(b-a)(c-a)}{2}}}&{\text{for }}c\geq {\frac {a+b}{2}},\\[6pt]b-{\sqrt {\frac {(b-a)(b-c)}{2}}}&{\text{for }}c\leq {\frac {a+b}{2}}.\end{cases}}$
Mode $c\,$
Variance ${\frac {a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}}$
Skewness ${\frac {{\sqrt {2}}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{\frac {3}{2}}}}$
Ex. kurtosis $-{\frac {3}{5}}$
Entropy ${\frac {1}{2}}+\ln \left({\frac {b-a}{2}}\right)$
MGF $2{\frac {(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}}{(b-a)(c-a)(b-c)t^{2}}}$
CF $-2{\frac {(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}}{(b-a)(c-a)(b-c)t^{2}}}$
Special cases
Mode at a bound
The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the PDF and CDF become:
$\left.{\begin{array}{rl}f(x)&=2x\\[8pt]F(x)&=x^{2}\end{array}}\right\}{\text{ for }}0\leq x\leq 1$
${\begin{aligned}\operatorname {E} (X)&={\frac {2}{3}}\\[8pt]\operatorname {Var} (X)&={\frac {1}{18}}\end{aligned}}$
Distribution of the absolute difference of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0 is the distribution of X = |X1 − X2|, where X1, X2 are two independent random variables with standard uniform distribution.
${\begin{aligned}f(x)&=2-2x{\text{ for }}0\leq x<1\\[6pt]F(x)&=2x-x^{2}{\text{ for }}0\leq x<1\\[6pt]E(X)&={\frac {1}{3}}\\[6pt]\operatorname {Var} (X)&={\frac {1}{18}}\end{aligned}}$
Symmetric triangular distribution
The symmetric case arises when c = (a + b) / 2. In this case, an alternate form of the distribution function is:
${\begin{aligned}f(x)&={\frac {(b-c)-|c-x|}{(b-c)^{2}}}\\[6pt]\end{aligned}}$
Distribution of the mean of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of X = (X1 + X2) / 2, where X1, X2 are two independent random variables with standard uniform distribution in [0, 1].[1] It is the case of the Bates distribution for two variables.
$f(x)={\begin{cases}4x&{\text{for }}0\leq x<{\frac {1}{2}}\\4(1-x)&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}$
$F(x)={\begin{cases}2x^{2}&{\text{for }}0\leq x<{\frac {1}{2}}\\2x^{2}-(2x-1)^{2}&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}$
${\begin{aligned}E(X)&={\frac {1}{2}}\\[6pt]\operatorname {Var} (X)&={\frac {1}{24}}\end{aligned}}$
Generating triangular-distributed random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
$X={\begin{cases}a+{\sqrt {U(b-a)(c-a)}}&{\text{ for }}0<U<F(c)\\&\\b-{\sqrt {(1-U)(b-a)(b-c)}}&{\text{ for }}F(c)\leq U<1\end{cases}}$[2]
where $F(c)=(c-a)/(b-a)$, has a triangular distribution with parameters $a,b$ and $c$. This can be obtained from the cumulative distribution function.
Use of the distribution
The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on a knowledge of the minimum and maximum and an "inspired guess"[3] as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution.
Business simulations
The triangular distribution is therefore often used in business decision making, particularly in simulations. Generally, when not much is known about the distribution of an outcome (say, only its smallest and largest values), it is possible to use the uniform distribution. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example under corporate finance.
Project management
The triangular distribution, along with the PERT distribution, is also widely used in project management (as an input into PERT and hence critical path method (CPM)) to model events which take place within an interval defined by a minimum and maximum value.
Audio dithering
The symmetric triangular distribution is commonly used in audio dithering, where it is called TPDF (triangular probability density function).
Beamforming
The triangular distribution has an application to beamforming and pattern synthesis.[4][5]
See also
• Trapezoidal distribution
• Thomas Simpson
• Three-point estimation
• Five-number summary
• Seven-number summary
• Triangular function
• Central limit theorem — The triangle distribution often occurs as a result of adding two uniform random variables together. In other words, the triangle distribution is often (not always) the result of the first iteration of the central limit theorem summing process (i.e. $ n=2$). In this sense, the triangle distribution can occasionally occur naturally. If this process of summing together more random variables continues (i.e. $ n\geq 3$), then the distribution will become increasingly bell-shaped.
• Irwin–Hall distribution — Using an Irwin–Hall distribution is an easy way to generate a triangle distribution.
• Bates distribution — Similar to the Irwin–Hall distribution, but with the values rescaled back into the 0 to 1 range. Useful for computation of a triangle distribution which can subsequently be rescaled and shifted to create other triangle distributions outside of the 0 to 1 range.
References
1. Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. Samuel Kotz and Johan René van Dorp. https://books.google.com/books?id=JO7ICgAAQBAJ&dq=chapter%201%20dig%20out%20suitable%20substitutes%20of%20the%20beta%20distribution%20one%20of%20our%20goals&pg=PA3
2. "Archived copy" (PDF). www.asianscientist.com. Archived from the original (PDF) on 7 April 2014. Retrieved 12 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)
3. "Archived copy" (PDF). Archived from the original (PDF) on 2006-09-23. Retrieved 2006-09-23.{{cite web}}: CS1 maint: archived copy as title (link)
4. Ma, Nam Nicholas; Buchanan, Kristopher; Jensen, Jeffrey; Huff, Gregory (2015). "Distributed beamforming from triangular planar random antenna arrays". MILCOM 2015 - 2015 IEEE Military Communications Conference. pp. 553–558. doi:10.1109/MILCOM.2015.7357501. ISBN 978-1-5090-0073-9. S2CID 3027268.
5. K. Buchanan, C. Flores-Molina, S. Wheeland, D. Overturf and T. Adeyemi, "Babinet's Principle Applied to Distributed Arrays," 2020 International Applied Computational Electromagnetics Society Symposium (ACES), 2020, pp. 1-2, doi: 10.23919/ACES49320.2020.9196157.
External links
• Weisstein, Eric W. "Triangular Distribution". MathWorld.
• Triangle Distribution, decisionsciences.org
• Triangular Distribution, brighton-webs.co.uk
• Proof for the variance of triangular distribution, math.stackexchange.com
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
Octahedral prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.
Octahedral prism
Schlegel diagram and skew orthogonal projection
TypePrismatic uniform 4-polytope
Uniform index51
Schläfli symbolt{2,3,4} or {3,4}×{}
t1,3{3,3,2} or r{3,3}×{}
s{2,6}×{}
sr{3,2}×{}
Coxeter diagram
Cells2 (3.3.3.3)
8 (3.4.4)
Faces16 {3}, 12 {4}
Edges30 (2×12+6)
Vertices12 (2×6)
Vertex figure
Square pyramid
Dual polytopeCubic bipyramid
Symmetry[3,4,2], order 96
[3,3,2], order 48
[6,2+,2], order 24
[(3,2)+,2], order 12
Propertiesconvex, Hanner polytope
Net
Alternative names
• Octahedral dyadic prism (Norman W. Johnson)
• Ope (Jonathan Bowers, for octahedral prism)
• Triangular antiprismatic prism
• Triangular antiprismatic hyperprism
Coordinates
It is a Hanner polytope with vertex coordinates, permuting first 3 coordinates:
([±1,0,0]; ±1)
Structure
The octahedral prism consists of two octahedra connected to each other via 8 triangular prisms. The triangular prisms are joined to each other via their square faces.
Projections
The octahedron-first orthographic projection of the octahedral prism into 3D space has an octahedral envelope. The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces.
The triangular-prism-first orthographic projection of the octahedral prism into 3D space has a hexagonal prismic envelope. The two octahedral cells project onto the two hexagonal faces. One triangular prismic cell projects onto a triangular prism at the center of the envelope, surrounded by the images of 3 other triangular prismic cells to cover the entire volume of the envelope. The remaining four triangular prismic cells are projected onto the entire volume of the envelope as well, in the same arrangement, except with opposite orientation.
Related polytopes
It is the second in an infinite series of uniform antiprismatic prisms.
Convex p-gonal antiprismatic prisms
Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{}
Coxeter
diagram
Image
Vertex
figure
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}
2 s{2,3}
2 {3}×{}
6 {3}×{}
2 s{2,4}
2 {4}×{}
8 {3}×{}
2 s{2,5}
2 {5}×{}
10 {3}×{}
2 s{2,6}
2 {6}×{}
12 {3}×{}
2 s{2,7}
2 {7}×{}
14 {3}×{}
2 s{2,8}
2 {8}×{}
16 {3}×{}
2 s{2,p}
2 {p}×{}
2p {3}×{}
Net
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
It is one of four four-dimensional Hanner polytopes; the other three are the tesseract, the 16-cell, and the dual of the octahedral prism (a cubical bipyramid).[1]
References
1. "Hanner polytopes".
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
External links
• 6. Convex uniform prismatic polychora - Model 51, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x x3o4o - ope".
| Wikipedia |
Triangular bifrustum
In geometry, the triangular bifrustum is the second in an infinite series of bifrustum polyhedra. It has 6 trapezoid and 2 triangle faces. It may also be called the truncated triangular bipyramid; however, that term is ambiguous, as it may also refer to polyhedra formed by truncating all five vertices of a triangular bipyramid.[1]
Triangular bifrustum
TypeBifrustum
Faces6 trapezoids,
2 triangles
Edges15
Vertices9
Symmetry groupD3h
Dual polyhedronElongated triangular bipyramid
Propertiesconvex
Net
This polyhedron can be constructed by taking a triangular bipyramid and truncating the polar axis vertices, making it into two end-to-end frustums. It appears as the form of certain nanocrystals.[2][3]
A truncated triangular bipyramid can be constructed by connecting two stacked regular octahedra with 3 pairs of tetrahedra around the sides. This represents a portion of the gyrated alternated cubic honeycomb.
References
1. For instance, Haji-Akbari et al. use it in the latter sense: see Haji-Akbari, Amir; Chen, Elizabeth R.; Engel, Michael; Glotzer, Sharon C. (2013), "Packing and self-assembly of truncated triangular bipyramids", Phys. Rev. E, 88 (1): 012127, arXiv:1304.3147, Bibcode:2013PhRvE..88a2127H, doi:10.1103/physreve.88.012127, PMID 23944434, S2CID 8184675.
2. Kharisov, Boris I.; Kharissova, Oxana Vasilievna; Ortiz-Mendez, Ubaldo (2012), Handbook of Less-Common Nanostructures, CRC Press, p. 466, ISBN 9781439853436.
3. Yoo, Hyojong; Millstone, Jill E.; Li, Shuzhou; Jang, Jae-Won; Wei, Wei; Wu, Jinsong; Schatz, George C.; Mirkin, Chad A. (2009), "Core–Shell Triangular Bifrustums", Nano Letters, 9 (8): 3038–3041, Bibcode:2009NanoL...9.3038Y, doi:10.1021/nl901513g, PMC 3930336, PMID 19603815.
External links
• Conway Notation for Polyhedra Try: t3dP3
| Wikipedia |
Cactus graph
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cactus) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle.
Properties
Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge.
The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph K4.[1]
Triangular cactus
A triangular cactus is a special type of cactus graph such that each cycle has length three and each edge belongs to a cycle. For instance, the friendship graphs, graphs formed from a collection of triangles joined together at a single shared vertex, are triangular cacti. As well as being cactus graphs the triangular cacti are also block graphs and locally linear graphs.
Triangular cactuses have the property that they remain connected if any matching is removed from them; for a given number of vertices, they have the fewest possible edges with this property. Every tree with an odd number of vertices may be augmented to a triangular cactus by adding edges to it, giving a minimal augmentation with the property of remaining connected after the removal of a matching.[2]
The largest triangular cactus in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular cactus graphs are planar graphs, the largest triangular cactus can be used as an approximation to the largest planar subgraph, an important subproblem in planarization. As an approximation algorithm, this method has approximation ratio 4/9, the best known for the maximum planar subgraph problem.[3]
The algorithm for finding the largest triangular cactus is associated with a theorem of Lovász and Plummer that characterizes the number of triangles in this largest cactus.[4] Lovász and Plummer consider pairs of partitions of the vertices and edges of the given graph into subsets, with the property that every triangle of the graph either has two vertices in a single class of the vertex partition or all three edges in a single class of the edge partition; they call a pair of partitions with this property valid. Then the number of triangles in the largest triangular cactus equals the maximum, over pairs of valid partitions ${\mathcal {P}}=\{V_{1},V_{2},\dots ,V_{k}\}$ and ${\mathcal {Q}}=\{E_{1},E_{2},\dots ,E_{m}\}$, of
$\sum _{i=1}^{m}{\frac {(u_{i}-1)}{2}}+n-k,$,
where $n$ is the number of vertices in the given graph and $u_{i}$ is the number of vertex classes met by edge class $E_{i}$.
Every plane graph $G$ contains a cactus subgraph $C\subseteq G$ which includes at least $1/6$ fraction of the triangular faces of $G$. This result implies a direct analysis of the 4/9 - approximation algorithm for maximum planar subgraph problem without using the above min-max formula.[5]
Rosa's Conjecture
An important conjecture related to the triangular cactus is the Rosa's Conjecture, named after Alexander Rosa, which says that all triangular cacti are graceful or nearly-graceful.[6] More precisely
All triangular cacti with t ≡ 0, 1 mod 4 blocks are graceful, and those with t ≡ 2, 3 mod 4 are near graceful.
Algorithms and applications
Some facility location problems which are NP-hard for general graphs, as well as some other graph problems, may be solved in polynomial time for cacti.[7][8]
Since cacti are special cases of outerplanar graphs, a number of combinatorial optimization problems on graphs may be solved for them in polynomial time.[9]
Cacti represent electrical circuits that have useful properties. An early application of cacti was associated with the representation of op-amps.[10][11][12]
Cacti have also been used in comparative genomics as a way of representing the relationship between different genomes or parts of genomes.[13]
If a cactus is connected, and each of its vertices belongs to at most two blocks, then it is called a Christmas cactus. Every polyhedral graph has a Christmas cactus subgraph that includes all of its vertices, a fact that plays an essential role in a proof by Leighton & Moitra (2010) that every polyhedral graph has a greedy embedding in the Euclidean plane, an assignment of coordinates to the vertices for which greedy forwarding succeeds in routing messages between all pairs of vertices.[14]
In topological graph theory, the graphs whose cellular embeddings are all planar are exactly the subfamily of the cactus graphs with the additional property that each vertex belongs to at most one cycle. These graphs have two forbidden minors, the diamond graph and the five-vertex friendship graph.[15]
History
Cacti were first studied under the name of Husimi trees, bestowed on them by Frank Harary and George Eugene Uhlenbeck in honor of previous work on these graphs by Kôdi Husimi.[16][17] The same Harary–Uhlenbeck paper reserves the name "cactus" for graphs of this type in which every cycle is a triangle, but now allowing cycles of all lengths is standard.
Meanwhile, the name Husimi tree commonly came to refer to graphs in which every block is a complete graph (equivalently, the intersection graphs of the blocks in some other graph). This usage had little to do with the work of Husimi, and the more pertinent term block graph is now used for this family; however, because of this ambiguity this phrase has become less frequently used to refer to cactus graphs.[18]
References
1. El-Mallah, Ehab; Colbourn, Charles J. (1988), "The complexity of some edge deletion problems", IEEE Transactions on Circuits and Systems, 35 (3): 354–362, doi:10.1109/31.1748
2. Farley, Arthur M.; Proskurowski, Andrzej (1982), "Networks immune to isolated line failures", Networks, 12 (4): 393–403, doi:10.1002/net.3230120404, MR 0686540
3. Călinescu, Gruia; Fernandes, Cristina G; Finkler, Ulrich; Karloff, Howard (2002), "A Better Approximation Algorithm for Finding Planar Subgraphs", Journal of Algorithms, 2, 27 (2): 269–302, CiteSeerX 10.1.1.47.4731, doi:10.1006/jagm.1997.0920, S2CID 8329680
4. Lovász, L.; Plummer, M.D. (2009), Matching Theory, AMS Chelsea Publishing Series, ISBN 9780821847596
5. Chalermsook, Parinya; Schmid, Andreas; Uniyal, Sumedha (2019), "A tight extremal bound on the Lovász cactus number in planar graphs", in Niedermeier, Rolf; Paul, Christophe (eds.), 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany, LIPIcs, vol. 126, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 19:1–19:14, arXiv:1804.03485, doi:10.4230/LIPIcs.STACS.2019.19, ISBN 9783959771009, S2CID 4751972
6. Rosa, A. (1988), "Cyclic Steiner Triple Systems and Labelings of Triangular Cacti", Scientia, 1: 87–95.
7. Ben-Moshe, Boaz; Bhattacharya, Binay; Shi, Qiaosheng (2005), "Efficient algorithms for the weighted 2-center problem in a cactus graph", Algorithms and Computation, 16th Int. Symp., ISAAC 2005, Lecture Notes in Computer Science, vol. 3827, Springer-Verlag, pp. 693–703, doi:10.1007/11602613_70, ISBN 978-3-540-30935-2
8. Zmazek, Blaz; Zerovnik, Janez (2005), "Estimating the traffic on weighted cactus networks in linear time", Ninth International Conference on Information Visualisation (IV'05), pp. 536–541, doi:10.1109/IV.2005.48, ISBN 978-0-7695-2397-2, S2CID 15963409
9. Korneyenko, N. M. (1994), "Combinatorial algorithms on a class of graphs", Discrete Applied Mathematics, 54 (2–3): 215–217, doi:10.1016/0166-218X(94)90022-1. Translated from Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci., (1984) no. 3, pp. 109-111 (in Russian)
10. Nishi, Tetsuo; Chua, Leon O. (1986), "Topological proof of the Nielsen-Willson theorem", IEEE Transactions on Circuits and Systems, 33 (4): 398–405, doi:10.1109/TCS.1986.1085935
11. Nishi, Tetsuo; Chua, Leon O. (1986), "Uniqueness of solution for nonlinear resistive circuits containing CCCS's or VCVS's whose controlling coefficients are finite", IEEE Transactions on Circuits and Systems, 33 (4): 381–397, doi:10.1109/TCS.1986.1085934
12. Nishi, Tetsuo (1991), "On the number of solutions of a class of nonlinear resistive circuit", Proceedings of the IEEE International Symposium on Circuits and Systems, Singapore, pp. 766–769
13. Paten, Benedict; Diekhans, Mark; Earl, Dent; St. John, John; Ma, Jian; Suh, Bernard; Haussler, David (2010), "Cactus Graphs for Genome Comparisons", Research in Computational Molecular Biology, Lecture Notes in Computer Science, vol. 6044, pp. 410–425, doi:10.1007/978-3-642-12683-3_27, ISBN 978-3-642-12682-6
14. Leighton, Tom; Moitra, Ankur (2010), "Some Results on Greedy Embeddings in Metric Spaces" (PDF), Discrete & Computational Geometry, 44 (3): 686–705, doi:10.1007/s00454-009-9227-6, S2CID 11186402.
15. Nordhaus, E. A.; Ringeisen, R. D.; Stewart, B. M.; White, A. T. (1972), "A Kuratowski-type theorem for the maximum genus of a graph", Journal of Combinatorial Theory, Series B, 12 (3): 260–267, doi:10.1016/0095-8956(72)90040-8, MR 0299523
16. Harary, Frank; Uhlenbeck, George E. (1953), "On the number of Husimi trees, I", Proceedings of the National Academy of Sciences, 39 (4): 315–322, Bibcode:1953PNAS...39..315H, doi:10.1073/pnas.39.4.315, MR 0053893, PMC 1063779, PMID 16589268
17. Husimi, Kodi (1950), "Note on Mayers' theory of cluster integrals", Journal of Chemical Physics, 18 (5): 682–684, Bibcode:1950JChPh..18..682H, doi:10.1063/1.1747725, MR 0038903
18. See, e.g., MR0659742, a 1983 review by Robert E. Jamison of a paper using the other definition, which attributes the ambiguity to an error in a book by Mehdi Behzad and Gary Chartrand.
External links
• Application of Cactus Graphs in Analysis and Design of Electronic Circuits
| Wikipedia |
Triangular cupola
In geometry, the triangular cupola is one of the Johnson solids (J3). It can be seen as half a cuboctahedron.
Triangular cupola
TypeJohnson
J2 – J3 – J4
Faces4 triangles
3 squares
1 hexagon
Edges15
Vertices9
Vertex configuration6(3.4.6)
3(3.4.3.4)
Symmetry groupC3v
Dual polyhedronhttps://levskaya.github.io/polyhedronisme/?recipe=C1000dJ3
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Formulae
The following formulae for the volume ($V$), the surface area ($A$) and the height ($H$) can be used if all faces are regular, with edge length a:[2][3]
$V=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\approx 1.17851...a^{3}$
$A=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\approx 7.33013...a^{2}$
$H={\frac {\sqrt {6}}{3}}a\approx 0.816496...a$
Dual polyhedron
The dual of the triangular cupola has 6 triangular and 3 kite faces:
Dual triangular cupola Net of dual
Related polyhedra and honeycombs
The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangles are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces.
The triangular cupola can form a tessellation of space with square pyramids and/or octahedra,[4] the same way octahedra and cuboctahedra can fill space.
The family of cupolae with regular polygons exists up to n=5 (pentagons), and higher if isosceles triangles are used in the cupolae.
Family of convex cupolae
n2345678
Schläfli symbol{2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6} {7} || t{7} {8} || t{8}
Cupola
Digonal cupola
Triangular cupola
Square cupola
Pentagonal cupola
Hexagonal cupola
(Flat)
Heptagonal cupola
(Non-regular face)
Octagonal cupola
(Non-regular face)
Related
uniform
polyhedra
Rhombohedron
Cuboctahedron
Rhombicuboctahedron
Rhombicosidodecahedron
Rhombitrihexagonal tiling
Rhombitriheptagonal tiling
Rhombitrioctagonal tiling
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Stephen Wolfram, "Triangular cupola" from Wolfram Alpha. Retrieved July 20, 2010.
3. Sapiña, R. "Area and volume of the Johnson solid J₃". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-08.
4. "J3 honeycomb".
External links
• Eric W. Weisstein, Triangular cupola (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
Triangular decomposition
In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one of the systems S1, ..., Se.
When the purpose is to describe the solution set of S in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficients of the polynomial systems S1, ..., Se are real numbers, then the real solutions of S can be obtained by a triangular decomposition into regular semi-algebraic systems. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology.
History
The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author, Wen-Tsun Wu, realized an implementation of this method and reported experimental data in his 1987 pioneer article titled "A zero structure theorem for polynomial equations solving".[1] To put this work into context, let us recall what was the common idea of an algebraic set decomposition at the time this article was written.
Let K be an algebraically closed field and k be a subfield of K. A subset V ⊂ Kn is an (affine) algebraic variety over k if there exists a polynomial set F ⊂ k[x1, ..., xn] such that the zero set V(F) ⊂ Kn of F equals V.
Recall that V is said irreducible if for all algebraic varieties V1, V2 ⊂ Kn the relation V = V1 ∪ V2 implies either V = V1 or V = V2. A first algebraic variety decomposition result is the famous Lasker–Noether theorem which implies the following.
Theorem (Lasker - Noether). For each algebraic variety V ⊂ Kn there exist finitely many irreducible algebraic varieties V1, ..., Ve ⊂ Kn such that we have
$V=V_{1}\cup \cdots \cup V_{e}.$
Moreover, if Vi ⊈ Vj holds for 1 ≤ i < j ≤ e then the set {V1, ..., Ve} is unique and forms the irreducible decomposition of V.
The varieties V1, ..., Ve in the above Theorem are called the irreducible components of V and can be regarded as a natural output for a decomposition algorithm, or, in other words, for an algorithm solving a system of equations in k[x1, ..., xn].
In order to lead to a computer program, this algorithm specification should prescribe how irreducible components are represented. Such an encoding is introduced by Joseph Ritt[2] through the following result.
Theorem (Ritt). If V ⊂ Kn is a non-empty and irreducible variety then one can compute a reduced triangular set C contained in the ideal $\langle F\rangle $ generated by F in k[x1, ..., xn] and such that all polynomials g in $\langle F\rangle $ reduces to zero by pseudo-division w.r.t C.
We call the set C in Ritt's Theorem a Ritt characteristic set of the ideal $\langle F\rangle $. Please refer to regular chain for the notion of a triangular set.
Joseph Ritt described a method for solving polynomial systems based on polynomial factorization over field extensions and computation of characteristic sets of prime ideals.
Deriving a practical implementation of this method, however, was and remains a difficult problem. In the 1980s, when the Characteristic set Method was introduced, polynomial factorization was an active research area and certain fundamental questions on this subject were solved recently[3]
Nowadays, decomposing an algebraic variety into irreducible components is not essential to process most application problems, since weaker notions of decompositions, less costly to compute, are sufficient.
The Characteristic Set Method relies on the following variant of Ritt's Theorem.
Theorem (Wen-Tsun Wu). For any finite polynomial set F ⊂ k[x1, ..., xn], one can compute a reduced triangular set $C\subset \langle F\rangle $ such that all polynomial g in F reduces to zero by pseudo-division w.r.t C.
Different concepts and algorithms extended the work of Wen-Tsun Wu. In the early 1990s, the notion of a regular chain, introduced independently by Michael Kalkbrener in 1991 in his PhD Thesis and, by Lu Yang and Jingzhong Zhang[4] led to important algorithmic discoveries.
In Kalkbrener's vision,[5] regular chains are used to represent the generic zeros of the irreducible components of an algebraic variety. In the original work of Yang and Zhang, they are used to decide whether a hypersurface intersects a quasi-variety (given by a regular chain). Regular chains have, in fact, several interesting properties and are the key notion in many algorithms for decomposing systems of algebraic or differential equations.
Regular chains have been investigated in many papers.[6][7][8]
The abundant literature on the subject can be explained by the many equivalent definitions of a regular chain. Actually, the original formulation of Kalkbrener is quite different from that of Yang and Zhang. A bridge between these two notions, the point of view of Kalkbrener and that of Yang and Zhang, appears in Dongming Wang's paper.[9]
There are various algorithms available for obtaining triangular decomposition of V(F) both in the sense of Kalkbrener and in the sense of Lazard and Wen-Tsun Wu. The Lextriangular Algorithm by Daniel Lazard[10] and the Triade Algorithm by Marc Moreno Maza[11] together with the Characteristic Set Method are available in various computer algebra systems, including Axiom and Maple.
Formal definitions
Let k be a field and x1 < ... < xn be ordered variables. We denote by R = k[x1, ..., xn] the corresponding polynomial ring. For F ⊂ R, regarded as a system of polynomial equations, there are two notions of a triangular decomposition over the algebraic closure of k. The first one is to decompose lazily, by representing only the generic points of the algebraic set V(F) in the so-called sense of Kalkbrener.
${\sqrt {(F)}}=\bigcap _{i=1}^{e}{\sqrt {\mathrm {sat} (T_{i})}}.$
The second is to describe explicitly all the points of V(F) in the so-called sense of in Lazard and Wen-Tsun Wu.
$V(F)=\bigcup _{i=1}^{e}W(T_{i}).$
In both cases T1, ..., Te are finitely many regular chains of R and ${\sqrt {\mathrm {sat} (T_{i})}}$ denotes the radical of the saturated ideal of Ti while W(Ti) denotes the quasi-component of Ti. Please refer to regular chain for definitions of these notions.
Assume from now on that k is a real closed field. Consider S a semi-algebraic system with polynomials in R. There exist[12] finitely many regular semi-algebraic systems S1, ..., Se such that we have
$Z_{\mathbf {k} }(S)=Z_{\mathbf {k} }(S_{1})\cup \cdots \cup Z_{\mathbf {k} }(S_{e})$
where Zk(S) denotes the points of kn which solve S. The regular semi-algebraic systems S1, ..., Se form a triangular decomposition of the semi-algebraic system S.
Examples
Denote Q the rational number field. In $Q[x,y,z]$ with variable ordering $x>y>z$, consider the following polynomial system:
$S={\begin{cases}x^{2}+y+z=1\\x+y^{2}+z=1\\x+y+z^{2}=1\end{cases}}$
According to the Maple code:
with(RegularChains):
R := PolynomialRing([x, y, z]):
sys := {x^2+y+z-1, x+y^2+z-1, x+y+z^2-1}:
l := Triangularize(sys, R):
map(Equations, l, R);
One possible triangular decompositions of the solution set of S with using RegularChains library is:
${\begin{cases}z=0\\y=1\\x=0\end{cases}}\cup {\begin{cases}z=0\\y=0\\x=1\end{cases}}\cup {\begin{cases}z=1\\y=0\\x=0\end{cases}}\cup {\begin{cases}z^{2}+2z-1=0\\y=z\\x=z\end{cases}}$
See also
• Wu's method of characteristic set
• Regular chain
• Regular semi-algebraic system
References
1. Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research Preprints, 1, 2–12
2. Ritt, J. (1966). Differential Algebra. New York, Dover Publications
3. A. M. Steel Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic
4. Yang, L., Zhang, J. (1994). Searching dependency between algebraic equations: an algorithm applied to automated reasoning. Artificial Intelligence in Mathematics, pp. 14715, Oxford University Press.
5. M. Kalkbrener: A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties. J. Symb. Comput. 15(2): 143–167 (1993)
6. S.C. Chou and X.S. Gao. On the dimension of an arbitrary ascending chain. Chinese Bull. of Sci., 38:799--804, 1991.
7. Michael Kalkbrener. Algorithmic properties of polynomial rings. J. Symb. Comput.}, 26(5):525--581, 1998.
8. P. Aubry, D. Lazard, M. Moreno Maza. On the theories of triangular sets. Journal of Symbolic Computation, 28(1–2):105–124, 1999.
9. D. Wang. Computing Triangular Systems and Regular Systems. Journal of Symbolic Computation 30(2) (2000) 221–236
10. D. Lazard, Solving zero-dimensional algebraic systems. Journal of Symbolic Computation 13, 1992
11. M. Moreno Maza: On triangular decomposition of algebraic varieties. MEGA 2000 (2000).
12. Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao. Triangular decomposition of semi-algebraic systems. Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187--194, 2010.
| Wikipedia |
Triangular function
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as the triangular function. Triangular functions are useful in signal processing and communication systems engineering as representations of idealized signals, and the triangular function specifically as an integral transform kernel function from which more realistic signals can be derived, for example in kernel density estimation. It also has applications in pulse-code modulation as a pulse shape for transmitting digital signals and as a matched filter for receiving the signals. It is also used to define the triangular window sometimes called the Bartlett window.
Not to be confused with Trigonometric functions or Schwarz triangle function.
Definitions
The most common definition is as a piecewise function:
${\begin{aligned}\operatorname {tri} (x)=\Lambda (x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-|x|,0{\big )}\\&={\begin{cases}1-|x|,&|x|<1;\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}$
Equivalently, it may be defined as the convolution of two identical unit rectangular functions:
${\begin{aligned}\operatorname {tri} (x)&=\operatorname {rect} (x)*\operatorname {rect} (x)\\&=\int _{-\infty }^{\infty }\operatorname {rect} (x-\tau )\cdot \operatorname {rect} (\tau )\,d\tau .\\\end{aligned}}$
The triangular function can also be represented as the product of the rectangular and absolute value functions:
$\operatorname {tri} (x)=\operatorname {rect} (x/2){\big (}1-|x|{\big )}.$
Note that some authors instead define the triangle function to have a base of width 1 instead of width 2:
${\begin{aligned}\operatorname {tri} (2x)=\Lambda (2x)\ &{\overset {\underset {\text{def}}{}}{=}}\ \max {\big (}1-2|x|,0{\big )}\\&={\begin{cases}1-2|x|,&|x|<{\tfrac {1}{2}};\\0&{\text{otherwise}}.\\\end{cases}}\end{aligned}}$
In its most general form a triangular function is any linear B-spline:[1]
$\operatorname {tri} _{j}(x)={\begin{cases}(x-x_{j-1})/(x_{j}-x_{j-1}),&x_{j-1}\leq x<x_{j};\\(x_{j+1}-x)/(x_{j+1}-x_{j}),&x_{j}\leq x<x_{j+1};\\0&{\text{otherwise}}.\end{cases}}$
Whereas the definition at the top is a special case
$\Lambda (x)=\operatorname {tri} _{j}(x),$
where $x_{j-1}=-1$, $x_{j}=0$, and $x_{j+1}=1$.
A linear B-spline is the same as a continuous piecewise linear function $f(x)$, and this general triangle function is useful to formally define $f(x)$ as
$f(x)=\sum _{j}y_{j}\cdot \operatorname {tri} _{j}(x),$
where $x_{j}<x_{j+1}$ for all integer $j$. The piecewise linear function passes through every point expressed as coordinates with ordered pair $(x_{j},y_{j})$, that is,
$f(x_{j})=y_{j}$.
Scaling
For any parameter $a\neq 0$:
${\begin{aligned}\operatorname {tri} \left({\tfrac {t}{a}}\right)&=\int _{-\infty }^{\infty }{\tfrac {1}{|a|}}\operatorname {rect} \left({\tfrac {\tau }{a}}\right)\cdot \operatorname {rect} \left({\tfrac {t-\tau }{a}}\right)\,d\tau \\&={\begin{cases}1-|t/a|,&|t|<|a|;\\0&{\text{otherwise}}.\end{cases}}\end{aligned}}$
Fourier transform
The transform is easily determined using the convolution property of Fourier transforms and the Fourier transform of the rectangular function:
${\begin{aligned}{\mathcal {F}}\{\operatorname {tri} (t)\}&={\mathcal {F}}\{\operatorname {rect} (t)*\operatorname {rect} (t)\}\\&={\mathcal {F}}\{\operatorname {rect} (t)\}\cdot {\mathcal {F}}\{\operatorname {rect} (t)\}\\&={\mathcal {F}}\{\operatorname {rect} (t)\}^{2}\\&=\mathrm {sinc} ^{2}(f),\end{aligned}}$
where $\operatorname {sinc} (x)=\sin(\pi x)/(\pi x)$ is the normalized sinc function.
See also
• Källén function, also known as triangle function
• Tent map
• Triangular distribution
• Triangle wave, a piecewise linear periodic function
• Trigonometric functions
References
1. "Basic properties of splines and B-splines" (PDF). INF-MAT5340 Lecture Notes. p. 38.
| Wikipedia |
Triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.
Triangular tiling
TypeRegular tiling
Vertex configuration3.3.3.3.3.3 (or 36)
Face configurationV6.6.6 (or V63)
Schläfli symbol(s){3,6}
{3[3]}
Wythoff symbol(s)6 | 3 2
3 | 3 3
| 3 3 3
Coxeter diagram(s)
=
Symmetryp6m, [6,3], (*632)
Rotation symmetryp6, [6,3]+, (632)
p3, [3[3]]+, (333)
DualHexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille.
It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.
Uniform colorings
There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.[1]
There is one class of Archimedean colorings, 111112, (marked with a *) which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.
111111 121212 111222 112122 111112(*)
p6m (*632) p3m1 (*333) cmm (2*22) p2 (2222) p2 (2222)
121213 111212 111112 121314 111213
p31m (3*3) p3 (333)
A2 lattice and circle packings
The vertex arrangement of the triangular tiling is called an A2 lattice.[2] It is the 2-dimensional case of a simplectic honeycomb.
The A*
2
lattice (also called A3
2
) can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice.
+ + = dual of =
The vertices of the triangular tiling are the centers of the densest possible circle packing.[3] Every circle is in contact with 6 other circles in the packing (kissing number). The packing density is π⁄√12 or 90.69%. The voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling, has a direct correspondence to the circle packings.
Geometric variations
Triangular tilings can be made with the equivalent {3,6} topology as the regular tiling (6 triangles around every vertex). With identical faces (face-transitivity) and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color.[4]
• Scalene triangle
p2 symmetry
• Scalene triangle
pmg symmetry
• Isosceles triangle
cmm symmetry
• Right triangle
cmm symmetry
• Equilateral triangle
p6m symmetry
Related polyhedra and tilings
The planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid. These can be expanded to Platonic solids: five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
*n32 symmetry mutation of regular tilings: {3,n}
Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i
It is also topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, and also continuing into the hyperbolic plane.
V3.6.6
V4.6.6
V5.6.6
V6.6.6
V7.6.6
Wythoff constructions from hexagonal and triangular tilings
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Uniform hexagonal/triangular tilings
Fundamental
domains
Symmetry: [6,3], (*632) [6,3]+, (632)
{6,3} t{6,3} r{6,3} t{3,6} {3,6} rr{6,3} tr{6,3} sr{6,3}
Config. 63 3.12.12 (6.3)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6
Triangular symmetry tilings
Wythoff 3 | 3 3 3 3 | 3 3 | 3 3 3 3 | 3 3 | 3 3 3 3 | 3 3 3 3 | | 3 3 3
Coxeter
Image
Vertex figure
(3.3)3
3.6.3.6
(3.3)3
3.6.3.6
(3.3)3
3.6.3.6
6.6.6
3.3.3.3.3.3
Related regular complex apeirogons
There are 4 regular complex apeirogons, sharing the vertices of the triangular tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.[5]
The first is made of 2-edges, and next two are triangular edges, and the last has overlapping hexagonal edges.
2{6}6 or 3{4}6 or 3{6}3 or 6{3}6 or
Other triangular tilings
There are also three Laves tilings made of single type of triangles:
Kisrhombille
30°-60°-90° right triangles
Kisquadrille
45°-45°-90° right triangles
Kisdeltile
30°-30°-120° isosceles triangles
See also
Wikimedia Commons has media related to Order-6 triangular tiling.
• Triangular tiling honeycomb
• Simplectic honeycomb
• Tilings of regular polygons
• List of uniform tilings
• Isogrid (structural design using triangular tiling)
References
1. Tilings and Patterns, p.102-107
2. "The Lattice A2".
3. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1
4. Tilings and Patterns, from list of 107 isohedral tilings, p.473-481
5. Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Grünbaum, Branko & Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65, Chapter 2.9 Archimedean and Uniform colorings pp. 102–107)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p35
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
External links
• Weisstein, Eric W. "Triangular Grid". MathWorld.
• Weisstein, Eric W. "Regular tessellation". MathWorld.
• Weisstein, Eric W. "Uniform tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings x3o6o - trat - O2".
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Lattice graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space $\mathbb {R} ^{n}$, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense.
Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid".
The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs.[1]
Square grid graph
A common type of a lattice graph (known under different names, such as square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 1, ..., n, y-coordinates being in the range 1, ..., m, and two vertices are connected by an edge whenever the corresponding points are at distance 1. In other words, it is a unit distance graph for the described point set.[2]
Properties
A square grid graph is a Cartesian product of graphs, namely, of two path graphs with n − 1 and m − 1 edges.[2] Since a path graph is a median graph, the latter fact implies that the square grid graph is also a median graph. All square grid graphs are bipartite, which is easily verified by the fact that one can color the vertices in a checkerboard fashion.
A path graph may also be considered to be a grid graph on the grid n times 1. A 2 × 2 grid graph is a 4-cycle.[2]
Every planar graph H is a minor of the h × h grid, where $h=2|V(H)|+4|E(H)|$.[3]
Grid graphs are fundamental objects in the theory of graph minors because of the grid exclusion theorem. They play a major role in bidimensionality theory.
Other kinds
A triangular grid graph is a graph that corresponds to a triangular grid.
A Hanan grid graph for a finite set of points in the plane is produced by the grid obtained by intersections of all vertical and horizontal lines through each point of the set.
The rook's graph (the graph that represents all legal moves of the rook chess piece on a chessboard) is also sometimes called the lattice graph, although this graph is strictly different than the lattice graph described in this article. The valid moves of fairy chess piece wazir form the square lattice graph.
See also
• Lattice path
• Pick's theorem
• Integer triangles in a 2D lattice
• Regular graph
References
1. Weisstein, Eric W. "Lattice graph". MathWorld.
2. Weisstein, Eric W. "Grid graph". MathWorld.
3. Robertson, N.; Seymour, P.; Thomas, R. (November 1994). "Quickly Excluding a Planar Graph". Journal of Combinatorial Theory, Series B. 62 (2): 323–348. doi:10.1006/jctb.1994.1073.
| Wikipedia |
Triangular hebesphenorotunda
In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J92).
Triangular hebesphenorotunda
TypeJohnson
J91 – J92 – J1
Faces13 triangles
3 squares
3 pentagons
1 hexagon
Edges36
Vertices18
Vertex configuration3(33.5)
6(3.4.3.5)
3(3.5.3.5)
2.3(32.4.6)
Symmetry groupC3v
Dual polyhedron-
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron.
The triangular hebesphenorotunda also has clusters of faces that can be aligned with corresponding faces of the rhombicosidodecahedron: the three lunes, each lune consisting of a square and two antipodal triangles adjacent to the square.
The faces around each (33.5) vertex can also be aligned with the corresponding faces of various diminished icosahedra.
Johnson uses the prefix hebespheno- to refer to a blunt wedge-like complex formed by three adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. The suffix (triangular) -rotunda refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears structural resemblance to the pentagonal rotunda.[1]
The triangular hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides.
Cartesian coordinates
Cartesian coordinates for the triangular hebesphenorotunda with edge length √5 – 1 are given by the union of the orbits of the points
$\left(0,-{\frac {2}{\tau {\sqrt {3}}}},{\frac {2\tau }{\sqrt {3}}}\right),\left(\tau ,{\frac {1}{{\sqrt {3}}\tau ^{2}}},{\frac {2}{\sqrt {3}}}\right),$
$\left(\tau ,-{\frac {\tau }{\sqrt {3}}},{\frac {2}{{\sqrt {3}}\tau }}\right),\left({\frac {2}{\tau }},0,0\right),$
under the action of the group generated by rotation by 120° around the z-axis and the reflection about the yz-plane.[2] Here, 𝜏 = √5 + 1/2 (sometimes written φ) is the golden ratio. The first point generates the triangle opposite the hexagon, the second point generates the bases of the triangles surrounding the previous triangle, the third point generates the tips of the pentagons opposite the first triangle, and the last point generates the hexagon.
One may then calculate the surface area of a triangular hebesphenorotunda of edge length a as
$A=\left(3+{\frac {1}{4}}{\sqrt {1308+90{\sqrt {5}}+114{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\approx 16.38867a^{2},$[3]
and its volume as
$V={\frac {1}{6}}\left(15+7{\sqrt {5}}\right)a^{3}\approx 5.10875a^{3}.$[4]
A second, inverted, triangular hebesphenorotunda can be obtained by negating the second and third coordinates of each point. This second polyhedron will be joined to the first at their common hexagonal face, and the pair will inscribe an icosidodecahedron. If the hexagonal face is scaled by the golden ratio, then the convex hull of the result will be the entire icosidodecahedron.
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, S2CID 122006114, Zbl 0132.14603.
2. Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 717. doi:10.1007/s10958-009-9655-0. S2CID 120114341.
3. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 92}, "SurfaceArea"] {{cite journal}}: Cite journal requires |journal= (help)
4. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 92}, "Volume"] {{cite journal}}: Cite journal requires |journal= (help)
External links
• Eric W. Weisstein, Triangular hebesphenorotunda (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
Hexagonal lattice
The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types.[1] The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths,
$|\mathbf {a} _{1}|=|\mathbf {a} _{2}|=a.$
Not to be confused with Hexagonal crystal family.
Hexagonal lattice Wallpaper group p6m Unit cell
The reciprocal lattice of the hexagonal lattice is a hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length
$g={\frac {4\pi }{a{\sqrt {3}}}}.$
Honeycomb point set
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis.[1] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb point set.
Crystal classes
The hexagonal lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
Geometric class, point group Arithmetic
class
Wallpaper groups
Schön.IntlOrb.Cox.
C33(33)[3]+ None p3
(333)
D33m(*33)[3] Between p3m1
(*333)
p31m
(3*3)
C66(66)[6]+ None p6
(632)
D66mm(*66)[6] Both p6m
(*632)
See also
• Square lattice
• Hexagonal tiling
• Close-packing
• Centered hexagonal number
• Eisenstein integer
• Voronoi diagram
References
1. Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18.
Crystal systems
• Bravais lattice
• Crystallographic point group
Seven 3D systems
• triclinic (anorthic)
• monoclinic
• orthorhombic
• tetragonal
• trigonal & hexagonal
• cubic (isometric)
Four 2D systems
• oblique
• rectangular
• square
• hexagonal
Wikimedia Commons has media related to Hexagonal lattices.
| Wikipedia |
Triangular matrix ring
In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.
Definition
If $T$ and $U$ are rings and $M$ is a $\left(U,T\right)$-bimodule, then the triangular matrix ring $R:=\left[{\begin{array}{cc}T&0\\M&U\\\end{array}}\right]$ consists of 2-by-2 matrices of the form $\left[{\begin{array}{cc}t&0\\m&u\\\end{array}}\right]$, where $t\in T,m\in M,$ and $u\in U,$ with ordinary matrix addition and matrix multiplication as its operations.
References
• Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422
| Wikipedia |
Triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...
(sequence A000217 in the OEIS)
Formula
The triangular numbers are given by the following explicit formulas:
$T_{n}=\sum _{k=1}^{n}k=1+2+3+\dotsb +n={\frac {n(n+1)}{2}}={\frac {n^{2}+n}{2}},{n+1 \choose 2}$
where $\textstyle {n+1 \choose 2}$, does not mean division, but is the notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".
The first equation can be illustrated using a visual proof.[1] For every triangular number $T_{n}$, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions $n\times (n+1)$, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: $T_{n}={\frac {n(n+1)}{2}}$. The example $T_{4}$ follows:
$2T_{4}=4(4+1)=20$ (green plus yellow) implies that $T_{4}={\frac {4(4+1)}{2}}=10$ (green).
This formula can be proven formally using mathematical induction.[2] It is clearly true for $1$:
$T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.$
Now assume that, for some natural number $m$, $T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}$. Adding $m+1$ to this yields
${\begin{aligned}\sum _{k=1}^{m}k+(m+1)&={\frac {m(m+1)}{2}}+m+1\\&={\frac {m(m+1)+2m+2}{2}}\\&={\frac {m^{2}+m+2m+2}{2}}\\&={\frac {m^{2}+3m+2}{2}}\\&={\frac {(m+1)(m+2)}{2}},\end{aligned}}$
so if the formula is true for $m$, it is true for $m+1$. Since it is clearly true for $1$, it is therefore true for $2$, $3$, and ultimately all natural numbers $n$ by induction.
The German mathematician and scientist, Carl Friedrich Gauss, is said to have found this relationship in his early youth, by multiplying n/2 pairs of numbers in the sum by the values of each pair n + 1.[3] However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans in the 5th century BC.[4] The two formulas were described by the Irish monk Dicuil in about 816 in his Computus.[5] An English translation of Dicuil's account is available.[6]
The triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of n people is Tn−1.[7] The function T is the additive analog of the factorial function, which is the products of integers from 1 to n.
This same function was coined as the "Termial function"[8] by Donald Knuth's The Art of Computer Programming and denoted n? (analog for the factorial notation n!)
For example, 10 termial is equivalent to:
$10?=1+2+3+4+5+6+7+8+9+10=55$
which of course, corresponds to the tenth triangular number.
The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation:
$L_{n}=3T_{n-1}=3{n \choose 2};~~~L_{n}=L_{n-1}+3(n-1),~L_{1}=0.$
In the limit, the ratio between the two numbers, dots and line segments is
$\lim _{n\to \infty }{\frac {T_{n}}{L_{n}}}={\frac {1}{3}}.$
Relations to other figurate numbers
Triangular numbers have a wide variety of relations to other figurate numbers.
Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically,
$T_{n}+T_{n-1}=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {\left(n-1\right)^{2}}{2}}+{\frac {n-1}{2}}\right)=\left({\frac {n^{2}}{2}}+{\frac {n}{2}}\right)+\left({\frac {n^{2}}{2}}-{\frac {n}{2}}\right)=n^{2}=(T_{n}-T_{n-1})^{2}.$
This fact can be demonstrated graphically by positioning the triangles in opposite directions to create a square:
6 + 10 = 16 10 + 15 = 25
The double of a triangular number, as in the visual proof from the above section § Formula, is called a pronic number.
There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula:
$S_{n+1}=4S_{n}\left(8S_{n}+1\right)$
with $S_{1}=1.$
All square triangular numbers are found from the recursion
$S_{n}=34S_{n-1}-S_{n-2}+2$
with $S_{0}=0$ and $S_{1}=1.$
Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n. This can also be expressed as
$\sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.$
The sum of the first n triangular numbers is the nth tetrahedral number:
$\sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}={\frac {n(n+1)(n+2)}{6}}.$
More generally, the difference between the nth m-gonal number and the nth (m + 1)-gonal number is the (n − 1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number; the nth centered k-gonal number is obtained by the formula
$Ck_{n}=kT_{n-1}+1$
where T is a triangular number.
The positive difference of two triangular numbers is a trapezoidal number.
The pattern found for triangular numbers $\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}$ and for tetrahedral numbers $\sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},$ which uses binomial coefficients, can be generalized. This leads to the formula:[9]
$\sum _{n_{k-1}=1}^{n_{k}}\sum _{n_{k-2}=1}^{n_{k-1}}\dots \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{k}+k-1}{k}}$
Other properties
Triangular numbers correspond to the first-degree case of Faulhaber's formula.
Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.
Every even perfect number is triangular (as well as hexagonal), given by the formula
$M_{p}2^{p-1}={\frac {M_{p}(M_{p}+1)}{2}}=T_{M_{p}}$
where Mp is a Mersenne prime. No odd perfect numbers are known; hence, all known perfect numbers are triangular.
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
0 = 9 × 0
1 = 9 × 0 + 1
3 = 9 × 0 + 3
6 = 9 × 0 + 6
10 = 9 × 1 + 1
15 = 9 × 1 + 6
21 = 9 × 2 + 3
28 = 9 × 3 + 1
36 = 9 × 4
45 = 9 × 5
55 = 9 × 6 + 1
66 = 9 × 7 + 3
78 = 9 × 8 + 6
91 = 9 × 10 + 1
...
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".
The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
If x is a triangular number, then ax + b is also a triangular number, given a is an odd square and b = a − 1/8. Note that b will always be a triangular number, because 8Tn + 1 = (2n + 1)2, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd square is the inverse of this operation. The first several pairs of this form (not counting 1x + 0) are: 9x + 1, 25x + 3, 49x + 6, 81x + 10, 121x + 15, 169x + 21, ... etc. Given x is equal to Tn, these formulas yield T3n + 1, T5n + 2, T7n + 3, T9n + 4, and so on.
The sum of the reciprocals of all the nonzero triangular numbers is
$\sum _{n=1}^{\infty }{1 \over {{n^{2}+n} \over 2}}=2\sum _{n=1}^{\infty }{1 \over {n^{2}+n}}=2.$
This can be shown by using the basic sum of a telescoping series:
$\sum _{n=1}^{\infty }{1 \over {n(n+1)}}=1.$
Two other formulas regarding triangular numbers are
$T_{a+b}=T_{a}+T_{b}+ab$
and
$T_{ab}=T_{a}T_{b}+T_{a-1}T_{b-1},$
both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.
In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including T0 = 0), writing in his diary his famous words, "ΕΥΡΗΚΑ! num = Δ + Δ + Δ". This theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10 + 0), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the Fermat polygonal number theorem.
The largest triangular number of the form 2k − 1 is 4095 (see Ramanujan–Nagell equation).
Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression. It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible and was later proven by Fang and Chen in 2007.[10][11]
Formulas involving expressing an integer as the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function.[12][13]
Applications
A fully connected network of n computing devices requires the presence of Tn − 1 cables or other connections; this is equivalent to the handshake problem mentioned above.
In a tournament format that uses a round-robin group stage, the number of matches that need to be played between n teams is equal to the triangular number Tn − 1. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.
One way of calculating the depreciation of an asset is the sum-of-years' digits method, which involves finding Tn, where n is the length in years of the asset's useful life. Each year, the item loses (b − s) × n − y/Tn, where b is the item's beginning value (in units of currency), s is its final salvage value, n is the total number of years the item is usable, and y the current year in the depreciation schedule. Under this method, an item with a usable life of n = 4 years would lose 4/10 of its "losable" value in the first year, 3/10 in the second, 2/10 in the third, and 1/10 in the fourth, accumulating a total depreciation of 10/10 (the whole) of the losable value.
Board game designers Geoffrey Engelstein and Isaac Shalev describe triangular numbers as having achieved "nearly the status of a mantra or koan among game designers", describing them as "deeply intuitive" and "featured in an enormous number of games, [proving] incredibly versatile at providing escalating rewards for larger sets without overly incentivizing specialization to the exclusion of all other strategies".[14]
Triangular roots and tests for triangular numbers
By analogy with the square root of x, one can define the (positive) triangular root of x as the number n such that Tn = x:[15]
$n={\frac {{\sqrt {8x+1}}-1}{2}}$
which follows immediately from the quadratic formula. So an integer x is triangular if and only if 8x + 1 is a square. Equivalently, if the positive triangular root n of x is an integer, then x is the nth triangular number.[15]
Alternative name
As stated, an alternative name proposed by Donald Knuth, by analogy to factorials, is "termial", with the notation n? for the nth triangular number.[16] However, although some other sources use this name and notation,[17] they are not in wide use.
See also
• 1 + 2 + 3 + 4 + ⋯
• Doubly triangular number, a triangular number whose position in the sequence of triangular numbers is also a triangular number
• Tetractys, an arrangement of ten points in a triangle, important in Pythagoreanism
References
1. "Triangular Number Sequence". Math Is Fun.
2. Spivak, Michael (2008). Calculus (4th ed.). Houston, Texas: Publish or Perish. pp. 21–22. ISBN 978-0-914098-91-1.
3. Hayes, Brian. "Gauss's Day of Reckoning". American Scientist. Computing Science. Archived from the original on 2015-04-02. Retrieved 2014-04-16.
4. Eves, Howard. "Webpage cites AN INTRODUCTION TO THE HISTORY OF MATHEMATICS". Mathcentral. Retrieved 28 March 2015.
5. Esposito, M. An unpublished astronomical treatise by the Irish monk Dicuil. Proceedings of the Royal Irish Academy, XXXVI C. Dublin, 1907, 378-446.
6. Ross, H.E. & Knott, B.I."Dicuil (9th century) on triangular and square numbers." British Journal for the History of Mathematics, 2019,34 (2), 79-94. https://doi.org/10.1080/26375451.2019.1598687.
7. "The Handshake Problem | National Association of Math Circles". MathCircles.org. Archived from the original on 10 March 2016. Retrieved 12 January 2022.
8. Knuth, Donald. The Art of Computer Programming. Vol. 1 (3rd ed.). p. 48.
9. Baumann, Michael Heinrich (2018-12-12). "Die k-dimensionale Champagnerpyramide" (PDF). Mathematische Semesterberichte (in German). 66: 89–100. doi:10.1007/s00591-018-00236-x. ISSN 1432-1815. S2CID 125426184.
10. Chen, Fang: Triangular numbers in geometric progression
11. Fang: Nonexistence of a geometric progression that contains four triangular numbers
12. Liu, Zhi-Guo (2003-12-01). "An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers". The Ramanujan Journal. 7 (4): 407–434. doi:10.1023/B:RAMA.0000012425.42327.ae. ISSN 1382-4090. S2CID 122221070.
13. Sun, Zhi-Hong (2016-01-24). "Ramanujan's theta functions and sums of triangular numbers". arXiv:1601.06378 [math.NT].
14. Engelstein, Geoffrey; Shalev, Isaac (2019-06-25). "Building Blocks of Tabletop Game Design". doi:10.1201/9780429430701. {{cite journal}}: Cite journal requires |journal= (help)
15. Euler, Leonhard; Lagrange, Joseph Louis (1810), Elements of Algebra, vol. 1 (2nd ed.), J. Johnson and Co., pp. 332–335
16. Donald E. Knuth (1997). The Art of Computer Programming: Volume 1: Fundamental Algorithms. 3rd Ed. Addison Wesley Longman, U.S.A. p. 48.
17. Stone, John David (2018), Algorithms for Functional Programming, Springer, p. 282, doi:10.1007/978-3-662-57970-1, ISBN 978-3-662-57968-8, S2CID 53079729
External links
Wikimedia Commons has media related to triangular numbers.
• "Arithmetic series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Triangular numbers at cut-the-knot
• There exist triangular numbers that are also square at cut-the-knot
• Weisstein, Eric W. "Triangular Number". MathWorld.
• Hypertetrahedral Polytopic Roots by Rob Hubbard, including the generalisation to triangular cube roots, some higher dimensions, and some approximate formulas
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| Wikipedia |
Hexagonal tiling-triangular tiling honeycomb
In the geometry of hyperbolic 3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal tiling cells, in a rhombitrihexagonal tiling vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
Hexagonal tiling-triangular tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbol{(3,6,3,6)} or {(6,3,6,3)}
Coxeter diagrams or or or
Cells{3,6}
{6,3}
r{6,3}
Facestriangular {3}
square {4}
hexagon {6}
Vertex figure
rhombitrihexagonal tiling
Coxeter group[(6,3)[2]]
PropertiesVertex-uniform, edge-uniform
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Symmetry
A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,6,3*)] symmetry, represented by a cube fundamental domain, and an octahedral Coxeter diagram .
Related honeycombs
The cyclotruncated octahedral-hexagonal tiling honeycomb, has a higher symmetry construction as the order-4 hexagonal tiling.
See also
• Uniform honeycombs in hyperbolic space
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
| Wikipedia |
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle measurements at known points, rather than measuring distances to the point directly as in trilateration; the use of both angles and distance measurements is referred to as triangulateration.
In computer vision
Computer stereo vision and optical 3D measuring systems use this principle to determine the spatial dimensions and the geometry of an item.[2] Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object's surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3D coordinate, is calculated from the triangular relations.
History
Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and, in the military, the gun direction, the trajectory and distribution of fire power of weapons.
The use of triangles to estimate distances dates to antiquity. In the 6th century BC, about 250 years prior to the establishment of the Ptolemaic dynasty, the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids of ancient Egypt. He measured the length of the pyramids' shadows and that of his own at the same moment, and compared the ratios to his height (intercept theorem).[3] Thales also estimated the distances to ships at sea as seen from a clifftop by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff.[4] Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe until in 1615 Snellius, after the work of Eratosthenes, reworked the technique for an attempt to measure the circumference of the earth. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances,[5] while Liu Hui (c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.[6][7]
See also
• Direction finding
• GSM localization
• Multilateration, where a point is calculated using the time-difference-of-arrival between other known points
• Parallax
• Resection (orientation)
• Stereopsis
• Tessellation, covering a polygon with triangles
• Trig point
• Wireless triangulation
References
1. "מה בתמונה? (תשובה: נקודת טריאנגולציה)" [what is in the picture? (Answer: Triangulation Point)]. Jeepolog.com forums (in Hebrew). 2007-07-08.
2. Thomas Luhmann; Stuart Robson; Stephen Kyle; Jan Boehm (27 November 2013). Close-Range Photogrammetry and 3D Imaging. De Gruyter. ISBN 978-3-11-030278-3.
3. Diogenes Laërtius, "Life of Thales", The Lives and Opinions of Eminent Philosophers, I, 27, retrieved 2008-02-22{{citation}}: CS1 maint: location (link)
4. Proclus, In Euclidem
5. Joseph Needham (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books Ltd. pp. 539–540
6. Liu Hui, Haidao Suanjing
7. Kurt Vogel (1983; 1997), A Surveying Problem Travels from China to Paris, in Yvonne Dold-Samplonius (ed.), From China to Paris, Proceedings of a conference held July, 1997, Mathematisches Forschungsinstitut, Oberwolfach, Germany. ISBN 3-515-08223-9.
Geography and cartography in the medieval Islamic world
Geographers
9th century
• Al-Khwarizmi
• Abu Hanifa Dinawari
• Ya'qubi
• Sulaiman al-Tajir
10th century
• Ibn Khordadbeh
• Ahmad ibn Rustah
• Ahmad ibn Fadlan
• Abu Zayd al-Balkhi
• Abu Muhammad al-Hasan al-Hamdani
• Al-Masudi
• Istakhri
• Khashkhash Ibn Saeed Ibn Aswad
• Ibn Hawqal
• Ibn al-Faqih
• Al-Muqaddasi
• Al-Ramhormuzi
• Qudama ibn Ja'far
11th century
• Abū Rayḥān al-Bīrūnī
• Abu Saʿīd Gardēzī
• Al-Bakri
• Mahmud al-Kashgari
• Domiyat
12th century
• al-Zuhri
• Muhammad al-Idrisi
• Abu'l Abbas al-Hijazi
13th century
• Ibn Jubayr
• Saadi Shirazi
• Yaqut al-Hamawi
• Ibn Said al-Maghribi
• Ibn al-Nafis
• Ibn al-Mujawir
14th century
• Al-Dimashqi
• Abu'l-Fida
• Ibn al-Wardi
• Hamdallah Mustawfi
• Ibn Battuta
• Lin Nu
15th century
• Abd-al-Razzāq Samarqandī
• Ghiyāth al-dīn Naqqāsh
• Ahmad ibn Mājid
• Zheng He
• Ma Huan
• Fei Xin
16th century
• Sulaiman Al Mahri
• Piri Reis
• Mir Ahmed Nasrallah Thattvi
• Amīn Rāzī
17th century
• Evliya Çelebi
Works
• Book of Roads and Kingdoms (al-Bakrī)
• Book of Roads and Kingdoms (ibn Khordadbeh)
• Tabula Rogeriana
• Kitab al-Rawd al-Mitar
• Mu'jam Al-Buldan
• Rihla
• The Meadows of Gold
• Piri Reis map
• Kitab al-Kharaj
Influences
• Geography (Ptolemy)
Authority control
National
• France
• BnF data
• Germany
• Israel
• United States
Other
• NARA
| Wikipedia |
Triangulated category
In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.
Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space X to complexes of sheaves, viewed as objects of the derived category of sheaves on X. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conjecture predicts that the derived category of a Calabi–Yau manifold is equivalent to the Fukaya category of its "mirror" symplectic manifold. Shift operator is a decategorified analogue of triangulated category.
History
Triangulated categories were introduced independently by Dieter Puppe (1962) and Jean-Louis Verdier (1963), although Puppe's axioms were less complete (lacking the octahedral axiom (TR 4)).[1] Puppe was motivated by the stable homotopy category. Verdier's key example was the derived category of an abelian category, which he also defined, developing ideas of Alexander Grothendieck. The early applications of derived categories included coherent duality and Verdier duality, which extends Poincaré duality to singular spaces.
Definition
A shift or translation functor on a category D is an additive automorphism (or for some authors, an auto-equivalence) $\Sigma $ from D to D. It is common to write $X[n]=\Sigma ^{n}X$ for integers n.
A triangle (X, Y, Z, u, v, w) consists of three objects X, Y, and Z, together with morphisms $u\colon X\to Y$, $v\colon Y\to Z$ and $w\colon Z\to X[1]$. Triangles are generally written in the unravelled form:
$X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1],$
or
$X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}$
for short.
A triangulated category is an additive category D with a translation functor and a class of triangles, called exact triangles[2] (or distinguished triangles), satisfying the following properties (TR 1), (TR 2), (TR 3) and (TR 4). (These axioms are not entirely independent, since (TR 3) can be derived from the others.[3])
TR 1
• For every object X, the following triangle is exact:
$X{\overset {\text{id}}{\to }}X\to 0\to X[1]$
• For every morphism $u\colon X\to Y$, there is an object Z (called a cone or cofiber of the morphism u) fitting into an exact triangle
$X{\xrightarrow {{} \atop u}}Y\to Z\to X[1]$
The name "cone" comes from the cone of a map of chain complexes, which in turn was inspired by the mapping cone in topology. It follows from the other axioms that an exact triangle (and in particular the object Z) is determined up to isomorphism by the morphism $X\to Y$, although not always up to a unique isomorphism.[4]
• Every triangle isomorphic to an exact triangle is exact. This means that if
$X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]$
is an exact triangle, and $f\colon X\to X'$, $g\colon Y\to Y'$, and $h\colon Z\to Z'$ are isomorphisms, then
$X'{\xrightarrow {guf^{-1}}}Y'{\xrightarrow {hvg^{-1}}}Z'{\xrightarrow {f[1]wh^{-1}}}X'[1]$
is also an exact triangle.
TR 2
If
$X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]$
is an exact triangle, then so are the two rotated triangles
$Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]{\xrightarrow {-u[1]}}Y[1]$
and
$Z[-1]{\xrightarrow {-w[-1]}}X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z.\ $
In view of the last triangle, the object Z[−1] is called a fiber of the morphism $X\to Y$.
The second rotated triangle has a more complex form when $[1]$ and $[-1]$ are not isomorphisms but only mutually inverse equivalences of categories, since $-w[-1]$ is a morphism from $Z[-1]$ to $(X[1])[-1]$, and to obtain a morphism to $[X]$ one must compose with the natural transformation $(X[1])[-1]{\xrightarrow {}}X$. This leads to complex questions about possible axioms one has to impose on the natural transformations making $[1]$ and $[-1]$ into a pair of inverse equivalences. Due to this issue, the assumption that $[1]$ and $[-1]$ are mutually inverse isomorphisms is the usual choice in the definition of a triangulated category.
TR 3
Given two exact triangles and a map between the first morphisms in each triangle, there exists a morphism between the third objects in each of the two triangles that makes everything commute. That is, in the following diagram (where the two rows are exact triangles and f and g are morphisms such that gu = u′f), there exists a map h (not necessarily unique) making all the squares commute:
TR 4: The octahedral axiom
Let $u\colon X\to Y$ and $v\colon Y\to Z$ be morphisms, and consider the composed morphism $vu\colon X\to Z$. Form exact triangles for each of these three morphisms according to TR 1. The octahedral axiom states (roughly) that the three mapping cones can be made into the vertices of an exact triangle so that "everything commutes".
More formally, given exact triangles
$X{\xrightarrow {u\,}}Y{\xrightarrow {j}}Z'{\xrightarrow {k}}X[1]$
$Y{\xrightarrow {v\,}}Z{\xrightarrow {l}}X'{\xrightarrow {i}}Y[1]$
$X{\xrightarrow {{} \atop vu}}Z{\xrightarrow {m}}Y'{\xrightarrow {n}}X[1]$,
there exists an exact triangle
$Z'{\xrightarrow {f}}Y'{\xrightarrow {g}}X'{\xrightarrow {h}}Z'[1]$
such that
$l=gm,\quad k=nf,\quad h=j[1]i,\quad ig=u[1]n,\quad fj=mv.$
This axiom is called the "octahedral axiom" because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are exact triangles. The presentation here is Verdier's own, and appears, complete with octahedral diagram, in (Hartshorne 1966). In the following diagram, u and v are the given morphisms, and the primed letters are the cones of various maps (chosen so that every exact triangle has an X, a Y, and a Z letter). Various arrows have been marked with [1] to indicate that they are of "degree 1"; e.g. the map from Z′ to X is in fact from Z′ to X[1]. The octahedral axiom then asserts the existence of maps f and g forming an exact triangle, and so that f and g form commutative triangles in the other faces that contain them:
Two different pictures appear in (Beilinson, Bernstein & Deligne 1982) (Gelfand and Manin (2006) also present the first one). The first presents the upper and lower pyramids of the above octahedron and asserts that given a lower pyramid, one can fill in an upper pyramid so that the two paths from Y to Y′, and from Y′ to Y, are equal (this condition is omitted, perhaps erroneously, from Hartshorne's presentation). The triangles marked + are commutative and those marked "d" are exact:
The second diagram is a more innovative presentation. Exact triangles are presented linearly, and the diagram emphasizes the fact that the four triangles in the "octahedron" are connected by a series of maps of triangles, where three triangles (namely, those completing the morphisms from X to Y, from Y to Z, and from X to Z) are given and the existence of the fourth is claimed. One passes between the first two by "pivoting" about X, to the third by pivoting about Z, and to the fourth by pivoting about X′. All enclosures in this diagram are commutative (both trigons and the square) but the other commutative square, expressing the equality of the two paths from Y′ to Y, is not evident. All the arrows pointing "off the edge" are degree 1:
This last diagram also illustrates a useful intuitive interpretation of the octahedral axiom. In triangulated categories, triangles play the role of exact sequences, and so it is suggestive to think of these objects as "quotients", $Z'=Y/X$ and $Y'=Z/X$. In those terms, the existence of the last triangle expresses on the one hand
$X'=Z/Y\ $ (looking at the triangle $Y\to Z\to X'\to $ ), and
$X'=Y'/Z'$ (looking at the triangle $Z'\to Y'\to X'\to $ ).
Putting these together, the octahedral axiom asserts the "third isomorphism theorem":
$(Z/X)/(Y/X)\cong Z/Y.$
If the triangulated category is the derived category D(A) of an abelian category A, and X, Y, Z are objects of A viewed as complexes concentrated in degree 0, and the maps $X\to Y$ and $Y\to Z$ are monomorphisms in A, then the cones of these morphisms in D(A) are actually isomorphic to the quotients above in A.
Finally, Neeman (2001) formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns. Beilinson, Bernstein, and Deligne (1982) also give generalizations of the octahedral axiom.
Properties
Here are some simple consequences of the axioms for a triangulated category D.
• Given an exact triangle
$X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]$
in D, the composition of any two successive morphisms is zero. That is, vu = 0, wv = 0, u[1]w = 0, and so on.[5]
• Given a morphism $u\colon X\to Y$, TR 1 guarantees the existence of a cone Z completing an exact triangle. Any two cones of u are isomorphic, but the isomorphism is not always uniquely determined.[4]
• Every monomorphism in D is the inclusion of a direct summand, $X\to X\oplus Y$, and every epimorphism is a projection $X\oplus Y\to X$.[6] A related point is that one should not talk about "injectivity" or "surjectivity" for morphisms in a triangulated category. Every morphism $X\to Y$ that is not an isomorphism has a nonzero "cokernel" Z (meaning that there is an exact triangle $X\to Y\to Z\to X[1]$) and also a nonzero "kernel", namely Z[−1].
Non-functoriality of the cone construction
One of the technical complications with triangulated categories is the fact the cone construction is not functorial. For example, given a ring $R$ and the partial map of distinguished triangles
${\begin{matrix}R&\to &0&\to &R[+1]&\to \\\downarrow &&\downarrow &&&\\0&\to &R[+1]&\to &R[+1]&\to \end{matrix}}$
in $D^{b}(R)$, there are two maps which complete this diagram. This could be the identity map, or the zero map
${\begin{aligned}{\text{id}}:&R[+1]\to R[+1]\\0:&R[+1]\to R[+1]\end{aligned}}$
both of which are commutative. The fact there exist two maps is a shadow of the fact that a triangulated category is a tool which encodes homotopy limits and colimit. One solution for this problem was proposed by Grothendieck where not only the derived category is considered, but the derived category of diagrams on this category. Such an object is called a Derivator.
Examples
1. Vector spaces over a field k form an elementary triangulated category in which X[1] = X for all X. An exact triangle is a sequence $X\to Y\to Z\to X\to Y$ of k-linear maps (writing the same map $X\to Y$ twice) which is exact at X, Y and Z.
2. If A is an additive category (for example, an abelian category), define the homotopy category $K(A)$ to have as objects the chain complexes in A, and as morphisms the homotopy classes of morphisms of complexes. Then $K(A)$ is a triangulated category.[7] The shift X[1] is the complex X moved one step to the left (and with differentials multiplied by −1). An exact triangle in $K(A)$ is a triangle which is isomorphic in $K(A)$ to the triangle $X\to Y\to {\text{cone}}(f)\to X[1]$ associated to some map $f\colon X\to Y$ of chain complexes. (Here ${\text{cone}}(f)$ denotes the mapping cone of a chain map.)
3. The derived category D(A) of an abelian category A is a triangulated category.[8] It is constructed from the category of complexes C(A) by localizing with respect to all quasi-isomorphisms. That is, formally adjoin an inverse morphism for every quasi-isomorphism. The objects of D(A) are unchanged; that is, they are chain complexes. An exact triangle in D(A) is a triangle which is isomorphic in D(A) to the triangle $X\to Y\to {\text{cone}}(f)\to X[1]$ associated to some map $f\colon X\to Y$ of chain complexes.
A key motivation for the derived category is that derived functors on A can be viewed as functors on the derived category.[9] Some natural subcategories of D(A) are also triangulated categories, for example the subcategory of complexes X whose cohomology objects $H^{i}(X)$ in A vanish for i sufficiently negative, sufficiently positive, or both, called $D^{+}(A),D^{-}(A),D^{\text{b}}(A)$, respectively.
4. In topology, the stable homotopy category $h{\cal {S}}$ is a triangulated category.[10] The objects are spectra, the shift X[1] is the suspension $\Sigma X$ (or equivalently the delooping $\Omega ^{-1}X$), and the exact triangles are the cofiber sequences. A distinctive feature of the stable homotopy category (compared to the unstable homotopy category) is that fiber sequences are the same as cofiber sequences. In fact, in any triangulated category, exact triangles can be viewed as fiber sequences and also as cofiber sequences.
5. In modular representation theory of a finite group G, the stable module category StMod(kG) is a triangulated category. Its objects are the representations of G over a field k, and the morphisms are the usual ones modulo those that factor via projective (or equivalently injective) kG-modules. More generally, the stable module category is defined for any Frobenius algebra in place of kG.
Are there better axioms?
Some experts suspect[11]pg 190 (see, for example, (Gelfand & Manin 2006, Introduction, Chapter IV)) that triangulated categories are not really the "correct" concept. The essential reason is that the cone of a morphism is unique only up to a non-unique isomorphism. In particular, the cone of a morphism does not in general depend functorially on the morphism (note the non-uniqueness in axiom (TR 3), for example). This non-uniqueness is a potential source of errors. The axioms work adequately in practice, however, and there is a great deal of literature devoted to their study.
Derivators
One alternative proposal is the theory of derivators proposed in Pursuing stacks by Grothendieck in the 80s[11]pg 191, and later developed in the 90s in his manuscript on the topic. Essentially, these are a system of homotopy categories given by the diagram categories $I\to M$ for a category with a class of weak equivalences $(M,W)$. These categories are then related by the morphisms of diagrams $I\to J$. This formalism has the advantage of being able to recover the homotopy limits and colimits, which replaces the cone construction.
Stable ∞-categories
Another alternative built is the theory of stable ∞-categories. The homotopy category of a stable ∞-category is canonically triangulated, and moreover mapping cones become essentially unique (in a precise homotopical sense). Moreover, a stable ∞-category naturally encodes a whole hierarchy of compatibilities for its homotopy category, at the bottom of which sits the octahedral axiom. Thus, it is strictly stronger to give the data of a stable ∞-category than to give the data of a triangulation of its homotopy category. Nearly all triangulated categories that arise in practice come from stable ∞-categories. A similar (but more special) enrichment of triangulated categories is the notion of a dg-category.
In some ways, stable ∞-categories or dg-categories work better than triangulated categories. One example is the notion of an exact functor between triangulated categories, discussed below. For a smooth projective variety X over a field k, the bounded derived category of coherent sheaves ${\text{D}}^{\text{b}}(X)$ comes from a dg-category in a natural way. For varieties X and Y, every functor from the dg-category of X to that of Y comes from a complex of sheaves on $X\times Y$ by the Fourier–Mukai transform.[12] By contrast, there is an example of an exact functor from ${\text{D}}^{\text{b}}(X)$ to ${\text{D}}^{\text{b}}(Y)$ that does not come from a complex of sheaves on $X\times Y$.[13] In view of this example, the "right" notion of a morphism between triangulated categories seems to be one that comes from a morphism of underlying dg-categories (or stable ∞-categories).
Another advantage of stable ∞-categories or dg-categories over triangulated categories appears in algebraic K-theory. One can define the algebraic K-theory of a stable ∞-category or dg-category C, giving a sequence of abelian groups $K_{i}(C)$ for integers i. The group $K_{0}(C)$ has a simple description in terms of the triangulated category associated to C. But an example shows that the higher K-groups of a dg-category are not always determined by the associated triangulated category.[14] Thus a triangulated category has a well-defined $K_{0}$ group, but in general not higher K-groups.
On the other hand, the theory of triangulated categories is simpler than the theory of stable ∞-categories or dg-categories, and in many applications the triangulated structure is sufficient. An example is the proof of the Bloch–Kato conjecture, where many computations were done at the level of triangulated categories, and the additional structure of ∞-categories or dg-categories was not required.
Cohomology in triangulated categories
Triangulated categories admit a notion of cohomology, and every triangulated category has a large supply of cohomological functors. A cohomological functor F from a triangulated category D to an abelian category A is a functor such that for every exact triangle
$X\to Y\to Z\to X[1],\ $
the sequence $F(X)\to F(Y)\to F(Z)$ in A is exact. Since an exact triangle determines an infinite sequence of exact triangles in both directions,
$\cdots \to Z[-1]\to X\to Y\to Z\to X[1]\to \cdots ,\ $
a cohomological functor F actually gives a long exact sequence in the abelian category A:
$\cdots \to F(Z[-1])\to F(X)\to F(Y)\to F(Z)\to F(X[1])\to \cdots .\ $
A key example is: for each object B in a triangulated category D, the functors $\operatorname {Hom} (B,{\text{-}})$ and $\operatorname {Hom} ({\text{-}},B)$ are cohomological, with values in the category of abelian groups.[15] (To be precise, the latter is a contravariant functor, which can be considered as a functor on the opposite category of D.) That is, an exact triangle $X\to Y\to Z\to X[1]$ determines two long exact sequences of abelian groups:
$\cdots \to \operatorname {Hom} (B,X[i])\to \operatorname {Hom} (B,Y[i])\to \operatorname {Hom} (B,Z[i])\to \operatorname {Hom} (B,X[i+1])\to \cdots $
and
$\cdots \to \operatorname {Hom} (Z,B[i])\to \operatorname {Hom} (Y,B[i])\to \operatorname {Hom} (X,B[i])\to \operatorname {Hom} (Z,B[i+1])\to \cdots .$
For particular triangulated categories, these exact sequences yield many of the important exact sequences in sheaf cohomology, group cohomology, and other areas of mathematics.
One may also use the notation
$\operatorname {Ext} ^{i}(B,X)=\operatorname {Hom} (B,X[i])$
for integers i, generalizing the Ext functor in an abelian category. In this notation, the first exact sequence above would be written:
$\cdots \to \operatorname {Ext} ^{i}(B,X)\to \operatorname {Ext} ^{i}(B,Y)\to \operatorname {Ext} ^{i}(B,Z)\to \operatorname {Ext} ^{i+1}(B,X)\to \cdots .\ $
For an abelian category A, another basic example of a cohomological functor on the derived category D(A) sends a complex X to the object $H^{0}(X)$ in A. That is, an exact triangle $X\to Y\to Z\to X[1]$ in D(A) determines a long exact sequence in A:
$\cdots \to H^{i}(X)\to H^{i}(Y)\to H^{i}(Z)\to H^{i+1}(X)\to \cdots ,$
using that $H^{0}(X[i])\cong H^{i}(X)$.
Exact functors and equivalences
An exact functor (also called triangulated functor) from a triangulated category D to a triangulated category E is an additive functor $F\colon D\to E$ which, loosely speaking, commutes with translation and sends exact triangles to exact triangles.[16]
In more detail, an exact functor comes with a natural isomorphism $\eta \colon F\Sigma \to \Sigma F$ (where the first $\Sigma $ denotes the translation functor of D and the second $\Sigma $ denotes the translation functor of E), such that whenever
$X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]$
is an exact triangle in D,
$F(X){\xrightarrow {F(u)}}F(Y){\xrightarrow {F(v)}}F(Z){\xrightarrow {\eta _{X}F(w)}}F(X)[1]$
is an exact triangle in E.
An equivalence of triangulated categories is an exact functor $F\colon D\to E$ that is also an equivalence of categories. In this case, there is an exact functor $G\colon E\to D$ such that FG and GF are naturally isomorphic to the respective identity functors.
Compactly generated triangulated categories
Let D be a triangulated category such that direct sums indexed by an arbitrary set (not necessarily finite) exist in D. An object X in D is called compact if the functor ${\text{Hom}}_{D}(X,{\text{-}})$ commutes with direct sums. Explicitly, this means that for every family of objects $Y_{i}$ in D indexed by a set S, the natural homomorphism of abelian groups $\oplus _{i\in S}\mathrm {Hom} _{D}(X,Y_{i})\to \mathrm {Hom} _{D}(X,\oplus _{i\in S}Y_{i})$ is an isomorphism. This is different from the general notion of a compact object in category theory, which involves all colimits rather than only coproducts.
For example, a compact object in the stable homotopy category $h{\cal {S}}$ is a finite spectrum.[17] A compact object in the derived category of a ring, or in the quasi-coherent derived category of a scheme, is a perfect complex. In the case of a smooth projective variety X over a field, the category Perf(X) of perfect complexes can also be viewed as the bounded derived category of coherent sheaves, $D_{\text{coh}}^{\text{b}}(X)$.
A triangulated category D is compactly generated if
• D has arbitrary (not necessarily finite) direct sums;
• There is a set S of compact objects in D such that for every nonzero object X in D, there is an object Y in S with a nonzero map $Y[n]\to X$ for some integer n.
Many naturally occurring "large" triangulated categories are compactly generated:
• The derived category of modules over a ring R is compactly generated by one object, the R-module R.
• The quasi-coherent derived category of a quasi-compact quasi-separated scheme is compactly generated by one object.[18]
• The stable homotopy category is compactly generated by one object, the sphere spectrum $S^{0}$.[19]
Amnon Neeman generalized the Brown representability theorem to any compactly generated triangulated category, as follows.[20] Let D be a compactly generated triangulated category, $H\colon D^{\text{op}}\to {\text{Ab}}$ a cohomological functor which takes coproducts to products. Then H is representable. (That is, there is an object W of D such that $H(X)\cong {\text{Hom}}(X,W)$ for all X.) For another version, let D be a compactly generated triangulated category, T any triangulated category. If an exact functor $F\colon D\to T$ sends coproducts to coproducts, then F has a right adjoint.
The Brown representability theorem can be used to define various functors between triangulated categories. In particular, Neeman used it to simplify and generalize the construction of the exceptional inverse image functor $f^{!}$ for a morphism f of schemes, the central feature of coherent duality theory.[21]
t-structures
For every abelian category A, the derived category D(A) is a triangulated category, containing A as a full subcategory (the complexes concentrated in degree zero). Different abelian categories can have equivalent derived categories, so that it is not always possible to reconstruct A from D(A) as a triangulated category.
Alexander Beilinson, Joseph Bernstein and Pierre Deligne described this situation by the notion of a t-structure on a triangulated category D.[22] A t-structure on D determines an abelian category inside D, and different t-structures on D may yield different abelian categories.
Localizing and thick subcategories
Let D be a triangulated category with arbitrary direct sums. A localizing subcategory of D is a strictly full triangulated subcategory that is closed under arbitrary direct sums.[23] To explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield localization functor $L\colon D\to D$ with kernel S.[24] (That is, for every object X in D there is an exact triangle $Y\to X\to LX\to Y[1]$ with Y in S and LX in the right orthogonal $S^{\perp }$.) For example, this construction includes the localization of a spectrum at a prime number, or the restriction from a complex of sheaves on a space to an open subset.
A parallel notion is more relevant for "small" triangulated categories: a thick subcategory of a triangulated category C is a strictly full triangulated subcategory that is closed under direct summands. (If C is idempotent-complete, a subcategory is thick if and only if it is also idempotent-complete.) A localizing subcategory is thick.[25] So if S is a localizing subcategory of a triangulated category D, then the intersection of S with the subcategory $D^{\text{c}}$ of compact objects is a thick subcategory of $D^{\text{c}}$.
For example, Devinatz–Hopkins–Smith described all thick subcategories of the triangulated category of finite spectra in terms of Morava K-theory.[26] The localizing subcategories of the whole stable homotopy category have not been classified.
See also
• Fourier–Mukai transform
• Six operations
• Perverse sheaf
• D-module
• Beilinson–Bernstein localization
• Module spectrum
• Semiorthogonal decomposition
• Bridgeland stability condition
Notes
1. Puppe (1962, 1967); Verdier (1963, 1967).
2. Weibel (1994), Definition 10.2.1.
3. J. Peter May, The axioms for triangulated categories.
4. Weibel (1994), Remark 10.2.2.
5. Weibel (1994), Exercise 10.2.1.
6. Gelfand & Manin (2006), Exercise IV.1.1.
7. Kashiwara & Schapira (2006), Theorem 11.2.6.
8. Weibel (1994), Corollary 10.4.3.
9. Weibel (1994), section 10.5.
10. Weibel (1994), Theorem 10.9.18.
11. Grothendieck. "Pursuing Stacks". thescrivener.github.io. Archived (PDF) from the original on 30 Jul 2020. Retrieved 2020-09-17.
12. Toën (2007), Theorem 8.15.
13. Rizzardo et al. (2019), Theorem 1.4.
14. Dugger & Shipley (2009), Remark 4.9.
15. Weibel (1994), Example 10.2.8.
16. Weibel (1994), Definition 10.2.6.
17. Neeman (2001), Remark D.1.5.
18. Stacks Project, Tag 09IS, Stacks Project, Tag 09M1.
19. Neeman (2001), Lemma D.1.3.
20. Neeman (1996), Theorems 3.1 and 4.1.
21. Neeman (1996), Example 4.2.
22. Beilinson et al. (1982), Definition 1.3.1.
23. Neeman (2001), Introduction, after Remark 1.4.
24. Krause (2010), Theorem, Introduction.
25. Neeman (2001), Remark 3.2.7.
26. Ravenel (1992), Theorem 3.4.3.
References
Some textbook introductions to triangulated categories are:
• Gelfand, Sergei; Manin, Yuri (2006), "IV. Triangulated Categories", Methods of homological algebra, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, doi:10.1007/978-3-662-12492-5, ISBN 978-3540435839, MR 1950475
• Kashiwara, Masaki; Schapira, Pierre (2006), Categories and sheaves, Grundlehren der mathematischen Wissenschaften, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-27950-4, ISBN 978-3-540-27949-5, MR 2182076
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
A concise summary with applications is:
• Kashiwara, Masaki; Schapira, Pierre (2002), "Chapter I. Homological Algebra", Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, doi:10.1007/978-3-662-02661-8, ISBN 978-3540518617, MR 1074006
Some more advanced references are:
• Beilinson, A.A.; Bernstein, J.; Deligne, P. (2018) [1982], "Faisceaux pervers", Astérisque, Société Mathématique de France, Paris, 100, ISBN 978-2-85629-878-7, MR 0751966
• Dugger, Daniel; Shipley, Brooke (2009), "A curious example of triangulated-equivalent model categories which are not Quillen equivalent", Algebraic and Geometric Topology, 9: 135–166, arXiv:0710.3070, doi:10.2140/agt.2009.9.135, MR 2482071
• Hartshorne, Robin (1966), "Chapter I. The Derived Category", Residues and duality, Lecture Notes in Mathematics 20, Springer-Verlag, pp. 20–48, doi:10.1007/BFb0080482, ISBN 978-3-540-03603-6, MR 0222093
• Krause, Henning (2010), "Localization theory for triangulated categories", Triangulated categories, London Mathematical Society Lecture Note Series, vol. 375, Cambridge University Press, pp. 161–235, arXiv:0806.1324, doi:10.1017/CBO9781139107075.005, MR 2681709
• Neeman, Amnon (1996), "The Grothendieck duality theorem via Bousfield's techniques and Brown representability", Journal of the American Mathematical Society, 9: 205–236, doi:10.1090/S0894-0347-96-00174-9, MR 1308405
• Neeman, Amnon (2001), Triangulated categories, Annals of Mathematics Studies, Princeton University Press, doi:10.1515/9781400837212, ISBN 978-0691086866, MR 1812507
• Puppe, Dieter (1962), "On the formal structure of stable homotopy theory", Colloquium on algebraic topology, Aarhus Universitet Matematisk Institute, pp. 65–71, Zbl 0139.41106
• Puppe, Dieter (1967), "Stabile Homotopietheorie. I.", Mathematische Annalen, 169: 243–274, doi:10.1007/BF01362348, MR 0211400
• Ravenel, Douglas (1992), Nilpotence and periodicity in stable homotopy theory, Princeton University Press, ISBN 9780691025728, MR 1192553
• Rizzardo, Alice; Van den Bergh, Michel; Neeman, Amnon (2019), "An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves", Inventiones Mathematicae, 216: 927–1004, arXiv:1410.4039, doi:10.1007/s00222-019-00862-9, MR 3955712
• Toën, Bertrand (2007), "The homotopy theory of dg-categories and derived Morita theory", Inventiones Mathematicae, 167: 615–667, arXiv:math/0408337, doi:10.1007/s00222-006-0025-y, MR 2276263
• Verdier, Jean-Louis (1977) [1963], "Catégories dérivées: quelques résultats (état 0)", Cohomologie étale (SGA 4 1/2) (PDF), Lecture Notes in Mathematics, vol. 569, Springer, pp. 262–311, doi:10.1007/BFb0091525, ISBN 978-3-540-08066-4, MR 3727440
• Verdier, Jean-Louis (1996) [1967], Des catégories dérivées des catégories abéliennes, Astérisque, vol. 239, Société Mathématique de France, MR 1453167
External links
• J. Peter May, The axioms for triangulated categories
• The Stacks Project Authors, The Stacks Project
| Wikipedia |
Triangulation (geometry)
In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it into tetrahedra packed together.
In most instances, the triangles of a triangulation are required to meet edge-to-edge and vertex-to-vertex.
Types
Different types of triangulations may be defined, depending both on what geometric object is to be subdivided and on how the subdivision is determined.
• A triangulation $T$ of $\mathbb {R} ^{d}$ is a subdivision of $\mathbb {R} ^{d}$ into $d$-dimensional simplices such that any two simplices in $T$ intersect in a common face (a simplex of any lower dimension) or not at all, and any bounded set in $\mathbb {R} ^{d}$ intersects only finitely many simplices in $T$. That is, it is a locally finite simplicial complex that covers the entire space.
• A point-set triangulation, i.e., a triangulation of a discrete set of points ${\mathcal {P}}\subset \mathbb {R} ^{d}$, is a subdivision of the convex hull of the points into simplices such that any two simplices intersect in a common face of any dimension or not at all and such that the set of vertices of the simplices are contained in ${\mathcal {P}}$. Frequently used and studied point set triangulations include the Delaunay triangulation (for points in general position, the set of simplices that are circumscribed by an open ball that contains no input points) and the minimum-weight triangulation (the point set triangulation minimizing the sum of the edge lengths).
• In cartography, a triangulated irregular network is a point set triangulation of a set of two-dimensional points together with elevations for each point. Lifting each point from the plane to its elevated height lifts the triangles of the triangulation into three-dimensional surfaces, which form an approximation of a three-dimensional landform.
• A polygon triangulation is a subdivision of a given polygon into triangles meeting edge-to-edge, again with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Polygon triangulations may be found in linear time and form the basis of several important geometric algorithms, including a simple approximate solution to the art gallery problem. The constrained Delaunay triangulation is an adaptation of the Delaunay triangulation from point sets to polygons or, more generally, to planar straight-line graphs.
• A triangulation of a surface consists of a net of triangles with points on a given surface covering the surface partly or totally.
• In the finite element method, triangulations are often used as the mesh (in this case, a triangle mesh) underlying a computation. In this case, the triangles must form a subdivision of the domain to be simulated, but instead of restricting the vertices to input points, it is allowed to add additional Steiner points as vertices. In order to be suitable as finite element meshes, a triangulation must have well-shaped triangles, according to criteria that depend on the details of the finite element simulation (see mesh quality); for instance, some methods require that all triangles be right or acute, forming nonobtuse meshes. Many meshing techniques are known, including Delaunay refinement algorithms such as Chew's second algorithm and Ruppert's algorithm.
• In more general topological spaces, triangulations of a space generally refer to simplicial complexes that are homeomorphic to the space.
Generalization
The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles -- polygons that, like triangles, have exactly three convex vertices. As in point set triangulations, pseudotriangulations are required to have their vertices at the given input points.
External links
• Weisstein, Eric W. "Simplicial complex". MathWorld.
• Weisstein, Eric W. "Triangulation". MathWorld.
| Wikipedia |
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling.
Motivation
On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object.
On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities rising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows now to assign such quantities to topological spaces.
Investigations concerning the existence and uniqueness of triangulations established a new branch in topology, namely the piecewise-linear-topology (short PL- topology). Its main purpose is topological properties of simplicial complexes and its generalization, cell-complexes.
Simplicial complexes
Abstract simplicial complexes
An abstract simplicial complex above a set $V$ is a system ${\mathcal {T}}\subset {\mathcal {P}}(V)$ of non-empty subsets such that:
• $\{v_{0}\}\in {\mathcal {T}}$ for each $v_{0}\in V$;
• if $E\in {\mathcal {T}}$ and $\emptyset \neq F\subset E$ $\Rightarrow $ $F\in {\mathcal {T}}$.
The elements of ${\mathcal {T}}$ are called simplices, the elements of $V$ are called vertices. A simplex with $n+1$ vertices has dimension $n$ by definition. The dimension of an abstract simplicial complex is defined as ${\text{dim}}({\mathcal {T}})={\text{sup}}\;\{{\text{dim}}(F):F\in {\mathcal {T}}\}\in \mathbb {N} \cup \infty $.[1]
Abstract simplicial complexes can be thought of as geometrical objects too. This requires the term of geometric simplex.
Geometric simplices
Let $p_{0},...p_{n}$ be $n+1$ affinely independent points in $\mathbb {R} ^{n}$, i.e. the vectors $(p_{1}-p_{0}),(p_{2}-p_{0}),\dots (p_{n}-p_{0})$are linearly independent. The set $ \Delta ={\Bigl \{}x\in \mathbb {R} ^{n}\;{\Big |}\;x=\sum _{i=0}^{n}t_{i}p_{i}\;with\;0\leq t_{i}\leq 1\;and\;\sum _{i=0}^{n}t_{i}=1{\Bigr \}}$ is said to be the simplex spanned by $p_{0},...p_{n}$. It has dimension $n$ by definition. The points $p_{0},...p_{n}$ are called the vertices of $\Delta $, the simplices spanned by $n$ of the $n+1$ vertices are called faces and the boundary $\partial \Delta $ is defined to be the union of its faces.
The $n$-dimensional standard-simplex is the simplex spanned by the unit vectors $e_{0},...e_{n}$[2]
Geometric simplicial complexes
A geometric simplicial complex ${\mathcal {S}}\subset \mathbb {R} ^{n}$ is a union of geometric simplices such that
• If $S$ is a simplex in ${\mathcal {S}}$, then all its faces are in ${\mathcal {S}}$.
• If $S,T$ are two distinct simplices in ${\mathcal {S}}$, their inners are disjoint.
The set ${\mathcal {S}}$ can be realized as a topological space $|{\mathcal {S}}|$ by choosing the closed sets to be ${\Bigl \{}A\subset |{\mathcal {S}}|\;{\Big |}\;A\cap \Delta $ is closed for all $\Delta \in {\mathcal {S}}{\Bigr \}}$. It should be mentioned, that in general, the simplicial complex won't provide the natural topology of $\mathbb {R} ^{n}$. In the case that each point in the complex lies only in finetly many simplices, both topologies coincide[2]
Each geometric complex can be associated with an abstract complex by choosing as a ground set $V$ the set of vertices that appear in any simplex of ${\mathcal {S}}$ and as system of subsets the subsets of $V$ which correspond to vertex sets of simplices in ${\mathcal {S}}$.
A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: Consider for instance abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex:
Let ${\mathcal {T}}$ be an abstract simplicial complex above a set $V$. Choose a union of simplices $(\Delta _{F})_{F\in {\mathcal {T}}}$, but each in $\mathbb {R} ^{N}$ of dimension sufficiently large, such that the geometric simplex $\Delta _{F}$ is of dimension $n$ if the abstract geometric simplex $F$ has dimension $n$. If $E\subset F$, $\Delta _{E}\subset \mathbb {R} ^{N}$can be identified with a face of $\Delta _{F}\subset \mathbb {R} ^{M}$ and the resulting topological space is the gluing $\Delta _{E}\cup _{i}\Delta _{F}$ Effectuating the gluing for each inclusion, one ends up with the desired topological space.
As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets being closed in the subspace topology of each simplex $\Delta _{F}$.
The simplicial complex ${\mathcal {T_{n}}}$, which consists of all simplices ${\mathcal {T}}$ of dimension $\leq n$ is called the $n$-th skeleton of ${\mathcal {T}}$.
A natural neighborhood of a vertex $V$ of a simplicial complex ${\mathcal {S}}$ is considered to be the star $star(K)={\Big \{}L\in {\mathcal {S}}\;|\;K\subset L{\Big \}}$ of a simplex, its boundary is the link
$lk(K)={\Big \{}M\in {\mathcal {S}}\;|\;M\cap K=\emptyset {\Big \}}$.
Simplicial maps
The maps considered in this category are simplicial maps: Let ${\mathcal {K}}$, ${\mathcal {L}}$ be abstract simplicial complexes above sets $V_{K}$, $V_{L}$. A simplicial map is a function $f:V_{K}\rightarrow V_{L}$ which maps each simplex in ${\mathcal {K}}$ onto a simplex in ${\mathcal {L}}$. By affine-linear extension on the simplices, $f$ induces a map between the geometric realizations of the complexes.[2]
Examples
• Let $W=\{a,b,c,d,e,f\}$ and let ${\mathcal {T}}={\Big \{}\{a\},\{b\},\{c\},\{d\},\{e\},\{f\},\{a,b\},\{a,c\},\{a,d\},\{a,e\},\{a,f\}{\Big \}}$. The associated geometric complex is a star with center $\{a\}$.
• Let $V=\{A,B,C,D\}$ and let ${\mathcal {S}}={\mathcal {P}}(V)$. Its geometric realization $|{\mathcal {S}}|$ is a tetrahedron.
• Let $V$ as above and let ${\mathcal {S}}'=\;{\mathcal {P}}({\mathcal {V}})\setminus \{A,B,C,D\}$. The geometric simplicial complex is the boundary of a tetrahedron $|{\mathcal {S'}}|=\partial |{\mathcal {S}}|$.
Definition
A triangulation of a topological space $X$ is a homeomorphism $t:|{\mathcal {T}}|\rightarrow X$ where ${\mathcal {T}}$ is a simplicial complex. Topological spaces do not necessarily admit a triangulation and if they do, it is not necessarily unique.
Examples
• Simplicial complexes can be triangulated by identity.
• Let ${\mathcal {S}},{\mathcal {S'}}$ be as in the examples seen above. The closed unit ball $\mathbb {D} ^{3}$ is homeomorphic to a tetrahedron so it admits a triangulation, namely the homeomorphism $t:|{\mathcal {S}}|\rightarrow \mathbb {D} ^{3}$. Restricting $t$ to $|{\mathcal {S}}'|$ yields a homeomorphism $t':|{\mathcal {S}}'|\rightarrow \mathbb {S} ^{2}$.
• The torus $\mathbb {T} ^{2}=\mathbb {S} ^{1}\times \mathbb {S} ^{1}$ admits a triangulation. To see this, consider the torus as a square where the parallel faces are glued together. This square can be triangulated as shown below:
• The projective plane $\mathbb {P} ^{2}$ admits a triangulation (see CW-complexes)
• One can show that differentiable manifolds admit triangulations.[3]
Invariants
Triangulations of spaces allow assigning combinatorial invariants rising from their dedicated simplicial complexes to spaces. These are characteristics that equal for complexes that are isomorphic via a simplicial map and thus have the same combinatorial pattern.
This data might be useful to classify topological spaces up to homeomorphism but only given that the characteristics are also topological invariants, meaning, they do not depend on the chosen triangulation. For the data listed here, this is the case.[4] For details and the link to singular homology, see topological invariance
Homology
Via triangulation, one can assign a chain complex to topological spaces that arise from its simplicial complex and compute its simplicial homology. Compact spaces always admit finite triangulations and therefore their homology groups are finitely generated and only finitely many of them do not vanish. Other data as Betti- Numbers or Euler characteristic can be derived from homology.
Betti- numbers and Euler-characteristics
Let $|{\mathcal {S}}|$ be a finite simplicial complex. The $n$- th Betti- number $b_{n}({\mathcal {S}})$ is defined to be the rank of the $n$- th simplicial homology- group of the spaces. These numbers encode geometric properties of the spaces: The Betti- Number $b_{0}({\mathcal {S}})$ for instance represents the number of connected components. For a triangulated, closed orientable surfaces $F$, $b_{1}(F)=2g$ holds where $g$ denotes the genus of the surface: Therefore its first Betti- number represents the doubled number of handles of the surface.[5]
With the comments above, for compact spaces all Betti- numbers are finite and almost all are zero. Therefore, one can form their alternating sum
$\sum _{k=0}^{\infty }(-1)^{k}b_{k}({\mathcal {L}})$
which is called the Euler Charakteristik of the complex, a catchy topological invariant.
Topological invariance
To use these invariants for the classification of topological spaces up to homeomorphism one needs invariance of the characteristics regarding homeomorphism.
A famous approach to the question was at the beginning of the 20th century the attempt to show that any two triangulations of the same topological space admit a common subdivision. This assumption is known as Hauptvermutung ( German: Main assumption). Let $|{\mathcal {L}}|\subset \mathbb {R} ^{N}$ be a simplicial complex. A complex $|{\mathcal {L'}}|\subset \mathbb {R} ^{N}$ is said to be a subdivision of ${\mathcal {L}}$ iff:
• every simplex of ${\mathcal {L'}}$ is contained in a simplex of ${\mathcal {L}}$ and
• every simplex of ${\mathcal {L}}$ is a finite union of simplices in ${\mathcal {L'}}$ .[2]
Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map $f:{\mathcal {K}}\rightarrow {\mathcal {L}}$ between simplicial complexes is said to be piecewise linear if there is a refinement ${\mathcal {K'}}$ of ${\mathcal {K}}$ such that $f$ is piecewise linear on each simplex of ${\mathcal {K}}$. Two complexes that correspond to another via piecewise linear bijection are said to be combinatorial isomorphic. In particular, two complexes that have a common refinement are combinatorially equivalent. Homology groups are invariant to combinatorial equivalence and therefore the Hauptvermutung would give the topological invariance of simplicial homology groups. In 1918, Alexander introduced the concept of singular homology. Henceforth, most of the invariants arising from triangulation were replaced by invariants arising from singular homology. For those new invariants, it can be shown that they were invariant regarding homeomorphism and even regarding homotopy equivalence.[6] Furthermore it was shown that singular and simplicial homology groups coincide.[6] This workaround has shown the invariance of the data to homeomorphism. Hauptvermutung lost in importance but it was initial for a new branch in topology: The piecewise linear (short PL- topology) topology examines topological properties of topological spaces.[7]
Hauptvermutung
The Hauptvermutung (German for main conjecture) states that two triangulations always admit a common subdivision. Originally, its purpose was to prove invariance of combinatorial invariants regarding homeomorphisms. The assumption that such subdivisions exist in general is intuitive, as subdivision are easy to construct for simple spaces, for instance for low dimensional manifolds. Indeed the assumption was proven for manifolds of dimension $\leq 3$ and for differentiable manifolds but it was disproved in general:[8] An important tool to show that triangulations do not admit a common subdivision. i. e their underlying complexes are not combinatorially isomorphic is the combinatorial invariant of Reidemeister Torsion.
Reidemeister-Torsion
To disprove the Hauptvermutung it is helpful to use combinatorial invariants which are not topological invariants. A famous example is Reidemeister-Torsion. It can be assigned to a tuple $(K,L)$ of CW- complexes: If $L=\emptyset $ this characteristic will be a topological invariant but if $L\neq \emptyset $ in general not. An approach to Hauptvermutung was to find homeomorphic spaces with different values of Reidemeister-Torsion. This invariant was used initially to classify lens- spaces and first counterexamples to the Hauptvermutung were built based on lens- spaces:[8]
Classification of lens- spaces
In its original formulation, Lens spaces are 3-manifolds, constructed as quotient spaces of the 3-sphere: Let $p,q$ be natural numbers, such that $p,q$ are coprime . The lens space $L(p,q)$ is defined to be the orbit space of the free group action
$\mathbb {Z} /p\mathbb {Z} \times S^{3}\to S^{3}$
$(k,(z_{1},z_{2}))\mapsto (z_{1}\cdot e^{2\pi ik/p},z_{2}\cdot e^{2\pi ikq/p})$.
For different tuples $(p,q)$, Lens spaces will be homotopy- equivalent but not homeomorphic. Therefore they can't be distinguished with the help of classical invariants as the fundamental group but by the use of Reidemeister-Torsion.
Two Lens spaces $L(p,q_{1}),L(p,q_{2})$are homeomorphic, if and only if $q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}$.[9] This is the case iff two Lens spaces are simple-homotopy-equivalent. The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces $L'_{1},L'_{2}$ derived from non-homeomorphic Lens spaces $L(p,q_{1}),L(p,q_{2})$having different Reidemeister torsion. Suppose further that the modification into $L'_{1},L'_{2}$ does not affect Reidemeister torsion but such that after modification $L'_{1}$ and $L'_{2}$ are homeomorphic. The resulting spaces will disprove the Hauptvermutung.
Existence of triangulation
Besides the question of concrete triangulations for computational issues, there are statements about spaces that are easier to prove given that they are simplicial complexes. Especially manifolds are of interest. Topological manifolds of dimension $\leq 3$ are always triangulable[10][11][1] but there are non-triangulable manifolds for dimension $n$, for $n$ arbitrary but greater than three.[12][13] Further, differentiable manifolds always admit triangulations.[3]
PL- Structures
Manifolds are an important class of spaces. It is natural to require them not only to be triangulable but moreover to admit a piecewise linear atlas, a PL- structure:
Let $|X|$ be a simplicial complex such that every point admits an open neighborhood $U$ such that there is a triangulation of $U$ and a piecewise linear homeomorphism $f:U\rightarrow \mathbb {R} ^{n}$. Then $|X|$ is said to be a piecewise linear (PL) manifold of dimension $n$ and the triangulation together with the PL- atlas is said to be a PL- structure on $|X|$.
An important lemma is the following:
Let $X$ be a topological space. It is equivalent
1. $X$ is an $n$-dimensional manifold and admits a PL- structure.
2. There is a triangulation of $X$ such that the link of each vertex is an $n-1$ sphere.
3. For each triangulation of $X$ the link of each vertex is an $n-1$ sphere.
The equivalence of the second and the third statement is because that the link of a vertex is independent of the chosen triangulation up to combinatorial isomorphism.[14] One can show that differentiable manifolds admit a PL- structure as well as manifolds of dimension $\leq 3$.[15] Counterexamples for the triangulation conjecture are counterexamples for the conjecture of the existence of PL- structure of course.
Moreover, there are examples for triangulated spaces which do not admit a PL- structure. Consider an $n-2$- dimensional PL- Homology-sphere $X$. The double suspension $S^{2}X$ is a topological $n$-sphere. Choosing a triangulation $t:|{\mathcal {S}}|\rightarrow S^{2}X$ obtained via the suspension operation on triangulations the resulting simplicial complex is not a PL- manifold, because there is a vertex $v$ such that $link(v)$ is not a $n-1$ sphere.[16]
A question arising with the definition is if PL-structures are always unique: Given two PL- structures for the same space $Y$, is there a there a homeomorphism $F:Y\rightarrow Y$ which is piecewise linear with respect to both PL- structures? The assumption is similar to the Hauptvermutung and indeed there are spaces which have different PL-structures which are not equivalent. Triangulation of PL- equivalent spaces can be transformed into one another via Pachner moves:
Pachner Moves
Pachner moves are a way to manipulate triangulations: Let ${\mathcal {S}}$ be a simplicial complex. For two simplices $K,L$ the Join
$K*L={\Big \{}tk+(1-t)l\;|\;k\in K,l\in L\;t\in [0,1]{\Big \}}$ are the points lying on straights between points in $K$ and in $L$. Choose $S\in {\mathcal {S}}$ such that $lk(S)=\partial K$ for any $K$ lying not in ${\mathcal {S}}$. A new complex ${\mathcal {S'}}$, can be obtained by replacing $S*\partial K$ by $\partial S*K$. This replacement is called a Pachner move. The theorem of Pachner states that whenever two triangulated manifolds are PL- equivalent, there is a series of Pachner moves transforming both into another.[17]
CW-complexes
A similar but more flexible construction than simplicial complexes is the one of CW-complexes. Its construction is as follows:
An $n$- cell is the closed $n$- dimensional unit-ball $B_{n}=[0,1]^{n}$, an open $n$-cell is its inner $B_{n}=[0,1]^{n}\setminus \mathbb {S} ^{n-1}$. Let $X$ be a topological space, let $f:\mathbb {S} ^{n-1}\rightarrow X$ be a continuous map. The gluing $X\cup _{f}B_{n}$ is said to be obtained by gluing on an $n$-cell.
A cell complex is a union $X=\cup _{n\geq 0}X_{n}$ of topological spaces such that
• $X_{0}$ is a discrete set
• each $X_{n}$ is obtained from $X_{n-1}$ by gluing on a family of $n$-cells.
Each simplicial complex is a CW-complex, the inverse is not true. The construction of CW- complexes can be used to define cellular homology and one can show that cellular homology and simplicial homology coincide.[18] For computational issues, it is sometimes easier to assume spaces to be CW- complexes and determine their homology via cellular decomposition, an example is the projective plane $\mathbb {P} ^{2}$: Its construction as a CW-complex needs three cells, whereas its simplicial complex consists of 54 simplices.
Other Applications
Classification of manifolds
By triangulating 1-dimensional manifolds, one can show that they are always homeomorphic to disjoint copies of the real line and the unit sphere $\mathbb {S} ^{1}$. Moreover, surfaces, i.e. 2-manifolds, can be classified completely: Let $S$ be a compact surface.
• If $S$ is orientable, it is homeomorphic to a 2-sphere with $n$ tori of dimension $2$ attached, for some $n\geq 0$.
• If $S$ is not orientable, it is homeomorphic to a Klein Bottle with $n$ tori of dimension $2$ attached, for some $n\geq 0$.
To prove this theorem one constructs a fundamental polygon of the surface: This can be done by using the simplicial structure obtained by the triangulation.[19]
Maps on simplicial complexes
Giving spaces the structure of a simplicial structure might help to understand maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem:
Simplicial approximation
Let ${\mathcal {K}}$, ${\mathcal {L}}$ be abstract simplicial complexes above sets $V_{K}$, $V_{L}$. A simplicial map is a function $f:V_{K}\rightarrow V_{L}$ which maps each simplex in ${\mathcal {K}}$ onto a simplex in ${\mathcal {L}}$. By affin-linear extension on the simplices, $f$ induces a map between the geometric realizations of the complexes. Each point in a geometric complex lies in the inner of exactly one simplex, its support. Consider now a continuous map $f:{\mathcal {K}}\rightarrow {\mathcal {L}}$. A simplicial map $g:{\mathcal {K}}\rightarrow {\mathcal {L}}$ is said to be a simplicial approximation of $f$ if and only if each $x\in {\mathcal {K}}$ is mapped by $g$ onto the support of $f(x)$ in ${\mathcal {L}}$. If such an approximation exists, one can construct a homotopy $H$ transforming $f$ into $g$ by defining it on each simplex; there it always exists, because simplices are contractible.
The simplicial approximation theorem guarantees for every continuous function $f:V_{K}\rightarrow V_{L}$ the existence of a simplicial approximation at least after refinement of ${\mathcal {K}}$, for instance by replacing ${\mathcal {K}}$ by its iterated barycentric subdivision.[2] The theorem plays an important role for certain statements in algebraic topology in order to reduce the behavior of continuous maps on those of simplicial maps, for instance in Lefschetz's fixed-point theorem.
Lefschetz's fixed-point theorem
The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points. This data is computed as follows: Suppose that $X$ and $Y$ are topological spaces that admit finite triangulations. A continuous map $f:X\rightarrow Y$ induces homomorphisms $f_{i}:H_{i}(X,K)\rightarrow H_{i}(Y,K)$ between its simplicial homology groups with coefficients in a field $K$. These are linear maps between $K$- vectorspaces, so their trace $tr_{i}$ can be determined and their alternating sum
$L_{K}(f)=\sum _{i}(-1)^{i}tr_{i}(f)\in K$
is called the Lefschetz number of $f$. If $f=id$, this number is the Euler characteristic of $K$. The fixpoint theorem states that whenever $L_{K}(f)\neq 0$, $f$ has a fixed-point. In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem. Brouwer's fixpoint theorem treats the case where $f:\mathbb {D} ^{n}\rightarrow \mathbb {D} ^{n}$ is an endomorphism of the unit-ball. For $k\geq 1$ all its homology groups $H_{k}(\mathbb {D} ^{n})$ vanishes, and $f_{0}$ is always the identity, so $L_{K}(f)=tr_{0}(f)=1\neq 0$, so $f$ has a fixpoint.[20]
Formula of Riemann-Hurwitz
The Riemann- Hurwitz formula allows to determine the gender of a compact, connected Riemann surface $X$ without using explicit triangulation. The proof needs the existence of triangulations for surfaces in an abstract sense: Let $F:X\rightarrow Y$ be a non-constant holomorphic function on a surface with known gender. The relation between the gender $g$ of the surfaces $X$ and $Y$ is
$2g(X)-2=deg(F)(2g(Y)-2)\sum _{x\in X}(ord(F)-1)$
where $deg(F)$ denotes the degree of the map. The sum is well defined as it counts only the ramifying points of the function.
The background of this formula is that holomorphic functions on Riemann surfaces are ramified coverings. The formula can be found by examining the image of the simplicial structure near to ramifiying points.[21]
Citations
1. John M. Lee (2000), Springer Verlag (ed.), Introduction to Topological manifolds (in German), New York/Berlin/Heidelberg: Springer Verlag, p. 92, ISBN 0-387-98759-2
2. James R. Munkres (1984), Elements of algebraic topology (in German), vol. 1984, Menlo Park, California: Addison Wesley, p. 83, ISBN 0-201-04586-9
3. J. H. C. Whitehead (1940), "On C1-Complexes", Annals of Mathematics (in German), vol. 41, no. 4, pp. 809–824, doi:10.2307/1968861, ISSN 0003-486X, JSTOR 1968861
4. J. W. Alexander (1926), "Combinatorial Analysis Situs", Transactions of the American Mathematical Society (in German), vol. 28, no. 2, pp. 301–329, doi:10.1090/S0002-9947-1926-1501346-5, ISSN 0002-9947, JSTOR 1989117
5. R. Stöcker, H. Zieschang (1994), Algebraische Topologie (in German) (2. überarbeitete ed.), Stuttgart: B.G.Teubner, p. 270, ISBN 3-519-12226-X
6. Allen Hatcher (2006), Algebraic Topologie (in German), Cambridge/New York/Melbourne: Cambridge University Press, p. 110, ISBN 0-521-79160--X
7. A.A.Ranicki. "One the Hauptvermutung" (PDF). The Hauptvermutung Book.
8. John Milnor (1961), "Two Complexes Which are Homeomorphic But Combinatorially Distinct", The Annals of Mathematics (in German), vol. 74, no. 3, p. 575, doi:10.2307/1970299, ISSN 0003-486X, JSTOR 1970299
9. Marshall M. Cohen (1973), "A Course in Simple-Homotopy Theory", Graduate Texts in Mathematics, Graduate Texts in Mathematics (in German), vol. 10, doi:10.1007/978-1-4684-9372-6, ISBN 978-0-387-90055-1, ISSN 0072-5285
10. Edwin Moise (1977), Geometric Topology in Dimensions 2 and 3 (in German), New York: Springer Verlag
11. Tibor Rado. "Über den Begriff der Riemannschen Fläche" (PDF).
12. R. C. Kirby, L. C. Siebenmann (1977-12-31), "Annex B. On The Triangulation of Manifolds and the Hauptvermutung", Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88) (in German), Princeton University Press, pp. 299–306
13. "Chapter IV; Casson's Invariant for Oriented Homology 3-spheres", Casson's Invariant for Oriented Homology Three-Spheres (in German), Princeton University Press, pp. 63–79, 1990-12-31
14. Toenniessen, Fridtjof (2017), Topologie | SpringerLink (PDF) (in German), doi:10.1007/978-3-662-54964-3, ISBN 978-3-662-54963-6, retrieved 2022-04-20
15. Edwin E. Moise (1952), "Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung", The Annals of Mathematics (in German), vol. 56, no. 1, p. 96, doi:10.2307/1969769, ISSN 0003-486X, JSTOR 1969769
16. Robert D. Edwards (2006-10-18), "Suspensions of homology spheres", arXiv:math/0610573 (in German)
17. W B R Lickorish (1999-11-20), "Simplicial moves on complexes and manifolds", Proceedings of the Kirbyfest (in German), Mathematical Sciences Publishers, arXiv:math/9911256, doi:10.2140/gtm.1999.2.299, S2CID 9765634
18. Toenniessen, Fridtjof (2017), Topologie | SpringerLink (PDF) (in German), p. 315, doi:10.1007/978-3-662-54964-3, ISBN 978-3-662-54963-6, retrieved 2022-04-20
19. Seifert, H. (Herbert), 1907-1996. (2003), Lehrbuch der Topologie (in German), AMS Chelsea Pub., ISBN 0-8218-3595-5{{citation}}: CS1 maint: multiple names: authors list (link)
20. Bredon, Glen E. (1993), Springer Verlag (ed.), Topology and Geometry (in German), Berlin/ Heidelberg/ New York, pp. 254 f, ISBN 3-540-97926-3{{citation}}: CS1 maint: location missing publisher (link)
21. Otto Forster (1977), "Kompakte Riemannsche Flächen", Heidelberger Taschenbücher (in German), Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 88–154, ISBN 978-3-540-08034-3
Literature
• Allen Hatcher: Algebraic Topology, Cambridge University Press, Cambridge/New York/Melbourne 2006, ISBN 0-521-79160-X
• James R. Munkres: . Band 1984. Addison Wesley, Menlo Park, California 1984, ISBN 0-201-04586-9
• Marshall M. Cohen: A course in Simple-Homotopy Theory . In: Graduate Texts in Mathematics. 1973, ISSN 0072-5285, doi:10.1007/978-1-4684-9372-6.
| Wikipedia |
Triapeirogonal tiling
In geometry, the triapeirogonal tiling (or trigonal-horocyclic tiling) is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,3}.
Triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(3.∞)2
Schläfli symbolr{∞,3} or ${\begin{Bmatrix}\infty \\3\end{Bmatrix}}$
Wythoff symbol2 | ∞ 3
Coxeter diagram or
Symmetry group[∞,3], (*∞32)
DualOrder-3-infinite rhombille tiling
PropertiesVertex-transitive edge-transitive
Uniform colorings
The half-symmetry form, , has two colors of triangles:
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform quasiregular polyhedra with vertex configurations (3.n.3.n), and [n,3] Coxeter group symmetry.
Quasiregular tilings: (3.n)2
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Figure
Figure
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2
Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i}
Coxeter
Dual uniform figures
Dual
conf.
V(3.3)2
V(3.4)2
V(3.5)2
V(3.6)2
V(3.7)2
V(3.8)2
V(3.∞)2
Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)
=
=
=
=
or
=
or
=
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3∞ V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞
Paracompact hyperbolic uniform tilings in [(∞,3,3)] family
Symmetry: [(∞,3,3)], (*∞33) [(∞,3,3)]+, (∞33)
(∞,∞,3) t0,1(∞,3,3) t1(∞,3,3) t1,2(∞,3,3) t2(∞,3,3) t0,2(∞,3,3) t0,1,2(∞,3,3) s(∞,3,3)
Dual tilings
V(3.∞)3 V3.∞.3.∞ V(3.∞)3 V3.6.∞.6 V(3.3)∞ V3.6.∞.6 V6.6.∞ V3.3.3.3.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-i-3-i.
• List of uniform planar tilings
• Tilings of regular polygons
• Uniform tilings in hyperbolic plane
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• http://bendwavy.org/klitzing/incmats/o3xinfino.htm
• Klitzing, Richard. "2D Non-Compact Tilings". o3x∞o
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Triaugmented triangular prism
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism,[1] tricapped trigonal prism,[2] tetracaidecadeltahedron,[3][4] or tetrakaidecadeltahedron;[1] these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
Triaugmented triangular prism
TypeDeltahedron,
Johnson
J50 – J51 – J52
Faces14 triangles
Edges21
Vertices9
Vertex configuration$3\times 3^{4}+6\times 3^{5}$
Symmetry group$D_{3\mathrm {h} }$
Dual polyhedronAssociahedron $K_{5}$
Propertiesconvex
Net
The edges and vertices of the triaugmented triangular prism form a maximal planar graph with 9 vertices and 21 edges, called the Fritsch graph. It was used by Rudolf and Gerda Fritsch to show that Alfred Kempe's attempted proof of the four color theorem was incorrect. The Fritsch graph is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle.
The dual polyhedron of the triaugmented triangular prism is an associahedron, a polyhedron with four quadrilateral faces and six pentagons whose vertices represent the 14 triangulations of a regular hexagon. In the same way, the nine vertices of the triaugmented triangular prism represent the nine diagonals of a hexagon, with two vertices connected by an edge when the corresponding two diagonals do not cross. Other applications of the triaugmented triangular prism appear in chemistry as the basis for the tricapped trigonal prismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem and Tammes problem.
Construction
The triaugmented triangular prism can be constructed by attaching equilateral square pyramids to each of the three square faces of a triangular prism, a process called augmentation.[5] These pyramids cover each square, replacing it with four equilateral triangles, so that the resulting polyhedron has 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the triaugmented triangular prism.[6][7] More generally, the convex polyhedra in which all faces are regular polygons are called the Johnson solids, and every convex deltahedron is a Johnson solid. The triaugmented triangular prism is numbered among the Johnson solids as $J_{51}$.[8]
One possible system of Cartesian coordinates for the vertices of a triaugmented triangular prism, giving it edge length 2, is:[1]
$\displaystyle {\begin{aligned}(0,{\frac {2}{\sqrt {3}}},\pm 1),\qquad &(\pm 1,-{\frac {1}{\sqrt {3}}},\pm 1),\\(0,-{\sqrt {2}}-{\frac {1}{\sqrt {3}}},0),\qquad &(\pm {\frac {1+{\sqrt {6}}}{2}},{\frac {1+{\sqrt {6}}}{2{\sqrt {3}}}},0).\\\end{aligned}}$
Properties
A triaugmented triangular prism with edge length $a$ has surface area[9]
${\frac {7{\sqrt {3}}}{2}}a^{2}\approx 6.062a^{2},$
the area of 14 equilateral triangles. Its volume,[9]
${\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}a^{3}\approx 1.140a^{3},$
can be derived by slicing it into a central prism and three square pyramids, and adding their volumes.[9]
It has the same three-dimensional symmetry group as the triangular prism, the dihedral group $D_{3\mathrm {h} }$ of order twelve. Its dihedral angles can be calculated by adding the angles of the component pyramids and prism. The prism itself has square-triangle dihedral angles $\pi /2$ and square-square angles $\pi /3$. The triangle-triangle angles on the pyramid are the same as in the regular octahedron, and the square-triangle angles are half that. Therefore, for the triaugmented triangular prism, the dihedral angles incident to the degree-four vertices, on the edges of the prism triangles, and on the square-to-square prism edges are, respectively,[10]
${\begin{aligned}\arccos \left(-{\frac {1}{3}}\right)&\approx 109.5^{\circ },\\{\frac {\pi }{2}}+{\frac {1}{2}}\arccos \left(-{\frac {1}{3}}\right)&\approx 144.7^{\circ },\\{\frac {\pi }{3}}+\arccos \left(-{\frac {1}{3}}\right)&\approx 169.5^{\circ }.\\\end{aligned}}$
Fritsch graph
The graph of the triaugmented triangular prism has 9 vertices and 21 edges. It was used by Fritsch & Fritsch (1998) as a small counterexample to Alfred Kempe's false proof of the four color theorem using Kempe chains, and its dual map was used as their book's cover illustration.[11] Therefore, this graph has subsequently been named the Fritsch graph.[12] An even smaller counterexample, called the Soifer graph, is obtained by removing one edge from the Fritsch graph (the bottom edge in the illustration here).[12][13]
The Fritsch graph is one of only six connected graphs in which the neighborhood of every vertex is a cycle of length four or five. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a topological surface called a Whitney triangulation. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive angular defect at every vertex. This makes them a combinatorial analogue of the positively curved smooth surfaces. They come from six of the eight deltahedra—excluding the two that have a vertex with a triangular neighborhood. As well as the Fritsch graph, the other five are the graphs of the regular octahedron, regular icosahedron, pentagonal bipyramid, snub disphenoid, and gyroelongated square bipyramid.[14]
Dual associahedron
The dual polyhedron of the triaugmented triangular prism has a face for each vertex of the triaugmented triangular prism, and a vertex for each face. It is an enneahedron (that is, a nine-sided polyhedron)[15] that can be realized with three non-adjacent square faces, and six more faces that are congruent irregular pentagons.[16] It is also known as an order-5 associahedron, a polyhedron whose vertices represent the 14 triangulations of a regular hexagon.[15] A less-symmetric form of this dual polyhedron, obtained by slicing a truncated octahedron into four congruent quarters by two planes that perpendicularly bisect two parallel families of its edges, is a space-filling polyhedron.[17]
More generally, when a polytope is the dual of an associahedron, its boundary (a simplicial complex of triangles, tetrahedra, or higher-dimensional simplices) is called a "cluster complex". In the case of the triaugmented triangular prism, it is a cluster complex of type $A_{3}$, associated with the $A_{3}$ Dynkin diagram , the $A_{3}$ root system, and the $A_{3}$ cluster algebra.[18] The connection with the associahedron provides a correspondence between the nine vertices of the triaugmented triangular prism and the nine diagonals of a hexagon. The edges of the triaugmented triangular prism correspond to pairs of diagonals that do not cross, and the triangular faces of the triaugmented triangular prism correspond to the triangulations of the hexagon (consisting of three non-crossing diagonals). The triangulations of other regular polygons correspond to polytopes in the same way, with dimension equal to the number of sides of the polygon minus three.[15]
Applications
In the geometry of chemical compounds, it is common to visualize an atom cluster surrounding a central atom as a polyhedron—the convex hull of the surrounding atoms' locations. The tricapped trigonal prismatic molecular geometry describes clusters for which this polyhedron is a triaugmented triangular prism, although not necessarily one with equilateral triangle faces.[2] For example, the lanthanides from lanthanum to dysprosium dissolve in water to form cations surrounded by nine water molecules arranged as a triaugmented triangular prism.[19]
In the Thomson problem, concerning the minimum-energy configuration of $n$ charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the minimum solution known for $n=9$ places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, inscribed in a sphere. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known.[20]
See also
Wikimedia Commons has media related to Triaugmented triangular prism.
• Császár polyhedron – Toroidal polyhedron with 14 triangle faces
• Steffen's polyhedron – Flexible polyhedron with 14 triangle faces
References
1. Sloane, N. J. A.; Hardin, R. H.; Duff, T. D. S.; Conway, J. H. (1995), "Minimal-energy clusters of hard spheres", Discrete & Computational Geometry, 14 (3): 237–259, doi:10.1007/BF02570704, MR 1344734, S2CID 26955765
2. Kepert, David L. (1982), "Polyhedra", Inorganic Chemistry Concepts, vol. 6, Springer, pp. 7–21, doi:10.1007/978-3-642-68046-5_2, ISBN 978-3-642-68048-9
3. Burgiel, Heidi (2015), "Unit origami: star-building on deltahedra", in Delp, Kelly; Kaplan, Craig S.; McKenna, Douglas; Sarhangi, Reza (eds.), Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, Phoenix, Arizona: Tessellations Publishing, pp. 585–588, ISBN 978-1-938664-15-1
4. Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, p. 31, ISBN 9780520030565; see table, line 35
5. Trigg, Charles W. (1978), "An infinite class of deltahedra", Mathematics Magazine, 51 (1): 55–57, doi:10.1080/0025570X.1978.11976675, JSTOR 2689647, MR 1572246
6. Freudenthal, H.; van der Waerden, B. L. (1947), "On an assertion of Euclid", Simon Stevin, 25: 115–121, MR 0021687
7. Cundy, H. Martyn (December 1952), "Deltahedra", The Mathematical Gazette, 36 (318): 263–266, doi:10.2307/3608204, JSTOR 3608204, MR 0051525, S2CID 250435684
8. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177
9. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245; see Table IV, line 71, p. 338
10. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/CJM-1966-021-8, MR 0185507, S2CID 122006114; see Table III, line 51
11. Fritsch, Rudolf; Fritsch, Gerda (1998), The Four-Color Theorem: History, Topological Foundations, and Idea of Proof, New York: Springer-Verlag, pp. 175–176, doi:10.1007/978-1-4612-1720-6, ISBN 0-387-98497-6, MR 1633950
12. Gethner, Ellen; Kallichanda, Bopanna; Mentis, Alexander; Braudrick, Sarah; Chawla, Sumeet; Clune, Andrew; Drummond, Rachel; Evans, Panagiota; Roche, William; Takano, Nao (October 2009), "How false is Kempe's proof of the Four Color Theorem? Part II", Involve: A Journal of Mathematics, Mathematical Sciences Publishers, 2 (3): 249–265, doi:10.2140/involve.2009.2.249
13. Soifer, Alexander (2008), The Mathematical Coloring Book, Springer-Verlag, pp. 181–182, ISBN 978-0-387-74640-1
14. Knill, Oliver (2019), A simple sphere theorem for graphs, arXiv:1910.02708
15. Fomin, Sergey; Reading, Nathan (2007), "Root systems and generalized associahedra", in Miller, Ezra; Reiner, Victor; Sturmfels, Bernd (eds.), Geometric combinatorics, IAS/Park City Mathematics Series, vol. 13, Providence, Rhode Island: American Mathematical Society, pp. 63–131, arXiv:math/0505518, doi:10.1090/pcms/013/03, MR 2383126, S2CID 11435731; see Definition 3.3, Figure 3.6, and related discussion
16. Amir, Yifat; Séquin, Carlo H. (2018), "Modular toroids constructed from nonahedra", in Torrence, Eve; Torrence, Bruce; Séquin, Carlo; Fenyvesi, Kristóf (eds.), Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, Phoenix, Arizona: Tessellations Publishing, pp. 131–138, ISBN 978-1-938664-27-4
17. Goldberg, Michael (1982), "On the space-filling enneahedra", Geometriae Dedicata, 12 (3): 297–306, doi:10.1007/BF00147314, MR 0661535, S2CID 120914105; see polyhedron 9-IV, p. 301
18. Barcelo, Hélène; Severs, Christopher; White, Jacob A. (2013), "The discrete fundamental group of the associahedron, and the exchange module", International Journal of Algebra and Computation, 23 (4): 745–762, arXiv:1012.2810, doi:10.1142/S0218196713400079, MR 3078054, S2CID 14722555
19. Persson, Ingmar (2022), "Structures of Hydrated Metal Ions in Solid State and Aqueous Solution", Liquids, 2 (3): 210–242, doi:10.3390/liquids2030014
20. Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611, doi:10.1080/00029890.1952.11988207, JSTOR 2306764, MR 0050303
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
Triaugmented hexagonal prism
In geometry, the triaugmented hexagonal prism is one of the Johnson solids (J57). As the name suggests, it can be constructed by triply augmenting a hexagonal prism by attaching square pyramids (J1) to three of its nonadjacent equatorial faces.
Triaugmented hexagonal prism
TypeJohnson
J56 – J57 – J58
Faces12 triangles
3 squares
2 hexagons
Edges30
Vertices15
Vertex configuration3(34)
12(32.4.6)
Symmetry groupD3h
Dual polyhedronalternate order-4 truncated hexagonal bipyramid
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
See also
• Hexagonal prism
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
External links
• Eric W. Weisstein, Triaugmented hexagonal prism (Johnson Solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
Generalizations of Fibonacci numbers
In mathematics, the Fibonacci numbers form a sequence defined recursively by:
$F_{n}={\begin{cases}0&n=0\\1&n=1\\F_{n-1}+F_{n-2}&n>1\end{cases}}$
That is, after two starting values, each number is the sum of the two preceding numbers.
The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.
Extension to negative integers
Using $F_{n-2}=F_{n}-F_{n-1}$, one can extend the Fibonacci numbers to negative integers. So we get:
... −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ...
and $F_{-n}=(-1)^{n+1}F_{n}$.[1]
See also Negafibonacci coding.
Extension to all real or complex numbers
There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio φ, and are based on Binet's formula
$F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.$
The analytic function
$\operatorname {Fe} (x)={\frac {\varphi ^{x}-\varphi ^{-x}}{\sqrt {5}}}$
has the property that $\operatorname {Fe} (n)=F_{n}$ for even integers $n$.[2] Similarly, the analytic function:
$\operatorname {Fo} (x)={\frac {\varphi ^{x}+\varphi ^{-x}}{\sqrt {5}}}$
satisfies $\operatorname {Fo} (n)=F_{n}$ for odd integers $n$.
Finally, putting these together, the analytic function
$\operatorname {Fib} (x)={\frac {\varphi ^{x}-\cos(x\pi )\varphi ^{-x}}{\sqrt {5}}}$
satisfies $\operatorname {Fib} (n)=F_{n}$ for all integers $n$.[3]
Since $\operatorname {Fib} (z+2)=\operatorname {Fib} (z+1)+\operatorname {Fib} (z)$ for all complex numbers $z$, this function also provides an extension of the Fibonacci sequence to the entire complex plane. Hence we can calculate the generalized Fibonacci function of a complex variable, for example,
$\operatorname {Fib} (3+4i)\approx -5248.5-14195.9i$
Vector space
The term Fibonacci sequence is also applied more generally to any function $g$ from the integers to a field for which $g(n+2)=g(n)+g(n+1)$. These functions are precisely those of the form $g(n)=F(n)g(1)+F(n-1)g(0)$, so the Fibonacci sequences form a vector space with the functions $F(n)$ and $F(n-1)$ as a basis.
More generally, the range of $g$ may be taken to be any abelian group (regarded as a Z-module). Then the Fibonacci sequences form a 2-dimensional Z-module in the same way.
Similar integer sequences
Fibonacci integer sequences
The 2-dimensional $\mathbb {Z} $-module of Fibonacci integer sequences consists of all integer sequences satisfying $g(n+2)=g(n)+g(n+1)$. Expressed in terms of two initial values we have:
$g(n)=F(n)g(1)+F(n-1)g(0)=g(1){\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}+g(0){\frac {\varphi ^{n-1}-(-\varphi )^{1-n}}{\sqrt {5}}},$
where $\varphi $ is the golden ratio.
The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is $(-\varphi )^{-1}$.
The sequence can be written in the form
$a\varphi ^{n}+b(-\varphi )^{-n},$
in which $a=0$ if and only if $b=0$. In this form the simplest non-trivial example has $a=b=1$, which is the sequence of Lucas numbers:
$L_{n}=\varphi ^{n}+(-\varphi )^{-n}.$
We have $L_{1}=1$ and $L_{2}=3$. The properties include:
${\begin{aligned}\varphi ^{n}&=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\!n}={\frac {L(n)+F(n){\sqrt {5}}}{2}},\\L(n)&=F(n-1)+F(n+1).\end{aligned}}$
Every nontrivial Fibonacci integer sequence appears (possibly after a shift by a finite number of positions) as one of the rows of the Wythoff array. The Fibonacci sequence itself is the first row, and a shift of the Lucas sequence is the second row.[4]
See also Fibonacci integer sequences modulo n.
Lucas sequences
A different generalization of the Fibonacci sequence is the Lucas sequences of the kind defined as follows:
${\begin{aligned}U(0)&=0\\U(1)&=1\\U(n+2)&=PU(n+1)-QU(n),\end{aligned}}$
where the normal Fibonacci sequence is the special case of $P=1$ and $Q=-1$. Another kind of Lucas sequence begins with $V(0)=2$, $V(1)=P$. Such sequences have applications in number theory and primality proving.
When $Q=-1$, this sequence is called P-Fibonacci sequence, for example, Pell sequence is also called 2-Fibonacci sequence.
The 3-Fibonacci sequence is
0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, ... (sequence A006190 in the OEIS)
The 4-Fibonacci sequence is
0, 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, 1762289, 7465176, 31622993, 133957148, 567451585, 2403763488, ... (sequence A001076 in the OEIS)
The 5-Fibonacci sequence is
0, 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, ... (sequence A052918 in the OEIS)
The 6-Fibonacci sequence is
0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, ... (sequence A005668 in the OEIS)
The n-Fibonacci constant is the ratio toward which adjacent $n$-Fibonacci numbers tend; it is also called the nth metallic mean, and it is the only positive root of $x^{2}-nx-1=0$. For example, the case of $n=1$ is ${\frac {1+{\sqrt {5}}}{2}}$, or the golden ratio, and the case of $n=2$ is $1+{\sqrt {2}}$, or the silver ratio. Generally, the case of $n$ is ${\frac {n+{\sqrt {n^{2}+4}}}{2}}$.
Generally, $U(n)$ can be called (P,−Q)-Fibonacci sequence, and V(n) can be called (P,−Q)-Lucas sequence.
The (1,2)-Fibonacci sequence is
0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, ... (sequence A001045 in the OEIS)
The (1,3)-Fibonacci sequence is
1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, 32689, 75316, 173383, 399331, 919480, 2117473, 4875913, 11228332, 25856071, 59541067, ... (sequence A006130 in the OEIS)
The (2,2)-Fibonacci sequence is
0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, ... (sequence A002605 in the OEIS)
The (3,3)-Fibonacci sequence is
0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893, 507627, 1924560, 7296561, 27663363, 104879772, 397629405, 1507527531, 5715470808, ... (sequence A030195 in the OEIS)
Fibonacci numbers of higher order
A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous $n$ elements (with the exception of the first $n$ elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases $n=3$ and $n=4$ have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most $n$ is a Fibonacci sequence of order $n$. The sequence of the number of strings of 0s and 1s of length $m$ that contain at most $n$ consecutive 0s is also a Fibonacci sequence of order $n$.
These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated by Mark Barr in 1913.[5]
Tribonacci numbers
The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are:
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, … (sequence A000073 in the OEIS)
The series was first described formally by Agronomof in 1914,[6] but its first unintentional use is in the Origin of Species by Charles R. Darwin. In the example of illustrating the growth of elephant population, he relied on the calculations made by his son, George H. Darwin.[7] The term tribonacci was suggested by Feinberg in 1963.[8]
The tribonacci constant
${\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}\approx 1.839286755214161,$ (sequence A058265 in the OEIS)
is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial $x^{3}-x^{2}-x-1=0$, and also satisfies the equation $x+x^{-3}=2$. It is important in the study of the snub cube.
The reciprocal of the tribonacci constant, expressed by the relation $\xi ^{3}+\xi ^{2}+\xi =1$, can be written as:
$\xi ={\frac {{\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{-17+3{\sqrt {33}}}}-1}{3}}={\frac {3}{1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}}\approx 0.543689012.$ (sequence A192918 in the OEIS)
The tribonacci numbers are also given by[9]
$T(n)=\left\lfloor 3b\,{\frac {\left({\frac {1}{3}}\left(a_{+}+a_{-}+1\right)\right)^{n}}{b^{2}-2b+4}}\right\rceil $
where $\lfloor \cdot \rceil $ denotes the nearest integer function and
${\begin{aligned}a_{\pm }&={\sqrt[{3}]{19\pm 3{\sqrt {33}}}}\,,\\b&={\sqrt[{3}]{586+102{\sqrt {33}}}}\,.\end{aligned}}$
Tetranacci numbers
The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are:
0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, … (sequence A000078 in the OEIS)
The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial $x^{4}-x^{3}-x^{2}-x-1=0$, approximately 1.927561975482925 (sequence A086088 in the OEIS), and also satisfies the equation $x+x^{-4}=2$.
The tetranacci constant can be expressed in terms of radicals by the following expression:[10]
$x={\frac {1}{4}}\!\left(1+{\sqrt {u}}+{\sqrt {11-u+{\frac {26}{\sqrt {u}}}}}\,\right)$
where,
$u={\frac {11}{12}}-{\frac {1}{3}}{\sqrt[{3}]{\frac {65+3{\sqrt {1689}}}{2}}}+{\frac {1}{3}}{\sqrt[{3}]{\frac {-65+3{\sqrt {1689}}}{2}}}$
and $u$ is the real root of the cubic equation $u^{3}-11u^{2}+115u-169.$
Higher orders
Pentanacci, hexanacci, heptanacci, octanacci and enneanacci numbers have been computed. The pentanacci numbers are:
0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, … (sequence A001591 in the OEIS)
Hexanacci numbers:
0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, … (sequence A001592 in the OEIS)
Heptanacci numbers:
0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, … (sequence A122189 in the OEIS)
Octanacci numbers:
0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, ... (sequence A079262 in the OEIS)
Enneanacci numbers:
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, ... (sequence A104144 in the OEIS)
The limit of the ratio of successive terms of an $n$-nacci series tends to a root of the equation $x+x^{-n}=2$ (OEIS: A103814, OEIS: A118427, OEIS: A118428).
An alternate recursive formula for the limit of ratio $r$ of two consecutive $n$-nacci numbers can be expressed as
$r=\sum _{k=0}^{n-1}r^{-k}$.
The special case $n=2$ is the traditional Fibonacci series yielding the golden section $\varphi =1+{\frac {1}{\varphi }}$.
The above formulas for the ratio hold even for $n$-nacci series generated from arbitrary numbers. The limit of this ratio is 2 as $n$ increases. An "infinacci" sequence, if one could be described, would after an infinite number of zeroes yield the sequence
[..., 0, 0, 1,] 1, 2, 4, 8, 16, 32, …
which are simply the powers of two.
The limit of the ratio for any $n>0$ is the positive root $r$ of the characteristic equation[10]
$x^{n}-\sum _{i=0}^{n-1}x^{i}=0.$
The root $r$ is in the interval $2(1-2^{-n})<r<2$. The negative root of the characteristic equation is in the interval (−1, 0) when $n$ is even. This root and each complex root of the characteristic equation has modulus $3^{-n}<|r|<1$.[10]
A series for the positive root $r$ for any $n>0$ is[10]
$2-2\sum _{i>0}{\frac {1}{i}}{\binom {(n+1)i-2}{i-1}}{\frac {1}{2^{(n+1)i}}}.$
There is no solution of the characteristic equation in terms of radicals when 5 ≤ n ≤ 11.[10]
The kth element of the n-nacci sequence is given by
$F_{k}^{(n)}=\left\lfloor {\frac {r^{k-1}(r-1)}{(n+1)r-2n}}\right\rceil \!,$
where $\lfloor \cdot \rceil $ denotes the nearest integer function and $r$ is the $n$-nacci constant, which is the root of $x+x^{-n}=2$ nearest to 2.
A coin-tossing problem is related to the $n$-nacci sequence. The probability that no $n$ consecutive tails will occur in $m$ tosses of an idealized coin is ${\frac {1}{2^{m}}}F_{m+2}^{(n)}$.[11]
Fibonacci word
Main article: Fibonacci word
In analogy to its numerical counterpart, the Fibonacci word is defined by:
$F_{n}:=F(n):={\begin{cases}{\text{b}}&n=0;\\{\text{a}}&n=1;\\F(n-1)+F(n-2)&n>1.\\\end{cases}}$
where $+$ denotes the concatenation of two strings. The sequence of Fibonacci strings starts:
b, a, ab, aba, abaab, abaababa, abaababaabaab, … (sequence A106750 in the OEIS)
The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number.
Fibonacci strings appear as inputs for the worst case in some computer algorithms.
If "a" and "b" represent two different materials or atomic bond lengths, the structure corresponding to a Fibonacci string is a Fibonacci quasicrystal, an aperiodic quasicrystal structure with unusual spectral properties.
Convolved Fibonacci sequences
A convolved Fibonacci sequence is obtained applying a convolution operation to the Fibonacci sequence one or more times. Specifically, define[12]
$F_{n}^{(0)}=F_{n}$
and
$F_{n}^{(r+1)}=\sum _{i=0}^{n}F_{i}F_{n-i}^{(r)}$
The first few sequences are
$r=1$: 0, 0, 1, 2, 5, 10, 20, 38, 71, … (sequence A001629 in the OEIS).
$r=2$: 0, 0, 0, 1, 3, 9, 22, 51, 111, … (sequence A001628 in the OEIS).
$r=3$: 0, 0, 0, 0, 1, 4, 14, 40, 105, … (sequence A001872 in the OEIS).
The sequences can be calculated using the recurrence
$F_{n+1}^{(r+1)}=F_{n}^{(r+1)}+F_{n-1}^{(r+1)}+F_{n}^{(r)}$
The generating function of the $r$th convolution is
$s^{(r)}(x)=\sum _{k=0}^{\infty }F_{n}^{(r)}x^{n}=\left({\frac {x}{1-x-x^{2}}}\right)^{r}.$
The sequences are related to the sequence of Fibonacci polynomials by the relation
$F_{n}^{(r)}=r!F_{n}^{(r)}(1)$
where $F_{n}^{(r)}(x)$ is the $r$th derivative of $F_{n}(x)$. Equivalently, $F_{n}^{(r)}$ is the coefficient of $(x-1)^{r}$ when $F_{x}(x)$ is expanded in powers of $(x-1)$.
The first convolution, $F_{n}^{(1)}$ can be written in terms of the Fibonacci and Lucas numbers as
$F_{n}^{(1)}={\frac {nL_{n}-F_{n}}{5}}$
and follows the recurrence
$F_{n+1}^{(1)}=2F_{n}^{(1)}+F_{n-1}^{(1)}-2F_{n-2}^{(1)}-F_{n-3}^{(1)}.$
Similar expressions can be found for $r>1$ with increasing complexity as $r$ increases. The numbers $F_{n}^{(1)}$ are the row sums of Hosoya's triangle.
As with Fibonacci numbers, there are several combinatorial interpretations of these sequences. For example $F_{n}^{(1)}$ is the number of ways $n-2$ can be written as an ordered sum involving only 0, 1, and 2 with 0 used exactly once. In particular $F_{4}^{(1)}=5$ and 2 can be written 0 + 1 + 1, 0 + 2, 1 + 0 + 1, 1 + 1 + 0, 2 + 0.[13]
Other generalizations
The Fibonacci polynomials are another generalization of Fibonacci numbers.
The Padovan sequence is generated by the recurrence $P(n)=P(n-2)+P(n-3)$.
The Narayana's cows sequence is generated by the recurrence $N(k)=N(k-1)+N(k-3)$.
A random Fibonacci sequence can be defined by tossing a coin for each position $n$ of the sequence and taking $F(n)=F(n-1)+F(n-2)$ if it lands heads and $F(n)=F(n-1)-F(n-2)$ if it lands tails. Work by Furstenberg and Kesten guarantees that this sequence almost surely grows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 by Divakar Viswanath. It is now known as Viswanath's constant.
A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc. The first few repfigits are:
14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, … (sequence A007629 in the OEIS)
Since the set of sequences satisfying the relation $S(n)=S(n-1)+S(n-2)$ is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we abbreviate such a sequence as $(S(0),S(1))$, the Fibonacci sequence $F(n)=(0,1)$ and the shifted Fibonacci sequence $F(n-1)=(1,0)$ are seen to form a canonical basis for this space, yielding the identity:
$S(n)=S(0)F(n-1)+S(1)F(n)$
for all such sequences S. For example, if S is the Lucas sequence 2, 1, 3, 4, 7, 11, ..., then we obtain
$L(n)=2F(n-1)+F(n)$.
N-generated Fibonacci sequence
We can define the N-generated Fibonacci sequence (where N is a positive rational number): if
$N=2^{a_{1}}\cdot 3^{a_{2}}\cdot 5^{a_{3}}\cdot 7^{a_{4}}\cdot 11^{a_{5}}\cdot 13^{a_{6}}\cdot \ldots \cdot p_{r}^{a_{r}},$
where pr is the rth prime, then we define
$F_{N}(n)=a_{1}F_{N}(n-1)+a_{2}F_{N}(n-2)+a_{3}F_{N}(n-3)+a_{4}F_{N}(n-4)+a_{5}F_{N}(n-5)+...$
If $n=r-1$, then $F_{N}(n)=1$, and if $n<r-1$, then $F_{N}(n)=0$.
Sequence N OEIS sequence
Fibonacci sequence 6 A000045
Pell sequence 12 A000129
Jacobsthal sequence 18 A001045
Tribonacci sequence 30 A000073
Tetranacci sequence 210 A000288
Padovan sequence 15 A000931
Narayana's cows sequence 10 A000930
Semi-Fibonacci sequence
The semi-Fibonacci sequence (sequence A030067 in the OEIS) is defined via the same recursion for odd-indexed terms $a(2n+1)=a(2n)+a(2n-1)$ and $a(1)=1$, but for even indices $a(2n)=a(n)$, $n\geq 1$. The bissection A030068 of odd-indexed terms $s(n)=a(2n-1)$ therefore verifies $s(n+1)=s(n)+a(n)$ and is strictly increasing. It yields the set of the semi-Fibonacci numbers
1, 2, 3, 5, 6, 9, 11, 16, 17, 23, 26, 35, 37, 48, 53, 69, 70, 87, 93, 116, 119, 145, 154, ... (sequence A030068 in the OEIS)
which occur as $s(n)=a(2^{k}(2n-1)),k=0,1,...\,.$
References
1. Triana, Juan. Negafibonacci numbers via matrices. Bulletin of TICMI, 2019, pp. 19–24.
2. "What is a Fibonacci Number? -- from Harry J. Smith". 2009-10-27. Archived from the original on 27 October 2009. Retrieved 2022-04-12.
3. Pravin Chandra and Eric W. Weisstein. "Fibonacci Number". MathWorld.
4. Morrison, D. R. (1980), "A Stolarsky array of Wythoff pairs", A Collection of Manuscripts Related to the Fibonacci Sequence (PDF), Santa Clara, CA: The Fibonacci Association, pp. 134–136, archived from the original (PDF) on 2016-03-04, retrieved 2012-07-15.
5. Gardner, Martin (1961). The Scientific American Book of Mathematical Puzzles and Diversions, Vol. II. Simon and Schuster. p. 101.
6. Agronomof, M. (1914). "Sur une suite récurrente". Mathesis. 4: 125–126.
7. Podani, János; Kun, Ádám; Szilágyi, András (2018). "How Fast Does Darwin's Elephant Population Grow?" (PDF). Journal of the History of Biology. 51 (2): 259–281. doi:10.1007/s10739-017-9488-5. PMID 28726021. S2CID 3988121.
8. Feinberg, M. (1963). "Fibonacci-Tribonacci". Fibonacci Quarterly. 1: 71–74.
9. Simon Plouffe, 1993
10. Wolfram, D.A. (1998). "Solving Generalized Fibonacci Recurrences" (PDF). Fib. Quart.
11. Eric W. Weisstein. "Coin Tossing". MathWorld.
12. V. E. Hoggatt, Jr. and M. Bicknell-Johnson, "Fibonacci Convolution Sequences", Fib. Quart., 15 (1977), pp. 117-122.
13. Sloane, N. J. A. (ed.). "Sequence A001629". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
External links
• "Tribonacci number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Fibonacci
Books
• Liber Abaci (1202)
• The Book of Squares (1225)
Theories
• Fibonacci sequence
• Greedy algorithm for Egyptian fractions
Related
• Fibonacci numbers in popular culture
• List of things named after Fibonacci
• Generalizations of Fibonacci numbers
• The Fibonacci Association
• Fibonacci Quarterly
| Wikipedia |
Rauzy fractal
In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution
$s(1)=12,\ s(2)=13,\ s(3)=1\,.$
It was studied in 1981 by Gérard Rauzy,[1] with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal set can be generalized to other maps over a 3-letter alphabet, generating other fractal sets with interesting properties, such as periodic tiling of the plane and self-similarity in three homothetic parts.
Definitions
Tribonacci word
The infinite tribonacci word is a word constructed by iteratively applying the Tribonacci or Rauzy map : $s(1)=12$, $s(2)=13$, $s(3)=1$.[2][3] It is an example of a morphic word. Starting from 1, the Tribonacci words are:[4]
• $t_{0}=1$
• $t_{1}=12$
• $t_{2}=1213$
• $t_{3}=1213121$
• $t_{4}=1213121121312$
We can show that, for $n>2$, $t_{n}=t_{n-1}t_{n-2}t_{n-3}$; hence the name "Tribonacci".
Fractal construction
Consider, now, the space $R^{3}$ with cartesian coordinates (x,y,z). The Rauzy fractal is constructed this way:[5]
1) Interpret the sequence of letters of the infinite Tribonacci word as a sequence of unitary vectors of the space, with the following rules (1 = direction x, 2 = direction y, 3 = direction z).
2) Then, build a "stair" by tracing the points reached by this sequence of vectors (see figure). For example, the first points are:
• $1\Rightarrow (1,0,0)$
• $2\Rightarrow (1,1,0)$
• $1\Rightarrow (2,1,0)$
• $3\Rightarrow (2,1,1)$
• $1\Rightarrow (3,1,1)$
etc...Every point can be colored according to the corresponding letter, to stress the self-similarity property.
3) Then, project those points on the contracting plane (plane orthogonal to the main direction of propagation of the points, none of those projected points escape to infinity).
Properties
• Can be tiled by three copies of itself, with area reduced by factors $k$, $k^{2}$ and $k^{3}$ with $k$ solution of $k^{3}+k^{2}+k-1=0$: $\scriptstyle {k={\frac {1}{3}}(-1-{\frac {2}{\sqrt[{3}]{17+3{\sqrt {33}}}}}+{\sqrt[{3}]{17+3{\sqrt {33}}}})=0.54368901269207636}$.
• Stable under exchanging pieces. We can obtain the same set by exchanging the place of the pieces.
• Connected and simply connected. Has no hole.
• Tiles the plane periodically, by translation.
• The matrix of the Tribonacci map has $x^{3}-x^{2}-x-1$ as its characteristic polynomial. Its eigenvalues are a real number $\beta =1.8392$, called the Tribonacci constant, a Pisot number, and two complex conjugates $\alpha $ and ${\bar {\alpha }}$ with $\alpha {\bar {\alpha }}=1/\beta $.
• Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of $2|\alpha |^{3s}+|\alpha |^{4s}=1$.[6]
Variants and generalization
For any unimodular substitution of Pisot type, which verifies a coincidence condition (apparently always verified), one can construct a similar set called "Rauzy fractal of the map". They all display self-similarity and generate, for the examples below, a periodic tiling of the plane.
• s(1)=12, s(2)=31, s(3)=1
• s(1)=12, s(2)=23, s(3)=312
• s(1)=123, s(2)=1, s(3)=31
• s(1)=123, s(2)=1, s(3)=1132
See also
• List of fractals
References
1. Rauzy, Gérard (1982). "Nombres algébriques et substitutions" (PDF). Bull. Soc. Math. Fr. (in French). 110: 147–178. Zbl 0522.10032.
2. Lothaire (2005) p.525
3. Pytheas Fogg (2002) p.232
4. Lothaire (2005) p.546
5. Pytheas Fogg (2002) p.233
6. Messaoudi, Ali (2000). "Frontière du fractal de Rauzy et système de numération complexe. (Boundary of the Rauzy fractal and complex numeration system)" (PDF). Acta Arith. (in French). 95 (3): 195–224. Zbl 0968.28005.
• Arnoux, Pierre; Harriss, Edmund (August 2014). "WHAT IS... a Rauzy Fractal?". Notices of the American Mathematical Society. 61 (7): 768–770. doi:10.1090/noti1144.
• Berthé, Valérie; Siegel, Anne; Thuswaldner, Jörg (2010). "Substitutions, Rauzy fractals and tilings". In Berthé, Valérie; Rigo, Michel (eds.). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. pp. 248–323. ISBN 978-0-521-51597-9. Zbl 1247.37015.
• Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and its Applications. Vol. 105. Cambridge University Press. ISBN 978-0-521-84802-2. MR 2165687. Zbl 1133.68067.
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
External links
Wikimedia Commons has media related to Rauzy fractals.
• Topological properties of Rauzy fractals
• Substitutions, Rauzy fractals and tilings, Anne Siegel, 2009
• Rauzy fractals for free group automorphisms, 2006
• Pisot Substitutions and Rauzy fractals
• Rauzy fractals
• Numberphile video about Rauzy fractals and Tribonacci numbers
| Wikipedia |
Law of trichotomy
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.[1]
More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exactly one of xRy, yRx and x = y holds. Writing R as <, this is stated in formal logic as:
$\forall x\in X\,\forall y\in X\,([x<y\,\land \,\lnot (y<x)\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,y<x\,\land \,\lnot (x=y)]\,\lor \,[\lnot (x<y)\,\land \,\lnot (y<x)\,\land \,x=y])\,.$
Properties
• A relation is trichotomous if, and only if, it is asymmetric and connected.
• If a trichotomous relation is also transitive, then it is a strict total order; this is a special case of a strict weak order.[2][3]
Examples
• On the set X = {a,b,c}, the relation R = { (a,b), (a,c), (b,c) } is transitive and trichotomous, and hence a strict total order.
• On the same set, the cyclic relation R = { (a,b), (b,c), (c,a) } is trichotomous, but not transitive; it is even antitransitive.
Trichotomy on numbers
A law of trichotomy on some set X of numbers usually expresses that some tacitly given ordering relation on X is a trichotomous one. An example is the law "For arbitrary real numbers x and y, exactly one of x < y, y < x, or x = y applies"; some authors even fix y to be zero,[1] relying on the real number's additive linearly ordered group structure. The latter is a group equipped with a trichotomous order.
In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between integers and between rational numbers. The law does not hold in general in intuitionistic logic.
In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).[4]
See also
• Begriffsschrift contains an early formulation of the law of trichotomy
• Dichotomy
• Law of noncontradiction
• Law of excluded middle
• Three-way comparison
References
1. Trichotomy Law at MathWorld
2. Jerrold E. Marsden & Michael J. Hoffman (1993) Elementary Classical Analysis, page 27, W. H. Freeman and Company ISBN 0-7167-2105-8
3. H.S. Bear (1997) An Introduction to Mathematical Analysis, page 11, Academic Press ISBN 0-12-083940-7
4. Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.
| Wikipedia |
Trichotomy theorem
In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher (1981, 1983) for rank 3 and by Gorenstein & Lyons (1983) for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), and groups of uniqueness type, where Aschbacher proved that there are no simple groups.
Not to be confused with Trichotomy (mathematics).
References
• Aschbacher, Michael (1981), "Finite groups of rank 3. I", Inventiones Mathematicae, 63 (3): 357–402, doi:10.1007/BF01389061, ISSN 0020-9910, MR 0620676
• Aschbacher, Michael (1983), "Finite groups of rank 3. II", Inventiones Mathematicae, 71 (1): 51–163, doi:10.1007/BF01393339, ISSN 0020-9910, MR 0688262
• Gorenstein, D.; Lyons, Richard (1983), "The local structure of finite groups of characteristic 2 type", Memoirs of the American Mathematical Society, 42 (276): vii+731, ISBN 978-0-8218-2276-0, ISSN 0065-9266, MR 0690900
| Wikipedia |
Shut the box
Shut the box (also called canoga,[1] batten down the hatches[1] or trick-track[2]) is a game of dice for one or more players, commonly played in a group of two to four for stakes. Traditionally, a counting box is used with tiles numbered 1 to 9 where each can be covered with a hinged or sliding mechanism, though the game can be played with only a pair of dice, pen, and paper. Variations exist where the box has 10 or 12 tiles. In 2018 the game had a renaissance in Liverpool, England, when it became the house game at Hobo Kiosk pub on the Baltic Triangle. It was popularized by DJ duo Coffee and Turntables and became the most played board game in Merseyside for 4 years in a row.
Shut the box
Other namesCanoga, batten down the hatches, trick-track
GenresDice-rolling
Solitaire
Players1 (Solitaire) or more
Setup time1 minute
Playing time2–3 minutes per player
ChanceHigh (Dice rolling)
SkillsRisk management
Arithmetic
Rules
At the start of the game all levers or tiles are "open" (cleared, up), showing the numerals 1 to 9.
During the game, each player plays in turn. A player begins their turn by throwing or rolling the die or dice into the box. If the player does not have 7, 8, or 9 still available, they may choose to either roll one die or the standard two. Otherwise, the player must roll both dice.
After throwing, the player adds up (or subtracts) the pips (dots) on the dice and then "shuts" (closes, covers) one of any combination of open numbers that sums to the total number of dots showing on the dice. For example, if the total number of dots is 8, the player may choose any of the following sets of numbers (as long as all of the numbers in the set are available to be covered):
• 8
• 7, 1
• 6, 2
• 5, 3
• 5, 2, 1
• 4, 3, 1
The player then rolls the dice again, aiming to shut more numbers. The player continues throwing the dice and shutting numbers until reaching a point at which, given the results produced by the dice, the player cannot shut any more numbers. At that point, the player scores the sum of the numbers that are still uncovered. For example, if the numbers 2, 3, and 5 are still open when the player throws a one, the player's score is 10 (2 + 3 + 5 = 10). Play then passes to the next player.
After every player has taken a turn, the player with the lowest score wins.
If a player succeeds in closing all of the numbers, that player is said to have "Shut the Box" – the player wins immediately and the game is over.
Traditional pub play
In English pubs, shut the box is traditionally played as a gambling game. Each player deposits an agreed amount of money into a pool at the beginning of the game, and the winner of the game collects the money in pool at the end of the game and in some cases the box as well.
Variants
Shut the box is a traditional game, and there are many local and traditional variations in the rules. In addition, due to the game's growing popularity, many variations of the game have developed in recent years.
Popular variants are:
• Golf – A player's score is the sum of the numbers remaining uncovered at the end of their turn. The player with the lowest score wins.
• Missionary – A player's score is how many of the tiles remain uncovered at the end of the player's turn. For example, a player scores 3 if, at the end of their turn, 3 tiles remain open. The player with the lowest score wins.
• Canoga – A gambling variant produced by the Pacific Game Company; the company also produced a 12-tile variant, Canoga XII. (Canoga can also be played using a regular game set using chips.)
1. Chips are divided evenly among all players.
2. Players decide on an ante to place in the kitty (a half-round pocket on the playing field).
3. Players roll to see who goes first; play then rotates clockwise.
4. Players play a traditional round, scoring as described in "Golf" above, resulting in a winner and loser(s).
5. Each loser pays their difference in score to the winner. For example, if the lowest (winning) score is 11, and a losing score is 15, the loser pays 4 to the winner. The winner is paid by each loser.
6. Bonus payout: if the winner "clears the board" (scores 0 or "shuts the box"), the payout is as above but doubled, and the winner takes the kitty.
7. If there are tied winners, total payout is either split between or among the winners or multiplied for each winner, depending on how the players agree to do this before starting the game.
The following are examples of known variations in play, setup, and scoring:
• 2 to go – Standard game, the numbers 1 to 9 start up. On the first roll, the number 2 must be one of the ones dropped. Any player who rolls a 4 on their first roll loses immediately.
• 3 to go – The same as "2 to go" but the number 3 must be dropped instead.
• 3 down extreme – Numbers 1–3 are pre-dropped, leaving numbers 4–9 up.
• Lucky number 7 – The only number up is 7, and the first person to roll a 7 wins.
• Unlucky number 7 – A standard game, when a 7 is rolled, the game stops.
• Against all odds – All odd numbers are up and evens down.
• Even Stevens – All even numbers are up and odds down.
• Full house – 12 numbers are up.
• The 300 – 2 boxes and 4 dice are used, with the second box representing numbers 13–24 (24 + 23 + 22 + ... + 2 + 1 = 300); in the absence of a second box, cards or dominoes can be used to represent tiles 13–24. A Double 12 Dominoes set can also be used with four dice for this variant and other domino sets can be used by themselves to, in the case of the Double 18 set, provide for the use of six dice by themselves without the counting box.
• Thai style (Jackpot) – Always roll two dice, but only cover one tile matching one of the dice or their sum. For example, if the dice show a 2 and a 3 you may cover one of 2, 3, or 5. The best strategy is to use the combined score for a high tile (7,8,9), if possible, otherwise choose the lowest tile. The success rate for this strategy is 7.9855%.[3]
• Digital – A player's score at the end of the turn is the number obtained by reading the up digits as a decimal number from left to right. For example, if 1, 2, and 5 are left up the score is 125. This is also known as "Say what you see", a reference to Roy Walker's catch phrase from the TV game show Catchphrase.
• 2012 – All 12 are up, but use a 20-sided die rather than the pair of 6 dice: 20-sided die playing 12 numbers.
It is also possible to play extended versions in which each game is a "round" of a longer game. Examples of such versions include:
• Tournament – Rounds are played with the Golf scoring method until a player reaches or exceeds a grand total of 100 points, at which time the player with the lowest point total is declared to be the winner. At the end of each round, each player's score for the round is added to the player's total score. When a player's score reaches 45, the player must drop out of the game. The last player remaining wins the game.
• Simplified variant for younger players – Needs at least a 2 player box. During the game, each player plays in turn. After rolling both dice, the player adds up the dots on the dice and then shuts the tile for either the total number of dots, or one or both of the numbers on the dice. For example, if the player rolled a 6 and a 2, they may close either the 8 tile, or both the 6 tile and the 2 tile, or just the 6 tile, or just the 2 tile (as long as the numbers are available to be covered). The player then rolls the dice again, aiming to shut more numbers. The player continues throwing the dice and shutting numbers. The first player to shut all the tiles wins.
Dominoes can also be used for the tiles – this also provides the option of using up to six dice if a Double 18 domino set is used. A deck of cards can also be used as tiles, and if so desired a complete conventional Western deck with the jokers (54 cards) can provide for the use of up to nine dice.
Played without dice
• Domino Non-Dice Variants – A non-dice variant of the game can be played with the dominoes from either Western or Chinese sets ranging from 1 and 1 to 6 and 6 pips being used and most effectively put into a small bag for drawing, and the double blank being included along with blank and 1, with the former being either a free turn of sorts as it adds to zero or ending the turn, and the latter effectively ending the turn if the 1 tile has already been used.
• Card Non-Dice Variants – Another variant using cards dealt from one or more decks using the A, 2, 3, 4, 5, and 6 (sometimes along with the 7, 8, 9, and 10), and two face cards agreed upon for the equivalent of dice rolls adding up to 11 and 12 pips.
History
Unconfirmed histories of the game suggest a variety of origins, including 12th century Normandy (northern France) as well as the mid 20th century Channel Islands (Jersey and Guernsey), which one source credits to a man known as 'Chalky' Towbridge.[4] A 1967 edition of Brewing Review describes the game as being native to the Channel Islands, and records it being played in Manchester pubs in the mid-1960s.[5]
Taylor in "Pub Games" from 1976 mentions a claim that the game dates back to at least Napoleonic times. He reports a revival in the United Kingdom in "the last fifteen years or so", that is from the 1960s. Canada Dry distributed them to many pubs as a publicity novelty "some years" prior to 1976.[6]
Shut the box is the basis of the American television quiz show High Rollers, which ran from 1974 to 1976 and 1978 to 1980 on NBC with Alex Trebek as the host. The show resurfaced from 1987 to 1988, this time hosted by Wink Martindale.
Versions of the game have also been played in Barotseland (Zambia, central Africa).
See also
• Pub games
• Partition (number theory)
References
1. Luck, Steve (2006). Classic Indoor Games: The Complete Guide. Aurum. ISBN 978-1-84513-164-7.
2. Parlett, David (1999). The Oxford History of Board Games. Oxford University Press. ISBN 978-0-19-212998-7.
3. "Jackpot". GitHub. Retrieved 2022-10-23.
4. Finn, Timothy (1975). Pub Games of England (New ed.). London: Queen Anne Press. ISBN 9780362002461.
5. "'Shut the Box' at Wilson's New House". Brewing Review. 1967.
6. Taylor, Arthur R. (1976). Pub Games. St. Albans: Mayflower. p. 188. ISBN 978-0-583-12650-2.
External links
• Shut the Box open source physics versions with options from Wikipedia
• Shut the Box – Online version. 9, 10, 11 and 12 tile versions.
• Shut the Slots – Online version. Variation based on spinning Slots for the tiles.
• Shut the Box – Online HTML5/Javascript version, rules and variants explained (accessible via application menu), MIT licensed.
• Shut the Box– Online PHP version, originally created as part of a Boy Scouts of America Programming merit badge project, includes two auto move algorithms with performance analysis.
Dice games
Traditional games
• Balut
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Gambling games
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Commercial games
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Portal:Games
| Wikipedia |
Tricolorability
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.
Rules of tricolorability
In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next.[1] A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:[2]
1. At least two colors must be used, and
2. At each crossing, the three incident strands are either all the same color or all different colors.
Some references state instead that all three colors must be used.[3] For a knot, this is equivalent to the definition above; however, for a link it is not.
"The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and figure-eight knot are not. If the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is."[2]
Examples
Here is an example of how to color a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue.
Example of a tricolorable knot
The granny knot is tricolorable. In this coloring the three strands at every crossing have three different colors. Coloring one but not both of the trefoil knots all red would also give an admissible coloring. The true lover's knot is also tricolorable.[4]
Tricolorable knots with less than nine crossings include 61, 74, 77, 85, 810, 811, 815, 818, 819, 820, and 821.
Example of a non-tricolorable knot
The figure-eight knot is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing. If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color. Since tricolorability is a knot invariant, none of its other diagrams can be tricolored either.
Isotopy invariant
Tricolorability is an isotopy invariant, which is a property of a knot or link that remains constant regardless of any ambient isotopy. This can be proven by examining Reidemeister moves. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant.
Reidemeister Move I is tricolorable.Reidemeister Move II is tricolorable.Reidemeister Move III is tricolorable.
Properties
Because tricolorability is a binary classification (a link is either tricolorable or not*), it is a relatively weak invariant. The composition of a tricolorable knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings. In this case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings (just color every arc the same color). In this case, a link is 3-colorable if it has more than three 3-colorings.
Any separable link with a tricolorable separable component is also tricolorable.
In torus knots
If the torus knot/link denoted by (m,n) is tricolorable, then so are (j*m,i*n) and (i*n,j*m) for any natural numbers i and j.
See also
• Fox n-coloring
• Graph coloring
Sources
1. Xaoyu Qiao, E. L., Knot Theory Week 2: Tricolorability (January 20, 2015), Section 3.
2. Weisstein, Eric W. (2010). CRC Concise Encyclopedia of Mathematics, Second Edition, p.3045. ISBN 9781420035223. quoted at Weisstein, Eric W. "Tricolorable". MathWorld. Accessed: May 5, 2013.
3. Gilbert, N.D. and Porter, T. (1994) Knots and Surfaces, p. 8
4. Bestvina, Mladen (February 2003). "Knots: a handout for mathcircles", Math.Utah.edu.
Further reading
• Weisstein, Eric W. "Three-Colorable Knot". MathWorld. Accessed: May 5, 2013.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
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Other
• Alexander's theorem
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| Wikipedia |
Tricomi–Carlitz polynomials
In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by Tricomi (1951) and Carlitz (1958) and Karlin and McGregor (1959), related to random walks on the positive integers.
They are given in terms of Laguerre polynomials by
$\ell _{n}(x)=(-1)^{n}L_{n}^{(x-n)}(x).$
They are special cases of the Chihara–Ismail polynomials.
References
• Carlitz, Leonard (1958), "On some polynomials of Tricomi", Boll. Un. Mat. Ital. (3), 13: 58–64, MR 0103303
• Karlin, Samuel; McGregor, James (1959), "Random walks", Illinois Journal of Mathematics, 3: 66–81, ISSN 0019-2082, MR 0100927
• Tricomi, Francesco G. (1951), "A class of non-orthogonal polynomials related to those of Laguerre", Journal d'Analyse Mathématique, 1: 209–231, doi:10.1007/BF02790089, ISSN 0021-7670, MR 0051351
| Wikipedia |
Tridecahedron
A tridecahedron, or triskaidecahedron, is a polyhedron with thirteen faces. There are numerous topologically distinct forms of a tridecahedron, for example the dodecagonal pyramid and hendecagonal prism. However, a tridecahedron cannot be a regular polyhedron, because there is no regular polygon that can form a regular tridecahedron, and there are only five known regular polyhedra.[notes 1][1]
Common tridecahedrons
Space-filling tridecahedron
Elongated hexagonal pyramid
Hendecagonal prism
Gyroelongated square pyramid
Convex
There are 96,262,938 topologically distinct convex tridecahedra, excluding mirror images, having at least 9 vertices.[2] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) There is a pseudo-space-filling tridecahedron that can fill all of 3-space together with its mirror-image.[3]
Common tridecahedrons
Examples
Name (vertex layout) Symbol Stereogram Expanded view Faces Edges Apexes
Hendecagonal prism t{2,11}
{11}x{}
13 square × 11
hendecagon × 2
3322
Dodecagonal pyramid ( )∨{12} 13 triangle × 12
dodecagon × 1
2413
Elongated hexagonal pyramid 13 triangle × 6
square × 6
hexagon × 1
2413
Space-filling tridecahedron 13quadrilateral × 6
pentagon × 6
hexagon × 1
3019
Gyroelongated square pyramid 13 triangle × 12
square × 1
20 9
truncated hexagonal trapohedron 13 1 hexagon base
6 pentagon sides
6 kite sides
30 19
Biaugmented pentagonal prism 13triangle × 8
square × 3
pentagon × 2
2312
Hendecagonal prism
Main article: Hendecagonal prism
A hendecagonal prism is a prism with a hendecagon base. It is a type of tridecahedron, which consists of 13 faces, 22 vertices, and 33 sides. A regular hendecagonal prism is a hendecagonal prism whose faces are regular hendecagons, and each of its vertices is a common vertex of 2 squares and 1 hendecagon. In a vertex figure a hendecagonal prism is represented by $4{.}4{.}11$; in Schläfly notation it can be represented by {11}×{} or t{2, 11}; can be used in a Coxeter-Dynkin diagram to represent it; its Wythoff symbol is 2 11 | 2; in Conway polyhedron notation it can be represented by P11. If the side length of the base of a regular hendecagonal prism is $s$ and the height is $h$, then its volume $V$ and surface area $S$ are:[4]
$V={\frac {11hs^{2}\cot {\frac {\pi }{11}}}{4}}\approx 9.36564hs^{2}$
$S=11s\left(h+{\frac {1}{2}}s\cot {\frac {\pi }{11}}\right)\approx 11s\left(h+1.70284s\right)$
Dodecagonal pyramid
See also: Pyramid (geometry)
A dodecagonal pyramid is a pyramid with a dodecagonal base. It is a type of tridecahedron, which has 13 faces, 24 edges, and 13 vertices, and its dual polyhedron is itself.[5] A regular dodecagonal pyramid is a dodecagonal pyramid whose base is a regular dodecagon. If the side length of the base of a regular twelve-sided pyramid is $s$ and the height is $h$, then its volume $V$ and surface area $S$ are:[5]
$V=\left(2+{\sqrt {3}}\right)hs^{2}\approx 3.73205hs^{2}$
$S=3s\left({\sqrt {4h^{2}+\left(7+4{\sqrt {3}}\right)s^{2}}}+\left(2+{\sqrt {3}}\right)s\right)\approx 3s\left({\sqrt {4h^{2}+13.9282s^{2}}}+3.73205s\right)$
Space-filling tridecahedron
Space-filling tridecahedron
Type6 trapezoids
6 pentagons
1 regular hexagons
Faces13
Edges30
Vertices19
A space-filling tridecahedron[6][7] is a tridecahedron that can completely fill three-dimensional space without leaving gaps. It has 13 faces, 30 edges, and 19 vertices. Among the thirteen faces, there are six trapezoids, six pentagons and one regular hexagon.[8]
Dual polyhedron
The polyhedron's dual polyhedron is an enneadecahedron. It is similar to a twisted half-cube, but one of its vertices is treated as a face before twisting.
Image Rotation animation Expanded view
Original polyhedron
tridecahedron
Dual polyhedron
enneadecahedron
Notes
1. Even if there were 13 faces that were all congruent, it would still not be considered a regular polyhedra. In addition to being congruent on each face of a regular polyhedron, the angles and sides on each face must be equal in size. Only regular polygons meet this condition, but the faces of a thirteen-sided shape do not, so there cannot be a regular tridecahedron.
References
1. proof of platonic solids Archived 2015-11-21 at the Wayback Machine mathsisfun.com [2016-1-10]
2. Counting polyhedra
3. Ludacer, Randy. "Honeycombs and Structural Package Design: More Ways of Taking Up Space". Beach Branding & Packaging Design. Archived from the original on 2016-03-07.
4. "Hendecagon prism". Wolfram Alpha Site.
5. "Dodecagon pyramid". Wolfram Alpha Site.
6. Oblate Rhombohedra Archived 2016-09-21 at the Wayback Machine science.unitn.it [2016-1-10]
7. Virtual Polyhedra, Greek Numerical Prefixes Archived 2016-01-15 at the Wayback Machine, 1996, George W. Hart, georgehart.com [2016-1-10]
8. A space-filling polyhedron with 13 faces Archived 2017-07-01 at the Wayback Machine science.unitn.it [2016-1-10]
External links
• Self-dual tridecahedra
• What Are Polyhedra?, with Greek Numerical Prefixes
Polyhedra
Listed by number of faces and type
1–10 faces
• Monohedron
• Dihedron
• Trihedron
• Tetrahedron
• Pentahedron
• Hexahedron
• Heptahedron
• Octahedron
• Enneahedron
• Decahedron
11–20 faces
• Hendecahedron
• Dodecahedron
• Tridecahedron
• Tetradecahedron
• Pentadecahedron
• Hexadecahedron
• Heptadecahedron
• Octadecahedron
• Enneadecahedron
• Icosahedron
>20 faces
• Icositetrahedron (24)
• Triacontahedron (30)
• Hexecontahedron (60)
• Enneacontahedron (90)
• Hectotriadiohedron (132)
• Apeirohedron (∞)
elemental things
• face
• edge
• vertex
• uniform polyhedron (two infinite groups and 75)
• regular polyhedron (9)
• quasiregular polyhedron (7)
• semiregular polyhedron (two infinite groups and 59)
convex polyhedron
• Platonic solid (5)
• Archimedean solid (13)
• Catalan solid (13)
• Johnson solid (92)
non-convex polyhedron
• Kepler–Poinsot polyhedron (4)
• Star polyhedron (infinite)
• Uniform star polyhedron (57)
prismatoids
• prism
• antiprism
• frustum
• cupola
• wedge
• pyramid
• parallelepiped
| Wikipedia |
Trident curve
In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula:
$xy+ax^{3}+bx^{2}+cx=d$
Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0, y = 1, z = 0; if we substitute x = x/z and y = 1/z into the equation of the trident curve, we get
$ax^{3}+bx^{2}z+cxz^{2}+xz=dz^{3},$
which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus zero.
References
• Lawrence, J. Dennis (1972). A Catalog of Special Plane Curves. Dover Publications. p. 110. ISBN 0-486-60288-5.
External links
• O'Connor, John J.; Robertson, Edmund F., "Trident of Newton", MacTutor History of Mathematics Archive, University of St Andrews
| Wikipedia |
Tridiagonal matrix
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main diagonal). For example, the following matrix is tridiagonal:
${\begin{pmatrix}1&4&0&0\\3&4&1&0\\0&2&3&4\\0&0&1&3\\\end{pmatrix}}.$
The determinant of a tridiagonal matrix is given by the continuant of its elements.[1]
An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be done with the Lanczos algorithm.
Properties
A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix.[2] In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n — the dimension of the tridiagonal. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties. Furthermore, if a real tridiagonal matrix A satisfies ak,k+1 ak+1,k > 0 for all k, so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, by a diagonal change of basis matrix. Hence, its eigenvalues are real. If we replace the strict inequality by ak,k+1 ak+1,k ≥ 0, then by continuity, the eigenvalues are still guaranteed to be real, but the matrix need no longer be similar to a Hermitian matrix.[3]
The set of all n × n tridiagonal matrices forms a 3n-2 dimensional vector space.
Many linear algebra algorithms require significantly less computational effort when applied to diagonal matrices, and this improvement often carries over to tridiagonal matrices as well.
Determinant
Main article: continuant (mathematics)
The determinant of a tridiagonal matrix A of order n can be computed from a three-term recurrence relation.[4] Write f1 = |a1| = a1 (i.e., f1 is the determinant of the 1 by 1 matrix consisting only of a1), and let
$f_{n}={\begin{vmatrix}a_{1}&b_{1}\\c_{1}&a_{2}&b_{2}\\&c_{2}&\ddots &\ddots \\&&\ddots &\ddots &b_{n-1}\\&&&c_{n-1}&a_{n}\end{vmatrix}}.$
The sequence (fi) is called the continuant and satisfies the recurrence relation
$f_{n}=a_{n}f_{n-1}-c_{n-1}b_{n-1}f_{n-2}$
with initial values f0 = 1 and f−1 = 0. The cost of computing the determinant of a tridiagonal matrix using this formula is linear in n, while the cost is cubic for a general matrix.
Inversion
The inverse of a non-singular tridiagonal matrix T
$T={\begin{pmatrix}a_{1}&b_{1}\\c_{1}&a_{2}&b_{2}\\&c_{2}&\ddots &\ddots \\&&\ddots &\ddots &b_{n-1}\\&&&c_{n-1}&a_{n}\end{pmatrix}}$
is given by
$(T^{-1})_{ij}={\begin{cases}(-1)^{i+j}b_{i}\cdots b_{j-1}\theta _{i-1}\phi _{j+1}/\theta _{n}&{\text{ if }}i<j\\\theta _{i-1}\phi _{j+1}/\theta _{n}&{\text{ if }}i=j\\(-1)^{i+j}c_{j}\cdots c_{i-1}\theta _{j-1}\phi _{i+1}/\theta _{n}&{\text{ if }}i>j\\\end{cases}}$
where the θi satisfy the recurrence relation
$\theta _{i}=a_{i}\theta _{i-1}-b_{i-1}c_{i-1}\theta _{i-2}\qquad i=2,3,\ldots ,n$
with initial conditions θ0 = 1, θ1 = a1 and the ϕi satisfy
$\phi _{i}=a_{i}\phi _{i+1}-b_{i}c_{i}\phi _{i+2}\qquad i=n-1,\ldots ,1$
with initial conditions ϕn+1 = 1 and ϕn = an.[5][6]
Closed form solutions can be computed for special cases such as symmetric matrices with all diagonal and off-diagonal elements equal[7] or Toeplitz matrices[8] and for the general case as well.[9][10]
In general, the inverse of a tridiagonal matrix is a semiseparable matrix and vice versa.[11]
Solution of linear system
Main article: tridiagonal matrix algorithm
A system of equations Ax = b for $b\in \mathbb {R} ^{n}$ can be solved by an efficient form of Gaussian elimination when A is tridiagonal called tridiagonal matrix algorithm, requiring O(n) operations.[12]
Eigenvalues
When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely:[13][14]
$a+2{\sqrt {bc}}\cos \left({\frac {k\pi }{n+1}}\right),\qquad k=1,\ldots ,n.$
A real symmetric tridiagonal matrix has real eigenvalues, and all the eigenvalues are distinct (simple) if all off-diagonal elements are nonzero.[15] Numerous methods exist for the numerical computation of the eigenvalues of a real symmetric tridiagonal matrix to arbitrary finite precision, typically requiring $O(n^{2})$ operations for a matrix of size $n\times n$, although fast algorithms exist which (without parallel computation) require only $O(n\log n)$.[16]
As a side note, an unreduced symmetric tridiagonal matrix is a matrix containing non-zero off-diagonal elements of the tridiagonal, where the eigenvalues are distinct while the eigenvectors are unique up to a scale factor and are mutually orthogonal.[17]
Similarity to symmetric tridiagonal matrix
For unsymmetric or nonsymmetric tridiagonal matrices one can compute the eigendecomposition using a similarity transformation. Given a real tridiagonal, nonsymmetric matrix
$T={\begin{pmatrix}a_{1}&b_{1}\\c_{1}&a_{2}&b_{2}\\&c_{2}&\ddots &\ddots \\&&\ddots &\ddots &b_{n-1}\\&&&c_{n-1}&a_{n}\end{pmatrix}}$
where $b_{i}\neq c_{i}$. Assume that each product of off-diagonal entries is strictly positive $b_{i}c_{i}>0$ and define a transformation matrix $D$ by
$D:=\operatorname {diag} (\delta _{1},\dots ,\delta _{n})\quad {\text{for}}\quad \delta _{i}:={\begin{cases}1&,\,i=1\\{\sqrt {\frac {c_{i-1}\dots c_{1}}{b_{i-1}\dots b_{1}}}}&,\,i=2,\dots ,n\,.\end{cases}}$
The similarity transformation $D^{-1}TD$ yields a symmetric tridiagonal matrix $J$ by:[18]
$J:=D^{-1}TD={\begin{pmatrix}a_{1}&\operatorname {sgn} b_{1}\,{\sqrt {b_{1}c_{1}}}\\\operatorname {sgn} b_{1}\,{\sqrt {b_{1}c_{1}}}&a_{2}&\operatorname {sgn} b_{2}\,{\sqrt {b_{2}c_{2}}}\\&\operatorname {sgn} b_{2}\,{\sqrt {b_{2}c_{2}}}&\ddots &\ddots \\&&\ddots &\ddots &\operatorname {sgn} b_{n-1}\,{\sqrt {b_{n-1}c_{n-1}}}\\&&&\operatorname {sgn} b_{n-1}\,{\sqrt {b_{n-1}c_{n-1}}}&a_{n}\end{pmatrix}}\,.$
Note that $T$ and $J$ have the same eigenvalues.
Computer programming
A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to tridiagonal form. So, many eigenvalue algorithms, when applied to a Hermitian matrix, reduce the input Hermitian matrix to (symmetric real) tridiagonal form as a first step.[19]
A tridiagonal matrix can also be stored more efficiently than a general matrix by using a special storage scheme. For instance, the LAPACK Fortran package stores an unsymmetric tridiagonal matrix of order n in three one-dimensional arrays, one of length n containing the diagonal elements, and two of length n − 1 containing the subdiagonal and superdiagonal elements.
Applications
The discretization in space of the one-dimensional diffusion or heat equation
${\frac {\partial u(t,x)}{\partial t}}=\alpha {\frac {\partial ^{2}u(t,x)}{\partial x^{2}}}$
using second order central finite differences results in
${\begin{pmatrix}{\frac {\partial u_{1}(t)}{\partial t}}\\{\frac {\partial u_{2}(t)}{\partial t}}\\\vdots \\{\frac {\partial u_{N}(t)}{\partial t}}\end{pmatrix}}={\frac {\alpha }{\Delta x}}{\begin{pmatrix}-2&1&0&\ldots &0\\1&-2&1&\ddots &\vdots \\0&\ddots &\ddots &\ddots &0\\\vdots &&1&-2&1\\0&\ldots &0&1&-2\end{pmatrix}}{\begin{pmatrix}u_{1}(t)\\u_{2}(t)\\\vdots \\u_{N}(t)\\\end{pmatrix}}$
with discretization constant $\Delta x$. The matrix is tridiagonal with $a_{i}=-2$ and $b_{i}=c_{i}=1$. Note: no boundary conditions are used here.
See also
• Pentadiagonal matrix
• Jacobi matrix (operator)
Notes
1. Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 516–525.
2. Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis. Cambridge University Press. p. 28. ISBN 0521386322.
3. Horn & Johnson, page 174
4. El-Mikkawy, M. E. A. (2004). "On the inverse of a general tridiagonal matrix". Applied Mathematics and Computation. 150 (3): 669–679. doi:10.1016/S0096-3003(03)00298-4.
5. Da Fonseca, C. M. (2007). "On the eigenvalues of some tridiagonal matrices". Journal of Computational and Applied Mathematics. 200: 283–286. doi:10.1016/j.cam.2005.08.047.
6. Usmani, R. A. (1994). "Inversion of a tridiagonal jacobi matrix". Linear Algebra and its Applications. 212–213: 413–414. doi:10.1016/0024-3795(94)90414-6.
7. Hu, G. Y.; O'Connell, R. F. (1996). "Analytical inversion of symmetric tridiagonal matrices". Journal of Physics A: Mathematical and General. 29 (7): 1511. doi:10.1088/0305-4470/29/7/020.
8. Huang, Y.; McColl, W. F. (1997). "Analytical inversion of general tridiagonal matrices". Journal of Physics A: Mathematical and General. 30 (22): 7919. doi:10.1088/0305-4470/30/22/026.
9. Mallik, R. K. (2001). "The inverse of a tridiagonal matrix". Linear Algebra and its Applications. 325: 109–139. doi:10.1016/S0024-3795(00)00262-7.
10. Kılıç, E. (2008). "Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions". Applied Mathematics and Computation. 197: 345–357. doi:10.1016/j.amc.2007.07.046.
11. Raf Vandebril; Marc Van Barel; Nicola Mastronardi (2008). Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems. JHU Press. Theorem 1.38, p. 41. ISBN 978-0-8018-8714-7.
12. Golub, Gene H.; Van Loan, Charles F. (1996). Matrix Computations (3rd ed.). The Johns Hopkins University Press. ISBN 0-8018-5414-8.
13. Noschese, S.; Pasquini, L.; Reichel, L. (2013). "Tridiagonal Toeplitz matrices: Properties and novel applications". Numerical Linear Algebra with Applications. 20 (2): 302. doi:10.1002/nla.1811.
14. This can also be written as $a+2{\sqrt {bc}}\cos(k\pi /{(n+1)})$ because $\cos(x)=-\cos(\pi -x)$, as is done in: Kulkarni, D.; Schmidt, D.; Tsui, S. K. (1999). "Eigenvalues of tridiagonal pseudo-Toeplitz matrices" (PDF). Linear Algebra and its Applications. 297: 63. doi:10.1016/S0024-3795(99)00114-7.
15. Parlett, B.N. (1980). The Symmetric Eigenvalue Problem. Prentice Hall, Inc.
16. Coakley, E.S.; Rokhlin, V. (2012). "A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices". Applied and Computational Harmonic Analysis. 34 (3): 379–414. doi:10.1016/j.acha.2012.06.003.
17. Dhillon, Inderjit Singh. A New O(n 2 ) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem (PDF). p. 8.
18. "www.math.hkbu.edu.hk math lecture" (PDF).
19. Eidelman, Yuli; Gohberg, Israel; Gemignani, Luca (2007-01-01). "On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms". Linear Algebra and its Applications. 420 (1): 86–101. doi:10.1016/j.laa.2006.06.028. ISSN 0024-3795.
External links
• Tridiagonal and Bidiagonal Matrices in the LAPACK manual.
• Moawwad El-Mikkawy, Abdelrahman Karawia (2006). "Inversion of general tridiagonal matrices" (PDF). Applied Mathematics Letters. 19 (8): 712–720. doi:10.1016/j.aml.2005.11.012. Archived from the original (PDF) on 2011-07-20.
• High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
• Tridiagonal linear system solver in C++
Matrix classes
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With specific applications
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| Wikipedia |
Tridiagonal matrix algorithm
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
$a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}=d_{i},$
where $a_{1}=0$ and $c_{n}=0$.
${\begin{bmatrix}b_{1}&c_{1}&&&0\\a_{2}&b_{2}&c_{2}&&\\&a_{3}&b_{3}&\ddots &\\&&\ddots &\ddots &c_{n-1}\\0&&&a_{n}&b_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}d_{1}\\d_{2}\\d_{3}\\\vdots \\d_{n}\end{bmatrix}}.$
For such systems, the solution can be obtained in $O(n)$ operations instead of $O(n^{3})$ required by Gaussian elimination. A first sweep eliminates the $a_{i}$'s, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation.
Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite;[1][2] for a more precise characterization of stability of Thomas' algorithm, see Higham Theorem 9.12.[3] If stability is required in the general case, Gaussian elimination with partial pivoting (GEPP) is recommended instead.[2]
Method
The forward sweep consists of the computation of new coefficients as follows, denoting the new coefficients with primes:
$c'_{i}={\begin{cases}{\cfrac {c_{i}}{b_{i}}},&i=1,\\{\cfrac {c_{i}}{b_{i}-a_{i}c'_{i-1}}},&i=2,3,\dots ,n-1\end{cases}}$
and
$d'_{i}={\begin{cases}{\cfrac {d_{i}}{b_{i}}},&i=1,\\{\cfrac {d_{i}-a_{i}d'_{i-1}}{b_{i}-a_{i}c'_{i-1}}},&i=2,3,\dots ,n.\end{cases}}$
The solution is then obtained by back substitution:
$x_{n}=d'_{n},$
$x_{i}=d'_{i}-c'_{i}x_{i+1},\quad i=n-1,n-2,\ldots ,1.$
The method above does not modify the original coefficient vectors, but must also keep track of the new coefficients. If the coefficient vectors may be modified, then an algorithm with less bookkeeping is:
For $i=2,3,\dots ,n,$ do
$w={\cfrac {a_{i}}{b_{i-1}}},$
$b_{i}:=b_{i}-wc_{i-1},$
$d_{i}:=d_{i}-wd_{i-1},$
followed by the back substitution
$x_{n}={\cfrac {d_{n}}{b_{n}}},$
$x_{i}={\cfrac {d_{i}-c_{i}x_{i+1}}{b_{i}}}\quad {\text{for }}i=n-1,n-2,\dots ,1.$
The implementation in a VBA subroutine without preserving the coefficient vectors:
Sub TriDiagonal_Matrix_Algorithm(N%, A#(), B#(), C#(), D#(), X#())
Dim i%, W#
For i = 2 To N
W = A(i) / B(i - 1)
B(i) = B(i) - W * C(i - 1)
D(i) = D(i) - W * D(i - 1)
Next i
X(N) = D(N) / B(N)
For i = N - 1 To 1 Step -1
X(i) = (D(i) - C(i) * X(i + 1)) / B(i)
Next i
End Sub
Derivation
The derivation of the tridiagonal matrix algorithm is a special case of Gaussian elimination.
Suppose that the unknowns are $x_{1},\ldots ,x_{n}$, and that the equations to be solved are:
${\begin{alignedat}{4}&&&b_{1}x_{1}&&+c_{1}x_{2}&&=d_{1}\\&a_{i}x_{i-1}&&+b_{i}x_{i}&&+c_{i}x_{i+1}&&=d_{i}\,,\quad i=2,\ldots ,n-1\\&a_{n}x_{n-1}&&+b_{n}x_{n}&&&&=d_{n}\,.\end{alignedat}}$
Consider modifying the second ($i=2$) equation with the first equation as follows:
$({\mbox{equation 2}})\cdot b_{1}-({\mbox{equation 1}})\cdot a_{2}$
which would give:
$(b_{2}b_{1}-c_{1}a_{2})x_{2}+c_{2}b_{1}x_{3}=d_{2}b_{1}-d_{1}a_{2}.$
Note that $x_{1}$ has been eliminated from the second equation. Using a similar tactic with the modified second equation on the third equation yields:
$(b_{3}(b_{2}b_{1}-c_{1}a_{2})-c_{2}b_{1}a_{3})x_{3}+c_{3}(b_{2}b_{1}-c_{1}a_{2})x_{4}=d_{3}(b_{2}b_{1}-c_{1}a_{2})-(d_{2}b_{1}-d_{1}a_{2})a_{3}.\,$
This time $x_{2}$ was eliminated. If this procedure is repeated until the $n^{th}$ row; the (modified) $n^{th}$ equation will involve only one unknown, $x_{n}$. This may be solved for and then used to solve the $(n-1)^{th}$ equation, and so on until all of the unknowns are solved for.
Clearly, the coefficients on the modified equations get more and more complicated if stated explicitly. By examining the procedure, the modified coefficients (notated with tildes) may instead be defined recursively:
${\tilde {a}}_{i}=0\,$
${\tilde {b}}_{1}=b_{1}\,$
${\tilde {b}}_{i}=b_{i}{\tilde {b}}_{i-1}-{\tilde {c}}_{i-1}a_{i}\,$
${\tilde {c}}_{1}=c_{1}\,$
${\tilde {c}}_{i}=c_{i}{\tilde {b}}_{i-1}\,$
${\tilde {d}}_{1}=d_{1}\,$
${\tilde {d}}_{i}=d_{i}{\tilde {b}}_{i-1}-{\tilde {d}}_{i-1}a_{i}.\,$
To further hasten the solution process, ${\tilde {b}}_{i}$ may be divided out (if there's no division by zero risk), the newer modified coefficients, each notated with a prime, will be:
$a'_{i}=0\,$
$b'_{i}=1\,$
$c'_{1}={\frac {c_{1}}{b_{1}}}\,$
$c'_{i}={\frac {c_{i}}{b_{i}-c'_{i-1}a_{i}}}\,$
$d'_{1}={\frac {d_{1}}{b_{1}}}\,$
$d'_{i}={\frac {d_{i}-d'_{i-1}a_{i}}{b_{i}-c'_{i-1}a_{i}}}.\,$
This gives the following system with the same unknowns and coefficients defined in terms of the original ones above:
${\begin{array}{lcl}x_{i}+c'_{i}x_{i+1}=d'_{i}\qquad &;&\ i=1,\ldots ,n-1\\x_{n}=d'_{n}\qquad &;&\ i=n.\\\end{array}}\,$
The last equation involves only one unknown. Solving it in turn reduces the next last equation to one unknown, so that this backward substitution can be used to find all of the unknowns:
$x_{n}=d'_{n}\,$
$x_{i}=d'_{i}-c'_{i}x_{i+1}\qquad ;\ i=n-1,n-2,\ldots ,1.$ ;\ i=n-1,n-2,\ldots ,1.}
Variants
In some situations, particularly those involving periodic boundary conditions, a slightly perturbed form of the tridiagonal system may need to be solved:
${\begin{alignedat}{4}&a_{1}x_{n}&&+b_{1}x_{1}&&+c_{1}x_{2}&&=d_{1}\\&a_{i}x_{i-1}&&+b_{i}x_{i}&&+c_{i}x_{i+1}&&=d_{i}\,,\quad i=2,\ldots ,n-1\\&a_{n}x_{n-1}&&+b_{n}x_{n}&&+c_{n}x_{1}&&=d_{n}\,.\end{alignedat}}$
In this case, we can make use of the Sherman–Morrison formula to avoid the additional operations of Gaussian elimination and still use the Thomas algorithm. The method requires solving a modified non-cyclic version of the system for both the input and a sparse corrective vector, and then combining the solutions. This can be done efficiently if both solutions are computed at once, as the forward portion of the pure tridiagonal matrix algorithm can be shared.
If we indicate by:
$A={\begin{bmatrix}b_{1}&c_{1}&&&a_{1}\\a_{2}&b_{2}&c_{2}&&\\&a_{3}&b_{3}&\ddots &\\&&\ddots &\ddots &c_{n-1}\\c_{n}&&&a_{n}&b_{n}\end{bmatrix}},x={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}},d={\begin{bmatrix}d_{1}\\d_{2}\\d_{3}\\\vdots \\d_{n}\end{bmatrix}}$
Then the system to be solved is:
$Ax=d$
In this case the coefficients $a_{1}$ and $c_{n}$ are, generally speaking, non-zero, so their presence does not allow to apply the Thomas algorithm directly. We can therefore consider $B\in \mathbb {R} ^{n\times n}$ and $u,v\in \mathbb {R} ^{n}$ as following:
$B={\begin{bmatrix}b_{1}-\gamma &c_{1}&&&0\\a_{2}&b_{2}&c_{2}&&\\&a_{3}&b_{3}&\ddots &\\&&\ddots &\ddots &c_{n-1}\\0&&&a_{n}&b_{n}-{\frac {c_{n}a_{1}}{\gamma }}\end{bmatrix}},u={\begin{bmatrix}\gamma \\0\\0\\\vdots \\c_{n}\end{bmatrix}},v={\begin{bmatrix}1\\0\\0\\\vdots \\a_{1}/\gamma \end{bmatrix}}.$
Where $\gamma \in \mathbb {R} $ is a parameter to be chosen. The matrix $A$ can be reconstructed as $A=B+uv^{\mathsf {T}}$. The solution is then obtained in the following way:[4] first we solve two tridiagonal systems of equations applying the Thomas algorithm:
$By=d\qquad \qquad Bq=u$
Then we reconstruct the solution $x$ using the Shermann-Morrison formula:
${\begin{aligned}x&=A^{-1}d=(B+uv^{T})^{-1}d=B^{-1}d-{\frac {B^{-1}uv^{T}B^{-1}}{1+v^{T}B^{-1}u}}d=y-{\frac {qv^{T}y}{1+v^{T}q}}\end{aligned}}$
The implementation in C without preserving the coefficient vectors:
typedef struct{
double A[n+2];
double B[n+2];
double C[n+2];
double D[n+2];
} COEFFICIENTS;
//Apply Thomas Alg., unknowns x[1],...,x[n]
void ThomasAlg(double x[n+1], COEFFICIENTS* coeff){
double u[n+1]={},v[n+1]={};
double y[n+1]={},q[n+1]={};
double* A=coeff->A, *B=coeff->B,*C=coeff->C,*D=coeff->D;
double Value=0;
double w;
int i;
u[1]=gamma;
u[n]=C[n];
v[1]=1;
v[n]=A[1]/gamma;
//create matrix B
A[1]=0;
B[1]=B[1]-gamma;
B[n]=B[n]-(C[n]*A[n])/gamma;
C[n]=0;
for(i=2;i<n+1;i++){
w=A[i]/B[i-1];
B[i]=B[i]-w*C[i-1];
D[i]=D[i]-w*D[i-1];
u[i]=u[i]-w*u[i-1];
}
y[n]=D[n]/B[n];
q[n]=u[n]/B[n];
for(i=n-1;i>0;i--){
y[i]=(D[i]-C[i]*y[i+1])/B[i];
q[i]=(u[i]-C[i]*q[i+1])/B[i];
}
Value=(v[1]*y[1]+v[n]*y[n])/(1+v[1]*q[1]+v[n]*q[n]);
for(i=1;i<n+1;i++){
x[i]=y[i]-q[i]*Value;
}
}
There is also another way to solve the slightly perturbed form of the tridiagonal system considered above.[5] Let us consider two auxiliary linear systems of dimension $(n-1)\times (n-1)$:
${\begin{aligned}\qquad \ \ \ \ \ b_{2}u_{2}+c_{2}u_{3}&=d_{2}\\a_{3}u_{2}+b_{3}u_{3}+c_{3}u_{4}&=d_{3}\\a_{i}u_{i-1}+b_{i}u_{i}+c_{i}u_{i+1}&=d_{i}\\\dots \\a_{n}u_{n-1}+b_{n}u_{n}\qquad &=d_{n}\,.\end{aligned}}\quad i=4,\ldots ,n-1\qquad \qquad {\begin{aligned}\qquad \ \ \ \ \ b_{2}v_{2}+c_{2}v_{3}&=-a_{2}\\a_{3}v_{2}+b_{3}v_{3}+c_{3}v_{4}&=0\\a_{i}u_{i-1}+b_{i}u_{i}+c_{i}u_{i+1}&=0\\\dots \\a_{n}v_{n-1}+b_{n}v_{n}\qquad &=-c_{n}\,.\end{aligned}}\quad i=4,\ldots ,n-1$
For convenience, we additionally define $u_{1}=0$ and $v_{1}=1$. We can now find the solutions $\{u_{2},u_{3}\dots ,u_{n}\}$ and $\{v_{2},v_{3}\dots ,v_{n}\}$ applying Thomas algorithm to the two auxiliary tridiagonal system.
The solution $\{x_{1},x_{2}\dots ,x_{n}\}$ can be then represented in the form:
$x_{i}=u_{i}+x_{1}v_{i}\qquad i=1,2,\dots ,n$
Indeed, multiplying each equation of the second auxiliary system by $x_{1}$, adding with the corresponding equation of the first auxiliary system and using the representation $x_{i}=u_{i}+x_{1}v_{i}$, we immediately see that equations number $2$ through $n$ of the original system are satisfied; it only remains to satisfy equation number $1$. To do so, consider formula for $i=2$ and $i=n$ and substitute $x_{2}=u_{2}+x_{1}v_{2}$and $x_{n}=u_{n}+x_{1}v_{n}$ into the first equation of the original system. This yields one scalar equation for $x_{1}$:
$b_{1}x_{1}+c_{1}(u_{2}+x_{1}v_{2})+a_{1}(u_{n}+x_{1}v_{n})=d_{1}$
As such, we find:
$x_{1}={\frac {d_{1}-a_{1}u_{n}-c_{1}u_{2}}{b_{1}+a_{1}v_{n}+c_{1}v_{2}}}$
The implementation in Dev-C++ without preserving the coefficient vectors:
typedef struct{
double A[n+2];
double B[n+2];
double C[n+2];
double D[n+2];
} COEFFICIENTS;
//Apply Thomas Alg., unknowns x[1],...,x[n]
void ThomasAlg(double x[n+1],COEFFICIENTS* coeff){
double u[n+1]={},v[n+1]={};
double* A=coeff->A, *B=coeff->B,*C=coeff->C,*D=coeff->D;
double w,F[n+1]={};
F[2]=-A[2];
F[n]=-C[n];
int i;
u[1]=0;
v[1]=1;
for(i=3;i<n+1;i++){
w=A[i]/B[i-1];
B[i]=B[i]-w*C[i-1];
D[i]=D[i]-w*D[i-1];
F[i]=F[i]-w*F[i-1];
}
u[n]=D[n]/B[n];
v[n]=F[n]/B[n];
for(i=n-1;i>1;i--){
u[i]=(D[i]-C[i]*u[i+1])/B[i];
v[i]=(F[i]-C[i]*v[i+1])/B[i];
}
x[1]=(D[1]-A[1]*u[n]-C[1]*u[2])/(B[1]+A[1]*v[n]+C[1]*v[2]);
for(i=2;i<n+1;i++){
x[i]=u[i]+x[1]*v[i];
}
}
In both cases the auxiliary systems to be solved are genuinely tri-diagonal, so the overall computational complexity of solving system $Ax=d$ remains linear with the respect to the dimension of the system $n$, that is $O(n)$ arithmetic operations.
In other situations, the system of equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g., the 2D Poisson problem). Simplified forms of Gaussian elimination have been developed for these situations.[6]
The textbook Numerical Mathematics by Quarteroni, Sacco and Saleri, lists a modified version of the algorithm which avoids some of the divisions (using instead multiplications), which is beneficial on some computer architectures.
Parallel tridiagonal solvers have been published for many vector and parallel architectures, including GPUs[7][8]
For an extensive treatment of parallel tridiagonal and block tridiagonal solvers see [9]
References
The Wikibook Algorithm Implementation has a page on the topic of: Tridiagonal matrix algorithm
1. Pradip Niyogi (2006). Introduction to Computational Fluid Dynamics. Pearson Education India. p. 76. ISBN 978-81-7758-764-7.
2. Biswa Nath Datta (2010). Numerical Linear Algebra and Applications, Second Edition. SIAM. p. 162. ISBN 978-0-89871-765-5.
3. Nicholas J. Higham (2002). Accuracy and Stability of Numerical Algorithms: Second Edition. SIAM. p. 175. ISBN 978-0-89871-802-7.
4. Batista, Milan; Ibrahim Karawia, Abdel Rahman A. (2009). "The use of the Sherman–Morrison–Woodbury formula to solve cyclic block tri-diagonal and cyclic block penta-diagonal linear systems of equations". Applied Mathematics and Computation. 210 (2): 558–563. doi:10.1016/j.amc.2009.01.003. ISSN 0096-3003.
5. Ryaben’kii, Victor S.; Tsynkov, Semyon V. (2006-11-02), "Introduction", A Theoretical Introduction to Numerical Analysis, Chapman and Hall/CRC, pp. 1–19, doi:10.1201/9781420011166-1, ISBN 978-0-429-14339-7, retrieved 2022-05-25
6. Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2007). "Section 3.8". Numerical Mathematics. Springer, New York. ISBN 978-3-540-34658-6.
7. Chang, L.-W.; Hwu, W,-M. (2014). "A guide for implementing tridiagonal solvers on GPUs". In V. Kidratenko (ed.). Numerical Computations with GPUs. Springer. ISBN 978-3-319-06548-9.{{cite conference}}: CS1 maint: multiple names: authors list (link)
8. Venetis, I.E.; Kouris, A.; Sobczyk, A.; Gallopoulos, E.; Sameh, A. (2015). "A direct tridiagonal solver based on Givens rotations for GPU architectures". Parallel Computing. 49: 101–116. doi:10.1016/j.parco.2015.03.008.
9. Gallopoulos, E.; Philippe, B.; Sameh, A.H. (2016). "Chapter 5". Parallelism in Matrix Computations. Springer. ISBN 978-94-017-7188-7.
• Conte, S. D.; de Boor, C. (1972). Elementary Numerical Analysis. McGraw-Hill, New York. ISBN 0070124469.
• This article incorporates text from the article Tridiagonal_matrix_algorithm_-_TDMA_(Thomas_algorithm) on CFD-Wiki that is under the GFDL license.
• Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). "Section 2.4". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
Numerical linear algebra
Key concepts
• Floating point
• Numerical stability
Problems
• System of linear equations
• Matrix decompositions
• Matrix multiplication (algorithms)
• Matrix splitting
• Sparse problems
Hardware
• CPU cache
• TLB
• Cache-oblivious algorithm
• SIMD
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Software
• MATLAB
• Basic Linear Algebra Subprograms (BLAS)
• LAPACK
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| Wikipedia |
Tridiminished icosahedron
In geometry, the tridiminished icosahedron is one of the Johnson solids (J63). The name refers to one way of constructing it, by removing three pentagonal pyramids (J2) from a regular icosahedron, which replaces three sets of five triangular faces from the icosahedron with three mutually adjacent pentagonal faces.
Tridiminished icosahedron
TypeJohnson
J62 – J63 – J64
Faces2+3 triangles
3 pentagons
Edges15
Vertices9
Vertex configuration2x3(3.52)
3(33.5)
Symmetry groupC3v
Dual polyhedronDual of tridiminished icosahedron (unnamed enneahedron)
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Related polytopes
The tridiminished icosahedron is the vertex figure of the snub 24-cell, a uniform 4-polytope (4-dimensional polytope).
See also
• Diminished icosahedron (J11)
• Metabidiminished icosahedron (J62)
External links
• Eric W. Weisstein, Tridiminished icosahedron (Johnson Solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
| Wikipedia |
Tridiminished rhombicosidodecahedron
In geometry, the tridiminished rhombicosidodecahedron is one of the Johnson solids (J83). It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae removed.
Tridiminished rhombicosidodecahedron
TypeJohnson
J82 – J83 – J84
Faces2+3 triangles
3×3+6 squares
3×3 pentagons
3 decagons
Edges75
Vertices45
Vertex configuration5×6(4.5.10)
3×3+6(3.4.5.4)
Symmetry groupC3v
Dual polyhedron-
PropertiesConvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Related Johnson solids are:
• J76: diminished rhombicosidodecahedron with one cupola removed,
• J80: parabidiminished rhombicosidodecahedron with two opposing cupolae removed, and
• J81: metabidiminished rhombicosidodecahedron with two non-opposing cupolae removed.
External links
• Eric W. Weisstein, Tridiminished rhombicosidodecahedron (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
| Wikipedia |
Tridyakis icosahedron
In geometry, the tridyakis icosahedron is the dual polyhedron of the nonconvex uniform polyhedron, icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
Tridyakis icosahedron
TypeStar polyhedron
Face
ElementsF = 120, E = 180
V = 44 (χ = −16)
Symmetry groupIh, [5,3], *532
Index referencesDU45
dual polyhedronIcositruncated dodecadodecahedron
Proportions
The triangles have one angle of $\arccos({\frac {3}{5}})\approx 53.130\,102\,354\,16^{\circ }$, one of $\arccos({\frac {1}{3}}+{\frac {4}{15}}{\sqrt {5}})\approx 21.624\,633\,927\,143^{\circ }$ and one of $\arccos({\frac {1}{3}}-{\frac {4}{15}}{\sqrt {5}})\approx 105.245\,263\,718\,70^{\circ }$. The dihedral angle equals $\arccos(-{\frac {7}{8}})\approx 151.044\,975\,628\,14^{\circ }$. Part of each triangle lies within the solid, hence is invisible in solid models.
See also
• Catalan solid Duals to convex uniform polyhedra
• Uniform polyhedra
• List of uniform polyhedra
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 Photo on page 96, Dorman Luke construction and stellation pattern on page 97.
• Weisstein, Eric W. "Tridyakis Icosahedron". MathWorld.
| Wikipedia |
Triebel–Lizorkin space
In the mathematical discipline known as functional analysis, a Triebel–Lizorkin space is a generalization of many standard function spaces such as Lp spaces and Sobolev spaces. It is named after Hans Triebel (born February 7th 1936 in Dessau) and Petr Ivanovich Lizorkin.
External links
• Homogeneity Property of Besov and Triebel-Lizorkin Spaces
References
• Peetre, Jaak (1975), "On spaces of Triebel–Lizorkin type", Arkiv för Matematik, 13 (1–2): 123–130, doi:10.1007/BF02386201.
• Kalton, N., Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin Spaces and Applications to Problems in Partial Differential Equations (PDF)
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
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• Hölder
• Nuclear
• Orlicz
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• Sobolev
• Topological vector
Properties
• Barrelled
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• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
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• Banach–Alaoglu
Operators
• Adjoint
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• Hilbert–Schmidt
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Algebras
• Banach algebra
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• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
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Open problems
• Invariant subspace problem
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Applications
• Hardy space
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• Noncommutative geometry
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• Distribution (or Generalized functions)
Advanced topics
• Approximation property
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• Mathematics portal
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| Wikipedia |
Proofs of trigonometric identities
There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.
Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine, or on the differential equation $f''+f=0$ to which they are solutions.
Elementary trigonometric identities
Definitions
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
$\sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}={\frac {a}{h}}$
$\cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}={\frac {b}{h}}$
$\tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {a}{b}}$
$\cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}={\frac {b}{a}}$
$\sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}={\frac {h}{b}}$
$\csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}={\frac {h}{a}}$
Ratio identities
In the case of angles smaller than a right angle, the following identities are direct consequences of above definitions through the division identity
${\frac {a}{b}}={\frac {\left({\frac {a}{h}}\right)}{\left({\frac {b}{h}}\right)}}.$
They remain valid for angles greater than 90° and for negative angles.
$\tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {\left({\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}\right)}{\left({\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}\right)}}={\frac {\sin \theta }{\cos \theta }}$
$\cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}={\frac {\left({\frac {\mathrm {adjacent} }{\mathrm {adjacent} }}\right)}{\left({\frac {\mathrm {opposite} }{\mathrm {adjacent} }}\right)}}={\frac {1}{\tan \theta }}={\frac {\cos \theta }{\sin \theta }}$
$\sec \theta ={\frac {1}{\cos \theta }}={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}$
$\csc \theta ={\frac {1}{\sin \theta }}={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}$
$\tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}={\frac {\left({\frac {\mathrm {opposite} \times \mathrm {hypotenuse} }{\mathrm {opposite} \times \mathrm {adjacent} }}\right)}{\left({\frac {\mathrm {adjacent} \times \mathrm {hypotenuse} }{\mathrm {opposite} \times \mathrm {adjacent} }}\right)}}={\frac {\left({\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}\right)}{\left({\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}\right)}}={\frac {\sec \theta }{\csc \theta }}$
Or
$\tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {\left({\frac {1}{\csc \theta }}\right)}{\left({\frac {1}{\sec \theta }}\right)}}={\frac {\left({\frac {\csc \theta \sec \theta }{\csc \theta }}\right)}{\left({\frac {\csc \theta \sec \theta }{\sec \theta }}\right)}}={\frac {\sec \theta }{\csc \theta }}$
$\cot \theta ={\frac {\csc \theta }{\sec \theta }}$
Complementary angle identities
Two angles whose sum is π/2 radians (90 degrees) are complementary. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining:
$\sin \left(\pi /2-\theta \right)=\cos \theta $
$\cos \left(\pi /2-\theta \right)=\sin \theta $
$\tan \left(\pi /2-\theta \right)=\cot \theta $
$\cot \left(\pi /2-\theta \right)=\tan \theta $
$\sec \left(\pi /2-\theta \right)=\csc \theta $
$\csc \left(\pi /2-\theta \right)=\sec \theta $
Pythagorean identities
Main article: Pythagorean trigonometric identity
Identity 1:
$\sin ^{2}\theta +\cos ^{2}\theta =1$
The following two results follow from this and the ratio identities. To obtain the first, divide both sides of $\sin ^{2}\theta +\cos ^{2}\theta =1$ by $\cos ^{2}\theta $; for the second, divide by $\sin ^{2}\theta $.
$\tan ^{2}\theta +1\ =\sec ^{2}\theta $
$\sec ^{2}\theta -\tan ^{2}\theta =1$
Similarly
$1\ +\cot ^{2}\theta =\csc ^{2}\theta $
$\csc ^{2}\theta -\cot ^{2}\theta =1$
Identity 2:
The following accounts for all three reciprocal functions.
$\csc ^{2}\theta +\sec ^{2}\theta -\cot ^{2}\theta =2\ +\tan ^{2}\theta $
Proof 2:
Refer to the triangle diagram above. Note that $a^{2}+b^{2}=h^{2}$ by Pythagorean theorem.
$\csc ^{2}\theta +\sec ^{2}\theta ={\frac {h^{2}}{a^{2}}}+{\frac {h^{2}}{b^{2}}}={\frac {a^{2}+b^{2}}{a^{2}}}+{\frac {a^{2}+b^{2}}{b^{2}}}=2\ +{\frac {b^{2}}{a^{2}}}+{\frac {a^{2}}{b^{2}}}$
Substituting with appropriate functions -
$2\ +{\frac {b^{2}}{a^{2}}}+{\frac {a^{2}}{b^{2}}}=2\ +\tan ^{2}\theta +\cot ^{2}\theta $
Rearranging gives:
$\csc ^{2}\theta +\sec ^{2}\theta -\cot ^{2}\theta =2\ +\tan ^{2}\theta $
Angle sum identities
See also: List of trigonometric identities § Angle sum and difference identities
Sine
Draw a horizontal line (the x-axis); mark an origin O. Draw a line from O at an angle $\alpha $ above the horizontal line and a second line at an angle $\beta $ above that; the angle between the second line and the x-axis is $\alpha +\beta $.
Place P on the line defined by $\alpha +\beta $ at a unit distance from the origin.
Let PQ be a line perpendicular to line OQ defined by angle $\alpha $, drawn from point Q on this line to point P. $\therefore $ OQP is a right angle.
Let QA be a perpendicular from point A on the x-axis to Q and PB be a perpendicular from point B on the x-axis to P. $\therefore $ OAQ and OBP are right angles.
Draw R on PB so that QR is parallel to the x-axis.
Now angle $RPQ=\alpha $ (because $OQA={\frac {\pi }{2}}-\alpha $, making $RQO=\alpha ,RQP={\frac {\pi }{2}}-\alpha $, and finally $RPQ=\alpha $)
$RPQ={\tfrac {\pi }{2}}-RQP={\tfrac {\pi }{2}}-({\tfrac {\pi }{2}}-RQO)=RQO=\alpha $
$OP=1$
$PQ=\sin \beta $
$OQ=\cos \beta $
${\frac {AQ}{OQ}}=\sin \alpha $, so $AQ=\sin \alpha \cos \beta $
${\frac {PR}{PQ}}=\cos \alpha $, so $PR=\cos \alpha \sin \beta $
$\sin(\alpha +\beta )=PB=RB+PR=AQ+PR=\sin \alpha \cos \beta +\cos \alpha \sin \beta $
By substituting $-\beta $ for $\beta $ and using Symmetry, we also get:
$\sin(\alpha -\beta )=\sin \alpha \cos(-\beta )+\cos \alpha \sin(-\beta )$
$\sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta $
Cosine
Using the figure above,
$OP=1$
$PQ=\sin \beta $
$OQ=\cos \beta $
${\frac {OA}{OQ}}=\cos \alpha $, so $OA=\cos \alpha \cos \beta $
${\frac {RQ}{PQ}}=\sin \alpha $, so $RQ=\sin \alpha \sin \beta $
$\cos(\alpha +\beta )=OB=OA-BA=OA-RQ=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta $
By substituting $-\beta $ for $\beta $ and using Symmetry, we also get:
$\cos(\alpha -\beta )=\cos \alpha \cos(-\beta )-\sin \alpha \sin(-\beta ),$
$\cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta $
Also, using the complementary angle formulae,
${\begin{aligned}\cos(\alpha +\beta )&=\sin \left(\pi /2-(\alpha +\beta )\right)\\&=\sin \left((\pi /2-\alpha )-\beta \right)\\&=\sin \left(\pi /2-\alpha \right)\cos \beta -\cos \left(\pi /2-\alpha \right)\sin \beta \\&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\end{aligned}}$
Tangent and cotangent
From the sine and cosine formulae, we get
$\tan(\alpha +\beta )={\frac {\sin(\alpha +\beta )}{\cos(\alpha +\beta )}}={\frac {\sin \alpha \cos \beta +\cos \alpha \sin \beta }{\cos \alpha \cos \beta -\sin \alpha \sin \beta }}$
Dividing both numerator and denominator by $\cos \alpha \cos \beta $, we get
$\tan(\alpha +\beta )={\frac {\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }}$
Subtracting $\beta $ from $\alpha $, using $\tan(-\beta )=-\tan \beta $,
$\tan(\alpha -\beta )={\frac {\tan \alpha +\tan(-\beta )}{1-\tan \alpha \tan(-\beta )}}={\frac {\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }}$
Similarly from the sine and cosine formulae, we get
$\cot(\alpha +\beta )={\frac {\cos(\alpha +\beta )}{\sin(\alpha +\beta )}}={\frac {\cos \alpha \cos \beta -\sin \alpha \sin \beta }{\sin \alpha \cos \beta +\cos \alpha \sin \beta }}$
Then by dividing both numerator and denominator by $\sin \alpha \sin \beta $, we get
$\cot(\alpha +\beta )={\frac {\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }}$
Or, using $\cot \theta ={\frac {1}{\tan \theta }}$,
$\cot(\alpha +\beta )={\frac {1-\tan \alpha \tan \beta }{\tan \alpha +\tan \beta }}={\frac {{\frac {1}{\tan \alpha \tan \beta }}-1}{{\frac {1}{\tan \alpha }}+{\frac {1}{\tan \beta }}}}={\frac {\cot \alpha \cot \beta -1}{\cot \alpha +\cot \beta }}$
Using $\cot(-\beta )=-\cot \beta $,
$\cot(\alpha -\beta )={\frac {\cot \alpha \cot(-\beta )-1}{\cot \alpha +\cot(-\beta )}}={\frac {\cot \alpha \cot \beta +1}{\cot \beta -\cot \alpha }}$
Double-angle identities
From the angle sum identities, we get
$\sin(2\theta )=2\sin \theta \cos \theta $
and
$\cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta $
The Pythagorean identities give the two alternative forms for the latter of these:
$\cos(2\theta )=2\cos ^{2}\theta -1$
$\cos(2\theta )=1-2\sin ^{2}\theta $
The angle sum identities also give
$\tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}={\frac {2}{\cot \theta -\tan \theta }}$
$\cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {\cot \theta -\tan \theta }{2}}$
It can also be proved using Euler's formula
$e^{i\varphi }=\cos \varphi +i\sin \varphi $
Squaring both sides yields
$e^{i2\varphi }=(\cos \varphi +i\sin \varphi )^{2}$
But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields
$e^{i2\varphi }=\cos 2\varphi +i\sin 2\varphi $
It follows that
$(\cos \varphi +i\sin \varphi )^{2}=\cos 2\varphi +i\sin 2\varphi $.
Expanding the square and simplifying on the left hand side of the equation gives
$i(2\sin \varphi \cos \varphi )+\cos ^{2}\varphi -\sin ^{2}\varphi \ =\cos 2\varphi +i\sin 2\varphi $.
Because the imaginary and real parts have to be the same, we are left with the original identities
$\cos ^{2}\varphi -\sin ^{2}\varphi \ =\cos 2\varphi $,
and also
$2\sin \varphi \cos \varphi =\sin 2\varphi $.
Half-angle identities
The two identities giving the alternative forms for cos 2θ lead to the following equations:
$\cos {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}},$
$\sin {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}.$
The sign of the square root needs to be chosen properly—note that if 2π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to use depends on the value of θ.
For the tan function, the equation is:
$\tan {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}.$
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:
$\tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}.$
Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
$\tan {\frac {\theta }{2}}={\frac {1-\cos \theta }{\sin \theta }}.$
This also gives:
$\tan {\frac {\theta }{2}}=\csc \theta -\cot \theta .$
Similar manipulations for the cot function give:
$\cot {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta .$
Miscellaneous – the triple tangent identity
If $\psi +\theta +\phi =\pi =$ half circle (for example, $\psi $, $\theta $ and $\phi $ are the angles of a triangle),
$\tan(\psi )+\tan(\theta )+\tan(\phi )=\tan(\psi )\tan(\theta )\tan(\phi ).$
Proof:[1]
${\begin{aligned}\psi &=\pi -\theta -\phi \\\tan(\psi )&=\tan(\pi -\theta -\phi )\\&=-\tan(\theta +\phi )\\&={\frac {-\tan \theta -\tan \phi }{1-\tan \theta \tan \phi }}\\&={\frac {\tan \theta +\tan \phi }{\tan \theta \tan \phi -1}}\\(\tan \theta \tan \phi -1)\tan \psi &=\tan \theta +\tan \phi \\\tan \psi \tan \theta \tan \phi -\tan \psi &=\tan \theta +\tan \phi \\\tan \psi \tan \theta \tan \phi &=\tan \psi +\tan \theta +\tan \phi \\\end{aligned}}$
Miscellaneous – the triple cotangent identity
If $\psi +\theta +\phi ={\tfrac {\pi }{2}}=$ quarter circle,
$\cot(\psi )+\cot(\theta )+\cot(\phi )=\cot(\psi )\cot(\theta )\cot(\phi )$.
Proof:
Replace each of $\psi $, $\theta $, and $\phi $ with their complementary angles, so cotangents turn into tangents and vice versa.
Given
$\psi +\theta +\phi ={\tfrac {\pi }{2}}$
$\therefore ({\tfrac {\pi }{2}}-\psi )+({\tfrac {\pi }{2}}-\theta )+({\tfrac {\pi }{2}}-\phi )={\tfrac {3\pi }{2}}-(\psi +\theta +\phi )={\tfrac {3\pi }{2}}-{\tfrac {\pi }{2}}=\pi $
so the result follows from the triple tangent identity.
Sum to product identities
• $\sin \theta \pm \sin \phi =2\sin \left({\frac {\theta \pm \phi }{2}}\right)\cos \left({\frac {\theta \mp \phi }{2}}\right)$
• $\cos \theta +\cos \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)$
• $\cos \theta -\cos \phi =-2\sin \left({\frac {\theta +\phi }{2}}\right)\sin \left({\frac {\theta -\phi }{2}}\right)$
Proof of sine identities
First, start with the sum-angle identities:
$\sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta $
$\sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta $
By adding these together,
$\sin(\alpha +\beta )+\sin(\alpha -\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta +\sin \alpha \cos \beta -\cos \alpha \sin \beta =2\sin \alpha \cos \beta $
Similarly, by subtracting the two sum-angle identities,
$\sin(\alpha +\beta )-\sin(\alpha -\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta -\sin \alpha \cos \beta +\cos \alpha \sin \beta =2\cos \alpha \sin \beta $
Let $\alpha +\beta =\theta $ and $\alpha -\beta =\phi $,
$\therefore \alpha ={\frac {\theta +\phi }{2}}$ and $\beta ={\frac {\theta -\phi }{2}}$
Substitute $\theta $ and $\phi $
$\sin \theta +\sin \phi =2\sin \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)$
$\sin \theta -\sin \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\sin \left({\frac {\theta -\phi }{2}}\right)=2\sin \left({\frac {\theta -\phi }{2}}\right)\cos \left({\frac {\theta +\phi }{2}}\right)$
Therefore,
$\sin \theta \pm \sin \phi =2\sin \left({\frac {\theta \pm \phi }{2}}\right)\cos \left({\frac {\theta \mp \phi }{2}}\right)$
Proof of cosine identities
Similarly for cosine, start with the sum-angle identities:
$\cos(\alpha +\beta )=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta $
$\cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta $
Again, by adding and subtracting
$\cos(\alpha +\beta )+\cos(\alpha -\beta )=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta +\cos \alpha \cos \beta +\sin \alpha \sin \beta =2\cos \alpha \cos \beta $
$\cos(\alpha +\beta )-\cos(\alpha -\beta )=\cos \alpha \cos \beta \ -\sin \alpha \sin \beta -\cos \alpha \cos \beta -\sin \alpha \sin \beta =-2\sin \alpha \sin \beta $
Substitute $\theta $ and $\phi $ as before,
$\cos \theta +\cos \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)$
$\cos \theta -\cos \phi =-2\sin \left({\frac {\theta +\phi }{2}}\right)\sin \left({\frac {\theta -\phi }{2}}\right)$
Inequalities
See also: List of triangle inequalities
The figure at the right shows a sector of a circle with radius 1. The sector is θ/(2π) of the whole circle, so its area is θ/2. We assume here that θ < π/2.
$OA=OD=1$
$AB=\sin \theta $
$CD=\tan \theta $
The area of triangle OAD is AB/2, or sin(θ)/2. The area of triangle OCD is CD/2, or tan(θ)/2.
Since triangle OAD lies completely inside the sector, which in turn lies completely inside triangle OCD, we have
$\sin \theta <\theta <\tan \theta .$
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property.[2] For the sine function, we can handle other values. If θ > π/2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin θ < θ. So we have
${\frac {\sin \theta }{\theta }}<1\ \ \ \mathrm {if} \ \ \ 0<\theta .$
For negative values of θ we have, by the symmetry of the sine function
${\frac {\sin \theta }{\theta }}={\frac {\sin(-\theta )}{-\theta }}<1.$
Hence
${\frac {\sin \theta }{\theta }}<1\quad {\text{if }}\quad \theta \neq 0,$
and
${\frac {\tan \theta }{\theta }}>1\quad {\text{if }}\quad 0<\theta <{\frac {\pi }{2}}.$
Identities involving calculus
Preliminaries
$\lim _{\theta \to 0}{\sin \theta }=0$
$\lim _{\theta \to 0}{\cos \theta }=1$
Sine and angle ratio identity
$\lim _{\theta \to 0}{\frac {\sin \theta }{\theta }}=1$
In other words, the function sine is differentiable at 0, and its derivative is 1.
Proof: From the previous inequalities, we have, for small angles
$\sin \theta <\theta <\tan \theta $,
Therefore,
${\frac {\sin \theta }{\theta }}<1<{\frac {\tan \theta }{\theta }}$,
Consider the right-hand inequality. Since
$\tan \theta ={\frac {\sin \theta }{\cos \theta }}$
$\therefore 1<{\frac {\sin \theta }{\theta \cos \theta }}$
Multiply through by $\cos \theta $
$\cos \theta <{\frac {\sin \theta }{\theta }}$
Combining with the left-hand inequality:
$\cos \theta <{\frac {\sin \theta }{\theta }}<1$
Taking $\cos \theta $ to the limit as $\theta \to 0$
$\lim _{\theta \to 0}{\cos \theta }=1$
Therefore,
$\lim _{\theta \to 0}{\frac {\sin \theta }{\theta }}=1$
Cosine and angle ratio identity
$\lim _{\theta \to 0}{\frac {1-\cos \theta }{\theta }}=0$
Proof:
${\begin{aligned}{\frac {1-\cos \theta }{\theta }}&={\frac {1-\cos ^{2}\theta }{\theta (1+\cos \theta )}}\\&={\frac {\sin ^{2}\theta }{\theta (1+\cos \theta )}}\\&=\left({\frac {\sin \theta }{\theta }}\right)\times \sin \theta \times \left({\frac {1}{1+\cos \theta }}\right)\\\end{aligned}}$
The limits of those three quantities are 1, 0, and 1/2, so the resultant limit is zero.
Cosine and square of angle ratio identity
$\lim _{\theta \to 0}{\frac {1-\cos \theta }{\theta ^{2}}}={\frac {1}{2}}$
Proof:
As in the preceding proof,
${\frac {1-\cos \theta }{\theta ^{2}}}={\frac {\sin \theta }{\theta }}\times {\frac {\sin \theta }{\theta }}\times {\frac {1}{1+\cos \theta }}.$
The limits of those three quantities are 1, 1, and 1/2, so the resultant limit is 1/2.
Proof of compositions of trig and inverse trig functions
All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function
$\sin[\arctan(x)]={\frac {x}{\sqrt {1+x^{2}}}}$
Proof:
We start from
$\sin ^{2}\theta +\cos ^{2}\theta =1$ (I)
Then we divide this equation (I) by $\cos ^{2}\theta $
$\cos ^{2}\theta ={\frac {1}{\tan ^{2}\theta +1}}$ (II)
$1-\sin ^{2}\theta ={\frac {1}{\tan ^{2}\theta +1}}$
Then use the substitution $\theta =\arctan(x)$:
$1-\sin ^{2}[\arctan(x)]={\frac {1}{\tan ^{2}[\arctan(x)]+1}}$
$\sin ^{2}[\arctan(x)]={\frac {\tan ^{2}[\arctan(x)]}{\tan ^{2}[\arctan(x)]+1}}$
Then we use the identity $\tan[\arctan(x)]\equiv x$
$\sin[\arctan(x)]={\frac {x}{\sqrt {x^{2}+1}}}$ (III)
And initial Pythagorean trigonometric identity proofed...
Similarly if we divide this equation (I) by $\sin ^{2}\theta $
$\sin ^{2}\theta ={\frac {\frac {1}{1}}{1+{\frac {1}{\tan ^{2}\theta }}}}$ (II)
$\sin ^{2}\theta ={\frac {\tan ^{2}\theta }{\tan ^{2}\theta +1}}$
Then use the substitution $\theta =\arctan(x)$:
$\sin ^{2}[\arctan(x)]={\frac {\tan ^{2}[\arctan(x)]}{\tan ^{2}[\arctan(x)]+1}}$
Then we use the identity $\tan[\arctan(x)]\equiv x$
$\sin[\arctan(x)]={\frac {x}{\sqrt {x^{2}+1}}}$ (III)
And initial Pythagorean trigonometric identity proofed...
$[\arctan(x)]=[\arcsin({\frac {x}{\sqrt {x^{2}+1}}})]$
$y={\frac {x}{\sqrt {x^{2}+1}}}$
$y^{2}={\frac {x^{2}}{x^{2}+1}}$ (IV)
Let we guess that we have to prove:
$x={\frac {y}{\sqrt {1-y^{2}}}}$
$x^{2}={\frac {y^{2}}{1-y^{2}}}$ (V)
Replacing (V) into (IV) :
$y^{2}={\frac {\frac {y^{2}}{(1-y^{2})}}{{\frac {y^{2}}{(1-y^{2})}}+1}}$
$y^{2}={\frac {\frac {y^{2}}{(1-y^{2})}}{\frac {1}{(1-y^{2})}}}$
So it's true: $y^{2}=y^{2}$ and guessing statement was true: $x={\frac {y}{\sqrt {1-y^{2}}}}$
$[\arctan(x)]=[\arcsin({\frac {x}{\sqrt {x^{2}+1}}})]=[\arcsin(y)]=[\arctan({\frac {y}{\sqrt {1-y^{2}}}})]$
Now y can be written as x ; and we have [arcsin] expressed through [arctan]...
$[\arcsin(x)]=[\arctan({\frac {x}{\sqrt {1-x^{2}}}})]$
Similarly if we seek :$[\arccos(x)]$...
$\cos[\arccos(x)]=x$
$\cos({\frac {\pi }{2}}-({\frac {\pi }{2}}-[\arccos(x)]))=x$
$\sin({\frac {\pi }{2}}-[\arccos(x)])=x$
${\frac {\pi }{2}}-[\arccos(x)]=[\arcsin(x)]$
$[\arccos(x)]={\frac {\pi }{2}}-[\arcsin(x)]$
From :$[\arcsin(x)]$...
$[\arccos(x)]={\frac {\pi }{2}}-[\arctan({\frac {x}{\sqrt {1-x^{2}}}})]$
$[\arccos(x)]={\frac {\pi }{2}}-[\operatorname {arccot}({\frac {\sqrt {1-x^{2}}}{x}})]$
And finally we have [arccos] expressed through [arctan]...
$[\arccos(x)]=[\arctan({\frac {\sqrt {1-x^{2}}}{x}})]$
See also
• List of trigonometric identities
• Bhaskara I's sine approximation formula
• Generating trigonometric tables
• Aryabhata's sine table
• Madhava's sine table
• Table of Newtonian series
• Madhava series
• Unit vector (explains direction cosines)
• Euler's formula
Notes
1. "Tangent Identity | Math 老师". Archived from the original on 2013-10-29. Retrieved 2013-10-30. dead link
2. Richman, Fred (March 1993). "A Circular Argument". The College Mathematics Journal. 24 (2): 160–162. doi:10.2307/2686787. JSTOR 2686787.
References
• E. T. Whittaker and G. N. Watson. A Course of Modern Analysis, Cambridge University Press, 1952
| Wikipedia |
Trigonometric tables
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices.
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics, where only modest accuracy may be required and speed is often paramount.
Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called twiddle factors) must be evaluated many times in a given transform, especially in the common case where many transforms of the same size are computed. In this case, calling generic library routines every time is unacceptably slow. One option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a regular sequence of values is required, is to use a recurrence formula to compute the trigonometric values on the fly. Significant research has been devoted to finding accurate, stable recurrence schemes in order to preserve the accuracy of the FFT (which is very sensitive to trigonometric errors).
On-demand computation
Modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles (Kantabutra, 1996). One common method, especially on higher-end processors with floating-point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Maintaining precision while performing such interpolation is nontrivial, but methods like Gal's accurate tables, Cody and Waite range reduction, and Payne and Hanek radian reduction algorithms can be used for this purpose. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.
The particular polynomial used to approximate a trigonometric function is generated ahead of time using some approximation of a minimax approximation algorithm.
For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral (Brent, 1976).
Trigonometric functions of angles that are rational multiples of 2π are algebraic numbers. The values for a/b·2π can be found by applying de Moivre's identity for n = a to a bth root of unity, which is also a root of the polynomial xb - 1 in the complex plane. For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts, respectively, of the 5th power of the 37th root of unity cos(2π/37) + sin(2π/37)i, which is a root of the degree-37 polynomial x37 − 1. For this case, a root-finding algorithm such as Newton's method is much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate. The latter algorithms are required for transcendental trigonometric constants, however.
Half-angle and angle-addition formulas
Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2) = 1, cos(π/2) = 0). This method was used by the ancient astronomer Ptolemy, who derived them in the Almagest, a treatise on astronomy. In modern form, the identities he derived are stated as follows (with signs determined by the quadrant in which x lies):
$\cos \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1+\cos x)}}$
$\sin \left({\frac {x}{2}}\right)=\pm {\sqrt {{\tfrac {1}{2}}(1-\cos x)}}$
$\sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)\,$
$\cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)\,$
These were used to construct Ptolemy's table of chords, which was applied to astronomical problems.
Various other permutations on these identities are possible: for example, some early trigonometric tables used not sine and cosine, but sine and versine.
A quick, but inaccurate, approximation
A quick, but inaccurate, algorithm for calculating a table of N approximations sn for sin(2πn/N) and cn for cos(2πn/N) is:
s0 = 0
c0 = 1
sn+1 = sn + d × cn
cn+1 = cn − d × sn
for n = 0,...,N − 1, where d = 2π/N.
This is simply the Euler method for integrating the differential equation:
$ds/dt=c$
$dc/dt=-s$
with initial conditions s(0) = 0 and c(0) = 1, whose analytical solution is s = sin(t) and c = cos(t).
Unfortunately, this is not a useful algorithm for generating sine tables because it has a significant error, proportional to 1/N.
For example, for N = 256 the maximum error in the sine values is ~0.061 (s202 = −1.0368 instead of −0.9757). For N = 1024, the maximum error in the sine values is ~0.015 (s803 = −0.99321 instead of −0.97832), about 4 times smaller. If the sine and cosine values obtained were to be plotted, this algorithm would draw a logarithmic spiral rather than a circle.
A better, but still imperfect, recurrence formula
A simple recurrence formula to generate trigonometric tables is based on Euler's formula and the relation:
$e^{i(\theta +\Delta )}=e^{i\theta }\times e^{i\Delta \theta }$
This leads to the following recurrence to compute trigonometric values sn and cn as above:
c0 = 1
s0 = 0
cn+1 = wr cn − wi sn
sn+1 = wi cn + wr sn
for n = 0, ..., N − 1, where wr = cos(2π/N) and wi = sin(2π/N). These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing Newton's method in the complex plane to solve for the primitive root of zN − 1).
This method would produce an exact table in exact arithmetic, but has errors in finite-precision floating-point arithmetic. In fact, the errors grow as O(ε N) (in both the worst and average cases), where ε is the floating-point precision.
A significant improvement is to use the following modification to the above, a trick (due to Singleton[1]) often used to generate trigonometric values for FFT implementations:
c0 = 1
s0 = 0
cn+1 = cn − (α cn + β sn)
sn+1 = sn + (β cn − α sn)
where α = 2 sin2(π/N) and β = sin(2π/N). The errors of this method are much smaller, O(ε √N) on average and O(ε N) in the worst case, but this is still large enough to substantially degrade the accuracy of FFTs of large sizes.
See also
• Aryabhata's sine table
• CORDIC
• Exact trigonometric values
• Madhava's sine table
• Numerical analysis
• Plimpton 322
• Prosthaphaeresis
References
1. Singleton 1967
• Carl B. Boyer (1991) A History of Mathematics, 2nd edition, John Wiley & Sons.
• Manfred Tasche and Hansmartin Zeuner (2002) "Improved roundoff error analysis for precomputed twiddle factors", Journal for Computational Analysis and Applications 4(1): 1–18.
• James C. Schatzman (1996) "Accuracy of the discrete Fourier transform and the fast Fourier transform", SIAM Journal on Scientific Computing 17(5): 1150–1166.
• Vitit Kantabutra (1996) "On hardware for computing exponential and trigonometric functions," IEEE Transactions on Computers 45(3): 328–339 .
• R. P. Brent (1976) "Fast Multiple-Precision Evaluation of Elementary Functions", Journal of the Association for Computing Machinery 23: 242–251.
• Singleton, Richard C (1967). "On Computing The Fast Fourier Transform". Communications of the ACM. 10 (10): 647–654. doi:10.1145/363717.363771. S2CID 6287781.
• William J. Cody Jr., William Waite, Software Manual for the Elementary Functions, Prentice-Hall, 1980, ISBN 0-13-822064-6.
• Mary H. Payne, Robert N. Hanek, Radian reduction for trigonometric functions, ACM SIGNUM Newsletter 18: 19-24, 1983.
• Gal, Shmuel and Bachelis, Boris (1991) "An accurate elementary mathematical library for the IEEE floating point standard", ACM Transactions on Mathematical Software.
| Wikipedia |
Trigenus
In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple $(g_{1},g_{2},g_{3})$. It is obtained by minimizing the genera of three orientable handle bodies — with no intersection between their interiors— which decompose the manifold as far as the Heegaard genus need only two.
That is, a decomposition $M=V_{1}\cup V_{2}\cup V_{3}$ with ${\rm {int}}V_{i}\cap {\rm {int}}V_{j}=\varnothing $ for $i,j=1,2,3$ and being $g_{i}$ the genus of $V_{i}$.
For orientable spaces, ${\rm {trig}}(M)=(0,0,h)$, where $h$ is $M$'s Heegaard genus.
For non-orientable spaces the ${\rm {trig}}$ has the form ${\rm {trig}}(M)=(0,g_{2},g_{3})\quad {\mbox{or}}\quad (1,g_{2},g_{3})$ depending on the image of the first Stiefel–Whitney characteristic class $w_{1}$ under a Bockstein homomorphism, respectively for $\beta (w_{1})=0\quad {\mbox{or}}\quad \neq 0.$
It has been proved that the number $g_{2}$ has a relation with the concept of Stiefel–Whitney surface, that is, an orientable surface $G$ which is embedded in $M$, has minimal genus and represents the first Stiefel–Whitney class under the duality map $D\colon H^{1}(M;{\mathbb {Z} }_{2})\to H_{2}(M;{\mathbb {Z} }_{2}),$, that is, $Dw_{1}(M)=[G]$. If $\beta (w_{1})=0\,$ then ${\rm {trig}}(M)=(0,2g,g_{3})\,$, and if $\beta (w_{1})\neq 0.\,$ then ${\rm {trig}}(M)=(1,2g-1,g_{3})\,$.
Theorem
A manifold S is a Stiefel–Whitney surface in M, if and only if S and M−int(N(S)) are orientable.
References
• J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel–Whitney surfaces and decompositions of 3-manifolds into handlebodies, Topology Appl. 60 (1994), 267–280.
• J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel–Whitney surfaces and the trigenus of non-orientable 3-manifolds, Manuscripta Math. 100 (1999), 405–422.
• "On the trigenus of surface bundles over $S^{1}$", 2005, Soc. Mat. Mex. | pdf
| Wikipedia |
Sedenion
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface S or blackboard bold $\mathbb {S} $. They are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the 32-ions or trigintaduonions.[1] It is possible to continue applying the Cayley–Dickson construction arbitrarily many times.
Sedenions
Symbol$\mathbb {S} $
Typenonassociative algebra
Unitse0, ..., e15
Multiplicative identitye0
Main propertiespower associativity
distributivity
Common systems
• $\mathbb {N} $ Natural numbers
• $\mathbb {Z} $ Integers
• $\mathbb {Q} $ Rational numbers
• $\mathbb {R} $ Real numbers
• $\mathbb {C} $ Complex numbers
• $\mathbb {H} $ Quaternions
Less common systems
Octonions ($\mathbb {O} $) Sedenions ($\mathbb {S} $)
The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied by Smith (1995).
Arithmetic
Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of power associativity, which can be stated as that, for any element x of $\mathbb {S} $, the power $x^{n}$ is well defined. They are also flexible.
Every sedenion is a linear combination of the unit sedenions $e_{0}$, $e_{1}$, $e_{2}$, $e_{3}$, ..., $e_{15}$, which form a basis of the vector space of sedenions. Every sedenion can be represented in the form
$x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\cdots +x_{14}e_{14}+x_{15}e_{15}.$
Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.
Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra they were constructed from. So, they contain the octonions (generated by $e_{0}$ to $e_{7}$ in the table below), and therefore also the quaternions (generated by $e_{0}$ to $e_{3}$), complex numbers (generated by $e_{0}$ and $e_{1}$) and real numbers (generated by $e_{0}$).
The sedenions have a multiplicative identity element $e_{0}$ and multiplicative inverses, but they are not a division algebra because they have zero divisors. This means that two nonzero sedenions can be multiplied to obtain zero: an example is $(e_{3}+e_{10})(e_{6}-e_{15})$. All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.
A sedenion multiplication table is shown below:
$e_{i}e_{j}$ $e_{j}$
$e_{0}$ $e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $e_{5}$ $e_{6}$ $e_{7}$ $e_{8}$ $e_{9}$ $e_{10}$ $e_{11}$ $e_{12}$ $e_{13}$ $e_{14}$ $e_{15}$
$e_{i}$ $e_{0}$ $e_{0}$ $e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $e_{5}$ $e_{6}$ $e_{7}$ $e_{8}$ $e_{9}$ $e_{10}$ $e_{11}$ $e_{12}$ $e_{13}$ $e_{14}$ $e_{15}$
$e_{1}$ $e_{1}$ $-e_{0}$ $e_{3}$ $-e_{2}$ $e_{5}$ $-e_{4}$ $-e_{7}$ $e_{6}$ $e_{9}$ $-e_{8}$ $-e_{11}$ $e_{10}$ $-e_{13}$ $e_{12}$ $e_{15}$ $-e_{14}$
$e_{2}$ $e_{2}$ $-e_{3}$ $-e_{0}$ $e_{1}$ $e_{6}$ $e_{7}$ $-e_{4}$ $-e_{5}$ $e_{10}$ $e_{11}$ $-e_{8}$ $-e_{9}$ $-e_{14}$ $-e_{15}$ $e_{12}$ $e_{13}$
$e_{3}$ $e_{3}$ $e_{2}$ $-e_{1}$ $-e_{0}$ $e_{7}$ $-e_{6}$ $e_{5}$ $-e_{4}$ $e_{11}$ $-e_{10}$ $e_{9}$ $-e_{8}$ $-e_{15}$ $e_{14}$ $-e_{13}$ $e_{12}$
$e_{4}$ $e_{4}$ $-e_{5}$ $-e_{6}$ $-e_{7}$ $-e_{0}$ $e_{1}$ $e_{2}$ $e_{3}$ $e_{12}$ $e_{13}$ $e_{14}$ $e_{15}$ $-e_{8}$ $-e_{9}$ $-e_{10}$ $-e_{11}$
$e_{5}$ $e_{5}$ $e_{4}$ $-e_{7}$ $e_{6}$ $-e_{1}$ $-e_{0}$ $-e_{3}$ $e_{2}$ $e_{13}$ $-e_{12}$ $e_{15}$ $-e_{14}$ $e_{9}$ $-e_{8}$ $e_{11}$ $-e_{10}$
$e_{6}$ $e_{6}$ $e_{7}$ $e_{4}$ $-e_{5}$ $-e_{2}$ $e_{3}$ $-e_{0}$ $-e_{1}$ $e_{14}$ $-e_{15}$ $-e_{12}$ $e_{13}$ $e_{10}$ $-e_{11}$ $-e_{8}$ $e_{9}$
$e_{7}$ $e_{7}$ $-e_{6}$ $e_{5}$ $e_{4}$ $-e_{3}$ $-e_{2}$ $e_{1}$ $-e_{0}$ $e_{15}$ $e_{14}$ $-e_{13}$ $-e_{12}$ $e_{11}$ $e_{10}$ $-e_{9}$ $-e_{8}$
$e_{8}$ $e_{8}$ $-e_{9}$ $-e_{10}$ $-e_{11}$ $-e_{12}$ $-e_{13}$ $-e_{14}$ $-e_{15}$ $-e_{0}$ $e_{1}$ $e_{2}$ $e_{3}$ $e_{4}$ $e_{5}$ $e_{6}$ $e_{7}$
$e_{9}$ $e_{9}$ $e_{8}$ $-e_{11}$ $e_{10}$ $-e_{13}$ $e_{12}$ $e_{15}$ $-e_{14}$ $-e_{1}$ $-e_{0}$ $-e_{3}$ $e_{2}$ $-e_{5}$ $e_{4}$ $e_{7}$ $-e_{6}$
$e_{10}$ $e_{10}$ $e_{11}$ $e_{8}$ $-e_{9}$ $-e_{14}$ $-e_{15}$ $e_{12}$ $e_{13}$ $-e_{2}$ $e_{3}$ $-e_{0}$ $-e_{1}$ $-e_{6}$ $-e_{7}$ $e_{4}$ $e_{5}$
$e_{11}$ $e_{11}$ $-e_{10}$ $e_{9}$ $e_{8}$ $-e_{15}$ $e_{14}$ $-e_{13}$ $e_{12}$ $-e_{3}$ $-e_{2}$ $e_{1}$ $-e_{0}$ $-e_{7}$ $e_{6}$ $-e_{5}$ $e_{4}$
$e_{12}$ $e_{12}$ $e_{13}$ $e_{14}$ $e_{15}$ $e_{8}$ $-e_{9}$ $-e_{10}$ $-e_{11}$ $-e_{4}$ $e_{5}$ $e_{6}$ $e_{7}$ $-e_{0}$ $-e_{1}$ $-e_{2}$ $-e_{3}$
$e_{13}$ $e_{13}$ $-e_{12}$ $e_{15}$ $-e_{14}$ $e_{9}$ $e_{8}$ $e_{11}$ $-e_{10}$ $-e_{5}$ $-e_{4}$ $e_{7}$ $-e_{6}$ $e_{1}$ $-e_{0}$ $e_{3}$ $-e_{2}$
$e_{14}$ $e_{14}$ $-e_{15}$ $-e_{12}$ $e_{13}$ $e_{10}$ $-e_{11}$ $e_{8}$ $e_{9}$ $-e_{6}$ $-e_{7}$ $-e_{4}$ $e_{5}$ $e_{2}$ $-e_{3}$ $-e_{0}$ $e_{1}$
$e_{15}$ $e_{15}$ $e_{14}$ $-e_{13}$ $-e_{12}$ $e_{11}$ $e_{10}$ $-e_{9}$ $e_{8}$ $-e_{7}$ $e_{6}$ $-e_{5}$ $-e_{4}$ $e_{3}$ $e_{2}$ $-e_{1}$ $-e_{0}$
Sedenion properties
From the above table, we can see that:
$e_{0}e_{i}=e_{i}e_{0}=e_{i}\,{\text{for all}}\,i,$
$e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0,$ and
$e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{with}}\,\,i,j\neq 0.$
Anti-associative
The sedenions are not fully anti-associative. Choose any four generators, $i,j,k$ and $l$. The following 5-cycle shows that these five relations cannot all be anti-associative.
$(ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl)=0$
In particular, in the table above, using $e_{1},e_{2},e_{4}$ and $e_{8}$ the last expression associates. $(e_{1}e_{2})e_{12}=e_{1}(e_{2}e_{12})=-e_{15}$
Quaternionic subalgebras
The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the octonions used in creating the sedenion through the Cayley–Dickson construction shown in bold:
The binary representations of the indices of these triples bitwise XOR to 0.
{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},
{2, 4, 6}, {2, 5, 7}, {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7},
{3, 6, 5}, {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},
{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }
Zero divisors
The list of 84 sets of zero divisors $\{e_{a},e_{b},e_{c},e_{d}\}$, where $(e_{a}+e_{b})\circ (e_{c}+e_{d})=0$:
${\begin{array}{c}{\text{Sedenion Zero Divisors}}\quad \{e_{a},e_{b},e_{c},e_{d}\}\\{\text{where}}~(e_{a}+e_{b})\circ (e_{c}+e_{d})=0\\{\begin{array}{ccc}1\leq a\leq 6,&c>a,&9\leq b\leq 15\\9\leq c\leq 15&&-9\geq d\geq -15\end{array}}\\{\begin{array}{ll}\{e_{1},e_{10},e_{5},e_{14}\}&\{e_{1},e_{10},e_{4},-e_{15}\}\\\{e_{1},e_{10},e_{7},e_{12}\}&\{e_{1},e_{10},e_{6},-e_{13}\}\\\{e_{1},e_{11},e_{4},e_{14}\}&\{e_{1},e_{11},e_{6},-e_{12}\}\\\{e_{1},e_{11},e_{5},e_{15}\}&\{e_{1},e_{11},e_{7},-e_{13}\}\\\{e_{1},e_{12},e_{2},e_{15}\}&\{e_{1},e_{12},e_{3},-e_{14}\}\\\{e_{1},e_{12},e_{6},e_{11}\}&\{e_{1},e_{12},e_{7},-e_{10}\}\\\{e_{1},e_{13},e_{6},e_{10}\}&\{e_{1},e_{13},e_{7},-e_{14}\}\\\{e_{1},e_{13},e_{7},e_{11}\}&\{e_{1},e_{13},e_{3},-e_{15}\}\\\{e_{1},e_{14},e_{2},e_{13}\}&\{e_{1},e_{14},e_{4},-e_{11}\}\\\{e_{1},e_{14},e_{3},e_{12}\}&\{e_{1},e_{14},e_{5},-e_{10}\}\\\{e_{1},e_{15},e_{3},e_{13}\}&\{e_{1},e_{15},e_{2},-e_{12}\}\\\{e_{1},e_{15},e_{4},e_{10}\}&\{e_{1},e_{15},e_{5},-e_{11}\}\\\{e_{2},e_{9},e_{4},e_{15}\}&\{e_{2},e_{9},e_{5},-e_{14}\}\\\{e_{2},e_{9},e_{6},e_{13}\}&\{e_{2},e_{9},e_{7},-e_{12}\}\\\{e_{2},e_{11},e_{5},e_{12}\}&\{e_{2},e_{11},e_{4},-e_{13}\}\\\{e_{2},e_{11},e_{6},e_{15}\}&\{e_{2},e_{11},e_{7},-e_{14}\}\\\{e_{2},e_{12},e_{3},e_{13}\}&\{e_{2},e_{12},e_{5},-e_{11}\}\\\{e_{2},e_{12},e_{7},e_{9}\}&\{e_{2},e_{13},e_{3},-e_{12}\}\\\{e_{2},e_{13},e_{4},e_{11}\}&\{e_{2},e_{13},e_{6},-e_{9}\}\\\{e_{2},e_{14},e_{5},e_{9}\}&\{e_{2},e_{14},e_{3},-e_{15}\}\\\{e_{2},e_{14},e_{3},e_{14}\}&\{e_{2},e_{15},e_{4},-e_{9}\}\\\{e_{2},e_{15},e_{3},e_{14}\}&\{e_{2},e_{15},e_{6},-e_{11}\}\\\{e_{3},e_{9},e_{6},e_{12}\}&\{e_{3},e_{9},e_{4},-e_{14}\}\\\{e_{3},e_{9},e_{7},e_{13}\}&\{e_{3},e_{9},e_{5},-e_{15}\}\\\{e_{3},e_{10},e_{4},e_{13}\}&\{e_{3},e_{10},e_{5},-e_{12}\}\\\{e_{3},e_{10},e_{7},e_{14}\}&\{e_{3},e_{10},e_{6},-e_{15}\}\\\{e_{3},e_{12},e_{5},e_{10}\}&\{e_{3},e_{12},e_{6},-e_{9}\}\\\{e_{3},e_{14},e_{4},e_{9}\}&\{e_{3},e_{13},e_{4},-e_{10}\}\\\{e_{3},e_{15},e_{5},e_{9}\}&\{e_{3},e_{13},e_{7},-e_{9}\}\\\{e_{3},e_{15},e_{6},e_{10}\}&\{e_{3},e_{14},e_{7},-e_{10}\}\\\{e_{4},e_{9},e_{7},e_{10}\}&\{e_{4},e_{9},e_{6},-e_{11}\}\\\{e_{4},e_{10},e_{5},e_{11}\}&\{e_{4},e_{10},e_{7},-e_{9}\}\\\{e_{4},e_{11},e_{6},e_{9}\}&\{e_{4},e_{11},e_{5},-e_{10}\}\\\{e_{4},e_{13},e_{6},e_{15}\}&\{e_{4},e_{13},e_{7},-e_{14}\}\\\{e_{4},e_{14},e_{7},e_{13}\}&\{e_{4},e_{14},e_{5},-e_{15}\}\\\{e_{4},e_{15},e_{5},e_{14}\}&\{e_{4},e_{15},e_{6},-e_{13}\}\\\{e_{5},e_{10},e_{6},e_{9}\}&\{e_{5},e_{9},e_{6},-e_{10}\}\\\{e_{5},e_{11},e_{7},e_{9}\}&\{e_{5},e_{9},e_{7},-e_{11}\}\\\{e_{5},e_{12},e_{7},e_{14}\}&\{e_{5},e_{12},e_{6},-e_{15}\}\\\{e_{5},e_{15},e_{6},e_{12}\}&\{e_{5},e_{14},e_{7},-e_{12}\}\\\{e_{6},e_{11},e_{7},e_{10}\}&\{e_{6},e_{10},e_{7},-e_{11}\}\\\{e_{6},e_{13},e_{7},e_{12}\}&\{e_{6},e_{10},e_{7},-e_{13}\}\end{array}}\end{array}}$
Applications
Moreno (1998) showed that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact form of the exceptional Lie group G2. (Note that in his paper, a "zero divisor" means a pair of elements that multiply to zero.)
Guillard & Gresnigt (2019) demonstrated that the three generations of leptons and quarks that are associated with unbroken gauge symmetry $\mathrm {SU(3)_{c}\times U(1)_{em}} $ can be represented using the algebra of the complexified sedenions $\mathbb {C\otimes S} $. Their reasoning follows that a primitive idempotent projector $\rho _{+}=1/2(1+ie_{15})$ — where $e_{15}$ is chosen as an imaginary unit akin to $e_{7}$ for $\mathbb {O} $ in the Fano plane — that acts on the standard basis of the sedenions uniquely divides the algebra into three sets of split basis elements for $\mathbb {C\otimes O} $, whose adjoint left actions on themselves generate three copies of the Clifford algebra $\mathrm {C} l(6)$ which in-turn contain minimal left ideals that describe a single generation of fermions with unbroken $\mathrm {SU(3)_{c}\times U(1)_{em}} $ gauge symmetry. In particular, they note that tensor products between normed division algebras generate zero divisors akin to those inside $\mathbb {S} $, where for $\mathbb {C\otimes O} $ the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and isomorphic to a Clifford algebra. Altogether, this permits three copies of $(\mathbb {C\otimes O} )_{L}\cong \mathrm {Cl(6)} $ to exist inside $\mathbb {(C\otimes S)} _{L}$. Furthermore, these three complexified octonion subalgebras are not independent; they share a common $\mathrm {C} l(2)$ subalgebra, which the authors note could form a theoretical basis for CKM and PMNS matrices that, respectively, describe quark mixing and neutrino oscillations.
Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.[3][4]
See also
• Pfister's sixteen-square identity
• Split-complex number
Notes
1. Raoul E. Cawagas, et al. (2009). "THE BASIC SUBALGEBRA STRUCTURE OF THE CAYLEY-DICKSON ALGEBRA OF DIMENSION 32 (TRIGINTADUONIONS)".
2. (Baez 2002, p. 6)
3. Saoud, Lyes Saad; Al-Marzouqi, Hasan (2020). "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm". IEEE Access. 8: 144823–144838. doi:10.1109/ACCESS.2020.3014690. ISSN 2169-3536.
4. Kopp, Michael; Kreil, David; Neun, Moritz; Jonietz, David; Martin, Henry; Herruzo, Pedro; Gruca, Aleksandra; Soleymani, Ali; Wu, Fanyou; Liu, Yang; Xu, Jingwei (2021-08-07). "Traffic4cast at NeurIPS 2020 – yet more on the unreasonable effectiveness of gridded geo-spatial processes". NeurIPS 2020 Competition and Demonstration Track. PMLR: 325–343.
References
• Imaeda, K.; Imaeda, M. (2000), "Sedenions: algebra and analysis", Applied Mathematics and Computation, 115 (2): 77–88, doi:10.1016/S0096-3003(99)00140-X, MR 1786945
• Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. New Series. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. MR 1886087. S2CID 586512.
• Biss, Daniel K.; Christensen, J. Daniel; Dugger, Daniel; Isaksen, Daniel C. (2007). "Large annihilators in Cayley-Dickson algebras II". Boletin de la Sociedad Matematica Mexicana. 3: 269–292. arXiv:math/0702075. Bibcode:2007math......2075B.
• Guillard, Adam B.; Gresnigt, Niels G. (2019). "Three fermion generations with two unbroken gauge symmetries from the complex sedenions". The European Physical Journal C. Springer. 79 (5): 1–11 (446). arXiv:1904.03186. Bibcode:2019EPJC...79..446G. doi:10.1140/epjc/s10052-019-6967-1. S2CID 102351250.
• Kinyon, M.K.; Phillips, J.D.; Vojtěchovský, P. (2007). "C-loops: Extensions and constructions". Journal of Algebra and Its Applications. 6 (1): 1–20. arXiv:math/0412390. CiteSeerX 10.1.1.240.6208. doi:10.1142/S0219498807001990. S2CID 48162304.
• Kivunge, Benard M.; Smith, Jonathan D. H (2004). "Subloops of sedenions" (PDF). Comment. Math. Univ. Carolinae. 45 (2): 295–302.
• Moreno, Guillermo (1998), "The zero divisors of the Cayley–Dickson algebras over the real numbers", Bol. Soc. Mat. Mexicana, Series 3, 4 (1): 13–28, arXiv:q-alg/9710013, Bibcode:1997q.alg....10013G, MR 1625585
• Smith, Jonathan D. H. (1995), "A left loop on the 15-sphere", Journal of Algebra, 176 (1): 128–138, doi:10.1006/jabr.1995.1237, MR 1345298
• L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020, doi: 10.1109/ACCESS.2020.3014690.
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
| Wikipedia |
Trigonometry
Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle', and μέτρον (métron) 'measure')[1] is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.[2] The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.[3]
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.[4]
Trigonometry is known for its many identities. These trigonometric identities[5][6] are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.[7]
History
Main article: History of trigonometry
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.[9] They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.[10]
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.[11] In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.[12] Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.[13] (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.
The modern definition of the sine is first attested in the Surya Siddhanta, and its properties were further documented in the 5th century (AD) by Indian mathematician and astronomer Aryabhata.[14] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. In 830 AD, Persian mathematician Habash al-Hasib al-Marwazi produced the first table of cotangents.[15][16] By the 10th century AD, in the work of Persian mathematician Abū al-Wafā' al-Būzjānī, all six trigonometric functions were used.[17] Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.[17] He also made important innovations in spherical trigonometry[18][19][20] The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.[21][22][23] He was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.[16] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.[24] Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.[25] One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years.[26] At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.[27] Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.[28] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[29] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[30] Also in the 18th century, Brook Taylor defined the general Taylor series.[31]
Trigonometric ratios
Main article: Trigonometric function
Trigonometric ratios are the ratios between edges of a right triangle. These ratios depend only on one acute angle of the right triangle, since any two right triangles with the same acute angle are similar.[32]
So, these ratios define functions of this angle that are called trigonometric functions. Explicitly, they are defined below as functions of the known angle A, where a, b and h refer to the lengths of the sides in the accompanying figure:
• Sine (denoted sin), defined as the ratio of the side opposite the angle to the hypotenuse.
$\sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{h}}.$
• Cosine (denoted cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
$\cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{h}}.$
• Tangent (denoted tan), defined as the ratio of the opposite leg to the adjacent leg.
$\tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{b}}={\frac {a/h}{b/h}}={\frac {\sin A}{\cos A}}.$
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under Mnemonics.
The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
$\csc A={\frac {1}{\sin A}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}={\frac {h}{a}},$
$\sec A={\frac {1}{\cos A}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}={\frac {h}{b}},$
$\cot A={\frac {1}{\tan A}}={\frac {\textrm {adjacent}}{\textrm {opposite}}}={\frac {\cos A}{\sin A}}={\frac {b}{a}}.$
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".[33]
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.[34] These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
Mnemonics
Main article: Mnemonics in trigonometry
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:[35]
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e. /ˌsoʊkəˈtoʊə/ SOH-kə-TOH-ə, similar to Krakatoa).[36] Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".[37]
The unit circle and common trigonometric values
Main article: Unit circle
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.[38] In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where $x=\cos A$ and $y=\sin A$.[38] This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:[39]
Function 0 $\pi /6$ $\pi /4$ $\pi /3$ $\pi /2$ $2\pi /3$ $3\pi /4$ $5\pi /6$ $\pi $
sine $0$ $1/2$ ${\sqrt {2}}/2$ ${\sqrt {3}}/2$ $1$ ${\sqrt {3}}/2$ ${\sqrt {2}}/2$ $1/2$ $0$
cosine $1$ ${\sqrt {3}}/2$ ${\sqrt {2}}/2$ $1/2$ $0$ $-1/2$ $-{\sqrt {2}}/2$ $-{\sqrt {3}}/2$ $-1$
tangent $0$ ${\sqrt {3}}/3$ $1$ ${\sqrt {3}}$ undefined $-{\sqrt {3}}$ $-1$ $-{\sqrt {3}}/3$ $0$
secant $1$ $2{\sqrt {3}}/3$ ${\sqrt {2}}$ $2$ undefined $-2$ $-{\sqrt {2}}$ $-2{\sqrt {3}}/3$ $-1$
cosecant undefined $2$ ${\sqrt {2}}$ $2{\sqrt {3}}/3$ $1$ $2{\sqrt {3}}/3$ ${\sqrt {2}}$ $2$ undefined
cotangent undefined ${\sqrt {3}}$ $1$ ${\sqrt {3}}/3$ $0$ $-{\sqrt {3}}/3$ $-1$ $-{\sqrt {3}}$ undefined
Trigonometric functions of real or complex variables
Main article: Trigonometric function
Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments[40] (see trigonometric function).
Graphs of trigonometric functions
The following table summarizes the properties of the graphs of the six main trigonometric functions:[41][42]
Function Period Domain Range Graph
sine $2\pi $ $(-\infty ,\infty )$ $[-1,1]$
cosine $2\pi $ $(-\infty ,\infty )$ $[-1,1]$
tangent $\pi $ $x\neq \pi /2+n\pi $ $(-\infty ,\infty )$
secant $2\pi $ $x\neq \pi /2+n\pi $ $(-\infty ,-1]\cup [1,\infty )$
cosecant $2\pi $ $x\neq n\pi $ $(-\infty ,-1]\cup [1,\infty )$
cotangent $\pi $ $x\neq n\pi $ $(-\infty ,\infty )$
Inverse trigonometric functions
Main article: Inverse trigonometric functions
Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible.[43]: 48ff
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:[43]: 48ff [44]: 521ff
Name Usual notation Definition Domain of x for real result Range of usual principal value
(radians)
Range of usual principal value
(degrees)
arcsiney = arcsin(x)x = sin(y)−1 ≤ x ≤ 1−π/2 ≤ y ≤ π/2−90° ≤ y ≤ 90°
arccosiney = arccos(x)x = cos(y)−1 ≤ x ≤ 10 ≤ y ≤ π0° ≤ y ≤ 180°
arctangenty = arctan(x)x = tan(y)all real numbers−π/2 < y < π/2−90° < y < 90°
arccotangenty = arccot(x)x = cot(y)all real numbers 0 < y < π0° < y < 180°
arcsecanty = arcsec(x)x = sec(y)x ≤ −1 or 1 ≤ x0 ≤ y < π/2 or π/2 < y ≤ π0° ≤ y < 90° or 90° < y ≤ 180°
arccosecanty = arccsc(x)x = csc(y)x ≤ −1 or 1 ≤ x−π/2 ≤ y < 0 or 0 < y ≤ π/2−90° ≤ y < 0° or 0° < y ≤ 90°
Power series representations
When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations:[45]
${\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\\end{aligned}}$
${\begin{aligned}\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.\end{aligned}}$
With these definitions the trigonometric functions can be defined for complex numbers.[46] When extended as functions of real or complex variables, the following formula holds for the complex exponential:
$e^{x+iy}=e^{x}(\cos y+i\sin y).$
This complex exponential function, written in terms of trigonometric functions, is particularly useful.[47][48]
Calculating trigonometric functions
Main article: Trigonometric tables
Trigonometric functions were among the earliest uses for mathematical tables.[49] Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy.[50] Slide rules had special scales for trigonometric functions.[51]
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).[52] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions.[53] The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.[54]
Other trigonometric functions
Main article: Trigonometric functions § History
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables[55]), the coversine (coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin2(θ/2)),[56] the exsecant (exsec(θ) = sec(θ) − 1), and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). See List of trigonometric identities for more relations between these functions.
Applications
Main article: Uses of trigonometry
Astronomy
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,[57] predicting eclipses, and describing the orbits of the planets.[58]
In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,[59] as well as in satellite navigation systems.[20]
Navigation
Main article: Navigation
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.[60]
Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.[61]
Surveying
In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.[62]
On a larger scale, trigonometry is used in geography to measure distances between landmarks.[63]
Periodic functions
Main articles: Fourier series and Fourier transform
The sine and cosine functions are fundamental to the theory of periodic functions,[64] such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.
Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics[65] and communications,[66] among other fields.
Optics and acoustics
Trigonometry is useful in many physical sciences,[67] including acoustics,[68] and optics.[68] In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.[69]
Other applications
Other fields that use trigonometry or trigonometric functions include music theory,[70] geodesy, audio synthesis,[71] architecture,[72] electronics,[70] biology,[73] medical imaging (CT scans and ultrasound),[74] chemistry,[75] number theory (and hence cryptology),[76] seismology,[68] meteorology,[77] oceanography,[78] image compression,[79] phonetics,[80] economics,[81] electrical engineering, mechanical engineering, civil engineering,[70] computer graphics,[82] cartography,[70] crystallography[83] and game development.[82]
Identities
Main article: List of trigonometric identities
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.[84]
Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,[85] relate both the sides and angles of a given triangle.
Triangle identities
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).
Law of sines
The law of sines (also known as the "sine rule") for an arbitrary triangle states:[86]
${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R={\frac {abc}{2\Delta }},$
where $\Delta $ is the area of the triangle and R is the radius of the circumscribed circle of the triangle:
$R={\frac {abc}{\sqrt {(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}}.$
Law of cosines
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:[86]
$c^{2}=a^{2}+b^{2}-2ab\cos C,$
or equivalently:
$\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$
Law of tangents
The law of tangents, developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables.[87] It is given by:
${\frac {a-b}{a+b}}={\frac {\tan \left[{\tfrac {1}{2}}(A-B)\right]}{\tan \left[{\tfrac {1}{2}}(A+B)\right]}}$
Area
Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:[86]
${\mbox{Area}}=\Delta ={\frac {1}{2}}ab\sin C$
Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
$s={\frac {1}{2}}(a+b+c),$
then the area of the triangle is:[88]
${\mbox{Area}}=\Delta ={\sqrt {s(s-a)(s-b)(s-c)}}={\frac {abc}{4R}}$,
where R is the radius of the circumcircle of the triangle.
Pythagorean identities
The following trigonometric identities are related to the Pythagorean theorem and hold for any value:[89]
$\sin ^{2}A+\cos ^{2}A=1\ $
$\tan ^{2}A+1=\sec ^{2}A\ $
$\cot ^{2}A+1=\csc ^{2}A\ $
The second and third equations are derived from dividing the first equation by $\cos ^{2}{A}$ and $\sin ^{2}{A}$, respectively.
Euler's formula
Euler's formula, which states that $e^{ix}=\cos x+i\sin x$, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
$\sin x={\frac {e^{ix}-e^{-ix}}{2i}},\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \tan x={\frac {i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}}.$
Other trigonometric identities
Main article: List of trigonometric identities
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.[32]
See also
• Aryabhata's sine table
• Generalized trigonometry
• Lénárt sphere
• List of triangle topics
• List of trigonometric identities
• Rational trigonometry
• Skinny triangle
• Small-angle approximation
• Trigonometric functions
• Unit circle
• Uses of trigonometry
References
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2. R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002)
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8. Boyer (1991), p. 162, "Greek Trigonometry and Mensuration".
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11. Thurston (1996), pp. 235–236, "Appendix 1: Hipparchus's Table of Chords".
12. Toomer, G. (1998), Ptolemy's Almagest, Princeton University Press, ISBN 978-0-691-00260-6
13. Thurston (1996), pp. 239–243, "Appendix 3: Ptolemy's Table of Chords".
14. Boyer (1991), p. 215.
15. Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media. ISBN 978-1-4020-0260-1.
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21. "Nasir al-Din al-Tusi". MacTutor History of Mathematics archive. Retrieved 2021-01-08. One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
22. Berggren, J. L. (October 2013). "Islamic Mathematics". the cambridge history of science. Vol. 2. Cambridge University Press. pp. 62–83. doi:10.1017/CHO9780511974007.004. ISBN 9780521594486.
23. "ṬUSI, NAṢIR-AL-DIN i. Biography". Encyclopaedia Iranica. Retrieved 2018-08-05. His major contribution in mathematics (Nasr, 1996, pp. 208–214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
24. Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
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39. W. Michael Kelley (2002). The Complete Idiot's Guide to Calculus. Alpha Books. p. 45. ISBN 978-0-02-864365-6.
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41. Mary P Attenborough (30 June 2003). Mathematics for Electrical Engineering and Computing. Elsevier. p. 418. ISBN 978-0-08-047340-6.
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43. Elizabeth G. Bremigan; Ralph J. Bremigan; John D. Lorch (2011). Mathematics for Secondary School Teachers. MAA. ISBN 978-0-88385-773-1.
44. Martin Brokate; Pammy Manchanda; Abul Hasan Siddiqi (3 August 2019). Calculus for Scientists and Engineers. Springer. ISBN 9789811384646.
45. Serge Lang (14 March 2013). Complex Analysis. Springer. p. 63. ISBN 978-3-642-59273-7.
46. Silvia Maria Alessio (9 December 2015). Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications. Springer. p. 339. ISBN 978-3-319-25468-5.
47. K. RAJA RAJESWARI; B. VISVESVARA RAO (24 March 2014). SIGNALS AND SYSTEMS. PHI Learning. p. 263. ISBN 978-81-203-4941-4.
48. John Stillwell (23 July 2010). Mathematics and Its History. Springer Science & Business Media. p. 313. ISBN 978-1-4419-6053-5.
49. Martin Campbell-Kelly; Mary Croarken; Raymond Flood; Eleanor Robson (2 October 2003). The History of Mathematical Tables: From Sumer to Spreadsheets. OUP Oxford. ISBN 978-0-19-850841-0.
50. George S. Donovan; Beverly Beyreuther Gimmestad (1980). Trigonometry with calculators. Prindle, Weber & Schmidt. ISBN 978-0-87150-284-1.
51. Ross Raymond Middlemiss (1945). Instructions for Post-trig and Mannheim-trig Slide Rules. Frederick Post Company.
52. "Calculator keys—what they do". Popular Science. Bonnier Corporation. April 1974. p. 125.
53. Steven S. Skiena; Miguel A. Revilla (18 April 2006). Programming Challenges: The Programming Contest Training Manual. Springer Science & Business Media. p. 302. ISBN 978-0-387-22081-9.
54. Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C (PDF). Intel. 2013.
55. Boyer (1991), pp. xxiii–xxiv.
56. Nielsen (1966), pp. xxiii–xxiv.
57. Olinthus Gregory (1816). Elements of Plane and Spherical Trigonometry: With Their Applications to Heights and Distances Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations. Baldwin, Cradock, and Joy.
58. Neugebauer, Otto (1948). "Mathematical methods in ancient astronomy". Bulletin of the American Mathematical Society. 54 (11): 1013–1041. doi:10.1090/S0002-9904-1948-09089-9.
59. Michael Seeds; Dana Backman (5 January 2009). Astronomy: The Solar System and Beyond. Cengage Learning. p. 254. ISBN 978-0-495-56203-0.
60. John Sabine (1800). The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc. p. 1.
61. Mordechai Ben-Ari; Francesco Mondada (2018). Elements of Robotics. Springer. p. 16. ISBN 978-3-319-62533-1.
62. George Roberts Perkins (1853). Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables. D. Appleton & Company.
63. Charles W. J. Withers; Hayden Lorimer (14 December 2015). Geographers: Biobibliographical Studies. A&C Black. p. 6. ISBN 978-1-4411-0785-5.
64. H. G. ter Morsche; J. C. van den Berg; E. M. van de Vrie (7 August 2003). Fourier and Laplace Transforms. Cambridge University Press. p. 61. ISBN 978-0-521-53441-3.
65. Bernd Thaller (8 May 2007). Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena. Springer Science & Business Media. p. 15. ISBN 978-0-387-22770-2.
66. M. Rahman (2011). Applications of Fourier Transforms to Generalized Functions. WIT Press. ISBN 978-1-84564-564-9.
67. Lawrence Bornstein; Basic Systems, Inc (1966). Trigonometry for the Physical Sciences. Appleton-Century-Crofts.
68. John J. Schiller; Marie A. Wurster (1988). College Algebra and Trigonometry: Basics Through Precalculus. Scott, Foresman. ISBN 978-0-673-18393-4.
69. Dudley H. Towne (5 May 2014). Wave Phenomena. Dover Publications. ISBN 978-0-486-14515-0.
70. E. Richard Heineman; J. Dalton Tarwater (1 November 1992). Plane Trigonometry. McGraw-Hill. ISBN 978-0-07-028187-5.
71. Mark Kahrs; Karlheinz Brandenburg (18 April 2006). Applications of Digital Signal Processing to Audio and Acoustics. Springer Science & Business Media. p. 404. ISBN 978-0-306-47042-4.
72. Kim Williams; Michael J. Ostwald (9 February 2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s. Birkhäuser. p. 260. ISBN 978-3-319-00137-1.
73. Dan Foulder (15 July 2019). Essential Skills for GCSE Biology. Hodder Education. p. 78. ISBN 978-1-5104-6003-4.
74. Luciano Beolchi; Michael H. Kuhn (1995). Medical Imaging: Analysis of Multimodality 2D/3D Images. IOS Press. p. 122. ISBN 978-90-5199-210-6.
75. Marcus Frederick Charles Ladd (2014). Symmetry of Crystals and Molecules. Oxford University Press. p. 13. ISBN 978-0-19-967088-8.
76. Gennady I. Arkhipov; Vladimir N. Chubarikov; Anatoly A. Karatsuba (22 August 2008). Trigonometric Sums in Number Theory and Analysis. Walter de Gruyter. ISBN 978-3-11-019798-3.
77. Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943. 1943.
78. Mary Sears; Daniel Merriman; Woods Hole Oceanographic Institution (1980). Oceanography, the past. Springer-Verlag. ISBN 978-0-387-90497-9.
79. "JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81)" (PDF). International Telecommunication Union. 1993. Retrieved 6 April 2019.
80. Kirsten Malmkjaer (4 December 2009). The Routledge Linguistics Encyclopedia. Routledge. p. 1. ISBN 978-1-134-10371-3.
81. Kamran Dadkhah (11 January 2011). Foundations of Mathematical and Computational Economics. Springer Science & Business Media. p. 46. ISBN 978-3-642-13748-8.
82. Christopher Griffith (12 November 2012). Real-World Flash Game Development: How to Follow Best Practices AND Keep Your Sanity. CRC Press. p. 153. ISBN 978-1-136-13702-0.
83. John Joseph Griffin (1841). A System of Crystallography, with Its Application to Mineralogy. R. Griffin. p. 119.
84. Dugopolski (July 2002). Trigonometry I/E Sup. Addison Wesley. ISBN 978-0-201-78666-8.
85. V&S EDITORIAL BOARD (6 January 2015). CONCISE DICTIONARY OF MATHEMATICS. V&S Publishers. p. 288. ISBN 978-93-5057-414-0.
86. Cynthia Y. Young (19 January 2010). Precalculus. John Wiley & Sons. p. 435. ISBN 978-0-471-75684-2.
87. Ron Larson (29 January 2010). Trigonometry. Cengage Learning. p. 331. ISBN 978-1-4390-4907-5.
88. Richard N. Aufmann; Vernon C. Barker; Richard D. Nation (5 February 2007). College Trigonometry. Cengage Learning. p. 306. ISBN 978-0-618-82507-3.
89. Peterson, John C. (2004). Technical Mathematics with Calculus (illustrated ed.). Cengage Learning. p. 856. ISBN 978-0-7668-6189-3. Extract of page 856
Bibliography
• Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
• Nielsen, Kaj L. (1966). Logarithmic and Trigonometric Tables to Five Places (2nd ed.). New York: Barnes & Noble. LCCN 61-9103.
• Thurston, Hugh (1996). Early Astronomy. Springer Science & Business Media. ISBN 978-0-387-94822-5.
Further reading
• "Trigonometric functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Linton, Christopher M. (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
• Weisstein, Eric W. "Trigonometric Addition Formulas". MathWorld.
External links
Library resources about
Trigonometry
• Resources in your library
• Khan Academy: Trigonometry, free online micro lectures
• Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented.
• Benjamin Banneker's Trigonometry Puzzle at Convergence
• Dave's Short Course in Trigonometry by David Joyce of Clark University
• Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License
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| Wikipedia |
Trigonal trapezohedral honeycomb
In geometry, the trigonal trapezohedral honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. Cells are identical trigonal trapezohedra or rhombohedra. Conway, Burgiel, and Goodman-Strauss call it an oblate cubille.[1]
Trigonal trapezohedral honeycomb
(No image)
TypeDual uniform honeycomb
Coxeter-Dynkin diagrams
Cell
Trigonal trapezohedron
(1/4 of rhombic dodecahedron)
Faces Rhombus
Space groupFd3m (227)
Coxeter group${\tilde {A}}_{3}$×2, 3[4] (double)
vertex figures
|
DualQuarter cubic honeycomb
PropertiesCell-transitive, Face-transitive
Related honeycombs and tilings
This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 4 trigonal trapezohedra or rhombohedra.
rhombic dodecahedral honeycomb
Rhombic dodecahedra dissection
Rhombic net
It is analogous to the regular hexagonal being dissectable into 3 rhombi and tiling the plane as a rhombille. The rhombille tiling is actually an orthogonal projection of the trigonal trapezohedral honeycomb. A different orthogonal projection produces the quadrille where the rhombi are distorted into squares.
Dual tiling
It is dual to the quarter cubic honeycomb with tetrahedral and truncated tetrahedral cells:
See also
• Architectonic and catoptric tessellation
References
1. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), The Symmetries of Things, Wellesley, Massachusetts: A K Peters, p. 294, ISBN 978-1-56881-220-5, MR 2410150
| Wikipedia |
CORDIC
CORDIC (for "coordinate rotation digital computer"), also known as Volder's algorithm, or: Digit-by-digit method Circular CORDIC (Jack E. Volder),[1][2] Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther),[3][4] and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.),[5][6] is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, and exponentials and logarithms with arbitrary base, typically converging with one digit (or bit) per iteration. CORDIC is therefore also an example of digit-by-digit algorithms. CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and FPGAs), as the only operations it requires are additions, subtractions, bitshift and lookup tables. As such, they all belong to the class of shift-and-add algorithms. In computer science, CORDIC is often used to implement floating-point arithmetic when the target platform lacks hardware multiply for cost or space reasons.
"Pseudo-division" redirects here. For polynomial pseudo-division, see Pseudo-remainder.
Trigonometry
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History
Similar mathematical techniques were published by Henry Briggs as early as 1624[7][8] and Robert Flower in 1771,[9] but CORDIC is better optimized for low-complexity finite-state CPUs.
CORDIC was conceived in 1956[10][11] by Jack E. Volder at the aeroelectronics department of Convair out of necessity to replace the analog resolver in the B-58 bomber's navigation computer with a more accurate and faster real-time digital solution.[11] Therefore, CORDIC is sometimes referred to as a digital resolver.[12][13]
In his research Volder was inspired by a formula in the 1946 edition of the CRC Handbook of Chemistry and Physics:[11]
${\begin{aligned}K_{n}R\sin(\theta \pm \varphi )&=R\sin(\theta )\pm 2^{-n}R\cos(\theta ),\\K_{n}R\cos(\theta \pm \varphi )&=R\cos(\theta )\mp 2^{-n}R\sin(\theta ),\\\end{aligned}}$
where $\varphi $ is such that $\tan(\varphi )=2^{-n}$, and $K_{n}:={\sqrt {1+2^{-2n}}}$.
His research led to an internal technical report proposing the CORDIC algorithm to solve sine and cosine functions and a prototypical computer implementing it.[10][11] The report also discussed the possibility to compute hyperbolic coordinate rotation, logarithms and exponential functions with modified CORDIC algorithms.[10][11] Utilizing CORDIC for multiplication and division was also conceived at this time.[11] Based on the CORDIC principle, Dan H. Daggett, a colleague of Volder at Convair, developed conversion algorithms between binary and binary-coded decimal (BCD).[11][14]
In 1958, Convair finally started to build a demonstration system to solve radar fix-taking problems named CORDIC I, completed in 1960 without Volder, who had left the company already.[1][11] More universal CORDIC II models A (stationary) and B (airborne) were built and tested by Daggett and Harry Schuss in 1962.[11][15]
Volder's CORDIC algorithm was first described in public in 1959,[1][2][11][13][16] which caused it to be incorporated into navigation computers by companies including Martin-Orlando, Computer Control, Litton, Kearfott, Lear-Siegler, Sperry, Raytheon, and Collins Radio.[11]
Volder teamed up with Malcolm McMillan to build Athena, a fixed-point desktop calculator utilizing his binary CORDIC algorithm.[17] The design was introduced to Hewlett-Packard in June 1965, but not accepted.[17] Still, McMillan introduced David S. Cochran (HP) to Volder's algorithm and when Cochran later met Volder he referred him to a similar approach John E. Meggitt (IBM[18]) had proposed as pseudo-multiplication and pseudo-division in 1961.[18][19] Meggitt's method also suggested the use of base 10[18] rather than base 2, as used by Volder's CORDIC so far. These efforts led to the ROMable logic implementation of a decimal CORDIC prototype machine inside of Hewlett-Packard in 1966,[20][19] built by and conceptually derived from Thomas E. Osborne's prototypical Green Machine, a four-function, floating-point desktop calculator he had completed in DTL logic[17] in December 1964.[21] This project resulted in the public demonstration of Hewlett-Packard's first desktop calculator with scientific functions, the HP 9100A in March 1968, with series production starting later that year.[17][21][22][23]
When Wang Laboratories found that the HP 9100A used an approach similar to the factor combining method in their earlier LOCI-1[24] (September 1964) and LOCI-2 (January 1965)[25][26] Logarithmic Computing Instrument desktop calculators,[27] they unsuccessfully accused Hewlett-Packard of infringement of one of An Wang's patents in 1968.[19][28][29][30]
John Stephen Walther at Hewlett-Packard generalized the algorithm into the Unified CORDIC algorithm in 1971, allowing it to calculate hyperbolic functions, natural exponentials, natural logarithms, multiplications, divisions, and square roots.[31][3][4][32] The CORDIC subroutines for trigonometric and hyperbolic functions could share most of their code.[28] This development resulted in the first scientific handheld calculator, the HP-35 in 1972.[28][33][34][35][36][37] Based on hyperbolic CORDIC, Yuanyong Luo et al. further proposed a Generalized Hyperbolic CORDIC (GH CORDIC) to directly compute logarithms and exponentials with an arbitrary fixed base in 2019.[5][6][38][39][40] Theoretically, Hyperbolic CORDIC is a special case of GH CORDIC.[5]
Originally, CORDIC was implemented only using the binary numeral system and despite Meggitt suggesting the use of the decimal system for his pseudo-multiplication approach, decimal CORDIC continued to remain mostly unheard of for several more years, so that Hermann Schmid and Anthony Bogacki still suggested it as a novelty as late as 1973[16][13][41][42][43] and it was found only later that Hewlett-Packard had implemented it in 1966 already.[11][13][20][28]
Decimal CORDIC became widely used in pocket calculators,[13] most of which operate in binary-coded decimal (BCD) rather than binary. This change in the input and output format did not alter CORDIC's core calculation algorithms. CORDIC is particularly well-suited for handheld calculators, in which low cost – and thus low chip gate count – is much more important than speed.
CORDIC has been implemented in the ARM-based STM32G4, Intel 8087,[43][44][45][46][47] 80287,[47][48] 80387[47][48] up to the 80486[43] coprocessor series as well as in the Motorola 68881[43][44] and 68882 for some kinds of floating-point instructions, mainly as a way to reduce the gate counts (and complexity) of the FPU sub-system.
Applications
CORDIC uses simple shift-add operations for several computing tasks such as the calculation of trigonometric, hyperbolic and logarithmic functions, real and complex multiplications, division, square-root calculation, solution of linear systems, eigenvalue estimation, singular value decomposition, QR factorization and many others. As a consequence, CORDIC has been used for applications in diverse areas such as signal and image processing, communication systems, robotics and 3D graphics apart from general scientific and technical computation.[49][50]
Hardware
The algorithm was used in the navigational system of the Apollo program's Lunar Roving Vehicle to compute bearing and range, or distance from the Lunar module.[51][52] CORDIC was used to implement the Intel 8087 math coprocessor in 1980, avoiding the need to implement hardware multiplication.[53]
CORDIC is generally faster than other approaches when a hardware multiplier is not available (e.g., a microcontroller), or when the number of gates required to implement the functions it supports should be minimized (e.g., in an FPGA or ASIC). In fact, CORDIC is a standard drop-in IP in FPGA development applications such as Vivado for Xilinx, while a power series implementation is not due to the specificity of such an IP, i.e. CORDIC can compute many different functions (general purpose) while a hardware multiplier configured to execute power series implementations can only compute the function it was designed for.
On the other hand, when a hardware multiplier is available (e.g., in a DSP microprocessor), table-lookup methods and power series are generally faster than CORDIC. In recent years, the CORDIC algorithm has been used extensively for various biomedical applications, especially in FPGA implementations.
The STM32G4 series and certain STM32H7 series of MCUs implement a CORDIC module to accelerate computations in various mixed signal applications such as graphics for human-machine interface and field oriented control of motors. While not as fast as a power series approximation, CORDIC is indeed faster than interpolating table based implementations such as the ones provided by the ARM CMSIS and C standard libraries.[54] Though the results may be slightly less accurate as the CORDIC modules provided only achieve 20 bits of precision in the result. For example, most of the performance difference compared to the ARM implementation is due to the overhead of the interpolation algorithm, which achieves full floating point precision (24 bits) and can likely achieve relative error to that precision.[55] Another benefit is that the CORDIC module is a coprocessor and can be run in parallel with other CPU tasks.
The issue with using Taylor series is that while they do provide small absolute error, they do not exhibit well behaved relative error.[56] Other means of polynomial approximation, such as minimax optimization, may be used to control both kinds of error.
Software
Many older systems with integer-only CPUs have implemented CORDIC to varying extents as part of their IEEE floating-point libraries. As most modern general-purpose CPUs have floating-point registers with common operations such as add, subtract, multiply, divide, sine, cosine, square root, log10, natural log, the need to implement CORDIC in them with software is nearly non-existent. Only microcontroller or special safety and time-constrained software applications would need to consider using CORDIC.
Modes of operation
Rotation mode
CORDIC can be used to calculate a number of different functions. This explanation shows how to use CORDIC in rotation mode to calculate the sine and cosine of an angle, assuming that the desired angle is given in radians and represented in a fixed-point format. To determine the sine or cosine for an angle $\beta $, the y or x coordinate of a point on the unit circle corresponding to the desired angle must be found. Using CORDIC, one would start with the vector $v_{0}$:
$v_{0}={\begin{bmatrix}1\\0\end{bmatrix}}.$
In the first iteration, this vector is rotated 45° counterclockwise to get the vector $v_{1}$. Successive iterations rotate the vector in one or the other direction by size-decreasing steps, until the desired angle has been achieved. Each step angle is $\gamma _{i}=\arctan {(2^{-i})}$ for $i=0,1,2,\dots $.
More formally, every iteration calculates a rotation, which is performed by multiplying the vector $v_{i}$ with the rotation matrix $R_{i}$:
$v_{i+1}=R_{i}v_{i}.$
The rotation matrix is given by
$R_{i}={\begin{bmatrix}\cos(\gamma _{i})&-\sin(\gamma _{i})\\\sin(\gamma _{i})&\cos(\gamma _{i})\end{bmatrix}}.$
Using the following two trigonometric identities:
${\begin{aligned}\cos(\gamma _{i})&={\frac {1}{\sqrt {1+\tan ^{2}(\gamma _{i})}}},\\\sin(\gamma _{i})&={\frac {\tan(\gamma _{i})}{\sqrt {1+\tan ^{2}(\gamma _{i})}}},\end{aligned}}$
the rotation matrix becomes
$R_{i}={\frac {1}{\sqrt {1+\tan ^{2}(\gamma _{i})}}}{\begin{bmatrix}1&-\tan(\gamma _{i})\\\tan(\gamma _{i})&1\end{bmatrix}}.$
The expression for the rotated vector $v_{i+1}=R_{i}v_{i}$ then becomes
${\begin{bmatrix}x_{i+1}\\y_{i+1}\end{bmatrix}}={\frac {1}{\sqrt {1+\tan ^{2}(\gamma _{i})}}}{\begin{bmatrix}1&-\tan(\gamma _{i})\\\tan(\gamma _{i})&1\end{bmatrix}}{\begin{bmatrix}x_{i}\\y_{i}\end{bmatrix}},$
where $x_{i}$ and $y_{i}$ are the components of $v_{i}$. Restricting the angles $\gamma _{i}$ such that $\tan(\gamma _{i})=\pm 2^{-i}$, the multiplication with the tangent can be replaced by a division by a power of two, which is efficiently done in digital computer hardware using a bit shift. The expression then becomes
${\begin{bmatrix}x_{i+1}\\y_{i+1}\end{bmatrix}}=K_{i}{\begin{bmatrix}1&-\sigma _{i}2^{-i}\\\sigma _{i}2^{-i}&1\end{bmatrix}}{\begin{bmatrix}x_{i}\\y_{i}\end{bmatrix}},$
where
$K_{i}={\frac {1}{\sqrt {1+2^{-2i}}}},$
and $\sigma _{i}$ is used to determine the direction of the rotation: if the angle $\gamma _{i}$ is positive, then $\sigma _{i}$ is +1, otherwise it is −1.
All $K_{i}$ factors can be ignored in the iterative process and then applied all at once afterwards with a scaling factor $K(n)$
$K(n)=\prod _{i=0}^{n-1}K_{i}=\prod _{i=0}^{n-1}{\frac {1}{\sqrt {1+2^{-2i}}}},$
which is calculated in advance and stored in a table or as a single constant, if the number of iterations is fixed. This correction could also be made in advance, by scaling $v_{0}$ and hence saving a multiplication. Additionally, it can be noted that[43]
$K=\lim _{n\to \infty }K(n)\approx 0.6072529350088812561694$
to allow further reduction of the algorithm's complexity. Some applications may avoid correcting for $K$ altogether, resulting in a processing gain $A$:[57]
$A={\frac {1}{K}}=\lim _{n\to \infty }\prod _{i=0}^{n-1}{\sqrt {1+2^{-2i}}}\approx 1.64676025812107.$
After a sufficient number of iterations, the vector's angle will be close to the wanted angle $\beta $. For most ordinary purposes, 40 iterations (n = 40) are sufficient to obtain the correct result to the 10th decimal place.
The only task left is to determine whether the rotation should be clockwise or counterclockwise at each iteration (choosing the value of $\sigma $). This is done by keeping track of how much the angle was rotated at each iteration and subtracting that from the wanted angle; then in order to get closer to the wanted angle $\beta $, if $\beta _{n+1}$ is positive, the rotation is clockwise, otherwise it is negative and the rotation is counterclockwise:
$\beta _{0}=\beta $
$\beta _{i+1}=\beta _{i}-\sigma _{i}\gamma _{i},\quad \gamma _{i}=\arctan(2^{-i}).$
The values of $\gamma _{n}$ must also be precomputed and stored. But for small angles, $\arctan(\gamma _{n})=\gamma _{n}$ in fixed-point representation, reducing table size.
As can be seen in the illustration above, the sine of the angle $\beta $ is the y coordinate of the final vector $v_{n},$ while the x coordinate is the cosine value.
Vectoring mode
The rotation-mode algorithm described above can rotate any vector (not only a unit vector aligned along the x axis) by an angle between −90° and +90°. Decisions on the direction of the rotation depend on $\beta _{i}$ being positive or negative.
The vectoring-mode of operation requires a slight modification of the algorithm. It starts with a vector whose x coordinate is positive whereas the y coordinate is arbitrary. Successive rotations have the goal of rotating the vector to the x axis (and therefore reducing the y coordinate to zero). At each step, the value of y determines the direction of the rotation. The final value of $\beta _{i}$ contains the total angle of rotation. The final value of x will be the magnitude of the original vector scaled by K. So, an obvious use of the vectoring mode is the transformation from rectangular to polar coordinates.
Implementation
Software example (MATLAB)
The following is a MATLAB/GNU Octave implementation of CORDIC that does not rely on any transcendental functions except in the precomputation of tables. If the number of iterations n is predetermined, then the second table can be replaced by a single constant. With MATLAB's standard double-precision arithmetic and "format long" printout, the results increase in accuracy for n up to about 48.
function v = cordic(alpha,iteration)
% This function computes v = [cos(alpha), sin(alpha)] (alpha in radians)
% using iteration increasing iteration value will increase the precision.
if alpha < -pi/2 || alpha > pi/2
if alpha < 0
v = cordic(alpha + pi, iteration);
else
v = cordic(alpha - pi, iteration);
end
end
% Initialization of tables of constants used by CORDIC
% need a table of arctangents of negative powers of two, in radians:
% angles = atan(2.^-(0:27));
angles = [ ...
0.78539816339745 0.46364760900081 0.24497866312686 0.12435499454676 ...
0.06241880999596 0.03123983343027 0.01562372862048 0.00781234106010 ...
0.00390623013197 0.00195312251648 0.00097656218956 0.00048828121119 ...
0.00024414062015 0.00012207031189 0.00006103515617 0.00003051757812 ...
0.00001525878906 0.00000762939453 0.00000381469727 0.00000190734863 ...
0.00000095367432 0.00000047683716 0.00000023841858 0.00000011920929 ...
0.00000005960464 0.00000002980232 0.00000001490116 0.00000000745058 ];
% and a table of products of reciprocal lengths of vectors [1, 2^-2j]:
% Kvalues = cumprod(1./sqrt(1 + 1j*2.^(-(0:23))))
Kvalues = [ ...
0.70710678118655 0.63245553203368 0.61357199107790 0.60883391251775 ...
0.60764825625617 0.60735177014130 0.60727764409353 0.60725911229889 ...
0.60725447933256 0.60725332108988 0.60725303152913 0.60725295913894 ...
0.60725294104140 0.60725293651701 0.60725293538591 0.60725293510314 ...
0.60725293503245 0.60725293501477 0.60725293501035 0.60725293500925 ...
0.60725293500897 0.60725293500890 0.60725293500889 0.60725293500888 ];
Kn = Kvalues(min(iteration, length(Kvalues)));
% Initialize loop variables:
v = [1;0]; % start with 2-vector cosine and sine of zero
poweroftwo = 1;
angle = angles(1);
% Iterations
for j = 0:iteration-1;
if alpha < 0
sigma = -1;
else
sigma = 1;
end
factor = sigma * poweroftwo;
% Note the matrix multiplication can be done using scaling by powers of two and addition subtraction
R = [1, -factor; factor, 1];
v = R * v; % 2-by-2 matrix multiply
alpha = alpha - sigma * angle; % update the remaining angle
poweroftwo = poweroftwo / 2;
% update the angle from table, or eventually by just dividing by two
if j+2 > length(angles)
angle = angle / 2;
else
angle = angles(j+2);
end
end
% Adjust length of output vector to be [cos(alpha), sin(alpha)]:
v = v * Kn;
return
endfunction
The two-by-two matrix multiplication can be carried out by a pair of simple shifts and adds.
x = v[0] - sigma * (v[1] * 2^(-j));
y = sigma * (v[0] * 2^(-j)) + v[1];
v = [x;y];
In Java the Math class has a scalb(double x,int scale) method to perform such a shift,[58] C has the ldexp function,[59] and the x86 class of processors have the fscale floating point operation.[60]
Software Example (Python)
from math import atan2, sqrt, sin, cos, pi, radians
ITERS = 16
theta_table = [atan2(1, 2**i) for i in range(ITERS)]
def compute_K(n):
"""
Compute K(n) for n = ITERS. This could also be
stored as an explicit constant if ITERS above is fixed.
"""
k = 1.0
for i in range(n):
k *= 1 / sqrt(1 + 2 ** (-2 * i))
return k
def CORDIC(alpha, n):
K_n = compute_K(n)
theta = 0.0
x = 1.0
y = 0.0
P2i = 1 # This will be 2**(-i) in the loop below
for arc_tangent in theta_table:
sigma = +1 if theta < alpha else -1
theta += sigma * arc_tangent
x, y = x - sigma * y * P2i, sigma * P2i * x + y
P2i /= 2
return x * K_n, y * K_n
if __name__ == "__main__":
# Print a table of computed sines and cosines, from -90° to +90°, in steps of 15°,
# comparing against the available math routines.
print(" x sin(x) diff. sine cos(x) diff. cosine ")
for x in range(-90, 91, 15):
cos_x, sin_x = CORDIC(radians(x), ITERS)
print(
f"{x:+05.1f}° {sin_x:+.8f} ({sin_x-sin(radians(x)):+.8f}) {cos_x:+.8f} ({cos_x-cos(radians(x)):+.8f})"
)
Output
$ python cordic.py
x sin(x) diff. sine cos(x) diff. cosine
-90.0° -1.00000000 (+0.00000000) -0.00001759 (-0.00001759)
-75.0° -0.96592181 (+0.00000402) +0.25883404 (+0.00001499)
-60.0° -0.86601812 (+0.00000729) +0.50001262 (+0.00001262)
-45.0° -0.70711776 (-0.00001098) +0.70709580 (-0.00001098)
-30.0° -0.50001262 (-0.00001262) +0.86601812 (-0.00000729)
-15.0° -0.25883404 (-0.00001499) +0.96592181 (-0.00000402)
+00.0° +0.00001759 (+0.00001759) +1.00000000 (-0.00000000)
+15.0° +0.25883404 (+0.00001499) +0.96592181 (-0.00000402)
+30.0° +0.50001262 (+0.00001262) +0.86601812 (-0.00000729)
+45.0° +0.70709580 (-0.00001098) +0.70711776 (+0.00001098)
+60.0° +0.86601812 (-0.00000729) +0.50001262 (+0.00001262)
+75.0° +0.96592181 (-0.00000402) +0.25883404 (+0.00001499)
+90.0° +1.00000000 (-0.00000000) -0.00001759 (-0.00001759)
Hardware example
The number of logic gates for the implementation of a CORDIC is roughly comparable to the number required for a multiplier as both require combinations of shifts and additions. The choice for a multiplier-based or CORDIC-based implementation will depend on the context. The multiplication of two complex numbers represented by their real and imaginary components (rectangular coordinates), for example, requires 4 multiplications, but could be realized by a single CORDIC operating on complex numbers represented by their polar coordinates, especially if the magnitude of the numbers is not relevant (multiplying a complex vector with a vector on the unit circle actually amounts to a rotation). CORDICs are often used in circuits for telecommunications such as digital down converters.
Double iterations CORDIC
In two of the publications by Vladimir Baykov,[61][62] it was proposed to use the double iterations method for the implementation of the functions: arcsine, arccosine, natural logarithm, exponential function, as well as for the calculation of the hyperbolic functions. Double iterations method consists in the fact that unlike the classical CORDIC method, where the iteration step value changes EVERY time, i.e. on each iteration, in the double iteration method, the iteration step value is repeated twice and changes only through one iteration. Hence the designation for the degree indicator for double iterations appeared: $i=0,0,1,1,2,2\dots $. Whereas with ordinary iterations: $i=0,1,2\dots $. The double iteration method guarantees the convergence of the method throughout the valid range of argument changes.
The generalization of the CORDIC convergence problems for the arbitrary positional number system with radix $R$ showed that for the functions sine, cosine, arctangent, it is enough to perform $R-1$ iterations for each value of i (i = 0 or 1 to n, where n is the number of digits), i.e. for each digit of the result. For the natural logarithm, exponential, hyperbolic sine, cosine and arctangent, $R$ iterations should be performed for each value $i$. For the functions arcsine and arccosine, two $R-1$ iterations should be performed for each number digit, i.e. for each value of $i$.[63]
For inverse hyperbolic sine and arcosine functions, the number of iterations will be $2R$ for each $i$, that is, for each result digit.
Related algorithms
CORDIC is part of the class of "shift-and-add" algorithms, as are the logarithm and exponential algorithms derived from Henry Briggs' work. Another shift-and-add algorithm which can be used for computing many elementary functions is the BKM algorithm, which is a generalization of the logarithm and exponential algorithms to the complex plane. For instance, BKM can be used to compute the sine and cosine of a real angle $x$ (in radians) by computing the exponential of $0+ix$, which is $\operatorname {cis} (x)=\cos(x)+i\sin(x)$. The BKM algorithm is slightly more complex than CORDIC, but has the advantage that it does not need a scaling factor (K).
See also
• Methods of computing square roots
• IEEE 754
• Floating-point units
• Digital Circuits/CORDIC in Wikibooks
References
1. Volder, Jack E. (1959-03-03). "The CORDIC Computing Technique" (PDF). Proceedings of the Western Joint Computer Conference (WJCC) (presentation). San Francisco, California, USA: National Joint Computer Committee (NJCC): 257–261. Retrieved 2016-01-02.
2. Volder, Jack E. (1959-05-25). "The CORDIC Trigonometric Computing Technique" (PDF). IRE Transactions on Electronic Computers. The Institute of Radio Engineers, Inc. (IRE) (published September 1959). 8 (3): 330–334 (reprint: 226–230). EC-8(3):330–334. Retrieved 2016-01-01.
3. Walther, John Stephen (May 1971). Written at Palo Alto, California, USA. "A unified algorithm for elementary functions" (PDF). Proceedings of the Spring Joint Computer Conference. Atlantic City, New Jersey, USA: Hewlett-Packard Company. 38: 379–385 – via American Federation of Information Processing Societies (AFIPS).
4. Walther, John Stephen (June 2000). "The Story of Unified CORDIC". The Journal of VLSI Signal Processing. Hingham, MA, USA: Kluwer Academic Publishers. 25 (2 (Special issue on CORDIC)): 107–112. doi:10.1023/A:1008162721424. ISSN 0922-5773. S2CID 26922158.
5. Luo, Yuanyong; Wang, Yuxuan; Ha, Yajun; Wang, Zhongfeng; Chen, Siyuan; Pan, Hongbing (September 2019). "Generalized Hyperbolic CORDIC and Its Logarithmic and Exponential Computation With Arbitrary Fixed Base". IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 27 (9): 2156–2169. doi:10.1109/TVLSI.2019.2919557. S2CID 196171166.
6. Luo, Yuanyong; Wang, Yuxuan; Ha, Yajun; Wang, Zhongfeng; Chen, Siyuan; Pan, Hongbing (September 2019). "Corrections to "Generalized Hyperbolic CORDIC and Its Logarithmic and Exponential Computation With Arbitrary Fixed Base"". IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 27 (9): 2222. doi:10.1109/TVLSI.2019.2932174. S2CID 201711001.
7. Briggs, Henry (1624). Arithmetica Logarithmica. London. (Translation: Archived 4 March 2016 at the Wayback Machine)
8. Laporte, Jacques (2014) [2005]. "Henry Briggs and the HP 35". Paris, France. Archived from the original on 2015-03-09. Retrieved 2016-01-02.
9. Flower, Robert (1771). The Radix. A new way of making logarithms. London: J. Beecroft. Retrieved 2016-01-02.
10. Volder, Jack E. (1956-06-15), Binary Computation Algorithms for Coordinate Rotation and Function Generation (internal report), Convair, Aeroelectronics group, IAR-1.148
11. Volder, Jack E. (June 2000). "The Birth of CORDIC" (PDF). Journal of VLSI Signal Processing. Hingham, MA, USA: Kluwer Academic Publishers. 25 (2 (Special issue on CORDIC)): 101–105. doi:10.1023/A:1008110704586. ISSN 0922-5773. S2CID 112881. Archived from the original (PDF) on 2016-03-04. Retrieved 2016-01-02.
12. Perle, Michael D. (June 1971), "CORDIC Technique Reduces Trigonometric Function Look-Up", Computer Design, Boston, MA, USA: Computer Design Publishing Corp.: 72–78 (NB. Some sources erroneously refer to this as by P. Z. Perle or in Component Design.)
13. Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. pp. 162, 165–176, 181–193. ISBN 0-89874-318-4. Retrieved 2016-01-03. (NB. At least some batches of this reprint edition were misprints with defective pages 115–146.)
14. Daggett, Dan H. (September 1959). "Decimal-Binary Conversions in CORDIC". IRE Transactions on Electronic Computers. The Institute of Radio Engineers, Inc. (IRE). 8 (3): 335–339. doi:10.1109/TEC.1959.5222694. ISSN 0367-9950. EC-8(3):335–339. Retrieved 2016-01-02.
15. Advanced Systems Group (1962-08-06), Technical Description of Fix-taking Tie-in Equipment (report), Fort Worth, Texas, USA: General Dynamics, FZE-052
16. Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons, Inc. pp. 162, 165–176, 181–193. ISBN 0-471-76180-X. Retrieved 2016-01-03. So far CORDIC has been known to be implemented only in binary form. But, as will be demonstrated here, the algorithm can be easily modified for a decimal system.* […] *In the meantime it has been learned that Hewlett-Packard and other calculator manufacturers employ the decimal CORDIC techniques in their scientific calculators.
17. Leibson, Steven (2010). "The HP 9100 Project: An Exothermic Reaction". Retrieved 2016-01-02.
18. Meggitt, John E. (1961-08-29). "Pseudo Division and Pseudo Multiplication Processes" (PDF). IBM Journal of Research and Development. Riverton, New Jersey, USA: IBM Corporation (published April 1962). 6 (2): 210–226, 287. doi:10.1147/rd.62.0210. Retrieved 2016-01-09. John E. Meggitt B.A., 1953; PhD, 1958, Cambridge University. Awarded the First Smith Prize at Cambridge in 1955 and elected a Research Fellowship at Emmanuel College. […] Joined IBM British Laboratory at Hursley, Winchester in 1958. Interests include error-correcting codes and small microprogrammed computers. (, )
19. Cochran, David S. (2010-11-19). "A Quarter Century at HP" (interview typescript). Computer History Museum / HP Memories. 7: Scientific Calculators, circa 1966. CHM X5992.2011. Retrieved 2016-01-02. I even flew down to Southern California to talk with Jack Volder who had implemented the transcendental functions in the Athena machine and talked to him for about an hour. He referred me to the original papers by Meggitt where he'd gotten the pseudo division, pseudo multiplication generalized functions. […] I did quite a bit of literary research leading to some very interesting discoveries. […] I found a treatise from 1624 by Henry Briggs discussing the calculation of common logarithms, interestingly used the same pseudo-division/pseudo-multiplication method that MacMillan and Volder used in Athena. […] We had purchased a LOCI-2 from Wang Labs and recognized that Wang Labs LOCI II used the same algorithm to do square root as well as log and exponential. After the introduction of the 9100 our legal department got a letter from Wang saying that we had infringed on their patent. And I just sent a note back with the Briggs reference in Latin and it said, "It looks like prior art to me." We never heard another word. ()
20. Cochran, David S. (1966-03-14). "About utilizing CORDIC for computing transcendental functions in BCD" (private communication with Jack E. Volder). {{cite journal}}: Cite journal requires |journal= (help)
21. Osborne, Thomas E. (2010) [1994]. "Tom Osborne's Story in His Own Words". Retrieved 2016-01-01.
22. Leibson, Steven (2010). "The HP 9100: The Initial Journey". Retrieved 2016-01-02.
23. Cochran, David S. (September 1968). "Internal Programming of the 9100A Calculator". Hewlett-Packard Journal. Palo Alto, California, USA: Hewlett-Packard: 14–16. Retrieved 2016-01-02. ()
24. Extend your Personal Computing Power with the new LOCI-1 Logarithmic Computing Instrument, Wang Laboratories, Inc., 1964, pp. 2–3, retrieved 2016-01-03
25. Bensene, Rick (2013-08-31) [1997]. "Wang LOCI-2". Old Calculator Web Museum. Beavercreek, Oregon City, Oregon, USA. Retrieved 2016-01-03.
26. "Wang LOCI Service Manual" (PDF). Wang Laboratories, Inc. 1967. L55-67. Retrieved 2018-09-14.
27. Bensene, Rick (2004-10-23) [1997]. "Wang Model 360SE Calculator System". Old Calculator Web Museum. Beavercreek, Oregon City, Oregon, USA. Retrieved 2016-01-03.
28. Cochran, David S. (June 2010). "The HP-35 Design, A Case Study in Innovation". HP Memory Project. Retrieved 2016-01-02. During the development of the desktop HP 9100 calculator I was responsible for developing the algorithms to fit the architecture suggested by Tom Osborne. Although the suggested methodology for the algorithms came from Malcolm McMillan I did considerable amount of reading to understand the core calculations […] Although Wang Laboratories had used similar methods of calculation, my study found prior art dated 1624 that read on their patents. […] This research enabled the adaption of the transcendental functions through the use of the algorithms to match the needs of the customer within the constraints of the hardware. This proved invaluable during the development of the HP-35, […] Power series, polynomial expansions, continued fractions, and Chebyshev polynomials were all considered for the transcendental functions. All were too slow because of the number of multiplications and divisions required. The generalized algorithm that best suited the requirements of speed and programming efficiency for the HP-35 was an iterative pseudo-division and pseudo-multiplication method first described in 1624 by Henry Briggs in 'Arithmetica Logarithmica' and later by Volder and Meggitt. This is the same type of algorithm that was used in previous HP desktop calculators. […] The complexity of the algorithms made multilevel programming a necessity. This meant the calculator had to have subroutine capability, […] To generate a transcendental function such as Arc-Hyperbolic-Tan required several levels of subroutines. […] Chris Clare later documented this as Algorithmic State Machine (ASM) methodology. Even the simple Sine or Cosine used the Tangent routine, and then calculated the Sine from trigonometric identities. These arduous manipulations were necessary to minimize the number of unique programs and program steps […] The arithmetic instruction set was designed specifically for a decimal transcendental-function calculator. The basic arithmetic operations are performed by a 10's complement adder-subtractor which has data paths to three of the registers that are used as working storage.
29. US patent 3402285A, Wang, An, "Calculating apparatus", published 1968-09-17, issued 1968-09-17, assigned to Wang Laboratories (, )
30. DE patent 1499281B1, Wang, An, "Rechenmaschine fuer logarithmische Rechnungen", published 1970-05-06, issued 1970-05-06, assigned to Wang Laboratories ()
31. Swartzlander, Jr., Earl E. (1990). Computer Arithmetic. Vol. 1 (2 ed.). Los Alamitos: IEEE Computer Society Press. ISBN 9780818689314. 0818689315. Retrieved 2016-01-02.
32. Petrocelli, Orlando R., ed. (1972), The Best Computer Papers of 1971, Auerbach Publishers, p. 71, ISBN 0877691274, retrieved 2016-01-02
33. Cochran, David S. (June 1972). "Algorithms and Accuracy in the HP-35" (PDF). Hewlett-Packard Journal. 23 (10): 10–11.
34. Laporte, Jacques (2005-12-06). "HP35 trigonometric algorithm". Paris, France. Archived from the original on 2015-03-09. Retrieved 2016-01-02.
35. Laporte, Jacques (February 2005) [1981]. "The secret of the algorithms". L'Ordinateur Individuel. Paris, France (24). Archived from the original on 2016-08-18. Retrieved 2016-01-02.
36. Laporte, Jacques (February 2012) [2006]. "Digit by digit methods". Paris, France. Archived from the original on 2016-08-18. Retrieved 2016-01-02.
37. Laporte, Jacques (February 2012) [2007]. "HP 35 Logarithm Algorithm". Paris, France. Archived from the original on 2016-08-18. Retrieved 2016-01-07.
38. Wang, Yuxuan; Luo, Yuanyong; Wang, Zhongfeng; Shen, Qinghong; Pan, Hongbing (January 2020). "GH CORDIC-Based Architecture for Computing Nth Root of Single-Precision Floating-Point Number". IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 28 (4): 864–875. doi:10.1109/TVLSI.2019.2959847. S2CID 212975618.
39. Mopuri, Suresh; Acharyya, Amit (September 2019). "Low Complexity Generic VLSI Architecture Design Methodology for Nth Root and Nth Power Computations". IEEE Transactions on Circuits and Systems I: Regular Papers. 66 (12): 4673–4686. doi:10.1109/TCSI.2019.2939720. S2CID 203992880.
40. Vachhani, Leena (November 2019). "CORDIC as a Switched Nonlinear System". Circuits, Systems and Signal Processing. 39 (6): 3234–3249. doi:10.1007/s00034-019-01295-8. S2CID 209904108.
41. Schmid, Hermann; Bogacki, Anthony (1973-02-20). "Use Decimal CORDIC for Generation of Many Transcendental Functions". EDN: 64–73.
42. Franke, Richard (1973-05-08). An Analysis of Algorithms for Hardware Evaluation of Elementary Functions (PDF). Monterey, California, USA: Department of the Navy, Naval Postgraduate School. NPS-53FE73051A. Retrieved 2016-01-03.
43. Muller, Jean-Michel (2006). Elementary Functions: Algorithms and Implementation (2 ed.). Boston: Birkhäuser. p. 134. ISBN 978-0-8176-4372-0. LCCN 2005048094. Retrieved 2015-12-01.
44. Nave, Rafi (March 1983). "Implementation of Transcendental Functions on a Numerics Processor". Microprocessing and Microprogramming. 11 (3–4): 221–225. doi:10.1016/0165-6074(83)90151-5.
45. Palmer, John F.; Morse, Stephen Paul (1984). The 8087 Primer (1 ed.). John Wiley & Sons Australia, Limited. ISBN 0471875694. 9780471875697. Retrieved 2016-01-02.
46. Glass, L. Brent (January 1990). "Math Coprocessors: A look at what they do, and how they do it". Byte. 15 (1): 337–348. ISSN 0360-5280.
47. Jarvis, Pitts (1990-10-01). "Implementing CORDIC algorithms – A single compact routine for computing transcendental functions". Dr. Dobb's Journal: 152–156. Archived from the original on 2016-03-04. Retrieved 2016-01-02.
48. Yuen, A. K. (1988). "Intel's Floating-Point Processors". Electro/88 Conference Record: 48/5/1–7.
49. Meher, Pramod Kumar; Valls, Javier; Juang, Tso-Bing; Sridharan, K.; Maharatna, Koushik (2008-08-22). "50 Years of CORDIC: Algorithms, Architectures and Applications" (PDF). IEEE Transactions on Circuits and Systems I: Regular Papers (published 2009-09-09). 56 (9): 1893–1907. doi:10.1109/TCSI.2009.2025803. S2CID 5465045.
50. Meher, Pramod Kumar; Park, Sang Yoon (February 2013). "Low Complexity Generic VLSI Architecture Design Methodology for Nth Root and Nth Power Computations". IEEE Transactions on Very Large Scale Integration (VLSI) Systems. 21 (2): 217–228. doi:10.1109/TVLSI.2012.2187080. S2CID 7059383.
51. Heffron, W. G.; LaPiana, F. (1970-12-11). "Technical Memorandum 70-2014-8: The Navigation System of the Lunar Roving Vehicle" (PDF). NASA. Washington, D.C., USA: Bellcomm. p. 14.
52. Smith, Earnest C.; Mastin, William C. (November 1973). "Technical Note D-7469: Lunar Roving Vehicle Navigation System Performance Review" (PDF). NASA. Huntsville, Alabama, USA: Marshall Space Flight Center. p. 17.
53. Shirriff, Ken (May 2020). "Extracting ROM constants from the 8087 math coprocessor's die". righto.com. Retrieved 2020-09-03. The ROM contains 16 arctangent values, the arctans of 2−n. It also contains 14 log values, the base-2 logs of (1+2−n). These may seem like unusual values, but they are used in an efficient algorithm called CORDIC, which was invented in 1958.
54. "Getting started with the CORDIC accelerator using STM32CubeG4 MCU Package" (PDF). STMicroelectronics. Retrieved 2021-01-01.
55. "CMSIS/CMSIS/DSP_Lib/Source/ControllerFunctions/arm_sin_cos_f32.c". Github. ARM. Retrieved 2021-01-01.
56. "Error bounds of Taylor Expansion for Sine". Math Stack Exchange. Retrieved 2021-01-01.
57. Andraka, Ray (1998). "A survey of CORDIC algorithms for FPGA based computers" (PDF). ACM. North Kingstown, RI, USA: Andraka Consulting Group, Inc. 0-89791-978-5/98/01. Retrieved 2016-05-08.
58. "Class Math". Java Platform Standard (8 ed.). Oracle Corporation. 2018 [1993]. Archived from the original on 2018-08-06. Retrieved 2018-08-06.
59. "ldexp, ldexpf, ldexpl". cppreference.com. 2015-06-11. Archived from the original on 2018-08-06. Retrieved 2018-08-06.
60. "Section 8.3.9 Logarithmic, Exponential, and Scale". Intel 64 and IA-32 Architectures Software Developer's Manual Volume 1: Basic Architecture (PDF). Intel Corporation. September 2016. pp. 8–22.
61. Baykov, Vladimir. "The outline (autoreferat) of my PhD, published in 1972". baykov.de. Retrieved 2023-05-03.
62. Vladimir, Baykov. "Hardware implementation of elementary functions in computers". baykov.de. Retrieved 2023-05-03.
63. "Special-purpose processors: iterative algorithms and structures". baykov.de. Retrieved 2023-05-03.
Further reading
• Parini, Joseph A. (1966-09-05). "DIVIC Gives Answer to Complex Navigation Questions". Electronics: 105–111. ISSN 0013-5070. (NB. DIVIC stands for DIgital Variable Increments Computer. Some sources erroneously refer to this as by J. M. Parini.)
• Anderson, Stanley F.; Earle, John G.; Goldschmidt, Robert Elliott; Powers, Don M. (1965-11-01). "The IBM System/360 Model 91: Floating-Point Execution Unit" (PDF). IBM Journal of Research and Development. Riverton, New Jersey, USA (published January 1967). 11 (1): 34–53. doi:10.1147/rd.111.0034. Retrieved 2016-01-02.
• Liccardo, Michael A. (September 1968). An Interconnect Processor with Emphasis on CORDIC Mode Operation (MSc thesis). Berkeley, CA, USA: University of California, Berkeley, Department of Electrical Engineering. OCLC 500565168.
• US patent 3576983A, Cochran, David S., "Digital calculator system for computing square roots", published 1971-05-04, issued 1971-05-04, assigned to Hewlett-Packard Co. ()
• Chen, Tien Chi (July 1972). "Automatic Computation of Exponentials, Logarithms, Ratios, and Square Roots" (PDF). IBM Journal of Research and Development. 16 (4): 380–388. doi:10.1147/rd.164.0380. ISSN 0018-8646. Retrieved 2016-01-02.
• Egbert, William E. (May 1977). "Personal Calculator Algorithms I: Square Roots" (PDF). Hewlett-Packard Journal. Palo Alto, California, USA: Hewlett-Packard. 28 (9): 22–24. Retrieved 2016-01-02. ()
• Egbert, William E. (June 1977). "Personal Calculator Algorithms II: Trigonometric Functions" (PDF). Hewlett-Packard Journal. Palo Alto, California, USA: Hewlett-Packard. 28 (10): 17–20. Retrieved 2016-01-02. ()
• Egbert, William E. (November 1977). "Personal Calculator Algorithms III: Inverse Trigonometric Functions" (PDF). Hewlett-Packard Journal. Palo Alto, California, USA: Hewlett-Packard. 29 (3): 22–23. Retrieved 2016-01-02. ()
• Egbert, William E. (April 1978). "Personal Calculator Algorithms IV: Logarithmic Functions" (PDF). Hewlett-Packard Journal. Palo Alto, California, USA: Hewlett-Packard. 29 (8): 29–32. Retrieved 2016-01-02. ()
• Senzig, Don (1975). "Calculator Algorithms". IEEE Compcon Reader Digest. IEEE: 139–141. IEEE Catalog No. 75 CH 0920-9C.
• Baykov, Vladimir D. (1972), Вопросы исследования вычисления элементарных функций по методу «цифра за цифрой» [Problems of elementary functions evaluation based on digit by digit (CORDIC) technique] (PhD thesis) (in Russian), Leningrad State University of Electrical Engineering
• Baykov, Vladimir D.; Smolov, Vladimir B. (1975). Apparaturnaja realizatsija elementarnikh funktsij v CVM Аппаратурная реализация элементарных функций в ЦВМ [Hardware implementation of elementary functions in computers] (in Russian). Leningrad State University. Archived from the original on 2019-03-02. Retrieved 2019-03-02.
• Baykov, Vladimir D.; Seljutin, S. A. (1982). Вычисление элементарных функций в ЭКВМ [Elementary functions evaluation in microcalculators] (in Russian). Moscow: Radio i svjaz (Радио и связь).
• Baykov, Vladimir D.; Smolov, Vladimir B. (1985). Специализированные процессоры: итерационные алгоритмы и структуры [Special-purpose processors: iterative algorithms and structures] (in Russian). Moscow: Radio i svjaz (Радио и связь).
• Coppens, Thomas, ed. (January 1980). "CORDIC constants in TI 58/59 ROM". Texas Instruments Software Exchange Newsletter. Kapellen, Belgium: TISOFT. 2 (2).
• Coppens, Thomas, ed. (April–June 1980). "Natural logarithm computation scheme / ex computing scheme / 1/x computing scheme". Texas Instruments Software Exchange Newsletter. Kapellen, Belgium: TISOFT. 2 (3). (about CORDIC in TI-58/TI-59)
• TI Graphic Products Team (1995) [1993]. "Transcendental function algorithms". Dallas, Texas, USA: Texas Instruments, Consumer Products. Archived from the original on 2016-03-17. Retrieved 2019-03-02.
• Jorke, Günter; Lampe, Bernhard; Wengel, Norbert (1989). Arithmetische Algorithmen der Mikrorechentechnik (in German) (1 ed.). Berlin, Germany: VEB Verlag Technik. pp. 219, 261, 271–296. ISBN 3341005153. EAN 9783341005156. MPN 5539165. License 201.370/4/89. Retrieved 2015-12-01.
• Zechmeister, M. (2021). Solving Kepler's equation with CORDIC double iterations. pp. 109–117. arXiv:2008.02894. doi:10.1093/mnras/staa2441. {{cite book}}: |journal= ignored (help)
• Frerking, Marvin E. (1994). Digital Signal Processing in Communication Systems (1 ed.).
• Kantabutra, Vitit (1996). "On hardware for computing exponential and trigonometric functions". IEEE Transactions on Computers. 45 (3): 328–339. doi:10.1109/12.485571.
• Johansson, Kenny (2008). "6.5 Sine and Cosine Functions". Low Power and Low Complexity Shift-and-Add Based Computations (PDF) (Dissertation thesis). Linköping Studies in Science and Technology (1 ed.). Linköping, Sweden: Department of Electrical Engineering, Linköping University. pp. 244–250. ISBN 978-91-7393-836-5. ISSN 0345-7524. No. 1201. Archived (PDF) from the original on 2017-08-13. Retrieved 2021-08-23. (x+268 pages)
• Banerjee, Ayan (2001). "FPGA realization of a CORDIC based FFT processor for biomedical signal processing". Microprocessors and Microsystems. Kharagpur, West Bengal, India. 25 (3): 131–142. doi:10.1016/S0141-9331(01)00106-5.
• Kahan, William Morton (2002-05-20). "Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials" (PDF). Berkeley, CA, USA: University of California. Archived from the original (PDF) on 2015-12-25. Retrieved 2016-01-15.
• Cockrum, Chris K. (Fall 2008). "Implementation of a CORDIC Algorithm in a Digital Down-Converter" (PDF).
• Lakshmi, Boppana; Dhar, Anindya Sundar (2009-10-06). "CORDIC Architectures: A Survey". VLSI Design. Kharagpur, West Bengal, India: Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology (published 2010-10-10). 2010: 1–19. doi:10.1155/2010/794891. 794891.
• Savard, John J. G. (2018) [2006]. "Advanced Arithmetic Techniques". quadibloc. Archived from the original on 2018-07-03. Retrieved 2018-07-16.
External links
Wikiversity has learning resources about CORDIC Hardware Implementations
• Wang, Shaoyun (July 2011), CORDIC Bibliography Site
• Soft CORDIC IP (verilog HDL code)
• CORDIC Bibliography Site
• BASIC Stamp, CORDIC math implementation
• CORDIC implementation in verilog
• CORDIC Vectoring with Arbitrary Target Value
• PicBasic Pro, Pic18 CORDIC math implementation
• Python CORDIC implementation
• Simple C code for fixed-point CORDIC
• Tutorial and MATLAB Implementation – Using CORDIC to Estimate Phase of a Complex Number
• Descriptions of hardware CORDICs in Arx with testbenches in C++ and VHDL
• An Introduction to the CORDIC algorithm
• Implementation of the CORDIC Algorithm in a Digital Down-Converter
• 50-th Anniversary of the CORDIC Algorithm
• Implementation of the CORDIC Algorithm: fixed point C code for trigonometric and hyperbolic functions, C code for test and performance verification
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| Wikipedia |
Eli Maor
Eli Maor (born 1937), an historian of mathematics, is the author of several books about the history of mathematics.[1] Eli Maor received his PhD at the Technion – Israel Institute of Technology. He teaches the history of mathematics at Loyola University Chicago.[2] Maor was the editor of the article on trigonometry for the Encyclopædia Britannica.[3]
Eli Maor
Born1937
Palestine
OccupationHistorian of mathematics
EmployerLoyola University Chicago
Asteroid 226861 Elimaor, discovered at the Jarnac Observatory in 2004, was named in his honor.[1] The official naming citation was published by the Minor Planet Center on 22 July 2013 (M.P.C. 84383).[4]
Selected works
• To Infinity and Beyond: A Cultural History of the Infinite, 1991, Princeton University Press. ISBN 978-0-691-02511-7
• e:The story of a Number, by Eli Maor, Princeton University Press (Princeton, New Jersey) (1994) ISBN 0-691-05854-7
• Venus in Transit, 2000, Princeton University Press. ISBN 0-691-04874-6
• Trigonometric Delights, Princeton University Press, 2002 ISBN 0-691-09541-8. Ebook version, in PDF format, full text presented.
• The Pythagorean Theorem: A 4,000-Year History, 2007, Princeton University Press, ISBN 978-0-691-12526-8
• The Facts on File Calculus Handbook (Facts on File, 2003), 2005, Checkmark Books, an encyclopedia of calculus concepts geared for high school and college students
• Music by the Numbers. Princeton University Press. 2018. ISBN 9780691176901.
References
1. "226861 Elimaor (2004 TV18)". Minor Planet Center. Retrieved 27 August 2019.
2. Eli Maor Archived 26 July 2009 at the Wayback Machine biography at Princeton University Press
3. Maor, Eli (2010). "Encyclopædia Britannica: Author". Encyclopædia Britannica. Retrieved 30 August 2010.(subscription required)
4. "MPC/MPO/MPS Archive". Minor Planet Center. Retrieved 27 August 2019.
Authority control
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| Wikipedia |
List of trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
Trigonometry
• Outline
• History
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• Generalized trigonometry
Reference
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• Tables
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Laws and theorems
• Sines
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Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Pythagorean identities
Main article: Pythagorean trigonometric identity
The basic relationship between the sine and cosine is given by the Pythagorean identity:
$\sin ^{2}\theta +\cos ^{2}\theta =1,$
where $\sin ^{2}\theta $ means $(\sin \theta )^{2}$ and $\cos ^{2}\theta $ means $(\cos \theta )^{2}.$
This can be viewed as a version of the Pythagorean theorem, and follows from the equation $x^{2}+y^{2}=1$ for the unit circle. This equation can be solved for either the sine or the cosine:
${\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}$
where the sign depends on the quadrant of $\theta .$
Dividing this identity by $\sin ^{2}\theta $, $\cos ^{2}\theta $, or both yields the following identities:
${\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}$
Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):
Each trigonometric function in terms of each of the other five.[1]
in terms of $\sin \theta $ $\csc \theta $ $\cos \theta $ $\sec \theta $ $\tan \theta $ $\cot \theta $
$\sin \theta =$ $\sin \theta $ ${\frac {1}{\csc \theta }}$ $\pm {\sqrt {1-\cos ^{2}\theta }}$ $\pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}$ $\pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}$ $\pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}$
$\csc \theta =$ ${\frac {1}{\sin \theta }}$ $\csc \theta $ $\pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}$ $\pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}$ $\pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}$ $\pm {\sqrt {1+\cot ^{2}\theta }}$
$\cos \theta =$ $\pm {\sqrt {1-\sin ^{2}\theta }}$ $\pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}$ $\cos \theta $ ${\frac {1}{\sec \theta }}$ $\pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}$ $\pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}$
$\sec \theta =$ $\pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}$ $\pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}$ ${\frac {1}{\cos \theta }}$ $\sec \theta $ $\pm {\sqrt {1+\tan ^{2}\theta }}$ $\pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}$
$\tan \theta =$ $\pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}$ $\pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}$ $\pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}$ $\pm {\sqrt {\sec ^{2}\theta -1}}$ $\tan \theta $ ${\frac {1}{\cot \theta }}$
$\cot \theta =$ $\pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}$ $\pm {\sqrt {\csc ^{2}\theta -1}}$ $\pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}$ $\pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}$ ${\frac {1}{\tan \theta }}$ $\cot \theta $
Reflections, shifts, and periodicity
By examining the unit circle, one can establish the following properties of the trigonometric functions.
Reflections
When the direction of a Euclidean vector is represented by an angle $\theta ,$ this is the angle determined by the free vector (starting at the origin) and the positive $x$-unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive $x$-axis. If a line (vector) with direction $\theta $ is reflected about a line with direction $\alpha ,$ then the direction angle $\theta ^{\prime }$ of this reflected line (vector) has the value
$\theta ^{\prime }=2\alpha -\theta .$
The values of the trigonometric functions of these angles $\theta ,\;\theta ^{\prime }$ for specific angles $\alpha $ satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.[2]
$\theta $ reflected in $\alpha =0$[3]
odd/even identities
$\theta $ reflected in $\alpha ={\frac {\pi }{4}}$ $\theta $ reflected in $\alpha ={\frac {\pi }{2}}$ $\theta $ reflected in $\alpha ={\frac {3\pi }{4}}$ $\theta $ reflected in $\alpha =\pi $
compare to $\alpha =0$
$\sin(-\theta )=-\sin \theta $ $\sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta $ $\sin(\pi -\theta )=+\sin \theta $ $\sin \left({\tfrac {3\pi }{2}}-\theta \right)=-\cos \theta $ $\sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )$
$\cos(-\theta )=+\cos \theta $ $\cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta $ $\cos(\pi -\theta )=-\cos \theta $ $\cos \left({\tfrac {3\pi }{2}}-\theta \right)=-\sin \theta $ $\cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )$
$\tan(-\theta )=-\tan \theta $ $\tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta $ $\tan(\pi -\theta )=-\tan \theta $ $\tan \left({\tfrac {3\pi }{2}}-\theta \right)=+\cot \theta $ $\tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )$
$\csc(-\theta )=-\csc \theta $ $\csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta $ $\csc(\pi -\theta )=+\csc \theta $ $\csc \left({\tfrac {3\pi }{2}}-\theta \right)=-\sec \theta $ $\csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )$
$\sec(-\theta )=+\sec \theta $ $\sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta $ $\sec(\pi -\theta )=-\sec \theta $ $\sec \left({\tfrac {3\pi }{2}}-\theta \right)=-\csc \theta $ $\sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )$
$\cot(-\theta )=-\cot \theta $ $\cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta $ $\cot(\pi -\theta )=-\cot \theta $ $\cot \left({\tfrac {3\pi }{2}}-\theta \right)=+\tan \theta $ $\cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )$
Shifts and periodicity
Shift by one quarter period Shift by one half period Shift by full periods[4] Period
$\sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta $ $\sin(\theta +\pi )=-\sin \theta $ $\sin(\theta +k\cdot 2\pi )=+\sin \theta $ $2\pi $
$\cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta $ $\cos(\theta +\pi )=-\cos \theta $ $\cos(\theta +k\cdot 2\pi )=+\cos \theta $ $2\pi $
$\csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta $ $\csc(\theta +\pi )=-\csc \theta $ $\csc(\theta +k\cdot 2\pi )=+\csc \theta $ $2\pi $
$\sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta $ $\sec(\theta +\pi )=-\sec \theta $ $\sec(\theta +k\cdot 2\pi )=+\sec \theta $ $2\pi $
$\tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}$ $\tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta $ $\tan(\theta +k\cdot \pi )=+\tan \theta $ $\pi $
$\cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}$ $\cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta $ $\cot(\theta +k\cdot \pi )=+\cot \theta $ $\pi $
Signs
The sign of trigonometric functions depends on quadrant of the angle. If ${-\pi }<\theta \leq \pi $ and sgn is the sign function,
${\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&{\text{if}}\ \ 0<\theta <\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <0\\0&{\text{if}}\ \ \theta \in \{0,\pi \}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&{\text{if}}\ \ {-{\tfrac {1}{2}}\pi }<\theta <{\tfrac {1}{2}}\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{or}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },{\tfrac {1}{2}}\pi {\bigr \}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{or}}\ \ 0<\theta <{\tfrac {1}{2}}\pi \\-1&{\text{if}}\ \ {-{\tfrac {1}{2}}\pi }<\theta <0\ \ {\text{or}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },0,{\tfrac {1}{2}}\pi ,\pi {\bigr \}}\end{cases}}\end{aligned}}$
The trigonometric functions are periodic with common period $2\pi ,$ so for values of θ outside the interval $({-\pi },\pi ],$ they take repeating values (see § Shifts and periodicity above).
Angle sum and difference identities
See also: Proofs of trigonometric identities § Angle sum identities, and Small-angle approximation § Angle sum and difference
These are also known as the angle addition and subtraction theorems (or formulae).
${\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}$
The angle difference identities for $\sin(\alpha -\beta )$ and $\cos(\alpha -\beta )$ can be derived from the angle sum versions by substituting $-\beta $ for $\beta $ and using the facts that $\sin(-\beta )=-\sin(\beta )$ and $\cos(-\beta )=\cos(\beta )$. They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.
These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.
Sine $\sin(\alpha \pm \beta )$ $=$ $\sin \alpha \cos \beta \pm \cos \alpha \sin \beta $[5][6]
Cosine $\cos(\alpha \pm \beta )$ $=$ $\cos \alpha \cos \beta \mp \sin \alpha \sin \beta $[6][7]
Tangent $\tan(\alpha \pm \beta )$ $=$ ${\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}$[6][8]
Cosecant $\csc(\alpha \pm \beta )$ $=$ ${\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}$[9]
Secant $\sec(\alpha \pm \beta )$ $=$ ${\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}$[9]
Cotangent $\cot(\alpha \pm \beta )$ $=$ ${\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}$[6][10]
Arcsine $\arcsin x\pm \arcsin y$ $=$ $\arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}}}\right)$[11]
Arccosine $\arccos x\pm \arccos y$ $=$ $\arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)$[12]
Arctangent $\arctan x\pm \arctan y$ $=$ $\arctan \left({\frac {x\pm y}{1\mp xy}}\right)$[13]
Arccotangent $\operatorname {arccot} x\pm \operatorname {arccot} y$ $=$ $\operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)$
Sines and cosines of sums of infinitely many angles
When the series $ \sum _{i=1}^{\infty }\theta _{i}$ converges absolutely then
${\begin{aligned}\sin \left(\sum _{i=1}^{\infty }\theta _{i}\right)&=\sum _{{\text{odd}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)\\\cos \left(\sum _{i=1}^{\infty }\theta _{i}\right)&=\sum _{{\text{even}}\ k\geq 0}~(-1)^{\frac {k}{2}}~~\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)\,.\end{aligned}}$
Because the series $ \sum _{i=1}^{\infty }\theta _{i}$ converges absolutely, it is necessarily the case that $ \lim _{i\to \infty }\theta _{i}=0,$ $ \lim _{i\to \infty }\sin \theta _{i}=0,$ and $ \lim _{i\to \infty }\cos \theta _{i}=1.$ In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.
When only finitely many of the angles $\theta _{i}$ are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
Tangents and cotangents of sums
Let $e_{k}$ (for $k=0,1,2,3,\ldots $) be the kth-degree elementary symmetric polynomial in the variables
$x_{i}=\tan \theta _{i}$
for $i=0,1,2,3,\ldots ,$ that is,
${\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum _{i<j<k}x_{i}x_{j}x_{k}&&=\sum _{i<j<k}\tan \theta _{i}\tan \theta _{j}\tan \theta _{k}\\&{}\ \ \vdots &&{}\ \ \vdots \end{aligned}}$
Then
${\begin{aligned}\tan \left(\sum _{i}\theta _{i}\right)&={\frac {\sin \left(\sum _{i}\theta _{i}\right)/\prod _{i}\cos \theta _{i}}{\cos \left(\sum _{i}\theta _{i}\right)/\prod _{i}\cos \theta _{i}}}\\&={\frac \sum _{{\text{odd}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}\sum _{{\text{even}}\ k\geq 0}~(-1)^{\frac {k}{2}}~~\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}}={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots }}\\\cot \left(\sum _{i}\theta _{i}\right)&={\frac {e_{0}-e_{2}+e_{4}-\cdots }{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}$
using the sine and cosine sum formulae above.
The number of terms on the right side depends on the number of terms on the left side.
For example:
${\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\tan \theta _{1}+\tan \theta _{2}}{1\ -\ \tan \theta _{1}\tan \theta _{2}}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}},\end{aligned}}$
and so on. The case of only finitely many terms can be proved by mathematical induction.[14]
Secants and cosecants of sums
${\begin{aligned}\sec \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}\\[8pt]\csc \left(\sum _{i}\theta _{i}\right)&={\frac {\prod _{i}\sec \theta _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}\end{aligned}}$
where $e_{k}$ is the kth-degree elementary symmetric polynomial in the n variables $x_{i}=\tan \theta _{i},$ $i=1,\ldots ,n,$ and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[15] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
For example,
${\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}$
Ptolemy's theorem
Main article: Ptolemy's theorem
See also: History of trigonometry § Classical antiquity
Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral $ABCD$, as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.[16] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
By Thales's theorem, $\angle DAB$ and $\angle DCB$ are both right angles. The right-angled triangles $DAB$ and $DCB$ both share the hypotenuse ${\overline {BD}}$ of length 1. Thus, the side ${\overline {AB}}=\sin \alpha $, ${\overline {AD}}=\cos \alpha $, ${\overline {BC}}=\sin \beta $ and ${\overline {CD}}=\cos \beta $.
By the inscribed angle theorem, the central angle subtended by the chord ${\overline {AC}}$ at the circle's center is twice the angle $\angle ADC$, i.e. $2(\alpha +\beta )$. Therefore, the symmetrical pair of red triangles each has the angle $\alpha +\beta $ at the center. Each of these triangles has a hypotenuse of length $ {\frac {1}{2}}$, so the length of ${\overline {AC}}$ is $ 2\times {\frac {1}{2}}\sin(\alpha +\beta )$, i.e. simply $\sin(\alpha +\beta )$. The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also $\sin(\alpha +\beta )$.
When these values are substituted into the statement of Ptolemy's theorem that $|{\overline {AC}}|\cdot |{\overline {BD}}|=|{\overline {AB}}|\cdot |{\overline {CD}}|+|{\overline {AD}}|\cdot |{\overline {BC}}|$, this yields the angle sum trigonometric identity for sine: $\sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta $. The angle difference formula for $\sin(\alpha -\beta )$ can be similarly derived by letting the side ${\overline {CD}}$ serve as a diameter instead of ${\overline {BD}}$.[17]
Multiple-angle formulae
Tn is the nth Chebyshev polynomial $\cos(n\theta )=T_{n}(\cos \theta )$[18]
de Moivre's formula, i is the imaginary unit $\cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}$[19]
Double-angle formulae
Formulae for twice an angle.[20]
• $\sin(2\theta )=2\sin \theta \cos \theta =(\sin \theta +\cos \theta )^{2}-1={\frac {2\tan \theta }{1+\tan ^{2}\theta }}$
• $\cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}$
• $\tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}$
• $\cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}={\frac {1-\tan ^{2}\theta }{2\tan \theta }}$
• $\sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}={\frac {1+\tan ^{2}\theta }{1-\tan ^{2}\theta }}$
• $\csc(2\theta )={\frac {\sec \theta \csc \theta }{2}}={\frac {1+\tan ^{2}\theta }{2\tan \theta }}$
Triple-angle formulae
Formulae for triple angles.[20]
• $\sin(3\theta )=3\sin \theta -4\sin ^{3}\theta =4\sin \theta \sin \left({\frac {\pi }{3}}-\theta \right)\sin \left({\frac {\pi }{3}}+\theta \right)$
• $\cos(3\theta )=4\cos ^{3}\theta -3\cos \theta =4\cos \theta \cos \left({\frac {\pi }{3}}-\theta \right)\cos \left({\frac {\pi }{3}}+\theta \right)$
• $\tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}=\tan \theta \tan \left({\frac {\pi }{3}}-\theta \right)\tan \left({\frac {\pi }{3}}+\theta \right)$
• $\cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}$
• $\sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}$
• $\csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}$
Multiple-angle and half-angle formulae
Formula for multiple angles.[21]
• ${\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \\{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned}}$
• $\cos(n\theta )=\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta $
• $\cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )$
• $\cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )$
• $\tan(n\theta )={\frac {\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\tan ^{k}\theta }}$
Chebyshev method
The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the $(n-1)$th and $(n-2)$th values.[22]
$\cos(nx)$ can be computed from $\cos((n-1)x)$, $\cos((n-2)x)$, and $\cos(x)$ with
$\cos(nx)=2\cos x\cos((n-1)x)-\cos((n-2)x).$
This can be proved by adding together the formulae
${\begin{aligned}\cos((n-1)x+x)&=\cos((n-1)x)\cos x-\sin((n-1)x)\sin x\\\cos((n-1)x-x)&=\cos((n-1)x)\cos x+\sin((n-1)x)\sin x\end{aligned}}$
It follows by induction that $\cos(nx)$ is a polynomial of $\cos x,$ the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.
Similarly, $\sin(nx)$ can be computed from $\sin((n-1)x),$ $\sin((n-2)x),$ and $\cos x$ with
$\sin(nx)=2\cos x\sin((n-1)x)-\sin((n-2)x)$
This can be proved by adding formulae for $\sin((n-1)x+x)$ and $\sin((n-1)x-x).$
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:
$\tan(nx)={\frac {\tan((n-1)x)+\tan x}{1-\tan((n-1)x)\tan x}}\,.$
Half-angle formulae
${\begin{aligned}\sin {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&={\frac {1-\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1+\cos \theta }}=\csc \theta -\cot \theta ={\frac {\tan \theta }{1+\sec {\theta }}}\\[6mu]&=\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {-1+\operatorname {sgn}(\cos \theta ){\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}\\[3pt]\cot {\frac {\theta }{2}}&={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\\sec {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1+\cos \theta }}}\\\csc {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1-\cos \theta }}}\\\end{aligned}}$
[23][24]
Also
${\begin{aligned}\tan {\frac {\eta \pm \theta }{2}}&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}\\[3pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\end{aligned}}$
Table
See also: Tangent half-angle formula
These can be shown by using either the sum and difference identities or the multiple-angle formulae.
SineCosineTangentCotangent
Double-angle formula[25][26] ${\begin{aligned}\sin(2\theta )&=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}$ ${\begin{aligned}\cos(2\theta )&=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}$ $\tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}$ $\cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}$
Triple-angle formula[18][27] ${\begin{aligned}\sin(3\theta )&=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned}}$ ${\begin{aligned}\cos(3\theta )&=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned}}$ $\tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}$ $\cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}$
Half-angle formula[23][24] ${\begin{aligned}&\sin {\frac {\theta }{2}}=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\\\&\left({\text{or }}\sin ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{2}}\right)\end{aligned}}$ ${\begin{aligned}&\cos {\frac {\theta }{2}}=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\\\&\left({\text{or }}\cos ^{2}{\frac {\theta }{2}}={\frac {1+\cos \theta }{2}}\right)\end{aligned}}$ ${\begin{aligned}\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}\\[5pt]\tan {\frac {\eta +\theta }{2}}&={\frac {\sin \eta +\sin \theta }{\cos \eta +\cos \theta }}\\[5pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[5pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\\[5pt]\tan {\frac {\theta }{2}}&={\frac {\tan \theta }{1+{\sqrt {1+\tan ^{2}\theta }}}}\\&{\text{for }}\theta \in \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)\end{aligned}}$ ${\begin{aligned}\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1-\cos \theta }}\\[4pt]&={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}$
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x3 − 3x + d = 0, where $x$ is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.
Power-reduction formulae
Obtained by solving the second and third versions of the cosine double-angle formula.
Sine Cosine Other
$\sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}$ $\cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2}}$ $\sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos(4\theta )}{8}}$
$\sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4}}$ $\cos ^{3}\theta ={\frac {3\cos \theta +\cos(3\theta )}{4}}$ $\sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32}}$
$\sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}$ $\cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8}}$ $\sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos(4\theta )+\cos(8\theta )}{128}}$
$\sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16}}$ $\cos ^{5}\theta ={\frac {10\cos \theta +5\cos(3\theta )+\cos(5\theta )}{16}}$ $\sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin(2\theta )-5\sin(6\theta )+\sin(10\theta )}{512}}$
In general terms of powers of $\sin \theta $ or $\cos \theta $ the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem .
if n is ... $\cos ^{n}\theta $ $\sin ^{n}\theta $
n is odd $\cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}$ $\sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{\left({\frac {n-1}{2}}-k\right)}{\binom {n}{k}}\sin {{\big (}(n-2k)\theta {\big )}}$
n is even $\cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}$ $\sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{\left({\frac {n}{2}}-k\right)}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}$
Product-to-sum and sum-to-product identities
The product-to-sum identities[28] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations.[29] See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.
Product-to-sum identities
• $\cos \theta \,\cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2}$
• $\sin \theta \,\sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2}$
• $\sin \theta \,\cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2}$
• $\cos \theta \,\sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2}$
• $\tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}$
• $\tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )}}$
• ${\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{where }}e=(e_{1},\cdots ,e_{n})\in S=\{1,-1\}^{n}\end{aligned}}$
• $\prod _{k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{2^{n}}}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;{\text{if}}\;n\;{\text{is even}},\\\displaystyle \sum _{e\in S}\sin(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;{\text{if}}\;n\;{\text{is odd}}\end{cases}}$
Sum-to-product identities
The sum-to-product identities are as follows:[30]
• $\sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)$
• $\cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)$
• $\cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2}}\right)\sin \left({\frac {\theta -\varphi }{2}}\right)$
• $\tan \theta \pm \tan \varphi ={\frac {\sin(\theta \pm \varphi )}{\cos \theta \,\cos \varphi }}$
Hermite's cotangent identity
Main article: Hermite's cotangent identity
Charles Hermite demonstrated the following identity.[31] Suppose $a_{1},\ldots ,a_{n}$ are complex numbers, no two of which differ by an integer multiple of π. Let
$A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})$
(in particular, $A_{1,1},$ being an empty product, is 1). Then
$\cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).$
The simplest non-trivial example is the case n = 2:
$\cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).$
Finite products of trigonometric functions
For coprime integers n, m
$\prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)$
where Tn is the Chebyshev polynomial.
The following relationship holds for the sine function
$\prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.$
More generally for an integer n > 0[32]
$\sin(nx)=2^{n-1}\prod _{k=0}^{n-1}\sin \left({\frac {k}{n}}\pi +x\right)=2^{n-1}\prod _{k=1}^{n}\sin \left({\frac {k}{n}}\pi -x\right).$
or written in terms of the chord function $ \operatorname {crd} x\equiv 2\sin {\tfrac {1}{2}}x$,
$\operatorname {crd} (nx)=\prod _{k=1}^{n}\operatorname {crd} \left({\frac {k}{n}}2\pi -x\right).$
This comes from the factorization of the polynomial $ z^{n}-1$ into linear factors (cf. root of unity): For a point z on the complex unit circle and an integer n > 0,
$z^{n}-1=\prod _{k=1}^{n}z-\exp {\Bigl (}{\frac {k}{n}}2\pi i{\Bigr )}.$
Linear combinations
For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of $c$ and $\varphi $.
Sine and cosine
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[33][34]
$a\cos x+b\sin x=c\cos(x+\varphi )$
where $c$ and $\varphi $ are defined as so:
${\begin{aligned}c&=\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}},\\\varphi &=\arctan \left(-{\frac {b}{a}}\right),\end{aligned}}$
given that $a\neq 0.$
Arbitrary phase shift
More generally, for arbitrary phase shifts, we have
$a\sin(x+\theta _{a})+b\sin(x+\theta _{b})=c\sin(x+\varphi )$
where $c$ and $\varphi $ satisfy:
${\begin{aligned}c^{2}&=a^{2}+b^{2}+2ab\cos \left(\theta _{a}-\theta _{b}\right),\\\tan \varphi &={\frac {a\sin \theta _{a}+b\sin \theta _{b}}{a\cos \theta _{a}+b\cos \theta _{b}}}.\end{aligned}}$
More than two sinusoids
The general case reads[34]
$\sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),$
where
$a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})$
and
$\tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.$
Lagrange's trigonometric identities
These identities, named after Joseph Louis Lagrange, are:[35][36][37]
${\begin{aligned}\sum _{k=0}^{n}\sin k\theta &={\frac {\cos {\tfrac {1}{2}}\theta -\cos \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\\[5pt]\sum _{k=0}^{n}\cos k\theta &={\frac {\sin {\tfrac {1}{2}}\theta +\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{2\sin {\tfrac {1}{2}}\theta }}\end{aligned}}$
for $\theta \not \equiv 0{\pmod {2\pi }}.$
A related function is the Dirichlet kernel:
$D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos k\theta ={\frac {\sin \left(\left(n+{\tfrac {1}{2}}\right)\theta \right)}{\sin {\tfrac {1}{2}}\theta }}.$
A similar identity is[38]
$\sum _{k=1}^{n}\cos(2k-1)\alpha ={\frac {\sin(2n\alpha )}{2\sin \alpha }}.$
The proof is the following. By using the angle sum and difference identities,
$\sin(A+B)-\sin(A-B)=2\cos A\sin B.$
Then let's examine the following formula,
$2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\ldots +2\sin \alpha \cos(2n-1)\alpha $
and this formula can be written by using the above identity,
${\begin{aligned}&2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha \\&\quad =\sum _{k=1}^{n}(\sin(2k\alpha )-\sin(2(k-1)\alpha ))\\&\quad =(\sin 2\alpha -\sin 0)+(\sin 4\alpha -\sin 2\alpha )+(\sin 6\alpha -\sin 4\alpha )+\ldots +(\sin(2n\alpha )-\sin(2(n-1)\alpha ))\\&\quad =\sin(2n\alpha ).\end{aligned}}$
So, dividing this formula with $2\sin \alpha $ completes the proof.
Certain linear fractional transformations
If $f(x)$ is given by the linear fractional transformation
$f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},$
and similarly
$g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},$
then
$f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.$
More tersely stated, if for all $\alpha $ we let $f_{\alpha }$ be what we called $f$ above, then
$f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.$
If $x$ is the slope of a line, then $f(x)$ is the slope of its rotation through an angle of $-\alpha .$
Relation to the complex exponential function
Main article: Euler's formula
Euler's formula states that, for any real number x:[39]
$e^{ix}=\cos x+i\sin x,$
where i is the imaginary unit. Substituting −x for x gives us:
$e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.$
These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,[40][41]
$\cos x={\frac {e^{ix}+e^{-ix}}{2}}$
$\sin x={\frac {e^{ix}-e^{-ix}}{2i}}$
These formulae are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = eiθ eiφ means that
cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm.
Function Inverse function[42]
$\sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}$ $\arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)$
$\cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}$ $\arccos x=-i\,\ln \left(x+\,{\sqrt {x^{2}-1}}\right)$
$\tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}$ $\arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)$
$\csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}$ $\operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)$
$\sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}$ $\operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}}\right)$
$\cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}$ $\operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)$
$\operatorname {cis} \theta =e^{i\theta }$ $\operatorname {arccis} x=-i\ln x$
Infinite product formulae
For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[43][44]
${\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)^{2}}}\right)\\\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)^{2}}}\right)\end{aligned}}$
Infinite sums
Using the identity $\sum _{k=1}^{\infty }{\frac {x^{k}}{k}}=-\ln(1-x)$, and substituting in $x=e^{\pm it}$, one can derive
$\sum _{k=1}^{\infty }{\frac {\cos kt}{k}}=-{\frac {1}{2}}\ln(2-2\cos t)\quad \quad \sum _{k=1}^{\infty }{\frac {\sin kt}{k}}={\frac {\pi -t}{2}}$
for $t\in [0,2\pi ]$. Both sides being periodic with period $2\pi $. Integrating with respect to $t$ yields
$\sum _{k=1}^{\infty }{\frac {\sin kt}{k^{2}}}=-{\frac {1}{2}}\int _{0}^{t}\ln(2-2\cos x)dx\quad \quad \sum _{k=1}^{\infty }{\frac {\cos kt}{k^{2}}}={\frac {\pi ^{2}}{6}}-{\frac {\pi t}{2}}+{\frac {t^{2}}{4}}$
and so on.
Inverse trigonometric functions
Main article: Inverse trigonometric functions
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[45]
${\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}$
Taking the multiplicative inverse of both sides of the each equation above results in the equations for $\csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ and }}\cot ={\frac {1}{\tan }}.$ The right hand side of the formula above will always be flipped. For example, the equation for $\cot(\arcsin x)$ is:
$\cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}$
while the equations for $\csc(\arccos x)$ and $\sec(\arccos x)$ are:
$\csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}\qquad {\text{ and }}\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.$
The following identities are implied by the reflection identities. They hold whenever $x,r,s,-x,-r,{\text{ and }}-s$ are in the domains of the relevant functions.
${\begin{alignedat}{9}{\frac {\pi }{2}}~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat}}$
Also,[46]
${\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{if }}x>0\\-{\frac {\pi }{2}},&{\text{if }}x<0\end{cases}}\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{if }}x>0\\{\frac {3\pi }{2}},&{\text{if }}x<0\end{cases}}\\\end{aligned}}$
$\arccos {\frac {1}{x}}=\operatorname {arcsec} x\qquad {\text{ and }}\qquad \operatorname {arcsec} {\frac {1}{x}}=\arccos x$
$\arcsin {\frac {1}{x}}=\operatorname {arccsc} x\qquad {\text{ and }}\qquad \operatorname {arccsc} {\frac {1}{x}}=\arcsin x$
The arctangent function can be expanded as a series:[47]
$\arctan(nx)=\sum _{m=1}^{n}\arctan {\frac {x}{1+(m-1)mx^{2}}}$
Identities without variables
In terms of the arctangent function we have[46]
$\arctan {\frac {1}{2}}=\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.$
The curious identity known as Morrie's law,
$\cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},$
is a special case of an identity that contains one variable:
$\prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.$
Similarly,
$\sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}$
is a special case of an identity with $x=20^{\circ }$:
$\sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.$
For the case $x=15^{\circ }$,
${\begin{aligned}\sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\sin 15^{\circ }\cdot \sin 75^{\circ }&={\frac {1}{4}}.\end{aligned}}$
For the case $x=10^{\circ }$,
$\sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8}}.$
The same cosine identity is
$\cos x\cdot \cos \left(60^{\circ }-x\right)\cdot \cos \left(60^{\circ }+x\right)={\frac {\cos 3x}{4}}.$
Similarly,
${\begin{aligned}\cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }&={\frac {\sqrt {3}}{8}},\\\cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }&={\frac {\sqrt {2}}{8}},\\\cos 15^{\circ }\cdot \cos 75^{\circ }&={\frac {1}{4}}.\end{aligned}}$
Similarly,
${\begin{aligned}\tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }&=\tan 80^{\circ },\\\tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }&=\tan 10^{\circ }.\end{aligned}}$
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
$\cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.$
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
$\cos {\frac {2\pi }{21}}+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.$
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Other cosine identities include:[48]
${\begin{aligned}2\cos {\frac {\pi }{3}}&=1,\\2\cos {\frac {\pi }{5}}\times 2\cos {\frac {2\pi }{5}}&=1,\\2\cos {\frac {\pi }{7}}\times 2\cos {\frac {2\pi }{7}}\times 2\cos {\frac {3\pi }{7}}&=1,\end{aligned}}$
and so forth for all odd numbers, and hence
$\cos {\frac {\pi }{3}}+\cos {\frac {\pi }{5}}\times \cos {\frac {2\pi }{5}}+\cos {\frac {\pi }{7}}\times \cos {\frac {2\pi }{7}}\times \cos {\frac {3\pi }{7}}+\dots =1.$
Many of those curious identities stem from more general facts like the following:[49]
$\prod _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}$
and
$\prod _{k=1}^{n-1}\cos {\frac {k\pi }{n}}={\frac {\sin {\frac {\pi n}{2}}}{2^{n-1}}}.$
Combining these gives us
$\prod _{k=1}^{n-1}\tan {\frac {k\pi }{n}}={\frac {n}{\sin {\frac {\pi n}{2}}}}$
If n is an odd number ($n=2m+1$) we can make use of the symmetries to get
$\prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}$
The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:
$\prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}$
Computing π
An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula:
${\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}$
or, alternatively, by using an identity of Leonhard Euler:
${\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}$
or by using Pythagorean triples:
$\pi =\arccos {\frac {4}{5}}+\arccos {\frac {5}{13}}+\arccos {\frac {16}{65}}=\arcsin {\frac {3}{5}}+\arcsin {\frac {12}{13}}+\arcsin {\frac {63}{65}}.$
Others include:[50][46]
${\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}},$
$\pi =\arctan 1+\arctan 2+\arctan 3,$
${\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}.$
Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = Σn−1
k=1
arctan tk ∈ (π/4, 3π/4)
, let tn = tan(π/2 − θn) = cot θn. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1). In particular, the computed tn will be rational whenever all the t1, ..., tn−1 values are rational. With these values,
${\begin{aligned}{\frac {\pi }{2}}&=\sum _{k=1}^{n}\arctan(t_{k})\\\pi &=\sum _{k=1}^{n}\operatorname {sgn}(t_{k})\arccos \left({\frac {1-t_{k}^{2}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arcsin \left({\frac {2t_{k}}{1+t_{k}^{2}}}\right)\\\pi &=\sum _{k=1}^{n}\arctan \left({\frac {2t_{k}}{1-t_{k}^{2}}}\right)\,,\end{aligned}}$
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the tk values is not within (−1, 1). Note that if t = p/q is rational, then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2 − p2, q2 + p2).
For example, for n = 3 terms,
${\frac {\pi }{2}}=\arctan \left({\frac {a}{b}}\right)+\arctan \left({\frac {c}{d}}\right)+\arctan \left({\frac {bd-ac}{ad+bc}}\right)$
for any a, b, c, d > 0.
An identity of Euclid
Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
$\sin ^{2}18^{\circ }+\sin ^{2}30^{\circ }=\sin ^{2}36^{\circ }.$
Ptolemy used this proposition to compute some angles in his table of chords in Book I, chapter 11 of Almagest.
Composition of trigonometric functions
These identities involve a trigonometric function of a trigonometric function:[51]
• $\cos(t\sin x)=J_{0}(t)+2\sum _{k=1}^{\infty }J_{2k}(t)\cos(2kx)$
• $\sin(t\sin x)=2\sum _{k=0}^{\infty }J_{2k+1}(t)\sin {\big (}(2k+1)x{\big )}$
• $\cos(t\cos x)=J_{0}(t)+2\sum _{k=1}^{\infty }(-1)^{k}J_{2k}(t)\cos(2kx)$
• $\sin(t\cos x)=2\sum _{k=0}^{\infty }(-1)^{k}J_{2k+1}(t)\cos {\big (}(2k+1)x{\big )}$
where Ji are Bessel functions.
Further "conditional" identities for the case α + β + γ = 180°
The following formulae apply to arbitrary plane triangles and follow from $\alpha +\beta +\gamma =180^{\circ },$ as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).
${\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\1&=\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta \\\cot \left({\frac {\alpha }{2}}\right)+\cot \left({\frac {\beta }{2}}\right)+\cot \left({\frac {\gamma }{2}}\right)&=\cot \left({\frac {\alpha }{2}}\right)\cot \left({\frac {\beta }{2}}\right)\cot \left({\frac {\gamma }{2}}\right)\\1&=\tan \left({\frac {\beta }{2}}\right)\tan \left({\frac {\gamma }{2}}\right)+\tan \left({\frac {\gamma }{2}}\right)\tan \left({\frac {\alpha }{2}}\right)+\tan \left({\frac {\alpha }{2}}\right)\tan \left({\frac {\beta }{2}}\right)\\\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)\\\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\sin \left({\frac {\beta }{2}}\right)\sin \left({\frac {\gamma }{2}}\right)+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)-1\\\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \sin \beta \sin \gamma \\-\sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )&=4\sin \alpha \cos \beta \cos \gamma \\\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos(2\alpha )+\cos(2\beta )+\cos(2\gamma )&=-4\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \\\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\\\sin ^{2}(2\alpha )+\sin ^{2}(2\beta )+\sin ^{2}(2\gamma )&=-2\cos(2\alpha )\cos(2\beta )\cos(2\gamma )+2\\\cos ^{2}(2\alpha )+\cos ^{2}(2\beta )+\cos ^{2}(2\gamma )&=2\cos(2\alpha )\,\cos(2\beta )\,\cos(2\gamma )+1\\1&=\sin ^{2}\left({\frac {\alpha }{2}}\right)+\sin ^{2}\left({\frac {\beta }{2}}\right)+\sin ^{2}\left({\frac {\gamma }{2}}\right)+2\sin \left({\frac {\alpha }{2}}\right)\,\sin \left({\frac {\beta }{2}}\right)\,\sin \left({\frac {\gamma }{2}}\right)\end{aligned}}$
Historical shorthands
Main articles: Versine and Exsecant
The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.
Miscellaneous
Relationship between all trigonometric ratios
The following identities each give a relationship between all the trigonometric ratios.
$(\sin \theta +\csc \theta )^{2}+(\cos \theta +\sec \theta )^{2}-(\tan \theta +\cot \theta )^{2}=5$
$(\sin \theta +\csc \theta )^{2}+(\cos \theta +\sec \theta )^{2}-(\tan \theta -\cot \theta )^{2}=9$
Similarly,
$(\sin \theta +\csc \theta )^{2}+(\cos \theta +\sec \theta )^{2}=\tan ^{2}\theta +\cot ^{2}\theta +7$
Dirichlet kernel
Main article: Dirichlet kernel
The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:
$1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.$
The convolution of any integrable function of period $2\pi $ with the Dirichlet kernel coincides with the function's $n$th-degree Fourier approximation. The same holds for any measure or generalized function.
Tangent half-angle substitution
Main article: Tangent half-angle substitution
If we set
$t=\tan {\frac {x}{2}},$
then[52]
$\sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}}$
where $e^{ix}=\cos x+i\sin x,$ sometimes abbreviated to cis x.
When this substitution of $t$ for tan x/2 is used in calculus, it follows that $\sin x$ is replaced by 2t/1 + t2, $\cos x$ is replaced by 1 − t2/1 + t2 and the differential dx is replaced by 2 dt/1 + t2. Thereby one converts rational functions of $\sin x$ and $\cos x$ to rational functions of $t$ in order to find their antiderivatives.
Viète's infinite product
See also: Viète's formula and Sinc function
$\cos {\frac {\theta }{2}}\cdot \cos {\frac {\theta }{4}}\cdot \cos {\frac {\theta }{8}}\cdots =\prod _{n=1}^{\infty }\cos {\frac {\theta }{2^{n}}}={\frac {\sin \theta }{\theta }}=\operatorname {sinc} \theta .$
See also
• Aristarchus's inequality
• Derivatives of trigonometric functions
• Exact trigonometric values (values of sine and cosine expressed in surds)
• Exsecant
• Half-side formula
• Hyperbolic function
• Laws for solution of triangles:
• Law of cosines
• Spherical law of cosines
• Law of sines
• Law of tangents
• Law of cotangents
• Mollweide's formula
• List of integrals of trigonometric functions
• Mnemonics in trigonometry
• Pentagramma mirificum
• Proofs of trigonometric identities
• Prosthaphaeresis
• Pythagorean theorem
• Tangent half-angle formula
• Trigonometric number
• Trigonometry
• Trigonometric constants expressed in real radicals
• Uses of trigonometry
• Versine and haversine
References
1. Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 4, eqn 4.3.45". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 73. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
2. Selby 1970, p. 188
3. Abramowitz and Stegun, p. 72, 4.3.13–15
4. Abramowitz and Stegun, p. 72, 4.3.7–9
5. Abramowitz and Stegun, p. 72, 4.3.16
6. Weisstein, Eric W. "Trigonometric Addition Formulas". MathWorld.
7. Abramowitz and Stegun, p. 72, 4.3.17
8. Abramowitz and Stegun, p. 72, 4.3.18
9. "Angle Sum and Difference Identities". www.milefoot.com. Retrieved 2019-10-12.
10. Abramowitz and Stegun, p. 72, 4.3.19
11. Abramowitz and Stegun, p. 80, 4.4.32
12. Abramowitz and Stegun, p. 80, 4.4.33
13. Abramowitz and Stegun, p. 80, 4.4.34
14. Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC '89 (Portland US-OR, 1989-07). New York: ACM. pp. 207–211. doi:10.1145/74540.74566. ISBN 0-89791-325-6.
15. Michael Hardy (August–September 2016). "On Tangents and Secants of Infinite Sums". American Mathematical Monthly. 123 (7): 701–703. doi:10.4169/amer.math.monthly.123.7.701. S2CID 126310545.
16. "Sine, Cosine, and Ptolemy's Theorem".
17. "Sine, Cosine, and Ptolemy's Theorem".
18. Weisstein, Eric W. "Multiple-Angle Formulas". MathWorld.
19. Abramowitz and Stegun, p. 74, 4.3.48
20. Selby 1970, pg. 190
21. Weisstein, Eric W. "Multiple-Angle Formulas". mathworld.wolfram.com. Retrieved 2022-02-06.
22. Ward, Ken. "Multiple angles recursive formula". Ken Ward's Mathematics Pages.
23. Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 4, eqn 4.3.20-22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 72. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
24. Weisstein, Eric W. "Half-Angle Formulas". MathWorld.
25. Abramowitz and Stegun, p. 72, 4.3.24–26
26. Weisstein, Eric W. "Double-Angle Formulas". MathWorld.
27. Abramowitz and Stegun, p. 72, 4.3.27–28
28. Abramowitz and Stegun, p. 72, 4.3.31–33
29. Eves, Howard (1990). An introduction to the history of mathematics (6th ed.). Philadelphia: Saunders College Pub. p. 309. ISBN 0-03-029558-0. OCLC 20842510.
30. Abramowitz and Stegun, p. 72, 4.3.34–39
31. Johnson, Warren P. (Apr 2010). "Trigonometric Identities à la Hermite". American Mathematical Monthly. 117 (4): 311–327. doi:10.4169/000298910x480784. S2CID 29690311.
32. "Product Identity Multiple Angle".
33. Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.
34. Weisstein, Eric W. "Harmonic Addition Theorem". MathWorld.
35. Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". American Journal of Physics. 21 (2): 140. Bibcode:1953AmJPh..21..140M. doi:10.1119/1.1933371.
36. Agarwal, Ravi P.; O'Regan, Donal (2008). Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems (illustrated ed.). Springer Science & Business Media. p. 185. ISBN 978-0-387-79146-3. Extract of page 185
37. Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". Handbook of Mathematical Formulas and Integrals (4th ed.). Academic Press. ISBN 978-0-12-374288-9.
38. Fay, Temple H.; Kloppers, P. Hendrik (2001). "The Gibbs' phenomenon". International Journal of Mathematical Education in Science and Technology. 32 (1): 73–89. doi:10.1080/00207390117151.
39. Abramowitz and Stegun, p. 74, 4.3.47
40. Abramowitz and Stegun, p. 71, 4.3.2
41. Abramowitz and Stegun, p. 71, 4.3.1
42. Abramowitz and Stegun, p. 80, 4.4.26–31
43. Abramowitz and Stegun, p. 75, 4.3.89–90
44. Abramowitz and Stegun, p. 85, 4.5.68–69
45. Abramowitz & Stegun 1972, p. 73, 4.3.45
46. Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
47. S. M. Abrarov, R. K. Jagpal, R. Siddiqui and B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", Mathematics, 9 (17), 2162, doi:10.3390/math9172162{{citation}}: CS1 maint: multiple names: authors list (link)
48. Humble, Steve (Nov 2004). "Grandma's identity". Mathematical Gazette. 88: 524–525. doi:10.1017/s0025557200176223. S2CID 125105552.
49. Weisstein, Eric W. "Sine". MathWorld.
50. Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, Proofs Without Words (1993, Mathematical Association of America), p. 39.
51. Milton Abramowitz and Irene Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972, formulae 9.1.42–9.1.45
52. Abramowitz and Stegun, p. 72, 4.3.23
Bibliography
• Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. ISBN 978-0-486-61272-0.
• Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York: Barnes & Noble, LCCN 61-9103
• Selby, Samuel M., ed. (1970), Standard Mathematical Tables (18th ed.), The Chemical Rubber Co.
External links
• Values of sin and cos, expressed in surds, for integer multiples of 3° and of 5+5/8°, and for the same angles csc and sec and tan
• Complete List of Trigonometric Formulas
| Wikipedia |
Fourier series
A Fourier series (/ˈfʊrieɪ, -iər/[1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series.[2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.
"Fourier's theorem" redirects here. For the number of real roots of a polynomial, see Budan's theorem § Fourier's theorem.
Fourier transforms
• Fourier transform
• Fourier series
• Discrete-time Fourier transform
• Discrete Fourier transform
• Discrete Fourier transform over a ring
• Fourier transform on finite groups
• Fourier analysis
• Related transforms
The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of a square wave.
• A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).
• The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums converge to (become more and more like) the square wave.
• Function $s_{6}(x)$ (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform $S(f)$ is a frequency-domain representation that reveals the amplitudes of the summed sine waves.
Fourier series are closely related to the Fourier transform, which can be used to find the frequency information for functions that are not periodic. Periodic functions can be identified with functions on a circle, for this reason Fourier series are the subject of Fourier analysis on a circle, usually denoted as $\mathbb {T} $ or $S_{1}$. The Fourier transform is also part of Fourier analysis, but is defined for functions on $\mathbb {R} ^{n}$.
Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.
Common forms of the Fourier series
The Fourier series can be represented in different forms. The sine-cosine form, exponential form, and amplitude-phase form are expressed here for a periodic function $s(x)$ with periodicity of P.
Sine-cosine form
The Fourier series coefficients[3] are defined by the integrals:
Fourier series coefficients
${\begin{aligned}A_{0}&={\frac {1}{P}}\int _{-P/2}^{P/2}s(x)\,dx\\A_{n}&={\frac {2}{P}}\int _{-P/2}^{P/2}s(x)\cos \left({\frac {2\pi nx}{P}}\right)\,dx\qquad {\text{for }}n\geq 1\qquad \\B_{n}&={\frac {2}{P}}\int _{-P/2}^{P/2}s(x)\sin \left({\frac {2\pi nx}{P}}\right)\,dx\qquad {\text{for }}n\geq 1\end{aligned}}$
(Eq. 1)
It is notable that $A_{0}$ is the average value of the function $s(x)$. This is a property that extends to similar transforms such as the Fourier transform.[upper-alpha 1]
With these coefficients defined the Fourier series is:
Fourier series
$s(x)\sim A_{0}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left({\frac {2\pi nx}{P}}\right)+B_{n}\sin \left({\frac {2\pi nx}{P}}\right)\right)$
(Eq. 2)
Many others use the $\sim $ symbol, because it is not always true that the sum of the Fourier series is equal to $s(x)$. It can fail to converge entirely, or converge to something that differs from $s(x)$. While these situations can occur, their differences are rarely a problem in science and engineering, and authors in these disciplines will sometimes write Eq. 2 with $\sim $ replaced by $=$.
The integer index $n$ in the Fourier series coefficients is the number of cycles the corresponding $\cos $ or $\sin $ from the series make in the function's period $P$. Therefore the terms corresponding to $A_{n}$ and $B_{n}$ have:
• a wavelength equal to ${\tfrac {P}{n}}$ and having the same units as $x$.
• a frequency equal to ${\tfrac {n}{P}}$ and having reciprocal units as $x$.
Example
Consider a sawtooth function:
$s(x)={\frac {x}{\pi }},\quad \mathrm {for} -\pi <x<\pi ,$
$s(x+2\pi k)=s(x),\quad \mathrm {for} -\pi <x<\pi {\text{ and }}k\in \mathbb {Z} .$
In this case, the Fourier coefficients are given by
${\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\cos(nx)\,dx=0,\quad n\geq 0.\\[4pt]B_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\[4pt]&=-{\frac {2}{\pi n}}\cos(n\pi )+{\frac {2}{\pi ^{2}n^{2}}}\sin(n\pi )\\[4pt]&={\frac {2\,(-1)^{n+1}}{\pi n}},\quad n\geq 1.\end{aligned}}$
It can be shown that the Fourier series converges to $s(x)$ at every point $x$ where $s$ is differentiable, and therefore:
${\begin{aligned}s(x)&=A_{0}+\sum _{n=1}^{\infty }\left(A_{n}\cos \left(nx\right)+B_{n}\sin \left(nx\right)\right)\\[4pt]&={\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin(nx),\quad \mathrm {for} \quad x-\pi \notin 2\pi \mathbb {Z} .\end{aligned}}$
(Eq.8)
When $x$ is an odd multiple of $\pi $, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of s at $x=\pi $. This is a particular instance of the Dirichlet theorem for Fourier series.
This example leads to a solution of the Basel problem.
Exponential form
It is possible to simplify the integrals for the Fourier series coefficients by using Euler's formula.
With the definitions
Complex Fourier series coefficients
${\begin{aligned}c_{0}&=A_{0}&\\c_{n}&=(A_{n}-iB_{n})/2\qquad &{\text{for }}n>0\\c_{n}&=(A_{-n}+iB_{-n})/2\qquad &{\text{for }}n<0\end{aligned}}$
(Eq. 3)
By substituting equation Eq. 1 into Eq. 3 it can be shown that:[4]
Complex Fourier series coefficients
$c_{n}={\frac {1}{P}}\int _{-P/2}^{P/2}s(x)e^{-{\frac {2\pi inx}{P}}}\,dx\qquad {\text{for all integers}}~n$
Given the complex Fourier series coefficients, it is possible to recover $A_{n}$ and $B_{n}$ from the formulas
Complex Fourier series coefficients
${\begin{aligned}A_{0}&=c_{0}&\\A_{n}&=c_{n}+c_{-n}\qquad &{\textrm {for}}~n>0\\B_{n}&=i(c_{n}-c_{-n})\qquad &{\textrm {for}}~n>0\end{aligned}}$
With these definitions the Fourier series is written as:
Fourier series, exponential form
$s(x)=\sum _{n=-\infty }^{\infty }c_{n}\cdot e^{\frac {2\pi inx}{P}}$
(Eq. 4)
This is the customary form for generalizing to complex-valued functions. Negative values of $n$ correspond to negative frequency. (Also see Fourier transform § Negative frequency).
Amplitude-phase form
The Fourier series in amplitude-phase form is:
Fourier series, amplitude-phase form
$s(x)\sim {\frac {A_{0}}{2}}+\sum _{n=1}^{\infty }A_{n}\cdot \cos \left(2\pi {\tfrac {n}{P}}x-\varphi _{n}\right)$
(Eq. 5)
• Its $n^{\text{th}}$ harmonic is $A_{n}\cdot \cos \left(2\pi {\tfrac {n}{P}}x-\varphi _{n}\right)$.
• $A_{n}$ is the $n^{\text{th}}$ harmonic's amplitude and $\varphi _{n}$ is its phase shift.
• The fundamental frequency of $s_{\scriptscriptstyle N}(x)$ is the term for when $n$ equals 1, and can be referred to as the $1^{\text{st}}$ harmonic.
• ${\tfrac {A_{0}}{2}}$ is sometimes called the $0^{\text{th}}$ harmonic or DC component. It is the mean value of $s(x)$.
Clearly Eq. 5 can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing, for those not yet familiar with this concept, is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the potentially infinite number of terms ($N$).
The coefficients $A_{n}$ and $\varphi _{n}$ can be understood and derived in terms of the cross-correlation between $s(x)$ and a sinusoid at frequency ${\tfrac {n}{P}}$. For a general frequency $f,$ and an analysis interval $[x_{0},x_{0}+P],$ the cross-correlation function:
$\mathrm {X} _{f}(\tau )={\tfrac {2}{P}}\int _{P}s(x)\cdot \cos \left(2\pi f(x-\tau )\right)\,dx;\quad \tau \in \left[0,{\tfrac {2\pi }{f}}\right]$
(Eq. 6)
is essentially a matched filter, with template $\cos(2\pi fx)$.[upper-alpha 2] Here $\int _{P}$ denotes $\int _{x_{0}}^{x_{0}+P}.$ If $s(x)$ is $P$-periodic, $x_{0}$ is arbitrary, often chosen to be $0$ or $-{\tfrac {P}{2}}.$ But in general, the Fourier series can also be used to represent a non-periodic function on just a finite interval, as depicted in Fig.1.
The maximum of $\mathrm {X} _{f}(\tau )$ is a measure of the amplitude $(a)$ of frequency $f$ in the function $s(x)$, and the value of $\tau $ at the maximum determines the phase $(\varphi )$ of that frequency. Figure 2 is an example, where $s(x)$ is a square wave (not shown), and frequency $f$ is the $4^{\text{th}}$ harmonic. It is also an example of deriving the maximum from just two samples, instead of searching the entire function. That is made possible by a trigonometric identity:
Equivalence of polar and rectangular forms
$\cos \left(2\pi {\tfrac {n}{P}}x-\varphi _{n}\right)\ \equiv \ \cos(\varphi _{n})\cdot \cos \left(2\pi {\tfrac {n}{P}}x\right)+\sin(\varphi _{n})\cdot \sin \left(2\pi {\tfrac {n}{P}}x\right)$
(Eq. 7)
Combining this with Eq. 6 gives:
${\begin{aligned}\mathrm {X} _{n}(\varphi )&={\tfrac {2}{P}}\int _{P}s(x)\cdot \cos \left(2\pi {\tfrac {n}{P}}x-\varphi \right)\,dx;\quad \varphi \in [0,2\pi ]\\&=\cos(\varphi )\cdot \underbrace {{\tfrac {2}{P}}\int _{P}s(x)\cdot \cos \left(2\pi {\tfrac {n}{P}}x\right)\,dx} _{A_{n}}+\sin(\varphi )\cdot \underbrace {{\tfrac {2}{P}}\int _{P}s(x)\cdot \sin \left(2\pi {\tfrac {n}{P}}x\right)\,dx} _{B_{n}}\\&=\cos(\varphi )\cdot A_{n}+\sin(\varphi )\cdot B_{n}\end{aligned}}$
which introduces the definitions of $A_{n}$ and $B_{n}$.[5] And we note for later reference that $a_{0}$ and $b_{0}$ can be simplified:
$A_{0}={\tfrac {2}{P}}\int _{P}s(x)dx~,\quad B_{0}=0~.$
The derivative of $\mathrm {X} _{n}(\varphi )$ is zero at the phase of maximum correlation.
$\mathrm {X} '_{n}(\varphi _{n})=\sin(\varphi _{n})\cdot A_{n}-\cos(\varphi _{n})\cdot B_{n}=0\quad \longrightarrow \quad \tan(\varphi _{n})={\frac {B_{n}}{A_{n}}}\quad \longrightarrow \quad \varphi _{n}=\arctan(B_{n},A_{n})$
And the correlation peak value is:
${\begin{aligned}a_{n}=\mathrm {X} _{n}(\varphi _{n})\ &=\cos(\varphi _{n})\cdot A_{n}+\sin(\varphi _{n})\cdot B_{n}\\&={\frac {A_{n}}{\sqrt {A_{n}^{2}+B_{n}^{2}}}}\cdot A_{n}+{\frac {B_{n}}{\sqrt {A_{n}^{2}+B_{n}^{2}}}}\cdot B_{n}={\frac {A_{n}^{2}+B_{n}^{2}}{\sqrt {A_{n}^{2}+B_{n}^{2}}}}&={\sqrt {A_{n}^{2}+B_{n}^{2}}}.\end{aligned}}$
Therefore $A_{n}$ and $B_{n}$ are the rectangular coordinates of a vector with polar coordinates $a_{n}$ and $\varphi _{n}.$
Extensions to non-periodic functions
Fourier series can also be applied to functions that are not necessarily periodic. The simplest extension occurs when the function $s(x)$ is defined only in a fixed interval $[x_{0},x_{0}+P]$. In this case the integrals defining the Fourier coefficients can be taken over this interval. In this case all of the convergence results will be the same as for the periodic extension of $s(x)$ to the whole real line. In particular, it may happen that for a continuous function $s(x)$ there is a discontinuity in the periodic extension of at $x_{0}$ and $x_{0}+P$. In this case, it is possible to see Gibbs phenomenon at the end points of the interval.
For functions which have compact support, meaning that values of $s(x)$ are defined everywhere but identically zero outside some fixed interval $[x_{0},x_{0}+P]$, the Fourier series can be taken on any interval containing the support $[x_{0},x_{0}+P]$.
For both the cases above, it is sometimes desirable to take an even or odd reflection of the function, or extend it by zero in the case the function is only defined on a finite interval. This allows one to prescribe desired properties for the Fourier coefficients. For example, by making the function even you ensure $B_{n}=0$. This is often known as a cosine series. One may similarly arrive at a sine series.
In the case where the function doesn't have compact support and is defined on entire real line, one can use the Fourier transform. Fourier series can be taken for a truncated version of the function or to the periodic summation.
Partial Sum Operator
Frequently when describing how Fourier series behave, authors introduce the partial sum operator $S_{n}$ for a function $f(x)$.[6]
$S_{N}(f)=\sum _{n=-N}^{N}c_{n}e^{\frac {2\pi inx}{P}}$
(Eq. 8)
Where $c_{n}$ are the Fourier coefficients of $f$. Unlike series in calculus, it is important that the partial sums are taken symmetrically for Fourier series, otherwise convergence results may not hold.
Convergence
Main article: Convergence of Fourier series
A proof that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions) is overviewed in § Fourier theorem proving convergence of Fourier series.
In engineering applications, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are better-behaved than functions encountered in other disciplines. In particular, if $s$ is continuous and the derivative of $s(x)$ (which may not exist everywhere) is square integrable, then the Fourier series of $s$ converges absolutely and uniformly to $s(x)$.[7] If a function is square-integrable on the interval $[x_{0},x_{0}+P]$, then the Fourier series converges to the function at almost everywhere. It is possible to define Fourier coefficients for more general functions or distributions, in which case point wise convergence often fails, and convergence in norm or weak convergence is usually studied.
• Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square wave improves as the number of terms increases (animation)
• Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a sawtooth wave improves as the number of terms increases (animation)
• Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections.
Other common notations
The notation $c_{n}$ is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function ($s$, in this case), such as ${\hat {s}}[n]$ or $S[n]$, and functional notation often replaces subscripting:
${\begin{aligned}s(x)&=\sum _{n=-\infty }^{\infty }{\hat {s}}(n)\cdot e^{2\pi inx/P}&&{\text{common mathematics notation}}\\&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i\,2\pi nx/P}&&{\text{common engineering notation}}\end{aligned}}$
In engineering, particularly when the variable $x$ represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
$S(f)\ \triangleq \ \sum _{n=-\infty }^{\infty }S[n]\cdot \delta \left(f-{\frac {n}{P}}\right),$
where $f$ represents a continuous frequency domain. When variable $x$ has units of seconds, $f$ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of ${\tfrac {1}{P}}$, which is called the fundamental frequency. $s_{\infty }(x)$ can be recovered from this representation by an inverse Fourier transform:
${\begin{aligned}{\mathcal {F}}^{-1}\{S(f)\}&=\int _{-\infty }^{\infty }\left(\sum _{n=-\infty }^{\infty }S[n]\cdot \delta \left(f-{\frac {n}{P}}\right)\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot \int _{-\infty }^{\infty }\delta \left(f-{\frac {n}{P}}\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i\,2\pi nx/P}\ \ \triangleq \ s_{\infty }(x).\end{aligned}}$
The constructed function $S(f)$ is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.[upper-alpha 3]
History
See also: Fourier analysis § History
The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.[upper-alpha 4] Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous[8] and later generalized to any piecewise-smooth[9]) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy.[10] Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles.
The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.
From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet[11] and Bernhard Riemann[12][13][14] expressed Fourier's results with greater precision and formality.
Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics,[15] shell theory,[16] etc.
Beginnings
Joseph Fourier wrote:
$\varphi (y)=a_{0}\cos {\frac {\pi y}{2}}+a_{1}\cos 3{\frac {\pi y}{2}}+a_{2}\cos 5{\frac {\pi y}{2}}+\cdots .$
Multiplying both sides by $\cos(2k+1){\frac {\pi y}{2}}$, and then integrating from $y=-1$ to $y=+1$ yields:
$a_{k}=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy.$
— Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides. (1807)[17][upper-alpha 5]
This immediately gives any coefficient ak of the trigonometrical series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral
${\begin{aligned}a_{k}&=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy\\&=\int _{-1}^{1}\left(a\cos {\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+a'\cos 3{\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+\cdots \right)\,dy\end{aligned}}$
can be carried out term-by-term. But all terms involving $\cos(2j+1){\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}$ for j ≠ k vanish when integrated from −1 to 1, leaving only the $k^{\text{th}}$ term.
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
Fourier's motivation
The Fourier series expansion of the sawtooth function (above) looks more complicated than the simple formula $s(x)={\tfrac {x}{\pi }}$, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose sides measure $\pi $ meters, with coordinates $(x,y)\in [0,\pi ]\times [0,\pi ]$. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by $y=\pi $, is maintained at the temperature gradient $T(x,\pi )=x$ degrees Celsius, for $x$ in $(0,\pi )$, then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by
$T(x,y)=2\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin(nx){\sinh(ny) \over \sinh(n\pi )}.$
Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of Eq.6 by $\sinh(ny)/\sinh(n\pi )$. While our example function $s(x)$ seems to have a needlessly complicated Fourier series, the heat distribution $T(x,y)$ is nontrivial. The function $T$ cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.
Complex Fourier series animation
An example of the ability of the complex Fourier series to trace any two dimensional closed figure is shown in the adjacent animation of the complex Fourier series tracing the letter 'e' (for exponential). Note that the animation uses the variable 't' to parameterize the letter 'e' in the complex plane, which is equivalent to using the parameter 'x' in this article's subsection on complex valued functions.
In the animation's back plane, the rotating vectors are aggregated in an order that alternates between a vector rotating in the positive (counter clockwise) direction and a vector rotating at the same frequency but in the negative (clockwise) direction, resulting in a single tracing arm with lots of zigzags. This perspective shows how the addition of each pair of rotating vectors (one rotating in the positive direction and one rotating in the negative direction) nudges the previous trace (shown as a light gray dotted line) closer to the shape of the letter 'e'.
In the animation's front plane, the rotating vectors are aggregated into two sets, the set of all the positive rotating vectors and the set of all the negative rotating vectors (the non-rotating component is evenly split between the two), resulting in two tracing arms rotating in opposite directions. The animation's small circle denotes the midpoint between the two arms and also the midpoint between the origin and the current tracing point denoted by '+'. This perspective shows how the complex Fourier series is an extension (the addition of an arm) of the complex geometric series which has just one arm. It also shows how the two arms coordinate with each other. For example, as the tracing point is rotating in the positive direction, the negative direction arm stays parked. Similarly, when the tracing point is rotating in the negative direction, the positive direction arm stays parked.
In between the animation's back and front planes are rotating trapezoids whose areas represent the values of the complex Fourier series terms. This perspective shows the amplitude, frequency, and phase of the individual terms of the complex Fourier series in relation to the series sum spatially converging to the letter 'e' in the back and front planes. The audio track's left and right channels correspond respectively to the real and imaginary components of the current tracing point '+' but increased in frequency by a factor of 3536 so that the animation's fundamental frequency (n=1) is a 220 Hz tone (A220).
Other applications
Another application is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.
Table of common Fourier series
Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.
• $s(x)$ designates a periodic function with period $P$.
• $A_{0},A_{n},B_{n}$ designate the Fourier series coefficients (sine-cosine form) of the periodic function $s(x)$.
Time domain
$s(x)$
Plot Frequency domain (sine-cosine form)
${\begin{aligned}&A_{0}\\&A_{n}\quad {\text{for }}n\geq 1\\&B_{n}\quad {\text{for }}n\geq 1\end{aligned}}$
Remarks Reference
$s(x)=A\left|\sin \left({\frac {2\pi }{P}}x\right)\right|\quad {\text{for }}0\leq x<P$ ${\begin{aligned}A_{0}=&{\frac {2A}{\pi }}\\A_{n}=&{\begin{cases}{\frac {-4A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\B_{n}=&0\\\end{aligned}}$ Full-wave rectified sine [19]: p. 193
$s(x)={\begin{cases}A\sin \left({\frac {2\pi }{P}}x\right)&\quad {\text{for }}0\leq x<P/2\\0&\quad {\text{for }}P/2\leq x<P\\\end{cases}}$ ${\begin{aligned}A_{0}=&{\frac {A}{\pi }}\\A_{n}=&{\begin{cases}{\frac {-2A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\B_{n}=&{\begin{cases}{\frac {A}{2}}&\quad n=1\\0&\quad n>1\end{cases}}\\\end{aligned}}$ Half-wave rectified sine [19]: p. 193
$s(x)={\begin{cases}A&\quad {\text{for }}0\leq x<D\cdot P\\0&\quad {\text{for }}D\cdot P\leq x<P\\\end{cases}}$ ${\begin{aligned}A_{0}=&AD\\A_{n}=&{\frac {A}{n\pi }}\sin \left(2\pi nD\right)\\B_{n}=&{\frac {2A}{n\pi }}\left(\sin \left(\pi nD\right)\right)^{2}\\\end{aligned}}$ $0\leq D\leq 1$
$s(x)={\frac {Ax}{P}}\quad {\text{for }}0\leq x<P$ ${\begin{aligned}A_{0}=&{\frac {A}{2}}\\A_{n}=&0\\B_{n}=&{\frac {-A}{n\pi }}\\\end{aligned}}$ [19]: p. 192
$s(x)=A-{\frac {Ax}{P}}\quad {\text{for }}0\leq x<P$ ${\begin{aligned}A_{0}=&{\frac {A}{2}}\\A_{n}=&0\\B_{n}=&{\frac {A}{n\pi }}\\\end{aligned}}$ [19]: p. 192
$s(x)={\frac {4A}{P^{2}}}\left(x-{\frac {P}{2}}\right)^{2}\quad {\text{for }}0\leq x<P$ ${\begin{aligned}A_{0}=&{\frac {A}{3}}\\A_{n}=&{\frac {4A}{\pi ^{2}n^{2}}}\\B_{n}=&0\\\end{aligned}}$ [19]: p. 193
Table of basic properties
This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:
• Complex conjugation is denoted by an asterisk.
• $s(x),r(x)$ designate $P$-periodic functions or functions defined only for $x\in [0,P].$
• $S[n],R[n]$ designate the Fourier series coefficients (exponential form) of $s$ and $r.$
Property Time domain Frequency domain (exponential form) Remarks Reference
Linearity $a\cdot s(x)+b\cdot r(x)$ $a\cdot S[n]+b\cdot R[n]$ $a,b\in \mathbb {C} $
Time reversal / Frequency reversal $s(-x)$ $S[-n]$ [20]: p. 610
Time conjugation $s^{*}(x)$ $S^{*}[-n]$ [20]: p. 610
Time reversal & conjugation $s^{*}(-x)$ $S^{*}[n]$
Real part in time $\operatorname {Re} {(s(x))}$ ${\frac {1}{2}}(S[n]+S^{*}[-n])$
Imaginary part in time $\operatorname {Im} {(s(x))}$ ${\frac {1}{2i}}(S[n]-S^{*}[-n])$
Real part in frequency ${\frac {1}{2}}(s(x)+s^{*}(-x))$ $\operatorname {Re} {(S[n])}$
Imaginary part in frequency ${\frac {1}{2i}}(s(x)-s^{*}(-x))$ $\operatorname {Im} {(S[n])}$
Shift in time / Modulation in frequency $s(x-x_{0})$ $S[n]\cdot e^{-i{\frac {2\pi x_{0}}{P}}n}$ $x_{0}\in \mathbb {R} $ [20]: p. 610
Shift in frequency / Modulation in time $s(x)\cdot e^{2\pi i{\frac {n_{0}}{P}}x}$ $S[n-n_{0}]\!$ $n_{0}\in \mathbb {Z} $ [20]: p. 610
Symmetry properties
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:[21]
${\begin{array}{rccccccccc}{\text{Time domain}}&s&=&s_{_{\text{RE}}}&+&s_{_{\text{RO}}}&+&is_{_{\text{IE}}}&+&\underbrace {i\ s_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\text{Frequency domain}}&S&=&S_{\text{RE}}&+&\overbrace {\,i\ S_{\text{IO}}\,} &+&iS_{\text{IE}}&+&S_{\text{RO}}\end{array}}$
From this, various relationships are apparent, for example:
• The transform of a real-valued function (sRE + sRO) is the even symmetric function SRE + i SIO. Conversely, an even-symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function (i sIE + i sIO) is the odd symmetric function SRO + i SIE, and the converse is true.
• The transform of an even-symmetric function (sRE + i sIO) is the real-valued function SRE + SRO, and the converse is true.
• The transform of an odd-symmetric function (sRO + i sIE) is the imaginary-valued function i SIE + i SIO, and the converse is true.
Other properties
Riemann–Lebesgue lemma
If $S$ is integrable, $ \lim _{|n|\to \infty }S[n]=0$, $ \lim _{n\to +\infty }a_{n}=0$ and $ \lim _{n\to +\infty }b_{n}=0.$ This result is known as the Riemann–Lebesgue lemma.
Parseval's theorem
Main article: Parseval's theorem
If $s$ belongs to $L^{2}(P)$ (periodic over an interval of length $P$) then: $ {\frac {1}{P}}\int _{P}|s(x)|^{2}\,dx=\sum _{n=-\infty }^{\infty }{\Bigl |}S[n]{\Bigr |}^{2}$
An extension of Parseval's theorem to $L^{4}(P)$; If $s$ belongs to $L^{4}(P)$ (periodic over an interval of length $P$), and $S[n]$ is of a finite-length $M$ then:[22]
for $S[n]\in \mathbb {C} $, then ${\frac {1}{P}}\int _{P}|s(x)|^{4}\,dx=\sum _{k=0}^{M-1}S[k]\sum _{l=0}^{M-1}S^{*}[l]{\Bigg [}{\underset {k\geq l}{\sum _{m=k-l}^{M-1}}}S^{*}[m]S[m-(k-l)]+{\underset {k<l}{\sum _{m=l-k}^{M-1}}}S^{*}[m-(l-k)]S[m]{\Bigg ]}$
and for $S[n]\in \mathbb {R} $, then ${\frac {1}{P}}\int _{P}|s(x)|^{4}\,dx=\sum _{k=0}^{M-1}S[k]\sum _{l=0}^{M-1}S[l]\sum _{m=|k-l|}^{M-1}S[m]S[m-|k-l|]$
Plancherel's theorem
Main article: Plancherel theorem
If $c_{0},\,c_{\pm 1},\,c_{\pm 2},\ldots $ are coefficients and $ \sum _{n=-\infty }^{\infty }|c_{n}|^{2}<\infty $ then there is a unique function $s\in L^{2}(P)$ such that $S[n]=c_{n}$ for every $n$.
Convolution theorems
Main article: Convolution theorem § Periodic convolution (Fourier series coefficients)
Given $P$-periodic functions, $s_{_{P}}$ and $r_{_{P}}$ with Fourier series coefficients $S[n]$ and $R[n],$ $n\in \mathbb {Z} ,$
• The pointwise product:
$h_{_{P}}(x)\triangleq s_{_{P}}(x)\cdot r_{_{P}}(x)$
is also $P$-periodic, and its Fourier series coefficients are given by the discrete convolution of the $S$ and $R$ sequences:
$H[n]=\{S*R\}[n].$
• The periodic convolution:
$h_{_{P}}(x)\triangleq \int _{P}s_{_{P}}(\tau )\cdot r_{_{P}}(x-\tau )\,d\tau $
is also $P$-periodic, with Fourier series coefficients:
$H[n]=P\cdot S[n]\cdot R[n].$
• A doubly infinite sequence $\left\{c_{n}\right\}_{n\in Z}$ in $c_{0}(\mathbb {Z} )$ is the sequence of Fourier coefficients of a function in $L^{1}([0,2\pi ])$ if and only if it is a convolution of two sequences in $\ell ^{2}(\mathbb {Z} )$. See [23]
Derivative property
We say that $s$ belongs to $C^{k}(\mathbb {T} )$ if $s$ is a 2π-periodic function on $\mathbb {R} $ which is $k$ times differentiable, and its $k^{\text{th}}$ derivative is continuous.
• If $s\in C^{1}(\mathbb {T} )$, then the Fourier coefficients ${\widehat {s'}}[n]$ of the derivative $s'$ can be expressed in terms of the Fourier coefficients ${\widehat {s}}[n]$ of the function $s$, via the formula ${\widehat {s'}}[n]=in{\widehat {s}}[n]$.
• If $s\in C^{k}(\mathbb {T} )$, then ${\widehat {s^{(k)}}}[n]=(in)^{k}{\widehat {s}}[n]$. In particular, since for a fixed $k\geq 1$ we have ${\widehat {s^{(k)}}}[n]\to 0$ as $n\to \infty $, it follows that $|n|^{k}{\widehat {s}}[n]$ tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n for any $k\geq 1$.
Compact groups
Main articles: Compact group, Lie group, and Peter–Weyl theorem
One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L2(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the [−π,π] case.
An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups.
Riemannian manifolds
Main articles: Laplace operator and Riemannian manifold
If the domain is not a group, then there is no intrinsically defined convolution. However, if $X$ is a compact Riemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold $X$. Then, by analogy, one can consider heat equations on $X$. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type $L^{2}(X)$, where $X$ is a Riemannian manifold. The Fourier series converges in ways similar to the $[-\pi ,\pi ]$ case. A typical example is to take $X$ to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.
Locally compact Abelian groups
Main article: Pontryagin duality
The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.
This generalizes the Fourier transform to $L^{1}(G)$ or $L^{2}(G)$, where $G$ is an LCA group. If $G$ is compact, one also obtains a Fourier series, which converges similarly to the $[-\pi ,\pi ]$ case, but if $G$ is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is $\mathbb {R} $.
Extensions
Fourier series on a square
We can also define the Fourier series for functions of two variables $x$ and $y$ in the square $[-\pi ,\pi ]\times [-\pi ,\pi ]$:
${\begin{aligned}f(x,y)&=\sum _{j,k\in \mathbb {Z} }c_{j,k}e^{ijx}e^{iky},\\[5pt]c_{j,k}&={\frac {1}{4\pi ^{2}}}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }f(x,y)e^{-ijx}e^{-iky}\,dx\,dy.\end{aligned}}$
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the JPEG image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.[24]
Fourier series of Bravais-lattice-periodic-function
A three-dimensional Bravais lattice is defined as the set of vectors of the form:
$\mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}$
where $n_{i}$ are integers and $\mathbf {a} _{i}$ are three linearly independent vectors. Assuming we have some function, $f(\mathbf {r} )$, such that it obeys the condition of periodicity for any Bravais lattice vector $\mathbf {R} $, $f(\mathbf {r} )=f(\mathbf {R} +\mathbf {r} )$, we could make a Fourier series of it. This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. It is useful to make the Fourier series of the potential when applying Bloch's theorem. First, we may write any arbitrary position vector $\mathbf {r} $ in the coordinate-system of the lattice:
$\mathbf {r} =x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},$
where $a_{i}\triangleq |\mathbf {a} _{i}|,$ meaning that $a_{i}$ is defined to be the magnitude of $\mathbf {a} _{i}$, so ${\hat {\mathbf {a} _{i}}}={\frac {\mathbf {a} _{i}}{a_{i}}}$ is the unit vector directed along $\mathbf {a} _{i}$.
Thus we can define a new function,
$g(x_{1},x_{2},x_{3})\triangleq f(\mathbf {r} )=f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right).$
This new function, $g(x_{1},x_{2},x_{3})$, is now a function of three-variables, each of which has periodicity $a_{1}$, $a_{2}$, and $a_{3}$ respectively:
$g(x_{1},x_{2},x_{3})=g(x_{1}+a_{1},x_{2},x_{3})=g(x_{1},x_{2}+a_{2},x_{3})=g(x_{1},x_{2},x_{3}+a_{3}).$
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers $m_{1},m_{2},m_{3}$. In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for $g$ on the interval $\left[0,a_{1}\right]$ for $x_{1}$, we can define the following:
$h^{\mathrm {one} }(m_{1},x_{2},x_{3})\triangleq {\frac {1}{a_{1}}}\int _{0}^{a_{1}}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\,dx_{1}$
And then we can write:
$g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }h^{\mathrm {one} }(m_{1},x_{2},x_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}$
Further defining:
${\begin{aligned}h^{\mathrm {two} }(m_{1},m_{2},x_{3})&\triangleq {\frac {1}{a_{2}}}\int _{0}^{a_{2}}h^{\mathrm {one} }(m_{1},x_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}\,dx_{2}\\[12pt]&={\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}\right)}\end{aligned}}$
We can write $g$ once again as:
$g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }\sum _{m_{2}=-\infty }^{\infty }h^{\mathrm {two} }(m_{1},m_{2},x_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\cdot e^{i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}$
Finally applying the same for the third coordinate, we define:
${\begin{aligned}h^{\mathrm {three} }(m_{1},m_{2},m_{3})&\triangleq {\frac {1}{a_{3}}}\int _{0}^{a_{3}}h^{\mathrm {two} }(m_{1},m_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{3}}{a_{3}}}x_{3}}\,dx_{3}\\[12pt]&={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}+{\frac {m_{3}}{a_{3}}}x_{3}\right)}\end{aligned}}$
We write $g$ as:
$g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }\sum _{m_{2}=-\infty }^{\infty }\sum _{m_{3}=-\infty }^{\infty }h^{\mathrm {three} }(m_{1},m_{2},m_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\cdot e^{i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}\cdot e^{i2\pi {\frac {m_{3}}{a_{3}}}x_{3}}$
Re-arranging:
$g(x_{1},x_{2},x_{3})=\sum _{m_{1},m_{2},m_{3}\in \mathbb {Z} }h^{\mathrm {three} }(m_{1},m_{2},m_{3})\cdot e^{i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}+{\frac {m_{3}}{a_{3}}}x_{3}\right)}.$
Now, every reciprocal lattice vector can be written (but does not mean that it is the only way of writing) as $\mathbf {G} =m_{1}\mathbf {g} _{1}+m_{2}\mathbf {g} _{2}+m_{3}\mathbf {g} _{3}$, where $m_{i}$ are integers and $\mathbf {g} _{i}$ are reciprocal lattice vectors to satisfy $\mathbf {g_{i}} \cdot \mathbf {a_{j}} =2\pi \delta _{ij}$ ($\delta _{ij}=1$ for $i=j$, and $\delta _{ij}=0$ for $i\neq j$). Then for any arbitrary reciprocal lattice vector $\mathbf {G} $ and arbitrary position vector $\mathbf {r} $ in the original Bravais lattice space, their scalar product is:
$\mathbf {G} \cdot \mathbf {r} =\left(m_{1}\mathbf {g} _{1}+m_{2}\mathbf {g} _{2}+m_{3}\mathbf {g} _{3}\right)\cdot \left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)=2\pi \left(x_{1}{\frac {m_{1}}{a_{1}}}+x_{2}{\frac {m_{2}}{a_{2}}}+x_{3}{\frac {m_{3}}{a_{3}}}\right).$
So it is clear that in our expansion of $g(x_{1},x_{2},x_{3})=f(\mathbf {r} )$, the sum is actually over reciprocal lattice vectors:
$f(\mathbf {r} )=\sum _{\mathbf {G} }h(\mathbf {G} )\cdot e^{i\mathbf {G} \cdot \mathbf {r} },$
where
$h(\mathbf {G} )={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}\,{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}\,{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}\,f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)\cdot e^{-i\mathbf {G} \cdot \mathbf {r} }.$
Assuming
$\mathbf {r} =(x,y,z)=x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},$
we can solve this system of three linear equations for $x$, $y$, and $z$ in terms of $x_{1}$, $x_{2}$ and $x_{3}$ in order to calculate the volume element in the original rectangular coordinate system. Once we have $x$, $y$, and $z$ in terms of $x_{1}$, $x_{2}$ and $x_{3}$, we can calculate the Jacobian determinant:
${\begin{vmatrix}{\dfrac {\partial x_{1}}{\partial x}}&{\dfrac {\partial x_{1}}{\partial y}}&{\dfrac {\partial x_{1}}{\partial z}}\\[12pt]{\dfrac {\partial x_{2}}{\partial x}}&{\dfrac {\partial x_{2}}{\partial y}}&{\dfrac {\partial x_{2}}{\partial z}}\\[12pt]{\dfrac {\partial x_{3}}{\partial x}}&{\dfrac {\partial x_{3}}{\partial y}}&{\dfrac {\partial x_{3}}{\partial z}}\end{vmatrix}}$
which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:
${\frac {a_{1}a_{2}a_{3}}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}$
(it may be advantageous for the sake of simplifying calculations, to work in such a rectangular coordinate system, in which it just so happens that $\mathbf {a} _{1}$ is parallel to the x axis, $\mathbf {a} _{2}$ lies in the xy-plane, and $\mathbf {a} _{3}$ has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitive-vectors $\mathbf {a} _{1}$, $\mathbf {a} _{2}$ and $\mathbf {a} _{3}$. In particular, we now know that
$dx_{1}\,dx_{2}\,dx_{3}={\frac {a_{1}a_{2}a_{3}}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}\cdot dx\,dy\,dz.$
We can write now $h(\mathbf {G} )$ as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the $x_{1}$, $x_{2}$ and $x_{3}$ variables:
$h(\mathbf {G} )={\frac {1}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}\int _{C}d\mathbf {r} f(\mathbf {r} )\cdot e^{-i\mathbf {G} \cdot \mathbf {r} }$
writing $d\mathbf {r} $ for the volume element $dx\,dy\,dz$; and where $C$ is the primitive unit cell, thus, $\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})$ is the volume of the primitive unit cell.
Hilbert space interpretation
Main article: Hilbert space
In the language of Hilbert spaces, the set of functions $\left\{e_{n}=e^{inx}:n\in \mathbb {Z} \right\}$ is an orthonormal basis for the space $L^{2}([-\pi ,\pi ])$ of square-integrable functions on $[-\pi ,\pi ]$. This space is actually a Hilbert space with an inner product given for any two elements $f$ and $g$ by:
$\langle f,\,g\rangle \;\triangleq \;{\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)g^{*}(x)\,dx,$ where $g^{*}(x)$ is the complex conjugate of $g(x).$
The basic Fourier series result for Hilbert spaces can be written as
$f=\sum _{n=-\infty }^{\infty }\langle f,e_{n}\rangle \,e_{n}.$
This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:
$\int _{-\pi }^{\pi }\cos(mx)\,\cos(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\cos((n-m)x)+\cos((n+m)x)\,dx=\pi \delta _{mn},\quad m,n\geq 1,$
$\int _{-\pi }^{\pi }\sin(mx)\,\sin(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\cos((n-m)x)-\cos((n+m)x)\,dx=\pi \delta _{mn},\quad m,n\geq 1$
(where δmn is the Kronecker delta), and
$\int _{-\pi }^{\pi }\cos(mx)\,\sin(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\sin((n+m)x)+\sin((n-m)x)\,dx=0;$
furthermore, the sines and cosines are orthogonal to the constant function $1$. An orthonormal basis for $L^{2}([-\pi ,\pi ])$ consisting of real functions is formed by the functions $1$ and ${\sqrt {2}}\cos(nx)$, ${\sqrt {2}}\sin(nx)$ with n= 1,2,.... The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.
Fourier theorem proving convergence of Fourier series
Main article: Convergence of Fourier series
These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.[25][26][27][28]
The earlier Eq.7
$s_{_{N}}(x)=\sum _{n=-N}^{N}S[n]\ e^{i{\tfrac {2\pi }{P}}nx},$
is a trigonometric polynomial of degree $N$ that can be generally expressed as:
$p_{_{N}}(x)=\sum _{n=-N}^{N}p[n]\ e^{i{\tfrac {2\pi }{P}}nx}.$
Least squares property
Parseval's theorem implies that:
Theorem — The trigonometric polynomial $s_{_{N}}$ is the unique best trigonometric polynomial of degree $N$ approximating $s(x)$, in the sense that, for any trigonometric polynomial $p_{_{N}}\neq s_{_{N}}$ of degree $N$, we have:
$\|s_{_{N}}-s\|_{2}<\|p_{_{N}}-s\|_{2},$
where the Hilbert space norm is defined as:
$\|g\|_{2}={\sqrt {{1 \over P}\int _{P}|g(x)|^{2}\,dx}}.$
Convergence theorems
See also: Gibbs phenomenon
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
Theorem — If $s$ belongs to $L^{2}(P)$ (an interval of length $P$), then $s_{\infty }$ converges to $s$ in $L^{2}(P)$, that is, $\|s_{_{N}}-s\|_{2}$ converges to 0 as $N\to \infty $.
We have already mentioned that if $s$ is continuously differentiable, then $(i\cdot n)S[n]$ is the $n^{\text{th}}$ Fourier coefficient of the derivative $s'$. It follows, essentially from the Cauchy–Schwarz inequality, that $s_{\infty }$ is absolutely summable. The sum of this series is a continuous function, equal to $s$, since the Fourier series converges in the mean to $s$:
Theorem — If $s\in C^{1}(\mathbb {T} )$, then $s_{\infty }$ converges to $s$ uniformly (and hence also pointwise.)
This result can be proven easily if $s$ is further assumed to be $C^{2}$, since in that case $n^{2}S[n]$ tends to zero as $n\rightarrow \infty $. More generally, the Fourier series is absolutely summable, thus converges uniformly to $s$, provided that $s$ satisfies a Hölder condition of order $\alpha >1/2$. In the absolutely summable case, the inequality:
$\sup _{x}|s(x)-s_{_{N}}(x)|\leq \sum _{|n|>N}|S[n]|$
proves uniform convergence.
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at $x$ if $s$ is differentiable at $x$, to Lennart Carleson's much more sophisticated result that the Fourier series of an $L^{2}$ function actually converges almost everywhere.
Divergence
Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact.
In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere (Katznelson 1976).
See also
• ATS theorem
• Carleson's theorem
• Dirichlet kernel
• Discrete Fourier transform
• Fast Fourier transform
• Fejér's theorem
• Fourier analysis
• Fourier sine and cosine series
• Fourier transform
• Gibbs phenomenon
• Half range Fourier series
• Laurent series – the substitution q = eix transforms a Fourier series into a Laurent series, or conversely. This is used in the q-series expansion of the j-invariant.
• Least-squares spectral analysis
• Multidimensional transform
• Spectral theory
• Sturm–Liouville theory
• Residue theorem integrals of f(z), singularities, poles
Notes
1. Some authors define $A_{0}$ differently so that the same integral can be used to define $A_{0}$ and $A_{n}$. This changes Eq. 2 so that the first term needs to be divided by 2, and is no longer the average value.
2. The scale factor ${\tfrac {2}{P}},$ which could be inserted later, results in a series that converges to $s(x)$ instead of ${\tfrac {P}{2}}s(x).$
3. Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as distributions. In this sense ${\mathcal {F}}\{e^{i{\frac {2\pi nx}{P}}}\}$ is a Dirac delta function, which is an example of a distribution.
4. These three did some important early work on the wave equation, especially D'Alembert. Euler's work in this area was mostly comtemporaneous/ in collaboration with Bernoulli, although the latter made some independent contributions to the theory of waves and vibrations. (See Fetter & Walecka 2003, pp. 209–210).
5. These words are not strictly Fourier's. Whilst the cited article does list the author as Fourier, a footnote indicates that the article was actually written by Poisson (that it was not written by Fourier is also clear from the consistent use of the third person to refer to him) and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.
References
1. "Fourier". Dictionary.com Unabridged (Online). n.d.
2. Zygmund, A. (2002). Trigonometric Series (3nd ed.). Cambridge, UK: Cambridge University Press. ISBN 0-521-89053-5.
3. Haberman, Richard (1987). Elementary Applied Partial Differential Equations (2nd ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 77. ISBN 0-13-252875-4.
4. Pinkus, Allan; Zafrany, Samy (1997). Fourier Series and Integral Transforms (1st ed.). Cambridge, UK: Cambridge University Press. pp. 42–44. ISBN 0-521-59771-4.
5. Dorf, Richard C.; Tallarida, Ronald J. (1993). Pocket Book of Electrical Engineering Formulas (1st ed.). Boca Raton,FL: CRC Press. pp. 171–174. ISBN 0849344735.
6. Katznelson, Yitzhak (1976). An introduction to Harmonic Analysis (2nd corrected ed.). New York, NY: Dover Publications, Inc. p. 46. ISBN 0-486-63331-4.
7. Tolstov, Georgi P. (1976). Fourier Series. Courier-Dover. ISBN 0-486-63317-9.
8. Stillwell, John (2013). "Logic and the philosophy of mathematics in the nineteenth century". In Ten, C. L. (ed.). Routledge History of Philosophy. Vol. VII: The Nineteenth Century. Routledge. p. 204. ISBN 978-1-134-92880-4.
9. Fasshauer, Greg (2015). "Fourier Series and Boundary Value Problems" (PDF). Math 461 Course Notes, Ch 3. Department of Applied Mathematics, Illinois Institute of Technology. Retrieved 6 November 2020.
10. Cajori, Florian (1893). A History of Mathematics. Macmillan. p. 283.
11. Lejeune-Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données" [On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits]. Journal für die reine und angewandte Mathematik (in French). 4: 157–169. arXiv:0806.1294.
12. "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" [About the representability of a function by a trigonometric series]. Habilitationsschrift, Göttingen; 1854. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867. Published posthumously for Riemann by Richard Dedekind (in German). Archived from the original on 20 May 2008. Retrieved 19 May 2008.
13. Mascre, D.; Riemann, Bernhard (1867), "Posthumous Thesis on the Representation of Functions by Trigonometric Series", in Grattan-Guinness, Ivor (ed.), Landmark Writings in Western Mathematics 1640–1940, Elsevier (published 2005), p. 49, ISBN 9780080457444
14. Remmert, Reinhold (1991). Theory of Complex Functions: Readings in Mathematics. Springer. p. 29. ISBN 9780387971957.
15. Nerlove, Marc; Grether, David M.; Carvalho, Jose L. (1995). Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics. Elsevier. ISBN 0-12-515751-7.
16. Wilhelm Flügge, Stresses in Shells (1973) 2nd edition. ISBN 978-3-642-88291-3. Originally published in German as Statik und Dynamik der Schalen (1937).
17. Fourier, Jean-Baptiste-Joseph (1888). Gaston Darboux (ed.). Oeuvres de Fourier [The Works of Fourier] (in French). Paris: Gauthier-Villars et Fils. pp. 218–219 – via Gallica.
18. Sepesi, G (13 February 2022). "Zeno's Enduring Example". Towards Data Science. pp. Appendix B.
19. Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (in German). Vieweg+Teubner Verlag. ISBN 978-3834807571.
20. Shmaliy, Y.S. (2007). Continuous-Time Signals. Springer. ISBN 978-1402062711.
21. Proakis, John G.; Manolakis, Dimitris G. (1996). Digital Signal Processing: Principles, Algorithms, and Applications (3rd ed.). Prentice Hall. p. 291. ISBN 978-0-13-373762-2.
22. Sharkas, Hesham (2022). "Solution of Integral of The Fourth Power of a Finite-Length Exponential Fourier Series". ResearchGate. doi:10.13140/RG.2.2.31527.83368/2.
23. "Characterizations of a linear subspace associated with Fourier series". MathOverflow. 2010-11-19. Retrieved 2014-08-08.
24. Vanishing of Half the Fourier Coefficients in Staggered Arrays
25. Siebert, William McC. (1985). Circuits, signals, and systems. MIT Press. p. 402. ISBN 978-0-262-19229-3.
26. Marton, L.; Marton, Claire (1990). Advances in Electronics and Electron Physics. Academic Press. p. 369. ISBN 978-0-12-014650-5.
27. Kuzmany, Hans (1998). Solid-state spectroscopy. Springer. p. 14. ISBN 978-3-540-63913-8.
28. Pribram, Karl H.; Yasue, Kunio; Jibu, Mari (1991). Brain and perception. Lawrence Erlbaum Associates. p. 26. ISBN 978-0-89859-995-4.
Further reading
• William E. Boyce; Richard C. DiPrima (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). New Jersey: John Wiley & Sons, Inc. ISBN 0-471-43338-1.
• Joseph Fourier, translated by Alexander Freeman (2003). The Analytical Theory of Heat. Dover Publications. ISBN 0-486-49531-0. 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822.
• Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly. 99 (5): 427–441. doi:10.2307/2325087. JSTOR 2325087.
• Fetter, Alexander L.; Walecka, John Dirk (2003). Theoretical Mechanics of Particles and Continua. Courier. ISBN 978-0-486-43261-8.
• Felix Klein, Development of mathematics in the 19th century. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert, Springer, Berlin, 1928.
• Walter Rudin (1976). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill, Inc. ISBN 0-07-054235-X.
• A. Zygmund (2002). Trigonometric Series (third ed.). Cambridge: Cambridge University Press. ISBN 0-521-89053-5. The first edition was published in 1935.
External links
• "Fourier series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Hobson, Ernest (1911). "Fourier's Series" . Encyclopædia Britannica. Vol. 10 (11th ed.). pp. 753–758.
• Weisstein, Eric W. "Fourier Series". MathWorld.
• Joseph Fourier – A site on Fourier's life which was used for the historical section of this article at the Wayback Machine (archived December 5, 2001)
This article incorporates material from example of Fourier series on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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| Wikipedia |
Trigonometric functions of matrices
The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations.[1] They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers:[2]
${\begin{aligned}\sin X&=X-{\frac {X^{3}}{3!}}+{\frac {X^{5}}{5!}}-{\frac {X^{7}}{7!}}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}X^{2n+1}\\\cos X&=I-{\frac {X^{2}}{2!}}+{\frac {X^{4}}{4!}}-{\frac {X^{6}}{6!}}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}X^{2n}\end{aligned}}$
with Xn being the nth power of the matrix X, and I being the identity matrix of appropriate dimensions.
Equivalently, they can be defined using the matrix exponential along with the matrix equivalent of Euler's formula, eiX = cos X + i sin X, yielding
${\begin{aligned}\sin X&={e^{iX}-e^{-iX} \over 2i}\\\cos X&={e^{iX}+e^{-iX} \over 2}.\end{aligned}}$
For example, taking X to be a standard Pauli matrix,
$\sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}~,$
one has
$\sin(\theta \sigma _{1})=\sin(\theta )~\sigma _{1},\qquad \cos(\theta \sigma _{1})=\cos(\theta )~I~,$
as well as, for the cardinal sine function,
$\operatorname {sinc} (\theta \sigma _{1})=\operatorname {sinc} (\theta )~I.$
See also: Axis–angle representation § Exponential map from so(3) to SO(3)
Properties
The analog of the Pythagorean trigonometric identity holds:[2]
$\sin ^{2}X+\cos ^{2}X=I$
If X is a diagonal matrix, sin X and cos X are also diagonal matrices with (sin X)nn = sin(Xnn) and (cos X)nn = cos(Xnn), that is, they can be calculated by simply taking the sines or cosines of the matrices's diagonal components.
The analogs of the trigonometric addition formulas are true if and only if XY = YX:[2]
${\begin{aligned}\sin(X\pm Y)=\sin X\cos Y\pm \cos X\sin Y\\\cos(X\pm Y)=\cos X\cos Y\mp \sin X\sin Y\end{aligned}}$
Other functions
The tangent, as well as inverse trigonometric functions, hyperbolic and inverse hyperbolic functions have also been defined for matrices:[3]
$\arcsin X=-i\ln \left(iX+{\sqrt {I-X^{2}}}\right)$ (see Inverse trigonometric functions#Logarithmic forms, Matrix logarithm, Square root of a matrix)
${\begin{aligned}\sinh X&={e^{X}-e^{-X} \over 2}\\\cosh X&={e^{X}+e^{-X} \over 2}\end{aligned}}$
and so on.
References
1. Gareth I. Hargreaves, Nicholas J. Higham (2005). "Efficient Algorithms for the Matrix Cosine and Sine" (PDF). Numerical Analysis Report. Manchester Centre for Computational Mathematics. 40 (461): 383. Bibcode:2005NuAlg..40..383H. doi:10.1007/s11075-005-8141-0. S2CID 1242875.{{cite journal}}: CS1 maint: uses authors parameter (link)
2. Nicholas J. Higham (2008). Functions of matrices: theory and computation. pp. 287f. ISBN 9780898717778.
3. Scilab trigonometry.
| Wikipedia |
Trigonometric moment problem
In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α0, ... αn }, does there exist a positive Borel measure μ on the interval [0, 2π] such that
$\alpha _{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ikt}\,d\mu (t).$
In other words, an affirmative answer to the problems means that {α0, ... αn } are the first n + 1 Fourier coefficients of some positive Borel measure μ on [0, 2π].
Characterization
The trigonometric moment problem is solvable, that is, {αk} is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Toeplitz matrix
$A=\left({\begin{matrix}\alpha _{0}&\alpha _{1}&\cdots &\alpha _{n}\\{\bar {\alpha _{1}}}&\alpha _{0}&\cdots &\alpha _{n-1}\\\vdots &\vdots &\ddots &\vdots \\{\bar {\alpha _{n}}}&{\bar {\alpha _{n-1}}}&\cdots &\alpha _{0}\\\end{matrix}}\right)$
is positive semidefinite.
The "only if" part of the claims can be verified by a direct calculation.
We sketch an argument for the converse. The positive semidefinite matrix A defines a sesquilinear product on Cn + 1, resulting in a Hilbert space
$({\mathcal {H}},\langle \;,\;\rangle )$
of dimensional at most n + 1, a typical element of which is an equivalence class denoted by [f]. The Toeplitz structure of A means that a "truncated" shift is a partial isometry on ${\mathcal {H}}$. More specifically, let { e0, ...en } be the standard basis of Cn + 1. Let ${\mathcal {E}}$ be the subspace generated by { [e0], ... [en - 1] } and ${\mathcal {F}}$ be the subspace generated by { [e1], ... [en] }. Define an operator
$V:{\mathcal {E}}\rightarrow {\mathcal {F}}$
by
$V[e_{k}]=[e_{k+1}]\quad {\mbox{for}}\quad k=0\ldots n-1.$
Since
$\langle V[e_{j}],V[e_{k}]\rangle =\langle [e_{j+1}],[e_{k+1}]\rangle =A_{j+1,k+1}=A_{j,k}=\langle [e_{j}],[e_{k}]\rangle ,$
V can be extended to a partial isometry acting on all of ${\mathcal {H}}$. Take a minimal unitary extension U of V, on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure m on the unit circle T such that for all integer k
$\langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\int _{\mathbf {T} }z^{k}dm.$
For k = 0,...,n, the left hand side is
$\langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle (V^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle [e_{n+1-k}],[e_{n+1}]\rangle =A_{n+1,n+1-k}={\bar {\alpha _{k}}}.$
So
$\int _{\mathbf {T} }z^{-k}dm=\int _{\mathbf {T} }{\bar {z}}^{k}dm=\alpha _{k}.$
Finally, parametrize the unit circle T by eit on [0, 2π] gives
${\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ikt}d\mu (t)=\alpha _{k}$
for some suitable measure μ.
Parametrization of solutions
The above discussion shows that the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix A is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry V.
References
• N.I. Akhiezer, The Classical Moment Problem, Olivier and Boyd, 1965.
• N.I. Akhiezer, M.G. Krein, Some Questions in the Theory of Moments, Amer. Math. Soc., 1962.
| Wikipedia |
Exact trigonometric values
In mathematics, the values of the trigonometric functions can be expressed approximately, as in $\cos(\pi /4)\approx 0.707$, or exactly, as in $\cos(\pi /4)={\sqrt {2}}/2$. While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots.
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
Common angles
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 90°.[1] For angles outside of this range, trigonometric values can be found by applying the reflection and shift identities. In the table below, $\infty $ stands for the ratio 1:0. These values can also be considered to be undefined (see division by zero).
RadiansDegreessincostancotseccsc
$0$$0^{\circ }$ $0$$1$$0$$\infty $$1$$\infty $
${\frac {\pi }{12}}$$15^{\circ }$ ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$$2-{\sqrt {3}}$$2+{\sqrt {3}}$${\sqrt {6}}-{\sqrt {2}}$${\sqrt {6}}+{\sqrt {2}}$
${\frac {\pi }{10}}$$18^{\circ }$ ${\frac {{\sqrt {5}}-1}{4}}$${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$ ${\frac {\sqrt {25-10{\sqrt {5}}}}{5}}$${\sqrt {5+2{\sqrt {5}}}}$${\frac {\sqrt {50-10{\sqrt {5}}}}{5}}$$1+{\sqrt {5}}$
${\frac {\pi }{8}}$$22.5^{\circ }$ ${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$${\sqrt {2}}-1$${\sqrt {2}}+1$${\sqrt {4-2{\sqrt {2}}}}$${\sqrt {4+2{\sqrt {2}}}}$
${\frac {\pi }{6}}$$30^{\circ }$ ${\frac {1}{2}}$${\frac {\sqrt {3}}{2}}$${\frac {\sqrt {3}}{3}}$${\sqrt {3}}$${\frac {2{\sqrt {3}}}{3}}$$2$
${\frac {\pi }{5}}$$36^{\circ }$ ${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$${\frac {1+{\sqrt {5}}}{4}}$${\sqrt {5-2{\sqrt {5}}}}$${\frac {\sqrt {25+10{\sqrt {5}}}}{5}}$${\sqrt {5}}-1$${\frac {\sqrt {50+10{\sqrt {5}}}}{5}}$
${\frac {\pi }{4}}$$45^{\circ }$ ${\frac {\sqrt {2}}{2}}$${\frac {\sqrt {2}}{2}}$$1$$1$${\sqrt {2}}$${\sqrt {2}}$
${\frac {3\pi }{10}}$$54^{\circ }$ ${\frac {1+{\sqrt {5}}}{4}}$${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$${\frac {\sqrt {25+10{\sqrt {5}}}}{5}}$${\sqrt {5-2{\sqrt {5}}}}$${\frac {\sqrt {50+10{\sqrt {5}}}}{5}}$${\sqrt {5}}-1$
${\frac {\pi }{3}}$$60^{\circ }$ ${\frac {\sqrt {3}}{2}}$${\frac {1}{2}}$${\sqrt {3}}$${\frac {\sqrt {3}}{3}}$$2$${\frac {2{\sqrt {3}}}{3}}$
${\frac {3\pi }{8}}$$67.5^{\circ }$ ${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$${\sqrt {2}}+1$${\sqrt {2}}-1$${\sqrt {4+2{\sqrt {2}}}}$${\sqrt {4-2{\sqrt {2}}}}$
${\frac {2\pi }{5}}$$72^{\circ }$ ${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$${\frac {{\sqrt {5}}-1}{4}}$${\sqrt {5+2{\sqrt {5}}}}$${\frac {\sqrt {25-10{\sqrt {5}}}}{5}}$$1+{\sqrt {5}}$${\frac {\sqrt {50-10{\sqrt {5}}}}{5}}$
${\frac {5\pi }{12}}$$75^{\circ }$ ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$$2+{\sqrt {3}}$$2-{\sqrt {3}}$${\sqrt {6}}+{\sqrt {2}}$${\sqrt {6}}-{\sqrt {2}}$
${\frac {\pi }{2}}$$90^{\circ }$ $1$$0$$\infty $$0$$\infty $$1$
Expressibility with square roots
Some exact trigonometric values, such as $\sin(60^{\circ })={\sqrt {3}}/2$, can be expressed in terms of a combination of arithmetic operations and square roots. Such numbers are called constructible, because one length can be constructed by compass and straightedge from another if and only if the ratio between the two lengths is such a number.[2] However, some trigonometric values, such as $\cos(20^{\circ })$, have been proven to not be constructible.
The sine and cosine of an angle are constructible if and only if the angle is constructible. If an angle is a rational multiple of π radians, whether or not it is constructible can be determined as follows. Let the angle be $a\pi /b$ radians, where a and b are relatively prime integers. Then it is constructible if and only if the prime factorization of the denominator, b, consists of any number of Fermat primes, each with an exponent of 1, times any power of two.[3] For example, $15^{\circ }$ and $24^{\circ }$ are constructible because they are equivalent to $\pi /12$ and $2\pi /15$ radians, respectively, and 12 is the product of 3 and 4, which are a Fermat prime and a power of two, and 15 is the product of Fermat primes 3 and 5. On the other hand, $20^{\circ }$ is not constructible because it corresponds to a denominator of 9 = 32, and the Fermat prime cannot be raised to a power greater than one. As another example, $(360/7)^{\circ }$ is not constructible, because the denominator of 7 is not a Fermat prime.[2]
Derivations of constructible values
The values of trigonometric numbers can be derived through a combination of methods. The values of sine and cosine of 30, 45, and 60 degrees are derived by analysis of the 30-60-90 and 90-45-45 triangles. If the angle is expressed in radians as $a\pi /b$, this takes care of the case where a is 1 and b is 2, 3, 4, or 6.
Half-angle formula
See also: Square root of 2 § Properties
If the denominator, b, is multiplied by additional factors of 2, the sine and cosine can be derived with the half-angle formulas. For example, 22.5° (π/8 rad) is half of 45°, so its sine and cosine are:
$\sin(22.5^{\circ })={\sqrt {\frac {1-\cos(45^{\circ })}{2}}}={\sqrt {\frac {1-{\frac {\sqrt {2}}{2}}}{2}}}={\frac {\sqrt {2-{\sqrt {2}}}}{2}}$
$\cos(22.5^{\circ })={\sqrt {\frac {1+\cos(45^{\circ })}{2}}}={\sqrt {\frac {1+{\frac {\sqrt {2}}{2}}}{2}}}={\frac {\sqrt {2+{\sqrt {2}}}}{2}}$
Repeated application of the cosine half-angle formula leads to nested square roots that continue in a pattern where each application adds a ${\sqrt {2+\cdots }}$ to the numerator and the denominator is 2. For example:
$\cos \left({\frac {\pi }{16}}\right)={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\qquad \cos \left({\frac {\pi }{32}}\right)={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$
$\cos \left({\frac {\pi }{12}}\right)={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}={\frac {\sqrt {2+{\sqrt {3}}}}{2}}\qquad \cos \left({\frac {\pi }{24}}\right)={\frac {\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}{2}}$
Sine of 18°
Cases where the denominator, b, is 5 times a power of 2 can start from the following derivation of $\sin(18^{\circ })$,[4] since $18^{\circ }=\pi /10$ radians. The derivation uses the multiple angle formulas for sine and cosine. By the double angle formula for sine:
$\sin(36^{\circ })=2\sin(18^{\circ })\cos(18^{\circ })$
By the triple angle formula for cosine:
$\cos(54^{\circ })=\cos ^{3}(18^{\circ })-3\sin ^{2}(18^{\circ })\cos(18^{\circ })=\cos(18^{\circ })(1-4\sin ^{2}(18^{\circ }))$
Since sin(36°) = cos(54°), we equate these two expressions and cancel a factor of cos(18°):
$2\sin(18^{\circ })=1-4\sin ^{2}(18^{\circ })$
This quadratic equation has only one positive root:
$\sin(18^{\circ })={\frac {{\sqrt {5}}-1}{4}}$
Using other identities
The sines and cosines of many other angles can be derived using the results described above and a combination of the multiple angle formulas and the sum and difference formulas. For example, for the case where b is 15 times a power of 2, since $24^{\circ }=60^{\circ }-36^{\circ }$, its cosine can be derived by the cosine difference formula:
${\begin{aligned}\cos(24^{\circ })&=\cos(60^{\circ })\cos(36^{\circ })+\sin(60^{\circ })\sin(36^{\circ })\\&={\frac {1}{2}}{\frac {{\sqrt {5}}+1}{4}}+{\frac {\sqrt {3}}{2}}{\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\\&={\frac {1+{\sqrt {5}}+{\sqrt {30-6{\sqrt {5}}}}}{8}}\end{aligned}}$
Denominator of 17
Main article: Heptadecagon
Since 17 is a Fermat prime, a regular 17-gon is constructible, which means that the sines and cosines of angles such as $2\pi /17$ radians can be expressed in terms of square roots. In particular, in 1796, Carl Friedrich Gauss showed that:[5][6]
$\cos \left({\frac {2\pi }{17}}\right)={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}$
The sines and cosines of other constructible angles with a denominator divisible by 17 can be derived from this one.
Roots of unity
Main article: Root of unity
An irrational number that can be expressed as the sine or cosine of a rational multiple of π radians is called a trigonometric number.[7]: ch. 5 Since $\sin(x)=\cos(x-\pi /2),$ the case of a sine can be omitted from this definition. Therefore any trigonometric number can be written as $\cos(2\pi k/n)$, where k and n are integers. This number can be thought of as the real part of the complex number $\cos(2\pi k/n)+i\sin(2\pi k/n)$. De Moivre's formula shows that numbers of this form are roots of unity:
$\left(\cos \left({\frac {2\pi k}{n}}\right)+i\sin \left({\frac {2\pi k}{n}}\right)\right)^{n}=\cos(2\pi k)+i\sin(2\pi k)=1$
Since the root of unity is a root of the polynomial xn − 1, it is algebraic. Since the trigonometric number is the average of the root of unity and its complex conjugate, and algebraic numbers are closed under arithmetic operations, every trigonometric number is algebraic.
The real part of any root of unity is trigonometric, unless it is rational. By Niven's theorem, the only rational numbers that can be expressed as the real part of a root of unity are 0, 1, −1, 1/2, and −1/2.[8]
Extended table of exact values: Until 360 degrees
Exact values of common angles[1][9]
RadianDegreesincostancotseccsc
$0$$0^{\circ }$ $0$$1$$0$$\infty $$1$$\infty $
${\frac {\pi }{24}}$$7.5^{\circ }$ ${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}$${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}$${\sqrt {6}}-{\sqrt {3}}+{\sqrt {2}}-2$${\sqrt {6}}+{\sqrt {3}}+{\sqrt {2}}+2$${\sqrt {2}}{\sqrt {8-3{\sqrt {6}}-{\sqrt {2(49-20{\sqrt {6}})}}}}$${\sqrt {2}}{\sqrt {8+3{\sqrt {6}}+{\sqrt {2(49+20{\sqrt {6}})}}}}$
${\frac {\pi }{12}}$$15^{\circ }$ ${\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$${\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$$2-{\sqrt {3}}$$2+{\sqrt {3}}$${\sqrt {2}}({\sqrt {3}}-1)$${\sqrt {2}}({\sqrt {3}}+1)$
${\frac {\pi }{10}}$$18^{\circ }$ ${\frac {{\sqrt {5}}-1}{4}}$${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$ ${\frac {\sqrt {25-10{\sqrt {5}}}}{5}}$${\sqrt {5+2{\sqrt {5}}}}$${\frac {\sqrt {50-10{\sqrt {5}}}}{5}}$$1+{\sqrt {5}}$
${\frac {\pi }{8}}$$22.5^{\circ }$ ${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$${\sqrt {2}}-1$${\sqrt {2}}+1$${\sqrt {4-2{\sqrt {2}}}}$${\sqrt {4+2{\sqrt {2}}}}$
${\frac {\pi }{6}}$$30^{\circ }$ ${\frac {1}{2}}$${\frac {\sqrt {3}}{2}}$${\frac {\sqrt {3}}{3}}$${\sqrt {3}}$${\frac {2{\sqrt {3}}}{3}}$$2$
${\frac {\pi }{5}}$$36^{\circ }$ ${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$${\frac {1+{\sqrt {5}}}{4}}$${\sqrt {5-2{\sqrt {5}}}}$${\frac {\sqrt {25+10{\sqrt {5}}}}{5}}$${\sqrt {5}}-1$${\frac {\sqrt {50+10{\sqrt {5}}}}{5}}$
${\frac {\pi }{4}}$$45^{\circ }$ ${\frac {\sqrt {2}}{2}}$${\frac {\sqrt {2}}{2}}$$1$$1$${\sqrt {2}}$${\sqrt {2}}$
${\frac {3\pi }{10}}$$54^{\circ }$ ${\frac {1+{\sqrt {5}}}{4}}$${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$${\frac {\sqrt {25+10{\sqrt {5}}}}{5}}$${\sqrt {5-2{\sqrt {5}}}}$${\frac {\sqrt {50+10{\sqrt {5}}}}{5}}$${\sqrt {5}}-1$
${\frac {\pi }{3}}$$60^{\circ }$ ${\frac {\sqrt {3}}{2}}$${\frac {1}{2}}$${\sqrt {3}}$${\frac {\sqrt {3}}{3}}$$2$${\frac {2{\sqrt {3}}}{3}}$
${\frac {3\pi }{8}}$$67.5^{\circ }$ ${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$${\sqrt {2}}+1$${\sqrt {2}}-1$${\sqrt {4+2{\sqrt {2}}}}$${\sqrt {4-2{\sqrt {2}}}}$
${\frac {2\pi }{5}}$$72^{\circ }$ ${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$${\frac {{\sqrt {5}}-1}{4}}$${\sqrt {5+2{\sqrt {5}}}}$${\frac {\sqrt {25-10{\sqrt {5}}}}{5}}$$1+{\sqrt {5}}$${\frac {\sqrt {50-10{\sqrt {5}}}}{5}}$
${\frac {5\pi }{12}}$$75^{\circ }$ ${\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$${\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$$2+{\sqrt {3}}$$2-{\sqrt {3}}$${\sqrt {2}}({\sqrt {3}}+1)$${\sqrt {2}}({\sqrt {3}}-1)$
${\frac {\pi }{2}}$$90^{\circ }$ $1$$0$$\infty $$0$$\infty $$1$
${\frac {7\pi }{12}}$$105^{\circ }$ ${\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$$-{\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$$-2-{\sqrt {3}}$$-2+{\sqrt {3}}$$-{\sqrt {2}}(1+{\sqrt {3}})$${\sqrt {2}}({\sqrt {3}}-1)$
${\frac {2\pi }{3}}$$120^{\circ }$ ${\frac {\sqrt {3}}{2}}$$-{\frac {1}{2}}$$-{\sqrt {3}}$$-{\frac {\sqrt {3}}{3}}$$-2$${\frac {2{\sqrt {3}}}{3}}$
${\frac {3\pi }{4}}$$135^{\circ }$ ${\frac {\sqrt {2}}{2}}$$-{\frac {\sqrt {2}}{2}}$$-1$$-1$$-{\sqrt {2}}$${\sqrt {2}}$
${\frac {5\pi }{6}}$$150^{\circ }$ ${\frac {1}{2}}$$-{\frac {\sqrt {3}}{2}}$$-{\frac {\sqrt {3}}{3}}$$-{\sqrt {3}}$$-{\frac {2{\sqrt {3}}}{3}}$$2$
${\frac {11\pi }{12}}$$165^{\circ }$ ${\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$$-{\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$$-2-{\sqrt {3}}$$-2+{\sqrt {3}}$$-{\sqrt {2}}({\sqrt {3}}-1)$${\sqrt {2}}({\sqrt {3}}+1)$
$\pi $$180^{\circ }$ $0$$-1$$0$$\infty $$-1$$\infty $
${\frac {13\pi }{12}}$$195^{\circ }$ $-{\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}$$-{\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}$$2-{\sqrt {3}}$$2+{\sqrt {3}}$$-{\sqrt {2}}({\sqrt {3}}-1)$$-{\sqrt {2}}(1+{\sqrt {3}})$
${\frac {7\pi }{6}}$$210^{\circ }$ $-{\frac {1}{2}}$$-{\frac {\sqrt {3}}{2}}$${\frac {\sqrt {3}}{3}}$${\sqrt {3}}$$-{\frac {2{\sqrt {3}}}{3}}$$-2$
${\frac {5\pi }{4}}$$225^{\circ }$ $-{\dfrac {\sqrt {2}}{2}}$$-{\dfrac {\sqrt {2}}{2}}$$1$$1$$-{\sqrt {2}}$$-{\sqrt {2}}$
${\frac {4\pi }{3}}$$240^{\circ }$ $-{\frac {\sqrt {3}}{2}}$$-{\frac {1}{2}}$${\sqrt {3}}$${\frac {\sqrt {3}}{3}}$$-2$$-{\frac {2{\sqrt {3}}}{3}}$
${\frac {17\pi }{12}}$$255^{\circ }$ $-{\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$$-{\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$$2+{\sqrt {3}}$$2-{\sqrt {3}}$$-{\sqrt {2}}({\sqrt {3}}+1)$$-{\sqrt {2}}({\sqrt {3}}-1)$
${\frac {3\pi }{2}}$$270^{\circ }$ $-1$$0$$\infty $$0$$\infty $$-1$
${\frac {19\pi }{12}}$$285^{\circ }$ $-{\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$${\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$$-2-{\sqrt {3}}$$-2+{\sqrt {3}}$${\sqrt {2}}({\sqrt {3}}+1)$$-{\sqrt {2}}({\sqrt {3}}-1)$
${\frac {5\pi }{3}}$$300^{\circ }$ $-{\frac {\sqrt {3}}{2}}$${\frac {1}{2}}$$-{\sqrt {3}}$$-{\frac {\sqrt {3}}{3}}$$2$$-{\frac {2{\sqrt {3}}}{3}}$
${\frac {7\pi }{4}}$$315^{\circ }$ $-{\frac {\sqrt {2}}{2}}$${\frac {\sqrt {2}}{2}}$$-1$$-1$${\sqrt {2}}$$-{\sqrt {2}}$
${\frac {11\pi }{6}}$$330^{\circ }$ $-{\frac {1}{2}}$${\frac {\sqrt {3}}{2}}$$-{\frac {\sqrt {3}}{3}}$$-{\sqrt {3}}$${\frac {2{\sqrt {3}}}{3}}$$-2$
${\frac {23\pi }{12}}$$345^{\circ }$ $-{\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$${\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$$-2+{\sqrt {3}}$$-2-{\sqrt {3}}$${\sqrt {2}}({\sqrt {3}}-1)$$-{\sqrt {2}}({\sqrt {3}}+1)$
See also
• List of trigonometric identities
References
1. Abramowitz & Stegun 1972, p. 74, 4.3.46
2. Fraleigh, John B. (1994), A First Course in Abstract Algebra (5th ed.), Addison Wesley, ISBN 978-0-201-53467-2, MR 0225619
3. Martin, George E. (1998), Geometric Constructions, Undergraduate Texts in Mathematics, Springer-Verlag, New York, p. 46, doi:10.1007/978-1-4612-0629-3, ISBN 0-387-98276-0, MR 1483895
4. "Exact Value of sin 18°". math-only-math.
5. Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, Abstract Algebra and Famous Impossibilities, Springer, 1991, ISBN 0387976612, p. 178.
6. Callagy, James J. "The central angle of the regular 17-gon", Mathematical Gazette 67, December 1983, 290–292.
7. Niven, Ivan. Numbers: Rational and Irrational, 1961. Random House. New Mathematical Library, Vol. 1. ISSN 0548-5932.
8. Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5 (1): 73–76. doi:10.2307/3026991. JSTOR 3026991.
9. Surgent, Scott (November 2018). "Exact Values of Sine and Cosine of Angles in Increments of 3 Degrees" (PDF). Scott Surgent's ASU Website. Wayback Machine. Archived from the original (PDF) on 2021-05-07.
Bibliography
• Lehmer, D. H. (1933). "A note on trigonometric algebraic numbers". American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023.
• Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. ISBN 978-0-486-61272-0.
• Watkins, William; Zeitlin, Joel (1993). "The minimal polynomial of cos(2*pi/n)". American Mathematical Monthly. 100 (5): 471–474. doi:10.2307/2324301. JSTOR 2324301.
• Girstmair, Kurt (1997). "Some linear relations between values of trigonometric functions at k*pi/n" (PDF). Acta Arithmetica. 81 (4): 387–498. doi:10.4064/aa-81-4-387-398. MR 1472818.
• Conway, John H.; Radin, Charles; Sadun, Lorenzo (1999). "On angles whose squared trigonometric functions are rational". Discrete & Computational Geometry. 22 (3): 321–332. arXiv:math-ph/9812019. doi:10.1007/PL00009463. MR 1706614. S2CID 563915.
• Bracken, Paul; Cizek, Jiri (2002). "evaluation of quanum mechanical perturbative sums in terms of quadratic surds and their use in the approximation of zeta(3)/pi^3". International Journal of Quantum Chemistry: 42–53. doi:10.1002/qua.1803.
• Servi, L. D. (2003). "Nested square roots of 2". American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. JSTOR 3647881.
• Beslin, Scott; de Angelis, Valerio (2004). "The minimal polynomials of sin(2*pi/p) and cos(2*pi/p)". Mathematics Magazine. 77 (2): 146–149. doi:10.1080/0025570X.2004.11953242. JSTOR 3219105. S2CID 118497912.
• Tangsupphathawat, Pinthira; Laohakosol, Vichian (2016). "Minimal polynomials of algebraic cosine values at rational multiples of pi". Journal of Integer Sequences. 19: 16.2.8.
Irrational numbers
• Chaitin's (Ω)
• Liouville
• Prime (ρ)
• Omega
• Cahen
• Logarithm of 2
• Gauss's (G)
• Twelfth root of 2
• Apéry's (ζ(3))
• Plastic (ρ)
• Square root of 2
• Supergolden ratio (ψ)
• Erdős–Borwein (E)
• Golden ratio (φ)
• Square root of 3
• Square root of pi (√π)
• Square root of 5
• Silver ratio (δS)
• Square root of 6
• Square root of 7
• Euler's (e)
• Pi (π)
• Schizophrenic
• Transcendental
• Trigonometric
| Wikipedia |
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series.
Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.
The term trigonometric polynomial for the real-valued case can be seen as using the analogy: the functions sin(nx) and cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of eix, Laurent polynomials in z under the change of variables z = eix.
Formal definition
Any function T of the form
$T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx)\qquad (x\in \mathbb {R} )$
with $a_{n},b_{n}\in \mathbb {C} $ for $0\leq n\leq N$, is called a complex trigonometric polynomial of degree N.[1] Using Euler's formula the polynomial can be rewritten as
$T(x)=\sum _{n=-N}^{N}c_{n}e^{inx}\qquad (x\in \mathbb {R} ).$
Analogously, letting $a_{n},b_{n}\in \mathbb {R} ,\quad 0\leq n\leq N$ and $a_{N}\neq 0$ or $b_{N}\neq 0$, then
$t(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx)\qquad (x\in \mathbb {R} )$
is called a real trigonometric polynomial of degree N.[2]
Properties
A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of 2π, or as a function on the unit circle.
A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm;[3] this is a special case of the Stone–Weierstrass theorem. More concretely, for every continuous function f and every ε > 0, there exists a trigonometric polynomial T such that |f(z) − T(z)| < ε for all z. Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of f converge uniformly to f, provided f is continuous on the circle, thus giving an explicit way to find an approximating trigonometric polynomial T.
A trigonometric polynomial of degree N has a maximum of 2N roots in any interval [a, a + 2π) with a in R, unless it is the zero function.[4]
Notes
1. Rudin 1987, p. 88
2. Powell 1981, p. 150
3. Rudin 1987, Thm 4.25
4. Powell 1981, p. 150
References
• Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7
• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157.
| Wikipedia |
Trigonometric Series
Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric series. The first edition was a single volume, published in 1935 (under the slightly different title Trigonometrical Series). The second edition of 1959 was greatly expanded, taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is similar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square-integrable functions.
This article is about the book. For the mathematical concept, see Trigonometric series.
Trigonometric Series
AuthorAntoni Zygmund
SubjectTrigonometric series
Published1935, 1959, 2002
Publication history
• Zygmund, Antoni (1935). Trigonometrical series. Monogr. Mat. Vol. 5. Warszawa, Lwow: Subwencji Fundusz Kultury Narodowej. Zbl 0011.01703. At icm.edu.pl: original archived
• Zygmund, Antoni (1952). Trigonometrical series. New York: Chelsea Publishing Co. MR 0076084.
• Zygmund, Antoni (1955). Trigonometrical series. New York: Dover Publications. MR 0072976.
• Zygmund, Antoni (1959). Trigonometric series (2nd ed.). Cambridge University Press. MR 0107776. Volume I, Volume II.
• Zygmund, Antoni (1968). Trigonometric series. Second edition, reprinted with corrections and some additions. Vol. I and II (2nd ed.). Cambridge University Press. MR 0236587.
• Zygmund, Antoni (1977). Trigonometric series. Vol. I and II. Cambridge University Press. ISBN 978-0-521-07477-3. MR 0617944.
• Zygmund, Antoni (1988). Trigonometric series. Cambridge Mathematical Library. Vol. I and II. Cambridge University Press. ISBN 978-0-521-35885-9. MR 0933759.
• Zygmund, Antoni (2002). Fefferman, Robert A. (ed.). Trigonometric series. Cambridge Mathematical Library. Vol. I and II (3rd ed.). Cambridge University Press. ISBN 978-0-521-89053-3. MR 1963498.
Reviews
• Kahane, Jean-Pierre (2004), "Book review: Trigonometric series, Vols. I, II", Bulletin of the American Mathematical Society, 41 (3): 377–390, doi:10.1090/s0273-0979-04-01013-4, ISSN 0002-9904
• Salem, Raphael (1960), "Book Review: Trigonometric series", Bulletin of the American Mathematical Society, 66 (1): 6–12, doi:10.1090/S0002-9904-1960-10362-X, ISSN 0002-9904, MR 1566029
• Tamarkin, J. D. (1936), "Zygmund on Trigonometric Series", Bull. Amer. Math. Soc., 42 (1): 11–13, doi:10.1090/s0002-9904-1936-06235-x
| Wikipedia |
Trigonometric series
In mathematics, a trigonometric series is an infinite series of the form
$A_{0}+\sum _{n=1}^{\infty }A_{n}\cos {nx}+B_{n}\sin {nx},$
This article is about the mathematical concept. For the book by Zygmund, see Trigonometric Series.
where $x$ is the variable and $\{A_{n}\}$ and $\{B_{n}\}$ are coefficients. It is an infinite version of a trigonometric polynomial.
A trigonometric series is called the Fourier series of the integrable function $ f$ if the coefficients have the form:
$A_{n}={\frac {1}{\pi }}\int _{0}^{2\pi }\!f(x)\cos {nx}\,dx$
$B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }\!f(x)\sin {nx}\,dx$
Examples
Every Fourier series gives an example of a trigonometric series. Let the function $f(x)=x$ on $[-\pi ,\pi ]$ be extended periodically (see sawtooth wave). Then its Fourier coefficients are:
${\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }x\cos {nx}\,dx=0,\quad n\geq 0.\\[4pt]B_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }x\sin {nx}\,dx\\[4pt]&=-{\frac {x}{n\pi }}\cos {nx}+{\frac {1}{n^{2}\pi }}\sin {nx}{\Bigg \vert }_{x=-\pi }^{\pi }\\[5mu]&={\frac {2\,(-1)^{n+1}}{n}},\quad n\geq 1.\end{aligned}}$
Which gives an example of a trigonometric series:
$2\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin {nx}=2\sin {x}-{\frac {2}{2}}\sin {2x}+{\frac {2}{3}}\sin {3x}-{\frac {2}{4}}\sin {4x}+\cdots $
The converse is false however, not every trigonometric series is a Fourier series. The series
$\sum _{n=2}^{\infty }{\frac {\sin {nx}}{\log {n}}}={\frac {\sin {2x}}{\log {2}}}+{\frac {\sin {3x}}{\log {3}}}+{\frac {\sin {4x}}{\log {4}}}+\cdots $
is a trigonometric series which converges for all $x$ but is not a Fourier series.[1] Here $B_{n}={\frac {1}{\log(n)}}$ for $n\geq 2$ and all other coefficients are zero.
Uniqueness of Trigonometric series
The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function $f(x)$ on the interval $[0,2\pi ]$, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.[2]
Later Cantor proved that even if the set S on which $f$ is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result. Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .[3]
Notes
1. Hardy, Godfrey Harold; Rogosinski, Werner Wolfgang (1956) [1st ed. 1944]. Fourier Series (3rd ed.). Cambridge University Press. pp. 4–5.
2. http://www.math.caltech.edu/papers/uniqueness.pdf
3. Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences, 45 (4): 281–334, doi:10.1007/BF01886630, S2CID 122744778.
References
• Bari, Nina Karlovna (1964). A Treatise on Trigonometric Series. Vol. 1. Translated by Mullins, Margaret F. Pergamon.
• Zygmund, Antoni (1968). Trigonometric Series. Vol. 1 and 2 (2nd, reprinted ed.). Cambridge University Press. MR 0236587.
See also
• Denjoy–Luzin theorem
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
History of trigonometry
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics.[1] Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy.[2] In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).
Trigonometry
• Outline
• History
• Usage
• Functions (inverse)
• Generalized trigonometry
Reference
• Identities
• Exact constants
• Tables
• Unit circle
Laws and theorems
• Sines
• Cosines
• Tangents
• Cotangents
• Pythagorean theorem
Calculus
• Trigonometric substitution
• Integrals (inverse functions)
• Derivatives
Etymology
The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".[3]
The modern words "sine" and "cosine" are derived from the Latin word sinus via mistranslation from Arabic (see Sine and cosine#Etymology). Particularly Fibonacci's sinus rectus arcus proved influential in establishing the term.[4]
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans "cutting" since the line cuts the circle.[5]
The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.[6][7]
The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae.[8] These roughly translate to "first small parts" and "second small parts".
Development
Ancient Near East
The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.[9]
The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.[10] Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants but does not work in this context as without using circles and angles in the situation modern trigonometric notations won't apply.[11] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.
The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.[10] The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:[10]
"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"
Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[10]
Classical antiquity
Ancient Greek and Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of one half the bisected angle, that is,[12]
$\mathrm {chord} \ \theta =2r\sin {\frac {\theta }{2}},$
and consequently the sine function is also known as the half-chord. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.[13][14]
Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas.[9] For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the law of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles.[9] To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the statement that, in modern notation, sin α/sin β < α/β < tan α/tan β whenever 0° < β < α < 90°, now known as Aristarchus's inequality.[15]
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry."[16] Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.[4][16]
Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 BC), since he measured an angle in terms of a fraction of a quadrant.[15] It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy.[17] In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts.[8] It is due to the Babylonian sexagesimal numeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.[8]
Menelaus of Alexandria (ca. 100 AD) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.[14] He established a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.[14] Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.[14] Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".[14] He further gave his famous "rule of six quantities".[18]
Later, Claudius Ptolemy (ca. 90 – ca. 168 AD) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. Ptolemy's table of chords gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees n in the corresponding arc of the circle, for n ranging from 1/2 to 180 by increments of 1/2.[19] The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity.[20] A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.[20] Ptolemy further derived the equivalent of the half-angle formula
$\sin ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos(x)}{2}}.$[20]
Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.[20]
Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.[21]
Pythagoras discovered many of the properties of what would become trigonometric functions. The Pythagorean Theorem, p2 + b2 = h2 is a representation of the fundamental trigonometric identity sin2(x) + cos2(x) = 1. The length 1 is the hypotenuse of any right triangle, and has legs length sin(x) and cos(x) with x being one of the two non-right angles. With this in mind, the identity upon which trigonometry is based turns out to be the Pythagorean Theorem.
Indian mathematics
See also: Indian Mathematics and Indian astronomy
Some of the early and very significant developments of trigonometry were in India. Influential works from the 4th–5th century AD, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta[22]) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine.[23] Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya.[24] The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.[25] They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.
In the 7th century, Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%:
$\sin x\approx {\frac {16x(\pi -x)}{5\pi ^{2}-4x(\pi -x)}},\qquad \left(0\leq x\leq \pi \right).$
Later in the 7th century, Brahmagupta redeveloped the formula
$\ 1-\sin ^{2}(x)=\cos ^{2}(x)=\sin ^{2}\left({\frac {\pi }{2}}-x\right)$
(also derived earlier, as mentioned above) and the Brahmagupta interpolation formula for computing sine values.[11]
Another later Indian author on trigonometry was Bhaskara II in the 12th century. Bhaskara II developed spherical trigonometry, and discovered many trigonometric results.
Bhaskara II was the one of the first to discover $\sin \left(a+b\right)$ and $\sin \left(a-b\right)$ trigonometric results like:
• $\sin \left(a+b\right)=\sin a\cos b+\cos a\sin b$
Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the power series expansions of sine, cosine, tangent, and arctangent.[26][27] Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the angle, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.[26][27]
No. Series Name Western discoverers of the series
and approximate dates of discovery[28]
1 sin x = x − x3 / 3! + x5 / 5! − x7 / 7! + ... Madhava's sine series Isaac Newton (1670) and Wilhelm Leibniz (1676)
2 cos x = 1 − x2 / 2! + x4 / 4! − x6 / 6! + ... Madhava's cosine series Isaac Newton (1670) and Wilhelm Leibniz (1676)
3 tan−1x = x − x3 / 3 + x5 / 5 − x7 / 7 + ... Madhava's arctangent series James Gregory (1671) and Wilhelm Leibniz (1676)
The Indian text the Yuktibhāṣā contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.
Chinese mathematics
In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty.[29] Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek, Hellenistic, Indian and Islamic worlds.[30] Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.[29] However, this embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.[29] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[29] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc s of a circle given the diameter d, sagitta v, and length c of the chord subtending the arc, the length of which he approximated as[31]
$s=c+{\frac {2v^{2}}{d}}.$
Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).[32] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[29][33] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:
Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).[34]
Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).[35]
Medieval Islamic world
Previous works were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."[36]
Methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems.[14][37] However, E. S. Kennedy points out that while it was possible in pre-Islamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice.[38] In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.[39]
In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry. In 830 AD, Habash al-Hasib al-Marwazi produced the first table of cotangents.[40][41] Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[41]
By the 10th century AD, in the work of Abū al-Wafā' al-Būzjānī, all six trigonometric functions were used.[42] Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values.[42] He also developed the following trigonometric formula:[43]
$\ \sin(2x)=2\sin(x)\cos(x)$ (a special case of Ptolemy's angle-addition formula; see above)
In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine minutes, and the result is half the sine of the double".[43] Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:[43]
$\sin(\alpha \pm \beta )={\sqrt {\sin ^{2}\alpha -(\sin \alpha \sin \beta )^{2}}}\pm {\sqrt {\sin ^{2}\beta -(\sin \alpha \sin \beta )^{2}}}$
$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta $
For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference".[43]
He also discovered the law of sines for spherical trigonometry:[40]
${\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.$
Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity:[44]
$\cos a\cos b={\frac {\cos(a+b)+\cos(a-b)}{2}}$
Al-Jayyani (989–1079) of al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry".[45] It "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.[45]
The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying[46] and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni himself introduced triangulation techniques to measure the size of the Earth and the distances between various places.[47] In the late 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.[41] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.[48] Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.[49][50][51]
In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a form suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi. He also gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg also gives accurate tables of sines and tangents correct to 8 decimal places around the same time.
European renaissance and afterwards
In 1342, Levi ben Gershon, known as Gersonides, wrote On Sines, Chords and Arcs, in particular proving the sine law for plane triangles and giving five-figure sine tables.[52]
A simplified trigonometric table, the "toleta de marteloio", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venetian captain Andrea Bianco.
Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline,[53] in his De triangulis omnimodis written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed. The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In the 17th century, Isaac Newton and James Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions.
In the 18th century, Leonhard Euler's Introduction in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "Euler's formula" eix = cos x + i sin x. Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Prior to this, Roger Cotes had computed the derivative of sine in his Harmonia Mensurarum (1722).[54] Also in the 18th century, Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.
See also
• Greek mathematics
• History of mathematics
• Trigonometric functions
• Trigonometry
• Ptolemy's table of chords
• Aryabhata's sine table
• Rational trigonometry
Citations and footnotes
1. Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag. p. 744. ISBN 978-3-540-06995-9.
2. Katz 1998, p. 212.
3. "trigonometry". Online Etymology Dictionary.
4. O'Connor, J.J.; Robertson, E.F. (1996). "Trigonometric functions". MacTutor History of Mathematics Archive. Archived from the original on 2007-06-04.
5. Oxford English Dictionary
6. Gunter, Edmund (1620). Canon triangulorum.
7. Roegel, Denis, ed. (6 December 2010). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. Archived from the original on 28 July 2017. Retrieved 28 July 2017.
8. Boyer 1991, pp. 166–167, Greek Trigonometry and Mensuration: "It should be recalled that form the days of Hipparchus until modern times there were no such things as trigonometric ratios. The Greeks, and after them the Hindus and the Arabs, used trigonometric lines. These at first took the form, as we have seen, of chords in a circle, and it became incumbent upon Ptolemy to associate numerical values (or approximations) with the chords. [...] It is not unlikely that the 260-degree measure was carried over from astronomy, where the zodiac had been divided into twelve "signs" or 36 "decans". A cycle of the seasons of roughly 360 days could readily be made to correspond to the system of zodiacal signs and decans by subdividing each sign into thirty parts and each decan into ten parts. Our common system of angle measure may stem from this correspondence. Moreover since the Babylonian position system for fractions was so obviously superior to the Egyptians unit fractions and the Greek common fractions, it was natural for Ptolemy to subdivide his degrees into sixty partes minutae primae, each of these latter into sixty partes minutae secundae, and so on. It is from the Latin phrases that translators used in this connection that our words "minute" and "second" have been derived. It undoubtedly was the sexagesimal system that led Ptolemy to subdivide the diameter of his trigonometric circle into 120 parts; each of these he further subdivided into sixty minutes and each minute of length sixty seconds."
9. Boyer 1991, pp. 158–159, Greek Trigonometry and Mensuration: "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry", or the measure of three sided polygons (trilaterals), than "trigonometry", the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the Elements, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles."
10. Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 978-0-691-09541-7.
11. Joseph 2000, pp. 383–384.
12. Katz 1998, p. 143.
13. As these historical calculations did not make use of a unit circle, the length of the radius was needed in the formula. Contrast this with the modern use of the crd function that assumes a unit circle in its definition.
14. Boyer 1991, p. 163, Greek Trigonometry and Mensuration: "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue – that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form – a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle)."
15. Boyer 1991, p. 159, Greek Trigonometry and Mensuration: "Instead we have an treatise, perhaps composed earlier (ca. 260 BC), On the Sizes and Distances of the Sun and Moon, which assumes a geocentric universe. In this work Aristarchus made the observation that when the moon is just half-full, the angle between the lines of sight to the sun and the moon is less than a right angle by one thirtieth of a quadrant. (The systematic introduction of the 360° circle came a little later. In trigonometric language of today this would mean that the ratio of the distance of the moon to that of the sun (the ration ME to SE in Fig. 10.1) is sin(3°). Trigonometric tables not having been developed yet, Aristarchus fell back upon a well-known geometric theorem of the time which now would be expressed in the inequalities sin α/ sin β < α/β < tan α/ tan β, for 0° < β < α < 90°.)"
16. Boyer 1991, p. 162, Greek Trigonometry and Mensuration: "For some two and a half centuries, from Hippocrates to Eratosthenes, Greek mathematicians had studied relationships between lines and circles and had applied these in a variety of astronomical problems, but no systematic trigonometry had resulted. Then, presumably during the second half of the 2nd century BC, the first trigonometric table apparently was compiled by the astronomer Hipparchus of Nicaea (ca. 180–ca. 125 BC), who thus earned the right to be known as "the father of trigonometry". Aristarchus had known that in a given circle the ratio of arc to chord decreases as the arc decreases from 180° to 0°, tending toward a limit of 1. However, it appears that not until Hipparchus undertook the task had anyone tabulated corresponding values of arc and chord for a whole series of angles."
17. Boyer 1991, p. 162, Greek Trigonometry and Mensuration: "It is not known just when the systematic use of the 360° circle came into mathematics, but it seems to be due largely to Hipparchus in connection with his table of chords. It is possible that he took over from Hypsicles, who earlier had divided the day into parts, a subdivision that may have been suggested by Babylonian astronomy."
18. Needham 1986, p. 108.
19. Toomer, Gerald J. (1998). Ptolemy's Almagest. Princeton University Press. ISBN 978-0-691-00260-6.
20. Boyer 1991, pp. 164–166, Greek Trigonometry and Mensuration: "The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. [...] Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from AD. 127 to 151 and, therefore, assume that he was born at the end of the 1st century. Suidas, a writer who lived in the 10th century, reported that Ptolemy was alive under Marcus Aurelius (emperor from AD 161 to 180).
Ptolemy's Almagest is presumed to be heavily indebted for its methods to the Chords in a Circle of Hipparchus, but the extent of the indebtedness cannot be reliably assessed. It is clear that in astronomy Ptolemy made use of the catalog of star positions bequeathed by Hipparchus, but whether or not Ptolemy's trigonometric tables were derived in large part from his distinguished predecessor cannot be determined. [...] Central to the calculation of Ptolemy's chords was a geometric proposition still known as "Ptolemy's theorem": [...] that is, the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. [...] A special case of Ptolemy's theorem had appeared in Euclid's Data (Proposition 93): [...] Ptolemy's theorem, therefore, leads to the result sin(α − β) = sin α cos β − cos α sin Β. Similar reasoning leads to the formula [...] These four sum-and-difference formulas consequently are often known today as Ptolemy's formulas.
It was the formula for sine of the difference – or, more accurately, chord of the difference – that Ptolemy found especially useful in building up his tables. Another formula that served him effectively was the equivalent of our half-angle formula."
21. Boyer 1991, pp. 158–168.
22. Boyer 1991, p. 208.
23. Boyer 1991, p. 209.
24. Boyer 1991, p. 210.
25. Boyer 1991, p. 215.
26. O'Connor, J.J.; Robertson, E.F. (2000). "Madhava of Sangamagramma". MacTutor History of Mathematics Archive.
27. Pearce, Ian G. (2002). "Madhava of Sangamagramma". MacTutor History of Mathematics Archive.
28. Charles Henry Edwards (1994). The historical development of the calculus. Springer Study Edition Series (3 ed.). Springer. p. 205. ISBN 978-0-387-94313-8.
29. Needham 1986, p. 109.
30. Needham 1986, pp. 108–109.
31. Katz 2007, p. 308.
32. Restivo 1992, p. 32.
33. Gauchet, L. (1917). Note Sur La Trigonométrie Sphérique de Kouo Cheou-King. p. 151.
34. Needham 1986, pp. 109–110.
35. Needham 1986, p. 110.
36. Kennedy, E. S. (1969). "The History of Trigonometry". 31st Yearbook. Washington DC: National Council of Teachers of Mathematics. (cf. Haq, Syed Nomanul (1996). "The Indian and Persian background". In Seyyed Hossein Nasr; Oliver Leaman (eds.). History of Islamic Philosophy. Routledge. pp. 52–70 [60–63]. ISBN 978-0-415-13159-9.)
37. O'Connor, John J.; Robertson, Edmund F., "Menelaus of Alexandria", MacTutor History of Mathematics Archive, University of St Andrews "Book 3 deals with spherical trigonometry and includes Menelaus's theorem".
38. Kennedy, E. S. (1969). "The History of Trigonometry". 31st Yearbook. Washington DC: National Council of Teachers of Mathematics: 337. (cf. Haq, Syed Nomanul (1996). "The Indian and Persian background". In Seyyed Hossein Nasr; Oliver Leaman (eds.). History of Islamic Philosophy. Routledge. pp. 52–70 [68]. ISBN 978-0-415-13159-9.)
39. Gingerich, Owen (April 1986). "Islamic astronomy". Scientific American. 254 (10): 74. Bibcode:1986SciAm.254d..74G. doi:10.1038/scientificamerican0486-74. Archived from the original on 2011-01-01. Retrieved 2008-05-18.
40. Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media. ISBN 978-1-4020-0260-1.
41. "trigonometry". Encyclopædia Britannica. Retrieved 2008-07-21.
42. Boyer 1991, p. 238.
43. Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1–56. doi:10.1017/S095742391000007X. S2CID 171015175.
44. William Charles Brice, 'An Historical atlas of Islam', p.413
45. O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics Archive, University of St Andrews
46. Donald Routledge Hill (1996), "Engineering", in Roshdi Rashed, Encyclopedia of the History of Arabic Science, Vol. 3, p. 751–795 [769].
47. O'Connor, John J.; Robertson, Edmund F., "Abu Arrayhan Muhammad ibn Ahmad al-Biruni", MacTutor History of Mathematics Archive, University of St Andrews
48. Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
49. "Al-Tusi_Nasir biography". www-history.mcs.st-andrews.ac.uk. Retrieved 2018-08-05. One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
50. Berggren, J. L. (October 2013). "Islamic Mathematics". The Cambridge History of Science. Cambridge University Press. pp. 62–83. doi:10.1017/CHO9780511974007.004. ISBN 978-0-511-97400-7.
51. electricpulp.com. "ṬUSI, NAṢIR-AL-DIN i. Biography – Encyclopaedia Iranica". www.iranicaonline.org. Retrieved 2018-08-05. His major contribution in mathematics (Nasr, 1996, pp. 208-214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
52. Charles G. Simonson (Winter 2000). "The Mathematics of Levi ben Gershon, the Ralbag" (PDF). Bekhol Derakhekha Daehu. Bar-Ilan University Press. 10: 5–21.
53. Boyer 1991, p. 274.
54. Katz, Victor J. (November 1987). "The calculus of the trigonometric functions". Historia Mathematica. 14 (4): 311–324. doi:10.1016/0315-0860(87)90064-4.. The proof of Cotes is mentioned on p. 315.
References
• Boyer, Carl Benjamin (1991). A History of Mathematics (2nd ed.). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
• Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd ed.). London: Penguin Books. ISBN 978-0-691-00659-8.
• Katz, Victor J. (1998). A History of Mathematics / An Introduction (2nd ed.). Addison Wesley. ISBN 978-0-321-01618-8.
• Katz, Victor J. (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press. ISBN 978-0-691-11485-9.
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
• Restivo, Sal (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ISBN 1-4020-0039-1.
Further reading
• Braunmühl, Anton von (1900–1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the History of Trigonometry] (in German). B. G. Teubner.
• Kennedy, Edward S. (1969). "The History of Trigonometry". Historical Topics for the Mathematics Classroom. NCTM Yearbooks. Vol. 31. National Council of Teachers of Mathematics. pp. 333–375.
• Maor, Eli (1998). Trigonometric Delights. Princeton University Press. doi:10.1515/9780691202204. ISBN 0691057540. Archived from the original on 2003-07-11.
• Ostermann, Alexander; Wanner, Gerhard (2012). "Trigonometry". Geometry by Its History. Springer. pp. 113–155. doi:10.1007/978-3-642-29163-0.
• Van Brummelen, Glen (2009). The Mathematics of the Heavens and the Earth: The Early History of Trigonometry. Princeton University Press.
• Van Brummelen, Glen (2021). The Doctrine of Triangles: A History of Modern Trigonometry. Princeton University Press.
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Mathematics in the medieval Islamic world
Mathematicians
9th century
• 'Abd al-Hamīd ibn Turk
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• Yusuf al-Khuri
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• Abu Said Gorgani
10th century
• Abu al-Wafa
• al-Khazin
• Al-Qabisi
• Abu Kamil
• Ahmad ibn Yusuf
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• Sinān ibn al-Fatḥ
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• Ibn Sahl
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• Abu al-Jud
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11th century
• Abu Nasr Mansur
• Alhazen
• Kushyar Gilani
• Al-Biruni
• Ibn al-Samh
• Abu Mansur al-Baghdadi
• Avicenna
• al-Jayyānī
• al-Nasawī
• al-Zarqālī
• ibn Hud
• Al-Isfizari
• Omar Khayyam
• Muhammad al-Baghdadi
12th century
• Jabir ibn Aflah
• Al-Kharaqī
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• Al-Samawal al-Maghribi
• al-Hassar
• Sharaf al-Din al-Tusi
• Ibn al-Yasamin
13th century
• Ibn al‐Ha'im al‐Ishbili
• Ahmad al-Buni
• Ibn Munim
• Alam al-Din al-Hanafi
• Ibn Adlan
• al-Urdi
• Nasir al-Din al-Tusi
• al-Abhari
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• al-Hasan al-Marrakushi
• Qutb al-Din al-Shirazi
• Shams al-Din al-Samarqandi
• Ibn al-Banna'
• Kamāl al-Dīn al-Fārisī
14th century
• Nizam al-Din al-Nisapuri
• Ibn al-Shatir
• Ibn al-Durayhim
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15th century
• Ibn al-Majdi
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• Ulugh Beg
• Ali Qushji
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16th century
• Al-Birjandi
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• Taqi ad-Din
• Ibn Hamza al-Maghribi
• Ahmad Ibn al-Qadi
Mathematical
works
• The Compendious Book on Calculation by Completion and Balancing
• De Gradibus
• Principles of Hindu Reckoning
• Book of Optics
• The Book of Healing
• Almanac
• Book on the Measurement of Plane and Spherical Figures
• Encyclopedia of the Brethren of Purity
• Toledan Tables
• Tabula Rogeriana
• Zij
Concepts
• Alhazen's problem
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Influences
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Influenced
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Related
• Hindu–Arabic numeral system
• Arabic numerals (Eastern Arabic numerals, Western Arabic numerals)
• Trigonometric functions
• History of trigonometry
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| Wikipedia |
Trigonometry of a tetrahedron
The trigonometry of a tetrahedron[1] explains the relationships between the lengths and various types of angles of a general tetrahedron.
Trigonometric quantities
Classical trigonometric quantities
The following are trigonometric quantities generally associated to a general tetrahedron:
• The 6 edge lengths - associated to the six edges of the tetrahedron.
• The 12 face angles - there are three of them for each of the four faces of the tetrahedron.
• The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge.
• The 4 solid angles - associated to each point of the tetrahedron.
Let $X={\overline {P_{1}P_{2}P_{3}P_{4}}}$ be a general tetrahedron, where $P_{1},P_{2},P_{3},P_{4}$ are arbitrary points in three-dimensional space.
Furthermore, let $e_{ij}$ be the edge that joins $P_{i}$ and $P_{j}$ and let $F_{i}$ be the face of the tetrahedron opposite the point $P_{i}$; in other words:
• $e_{ij}={\overline {P_{i}P_{j}}}$
• $F_{i}={\overline {P_{j}P_{k}P_{l}}}$
where $i,j,k,l\in \{1,2,3,4\}$ and $i\neq j\neq k\neq l$.
Define the following quantities:
• $d_{ij}$ = the length of the edge $e_{ij}$
• $\alpha _{i,j}$ = the face angle at the point $P_{i}$ on the face $F_{j}$
• $\theta _{ij}$ = the dihedral angle between two faces adjacent to the edge $e_{ij}$
• $\Omega _{i}$ = the solid angle at the point $P_{i}$
Area and volume
Let $\Delta _{i}$ be the area of the face $F_{i}$. Such area may be calculated by Heron's formula (if all three edge lengths are known):
$\Delta _{i}={\sqrt {\frac {(d_{jk}+d_{jl}+d_{kl})(-d_{jk}+d_{jl}+d_{kl})(d_{jk}-d_{jl}+d_{kl})(d_{jk}+d_{jl}-d_{kl})}{16}}}$
or by the following formula (if an angle and two corresponding edges are known):
$\Delta _{i}={\frac {1}{2}}d_{jk}d_{jl}\sin \alpha _{j,i}$
Let $h_{i}$ be the altitude from the point $P_{i}$ to the face $F_{i}$. The volume $V$ of the tetrahedron $X$ is given by the following formula:
$V={\frac {1}{3}}\Delta _{i}h_{i}$
It satisfies the following relation:[2]
$288V^{2}={\begin{vmatrix}2Q_{12}&Q_{12}+Q_{13}-Q_{23}&Q_{12}+Q_{14}-Q_{24}\\Q_{12}+Q_{13}-Q_{23}&2Q_{13}&Q_{13}+Q_{14}-Q_{34}\\Q_{12}+Q_{14}-Q_{24}&Q_{13}+Q_{14}-Q_{34}&2Q_{14}\end{vmatrix}}$
where $Q_{ij}=d_{ij}^{2}$ are the quadrances (length squared) of the edges.
Basic statements of trigonometry
Affine triangle
Take the face $F_{i}$; the edges will have lengths $d_{jk},d_{jl},d_{kl}$ and the respective opposite angles are given by $\alpha _{l,i},\alpha _{k,i},\alpha _{j,i}$.
The usual laws for planar trigonometry of a triangle hold for this triangle.
Projective triangle
Consider the projective (spherical) triangle at the point $P_{i}$; the vertices of this projective triangle are the three lines that join $P_{i}$ with the other three vertices of the tetrahedron. The edges will have spherical lengths $\alpha _{i,j},\alpha _{i,k},\alpha _{i,l}$ and the respective opposite spherical angles are given by $\theta _{ij},\theta _{ik},\theta _{il}$.
The usual laws for spherical trigonometry hold for this projective triangle.
Laws of trigonometry for the tetrahedron
Alternating sines theorem
Take the tetrahedron $X$, and consider the point $P_{i}$ as an apex. The Alternating sines theorem is given by the following identity:
$\sin(\alpha _{j,l})\sin(\alpha _{k,j})\sin(\alpha _{l,k})=\sin(\alpha _{j,k})\sin(\alpha _{k,l})\sin(\alpha _{l,j})$
One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.
The space of all shapes of tetrahedra
Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent; if the "clockwise" sides of three of the four identities are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.
Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by the sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.[3]
Law of sines for the tetrahedron
See: Law of sines
Law of cosines for the tetrahedron
The law of cosines for the tetrahedron[4] relates the areas of each face of the tetrahedron and the dihedral angles about a point. It is given by the following identity:
$\Delta _{i}^{2}=\Delta _{j}^{2}+\Delta _{k}^{2}+\Delta _{l}^{2}-2(\Delta _{j}\Delta _{k}\cos \theta _{il}+\Delta _{j}\Delta _{l}\cos \theta _{ik}+\Delta _{k}\Delta _{l}\cos \theta _{ij})$
Relationship between dihedral angles of tetrahedron
Take the general tetrahedron $X$ and project the faces $F_{i},F_{j},F_{k}$ onto the plane with the face $F_{l}$. Let $c_{ij}=\cos \theta _{ij}$.
Then the area of the face $F_{l}$ is given by the sum of the projected areas, as follows:
$\Delta _{l}=\Delta _{i}c_{jk}+\Delta _{j}c_{ik}+\Delta _{k}c_{ij}$
By substitution of $i,j,k,l\in \{1,2,3,4\}$ with each of the four faces of the tetrahedron, one obtains the following homogeneous system of linear equations:
${\begin{cases}-\Delta _{1}+\Delta _{2}c_{34}+\Delta _{3}c_{24}+\Delta _{4}c_{23}=0\\\Delta _{1}c_{34}-\Delta _{2}+\Delta _{3}c_{14}+\Delta _{4}c_{13}=0\\\Delta _{1}c_{24}+\Delta _{2}c_{14}-\Delta _{3}+\Delta _{4}c_{12}=0\\\Delta _{1}c_{23}+\Delta _{2}c_{13}+\Delta _{3}c_{12}-\Delta _{4}=0\end{cases}}$
This homogeneous system will have solutions precisely when:
${\begin{vmatrix}-1&c_{34}&c_{24}&c_{23}\\c_{34}&-1&c_{14}&c_{13}\\c_{24}&c_{14}&-1&c_{12}\\c_{23}&c_{13}&c_{12}&-1\end{vmatrix}}=0$
By expanding this determinant, one obtains the relationship between the dihedral angles of the tetrahedron,[1] as follows:
$1-\sum _{1\leq i<j\leq 4}c_{ij}^{2}+\sum _{j=2 \atop k\neq l\neq j}^{4}c_{1j}^{2}c_{kl}^{2}=2\left(\sum _{i=1 \atop j\neq k\neq l\neq i}^{4}c_{ij}c_{ik}c_{il}+\sum _{2\leq j<k\leq 4 \atop l\neq j,k}c_{1j}c_{1k}c_{jl}c_{kl}\right)$
Skew distances between edges of tetrahedron
Take the general tetrahedron $X$ and let $P_{ij}$ be the point on the edge $e_{ij}$ and $P_{kl}$ be the point on the edge $e_{kl}$ such that the line segment ${\overline {P_{ij}P_{kl}}}$ is perpendicular to both $e_{ij}$ & $e_{kl}$. Let $R_{ij}$ be the length of the line segment ${\overline {P_{ij}P_{kl}}}$.
To find $R_{ij}$:[1]
First, construct a line through $P_{k}$ parallel to $e_{il}$ and another line through $P_{i}$ parallel to $e_{kl}$. Let $O$ be the intersection of these two lines. Join the points $O$ and $P_{j}$. By construction, ${\overline {OP_{i}P_{l}P_{k}}}$ is a parallelogram and thus ${\overline {OP_{k}P_{i}}}$ and ${\overline {OP_{l}P_{i}}}$ are congruent triangles. Thus, the tetrahedron $X$ and $Y={\overline {OP_{i}P_{j}P_{k}}}$ are equal in volume.
As a consequence, the quantity $R_{ij}$ is equal to the altitude from the point $P_{k}$ to the face ${\overline {OP_{i}P_{j}}}$ of the tetrahedron $Y$; this is shown by translation of the line segment ${\overline {P_{ij}P_{kl}}}$.
By the volume formula, the tetrahedron $Y$ satisfies the following relation:
$3V=R_{ij}\times \Delta ({\overline {OP_{i}P_{j}}})$
where $\Delta ({\overline {OP_{i}P_{j}}})$ is the area of the triangle ${\overline {OP_{i}P_{j}}}$. Since the length of the line segment ${\overline {OP_{i}}}$ is equal to $d_{kl}$ (as ${\overline {OP_{i}P_{l}P_{k}}}$ is a parallelogram):
$\Delta ({\overline {OP_{i}P_{j}}})={\frac {1}{2}}d_{ij}d_{kl}\sin \lambda $
where $\lambda =\angle OP_{i}P_{j}$. Thus, the previous relation becomes:
$6V=R_{ij}d_{ij}d_{kl}\sin \lambda $
To obtain $\sin \lambda $, consider two spherical triangles:
1. Take the spherical triangle of the tetrahedron $X$ at the point $P_{i}$; it will have sides $\alpha _{i,j},\alpha _{i,k},\alpha _{i,l}$ and opposite angles $\theta _{ij},\theta _{ik},\theta _{il}$. By the spherical law of cosines:
$\cos \alpha _{i,k}=\cos \alpha _{i,j}\cos \alpha _{i,l}+\sin \alpha _{i,j}\sin \alpha _{i,l}\cos \theta _{ik}$
2. Take the spherical triangle of the tetrahedron $X$ at the point $P_{i}$. The sides are given by $\alpha _{i,l},\alpha _{k,j},\lambda $ and the only known opposite angle is that of $\lambda $, given by $\pi -\theta _{ik}$. By the spherical law of cosines:
$\cos \lambda =\cos \alpha _{i,l}\cos \alpha _{k,j}-\sin \alpha _{i,l}\sin \alpha _{k,j}\cos \theta _{ik}$
Combining the two equations gives the following result:
$\cos \alpha _{i,k}\sin \alpha _{k,j}+\cos \lambda \sin \alpha _{i,j}=\cos \alpha _{i,l}\left(\cos \alpha _{i,j}\sin \alpha _{k,j}+\sin \alpha _{i,j}\cos \alpha _{k,j}\right)=\cos \alpha _{i,l}\sin \alpha _{l,j}$
Making $\cos \lambda $ the subject:
$\cos \lambda =\cos \alpha _{i,l}{\frac {\sin \alpha _{l,j}}{\sin \alpha _{i,j}}}-\cos \alpha _{i,k}{\frac {\sin \alpha _{k,j}}{\sin \alpha _{i,j}}}$
Thus, using the cosine law and some basic trigonometry:
$\cos \lambda ={\frac {d_{ij}^{2}+d_{ik}^{2}-d_{jk}^{2}}{2d_{ij}d_{ik}}}{\frac {d_{ik}}{d_{kl}}}-{\frac {d_{ij}^{2}+d_{il}^{2}-d_{jl}^{2}}{2d_{ij}d_{il}}}{\frac {d_{il}}{d_{kl}}}={\frac {d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2}}{2d_{ij}d_{kl}}}$
Thus:
$\sin \lambda ={\sqrt {1-\left({\frac {d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2}}{2d_{ij}d_{kl}}}\right)^{2}}}={\frac {\sqrt {4d_{ij}^{2}d_{kl}^{2}-(d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2})^{2}}}{2d_{ij}d_{kl}}}$
So:
$R_{ij}={\frac {12V}{\sqrt {4d_{ij}^{2}d_{kl}^{2}-(d_{ik}^{2}+d_{jl}^{2}-d_{il}^{2}-d_{jk}^{2})^{2}}}}$
$R_{ik}$ and $R_{il}$ are obtained by permutation of the edge lengths.
Note that the denominator is a re-formulation of the Bretschneider-von Staudt formula, which evaluates the area of a general convex quadrilateral.
References
1. Richardson, G. (1902-03-01). "The Trigonometry of the Tetrahedron". The Mathematical Gazette. 2 (32): 149–158. doi:10.2307/3603090. JSTOR 3603090.
2. 100 Great Problems of Elementary Mathematics. New York: Dover Publications. 1965-06-01. ISBN 9780486613482.
3. Rassat, André; Fowler, Patrick W. (2004). "Is There a "Most Chiral Tetrahedron"?". Chemistry: A European Journal. 10 (24): 6575–6580. doi:10.1002/chem.200400869. PMID 15558830.
4. Lee, Jung Rye (June 1997). "The law of cosines in a tetrahedron". J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 4 (1): 1–6. ISSN 1226-0657.
| Wikipedia |
Trigyrate rhombicosidodecahedron
In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (J75). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron.
Trigyrate rhombicosidodecahedron
TypeJohnson
J74 – J75 – J76
Faces2+2x3+2x6 triangles
4x3+3x6 squares
4x3 pentagons
Edges120
Vertices60
Vertex configuration5x6(3.42.5)
4x3+3x6(3.4.5.4)
Symmetry groupC3v
Dual polyhedron-
Propertiesconvex, canonical
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees. Related Johnson solids are:
• The gyrate rhombicosidodecahedron (J72) where one cupola is rotated;
• The parabigyrate rhombicosidodecahedron (J73) where two opposing cupolae are rotated;
• And the metabigyrate rhombicosidodecahedron (J74) where two non-opposing cupolae are rotated.
References
• Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.
External links
• Eric W. Weisstein, Trigyrate rhombicosidodecahedron (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
| Wikipedia |
Tridecagon
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.
Regular tridecagon
A regular tridecagon
TypeRegular polygon
Edges and vertices13
Schläfli symbol{13}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D13), order 2×13
Internal angle (degrees)≈152.308°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular tridecagon
A regular tridecagon is represented by Schläfli symbol {13}.
The measure of each internal angle of a regular tridecagon is approximately 152.308 degrees, and the area with side length a is given by
$A={\frac {13}{4}}a^{2}\cot {\frac {\pi }{13}}\simeq 13.1858\,a^{2}.$
Construction
As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisector.
The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle ${\overline {OA}}=12,$ according to Andrew M. Gleason,[1] based on the angle trisection by means of the Tomahawk (light blue).
An approximate construction of a regular tridecagon using straightedge and compass is shown here.
Another possible animation of an approximate construction, also possible with using straightedge and compass.
Based on the unit circle r = 1 [unit of length]
• Constructed side length in GeoGebra $a=0.478631328575115\;[{\text{unit of length}}]$
• Side length of the tridecagon $a_{\text{target}}=r\cdot 2\cdot \sin \left({\frac {180^{\circ }}{13}}\right)=0.478631328575115\ldots \;[{\text{unit of length}}]$
• Absolute error of the constructed side length:
Up to the maximum precision of 15 decimal places, the absolute error is $F_{a}=a-a_{\text{target}}=0.0\;[{\text{unit of length}}]$
• Constructed central angle of the tridecagon in GeoGebra (display significant 13 decimal places, rounded) $\mu =27.6923076923077^{\circ }$
• Central angle of tridecagon $\mu _{\text{target}}=\left({\frac {360^{\circ }}{13}}\right)=27.{\overline {692307}}^{\circ }$
• Absolute angular error of the constructed central angle:
Up to 13 decimal places, the absolute error is $F_{\mu }=\mu -\mu _{\text{target}}=0.0^{\circ }$
Example to illustrate the error
At a circumscribed circle of radius r = 1 billion km (a distance which would take light approximately 55 minutes to travel), the absolute error on the side length constructed would be less than 1 mm.
Symmetry
The regular tridecagon has Dih13 symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z13, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order.[2] Full symmetry of the regular form is r26 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g13 subgroup has no degrees of freedom but can seen as directed edges.
Numismatic use
The regular tridecagon is used as the shape of the Czech 20 korun coin.[3]
Related polygons
A tridecagram is a 13-sided star polygon. There are 5 regular forms given by Schläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}. Since 13 is prime, none of the tridecagrams are compound figures.
Tridecagrams
Picture
{13/2}
{13/3}
{13/4}
{13/5}
{13/6}
Internal angle ≈124.615°≈96.9231°≈69.2308°≈41.5385°≈13.8462°
Petrie polygons
The regular tridecagon is the Petrie polygon 12-simplex:
A12
12-simplex
References
1. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon p. 192–194 (p. 193 Fig.4)" (PDF). The American Mathematical Monthly. 95 (3): 186–194. doi:10.2307/2323624. Archived from the original (PDF) on 2015-12-19. Retrieved 24 December 2015.
2. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
3. Colin R. Bruce, II, George Cuhaj, and Thomas Michael, 2007 Standard Catalog of World Coins, Krause Publications, 2006, ISBN 0896894290, p. 81.
External links
• Weisstein, Eric W. "Tridecagon". MathWorld.
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
| Wikipedia |
Multilinear form
In abstract algebra and multilinear algebra, a multilinear form on a vector space $V$ over a field $K$ is a map
$f\colon V^{k}\to K$
that is separately $K$-linear in each of its $k$ arguments.[1] More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.
A multilinear $k$-form on $V$ over $\mathbb {R} $ is called a (covariant) ${\boldsymbol {k}}$-tensor, and the vector space of such forms is usually denoted ${\mathcal {T}}^{k}(V)$ or ${\mathcal {L}}^{k}(V)$.[2]
Tensor product
Given a $k$-tensor $f\in {\mathcal {T}}^{k}(V)$ and an $\ell $-tensor $g\in {\mathcal {T}}^{\ell }(V)$, a product $f\otimes g\in {\mathcal {T}}^{k+\ell }(V)$, known as the tensor product, can be defined by the property
$(f\otimes g)(v_{1},\ldots ,v_{k},v_{k+1},\ldots ,v_{k+\ell })=f(v_{1},\ldots ,v_{k})g(v_{k+1},\ldots ,v_{k+\ell }),$
for all $v_{1},\ldots ,v_{k+\ell }\in V$. The tensor product of multilinear forms is not commutative; however it is bilinear and associative:
$f\otimes (ag_{1}+bg_{2})=a(f\otimes g_{1})+b(f\otimes g_{2})$, $(af_{1}+bf_{2})\otimes g=a(f_{1}\otimes g)+b(f_{2}\otimes g),$
and
$(f\otimes g)\otimes h=f\otimes (g\otimes h).$
If $(v_{1},\ldots ,v_{n})$ forms a basis for an $n$-dimensional vector space $V$ and $(\phi ^{1},\ldots ,\phi ^{n})$ is the corresponding dual basis for the dual space $V^{*}={\mathcal {T}}^{1}(V)$, then the products $\phi ^{i_{1}}\otimes \cdots \otimes \phi ^{i_{k}}$, with $1\leq i_{1},\ldots ,i_{k}\leq n$ form a basis for ${\mathcal {T}}^{k}(V)$. Consequently, ${\mathcal {T}}^{k}(V)$ has dimensionality $n^{k}$.
Examples
Bilinear forms
Main article: Bilinear form
If $k=2$, $f:V\times V\to K$ is referred to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot product) of vectors.
Alternating multilinear forms
Main article: Alternating multilinear map
An important class of multilinear forms are the alternating multilinear forms, which have the additional property that[3]
$f(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\operatorname {sgn}(\sigma )f(x_{1},\ldots ,x_{k}),$
where $\sigma :\mathbf {N} _{k}\to \mathbf {N} _{k}$ :\mathbf {N} _{k}\to \mathbf {N} _{k}} is a permutation and $\operatorname {sgn}(\sigma )$ denotes its sign (+1 if even, –1 if odd). As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., $\sigma (p)=q,\sigma (q)=p$ and $\sigma (i)=i,1\leq i\leq k,i\neq p,q$):
$f(x_{1},\ldots ,x_{p},\ldots ,x_{q},\ldots ,x_{k})=-f(x_{1},\ldots ,x_{q},\ldots ,x_{p},\ldots ,x_{k}).$
With the additional hypothesis that the characteristic of the field $K$ is not 2, setting $x_{p}=x_{q}=x$ implies as a corollary that $f(x_{1},\ldots ,x,\ldots ,x,\ldots ,x_{k})=0$; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors[4] use this last condition as the defining property of alternating forms. This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when $\operatorname {char} (K)\neq 2$.
An alternating multilinear $k$-form on $V$ over $\mathbb {R} $ is called a multicovector of degree ${\boldsymbol {k}}$ or ${\boldsymbol {k}}$-covector, and the vector space of such alternating forms, a subspace of ${\mathcal {T}}^{k}(V)$, is generally denoted ${\mathcal {A}}^{k}(V)$, or, using the notation for the isomorphic kth exterior power of $V^{*}$(the dual space of $V$), $ \bigwedge ^{k}V^{*}$.[5] Note that linear functionals (multilinear 1-forms over $\mathbb {R} $) are trivially alternating, so that ${\mathcal {A}}^{1}(V)={\mathcal {T}}^{1}(V)=V^{*}$, while, by convention, 0-forms are defined to be scalars: ${\mathcal {A}}^{0}(V)={\mathcal {T}}^{0}(V)=\mathbb {R} $.
The determinant on $n\times n$ matrices, viewed as an $n$ argument function of the column vectors, is an important example of an alternating multilinear form.
Exterior product
The tensor product of alternating multilinear forms is, in general, no longer alternating. However, by summing over all permutations of the tensor product, taking into account the parity of each term, the exterior product ($\wedge $, also known as the wedge product) of multicovectors can be defined, so that if $f\in {\mathcal {A}}^{k}(V)$ and $g\in {\mathcal {A}}^{\ell }(V)$, then $f\wedge g\in {\mathcal {A}}^{k+\ell }(V)$:
$(f\wedge g)(v_{1},\ldots ,v_{k+\ell })={\frac {1}{k!\ell !}}\sum _{\sigma \in S_{k+\ell }}(\operatorname {sgn}(\sigma ))f(v_{\sigma (1)},\ldots ,v_{\sigma (k)})g(v_{\sigma (k+1)},\ldots ,v_{\sigma (k+\ell )}),$ !}}\sum _{\sigma \in S_{k+\ell }}(\operatorname {sgn}(\sigma ))f(v_{\sigma (1)},\ldots ,v_{\sigma (k)})g(v_{\sigma (k+1)},\ldots ,v_{\sigma (k+\ell )}),}
where the sum is taken over the set of all permutations over $k+\ell $ elements, $S_{k+\ell }$. The exterior product is bilinear, associative, and graded-alternating: if $f\in {\mathcal {A}}^{k}(V)$ and $g\in {\mathcal {A}}^{\ell }(V)$ then $f\wedge g=(-1)^{k\ell }g\wedge f$.
Given a basis $(v_{1},\ldots ,v_{n})$ for $V$ and dual basis $(\phi ^{1},\ldots ,\phi ^{n})$ for $V^{*}={\mathcal {A}}^{1}(V)$, the exterior products $\phi ^{i_{1}}\wedge \cdots \wedge \phi ^{i_{k}}$, with $1\leq i_{1}<\cdots <i_{k}\leq n$ form a basis for ${\mathcal {A}}^{k}(V)$. Hence, the dimensionality of ${\mathcal {A}}^{k}(V)$ for n-dimensional $V$ is $ {\tbinom {n}{k}}={\frac {n!}{(n-k)!\,k!}}$.
Differential forms
Main article: Differential form
Differential forms are mathematical objects constructed via tangent spaces and multilinear forms that behave, in many ways, like differentials in the classical sense. Though conceptually and computationally useful, differentials are founded on ill-defined notions of infinitesimal quantities developed early in the history of calculus. Differential forms provide a mathematically rigorous and precise framework to modernize this long-standing idea. Differential forms are especially useful in multivariable calculus (analysis) and differential geometry because they possess transformation properties that allow them be integrated on curves, surfaces, and their higher-dimensional analogues (differentiable manifolds). One far-reaching application is the modern statement of Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions.
The synopsis below is primarily based on Spivak (1965)[6] and Tu (2011).[3]
Definition of differential k-forms and construction of 1-forms
To define differential forms on open subsets $U\subset \mathbb {R} ^{n}$, we first need the notion of the tangent space of $\mathbb {R} ^{n}$at $p$, usually denoted $T_{p}\mathbb {R} ^{n}$ or $\mathbb {R} _{p}^{n}$. The vector space $\mathbb {R} _{p}^{n}$ can be defined most conveniently as the set of elements $v_{p}$ ($v\in \mathbb {R} ^{n}$, with $p\in \mathbb {R} ^{n}$ fixed) with vector addition and scalar multiplication defined by $v_{p}+w_{p}:=(v+w)_{p}$ and $a\cdot (v_{p}):=(a\cdot v)_{p}$, respectively. Moreover, if $(e_{1},\ldots ,e_{n})$ is the standard basis for $\mathbb {R} ^{n}$, then $((e_{1})_{p},\ldots ,(e_{n})_{p})$ is the analogous standard basis for $\mathbb {R} _{p}^{n}$. In other words, each tangent space $\mathbb {R} _{p}^{n}$ can simply be regarded as a copy of $\mathbb {R} ^{n}$ (a set of tangent vectors) based at the point $p$. The collection (disjoint union) of tangent spaces of $\mathbb {R} ^{n}$ at all $p\in \mathbb {R} ^{n}$ is known as the tangent bundle of $\mathbb {R} ^{n}$ and is usually denoted $ T\mathbb {R} ^{n}:=\bigcup _{p\in \mathbb {R} ^{n}}\mathbb {R} _{p}^{n}$. While the definition given here provides a simple description of the tangent space of $\mathbb {R} ^{n}$, there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general (see the article on tangent spaces for details).
A differential ${\boldsymbol {k}}$-form on $U\subset \mathbb {R} ^{n}$ is defined as a function $\omega $ that assigns to every $p\in U$ a $k$-covector on the tangent space of $\mathbb {R} ^{n}$at $p$, usually denoted $\omega _{p}:=\omega (p)\in {\mathcal {A}}^{k}(\mathbb {R} _{p}^{n})$. In brief, a differential $k$-form is a $k$-covector field. The space of $k$-forms on $U$ is usually denoted $\Omega ^{k}(U)$; thus if $\omega $ is a differential $k$-form, we write $\omega \in \Omega ^{k}(U)$. By convention, a continuous function on $U$ is a differential 0-form: $f\in C^{0}(U)=\Omega ^{0}(U)$.
We first construct differential 1-forms from 0-forms and deduce some of their basic properties. To simplify the discussion below, we will only consider smooth differential forms constructed from smooth ($C^{\infty }$) functions. Let $f:\mathbb {R} ^{n}\to \mathbb {R} $ be a smooth function. We define the 1-form $df$ on $U$ for $p\in U$ and $v_{p}\in \mathbb {R} _{p}^{n}$ by $(df)_{p}(v_{p}):=Df|_{p}(v)$, where $Df|_{p}:\mathbb {R} ^{n}\to \mathbb {R} $ is the total derivative of $f$ at $p$. (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions) $\pi ^{i}:\mathbb {R} ^{n}\to \mathbb {R} $, defined by $x\mapsto x^{i}$, where $x^{i}$ is the ith standard coordinate of $x\in \mathbb {R} ^{n}$. The 1-forms $d\pi ^{i}$ are known as the basic 1-forms; they are conventionally denoted $dx^{i}$. If the standard coordinates of $v_{p}\in \mathbb {R} _{p}^{n}$ are $(v^{1},\ldots ,v^{n})$, then application of the definition of $df$ yields $dx_{p}^{i}(v_{p})=v^{i}$, so that $dx_{p}^{i}((e_{j})_{p})=\delta _{j}^{i}$, where $\delta _{j}^{i}$ is the Kronecker delta.[7] Thus, as the dual of the standard basis for $\mathbb {R} _{p}^{n}$, $(dx_{p}^{1},\ldots ,dx_{p}^{n})$ forms a basis for ${\mathcal {A}}^{1}(\mathbb {R} _{p}^{n})=(\mathbb {R} _{p}^{n})^{*}$. As a consequence, if $\omega $ is a 1-form on $U$, then $\omega $ can be written as $ \sum a_{i}\,dx^{i}$ for smooth functions $a_{i}:U\to \mathbb {R} $. Furthermore, we can derive an expression for $df$ that coincides with the classical expression for a total differential:
$df=\sum _{i=1}^{n}D_{i}f\;dx^{i}={\partial f \over \partial x^{1}}\,dx^{1}+\cdots +{\partial f \over \partial x^{n}}\,dx^{n}.$
[Comments on notation: In this article, we follow the convention from tensor calculus and differential geometry in which multivectors and multicovectors are written with lower and upper indices, respectively. Since differential forms are multicovector fields, upper indices are employed to index them.[3] The opposite rule applies to the components of multivectors and multicovectors, which instead are written with upper and lower indices, respectively. For instance, we represent the standard coordinates of vector $v\in \mathbb {R} ^{n}$ as $(v^{1},\ldots ,v^{n})$, so that $ v=\sum _{i=1}^{n}v^{i}e_{i}$ in terms of the standard basis $(e_{1},\ldots ,e_{n})$. In addition, superscripts appearing in the denominator of an expression (as in $ {\frac {\partial f}{\partial x^{i}}}$) are treated as lower indices in this convention. When indices are applied and interpreted in this manner, the number of upper indices minus the number of lower indices in each term of an expression is conserved, both within the sum and across an equal sign, a feature that serves as a useful mnemonic device and helps pinpoint errors made during manual computation.]
Basic operations on differential k-forms
The exterior product ($\wedge $) and exterior derivative ($d$) are two fundamental operations on differential forms. The exterior product of a $k$-form and an $\ell $-form is a $(k+\ell )$-form, while the exterior derivative of a $k$-form is a $(k+1)$-form. Thus, both operations generate differential forms of higher degree from those of lower degree.
The exterior product $\wedge :\Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U)$ :\Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U)} of differential forms is a special case of the exterior product of multicovectors in general (see above). As is true in general for the exterior product, the exterior product of differential forms is bilinear, associative, and is graded-alternating.
More concretely, if $\omega =a_{i_{1}\ldots i_{k}}\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}$ and $\eta =a_{j_{1}\ldots i_{\ell }}dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}$, then
$\omega \wedge \eta =a_{i_{1}\ldots i_{k}}a_{j_{1}\ldots j_{\ell }}\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}\wedge dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}.$
Furthermore, for any set of indices $\{\alpha _{1}\ldots ,\alpha _{m}\}$,
$dx^{\alpha _{1}}\wedge \cdots \wedge dx^{\alpha _{p}}\wedge \cdots \wedge dx^{\alpha _{q}}\wedge \cdots \wedge dx^{\alpha _{m}}=-dx^{\alpha _{1}}\wedge \cdots \wedge dx^{\alpha _{q}}\wedge \cdots \wedge dx^{\alpha _{p}}\wedge \cdots \wedge dx^{\alpha _{m}}.$
If $I=\{i_{1},\ldots ,i_{k}\}$, $J=\{j_{1},\ldots ,j_{\ell }\}$, and $I\cap J=\varnothing $, then the indices of $\omega \wedge \eta $ can be arranged in ascending order by a (finite) sequence of such swaps. Since $dx^{\alpha }\wedge dx^{\alpha }=0$, $I\cap J\neq \varnothing $ implies that $\omega \wedge \eta =0$. Finally, as a consequence of bilinearity, if $\omega $ and $\eta $ are the sums of several terms, their exterior product obeys distributivity with respect to each of these terms.
The collection of the exterior products of basic 1-forms $\{dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}\mid 1\leq i_{1}<\cdots <i_{k}\leq n\}$ constitutes a basis for the space of differential k-forms. Thus, any $\omega \in \Omega ^{k}(U)$ can be written in the form
$\omega =\sum _{i_{1}<\cdots <i_{k}}a_{i_{1}\ldots i_{k}}\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}},\qquad (*)$
where $a_{i_{1}\ldots i_{k}}:U\to \mathbb {R} $ are smooth functions. With each set of indices $\{i_{1},\ldots ,i_{k}\}$ placed in ascending order, (*) is said to be the standard presentation of $\omega $.
In the previous section, the 1-form $df$ was defined by taking the exterior derivative of the 0-form (continuous function) $f$. We now extend this by defining the exterior derivative operator $d:\Omega ^{k}(U)\to \Omega ^{k+1}(U)$ for $k\geq 1$. If the standard presentation of $k$-form $\omega $ is given by (*), the $(k+1)$-form $d\omega $ is defined by
$d\omega :=\sum _{i_{1}<\ldots <i_{k}}da_{i_{1}\ldots i_{k}}\wedge dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}.$ :=\sum _{i_{1}<\ldots <i_{k}}da_{i_{1}\ldots i_{k}}\wedge dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}.}
A property of $d$ that holds for all smooth forms is that the second exterior derivative of any $\omega $ vanishes identically: $d^{2}\omega =d(d\omega )\equiv 0$. This can be established directly from the definition of $d$ and the equality of mixed second-order partial derivatives of $C^{2}$ functions (see the article on closed and exact forms for details).
Integration of differential forms and Stokes' theorem for chains
To integrate a differential form over a parameterized domain, we first need to introduce the notion of the pullback of a differential form. Roughly speaking, when a differential form is integrated, applying the pullback transforms it in a way that correctly accounts for a change-of-coordinates.
Given a differentiable function $f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ and $k$-form $\eta \in \Omega ^{k}(\mathbb {R} ^{m})$, we call $f^{*}\eta \in \Omega ^{k}(\mathbb {R} ^{n})$ the pullback of $\eta $ by $f$ and define it as the $k$-form such that
$(f^{*}\eta )_{p}(v_{1p},\ldots ,v_{kp}):=\eta _{f(p)}(f_{*}(v_{1p}),\ldots ,f_{*}(v_{kp})),$
for $v_{1p},\ldots ,v_{kp}\in \mathbb {R} _{p}^{n}$, where $f_{*}:\mathbb {R} _{p}^{n}\to \mathbb {R} _{f(p)}^{m}$ is the map $v_{p}\mapsto (Df|_{p}(v))_{f(p)}$.
If $\omega =f\,dx^{1}\wedge \cdots \wedge dx^{n}$ is an $n$-form on $\mathbb {R} ^{n}$ (i.e., $\omega \in \Omega ^{n}(\mathbb {R} ^{n})$), we define its integral over the unit $n$-cell as the iterated Riemann integral of $f$:
$\int _{[0,1]^{n}}\omega =\int _{[0,1]^{n}}f\,dx^{1}\wedge \cdots \wedge dx^{n}:=\int _{0}^{1}\cdots \int _{0}^{1}f\,dx^{1}\cdots dx^{n}.$
Next, we consider a domain of integration parameterized by a differentiable function $c:[0,1]^{n}\to A\subset \mathbb {R} ^{m}$, known as an n-cube. To define the integral of $\omega \in \Omega ^{n}(A)$ over $c$, we "pull back" from $A$ to the unit n-cell:
$\int _{c}\omega :=\int _{[0,1]^{n}}c^{*}\omega .$ :=\int _{[0,1]^{n}}c^{*}\omega .}
To integrate over more general domains, we define an ${\boldsymbol {n}}$-chain $ C=\sum _{i}n_{i}c_{i}$ as the formal sum of $n$-cubes and set
$\int _{C}\omega :=\sum _{i}n_{i}\int _{c_{i}}\omega .$ :=\sum _{i}n_{i}\int _{c_{i}}\omega .}
An appropriate definition of the $(n-1)$-chain $\partial C$, known as the boundary of $C$,[8] allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of $\mathbb {R} ^{m}$:
If $\omega $ is a smooth $(n-1)$-form on an open set $A\subset \mathbb {R} ^{m}$ and $C$ is a smooth $n$-chain in $A$, then$\int _{C}d\omega =\int _{\partial C}\omega $.
Using more sophisticated machinery (e.g., germs and derivations), the tangent space $T_{p}M$ of any smooth manifold $M$ (not necessarily embedded in $\mathbb {R} ^{m}$) can be defined. Analogously, a differential form $\omega \in \Omega ^{k}(M)$ on a general smooth manifold is a map $\omega :p\in M\mapsto \omega _{p}\in {\mathcal {A}}^{k}(T_{p}M)$. Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary and even certain "rough" domains (see the article on Stokes' theorem for details).
See also
• Bilinear map
• Exterior algebra
• Homogeneous polynomial
• Linear form
• Multilinear map
References
1. Weisstein, Eric W. "Multilinear Form". MathWorld.
2. Many authors use the opposite convention, writing ${\mathcal {T}}^{k}(V)$ to denote the contravariant k-tensors on $V$ and ${\mathcal {T}}_{k}(V)$ to denote the covariant k-tensors on $V$.
3. Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). Springer. pp. 22–23. ISBN 978-1-4419-7399-3.
4. Halmos, Paul R. (1958). Finite-Dimensional Vector Spaces (2nd ed.). Van Nostrand. p. 50. ISBN 0-387-90093-4.
5. Spivak uses $\Omega ^{k}(V)$ for the space of $k$-covectors on $V$. However, this notation is more commonly reserved for the space of differential $k$-forms on $V$. In this article, we use $\Omega ^{k}(V)$ to mean the latter.
6. Spivak, Michael (1965). Calculus on Manifolds. W. A. Benjamin, Inc. pp. 75–146. ISBN 0805390219.
7. The Kronecker delta is usually denoted by $\delta _{ij}=\delta (i,j)$ and defined as $ \delta :X\times X\to \{0,1\},\ (i,j)\mapsto {\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}$. Here, the notation $\delta _{j}^{i}$ is used to conform to the tensor calculus convention on the use of upper and lower indices.
8. The formal definition of the boundary of a chain is somewhat involved and is omitted here (see Spivak 1965, pp. 98–99 for a discussion). Intuitively, if $C$ maps to a square, then $\partial C$ is a linear combination of functions that maps to its edges in a counterclockwise manner. The boundary of a chain is distinct from the notion of a boundary in point-set topology.
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Multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
$f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}$
For multilinear maps used in cryptography, see Cryptographic multilinear map.
where $V_{1},\ldots ,V_{n}$ and $W$ are vector spaces (or modules over a commutative ring), with the following property: for each $i$, if all of the variables but $v_{i}$ are held constant, then $f(v_{1},\ldots ,v_{i},\ldots ,v_{n})$ is a linear function of $v_{i}$.[1]
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
Examples
• Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in $\mathbb {R} ^{3}$.
• The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
• If $F\colon \mathbb {R} ^{m}\to \mathbb {R} ^{n}$ is a Ck function, then the $k\!$th derivative of $F\!$ at each point $p$ in its domain can be viewed as a symmetric $k$-linear function $D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}$.
Coordinate representation
Let
$f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}$
be a multilinear map between finite-dimensional vector spaces, where $V_{i}\!$ has dimension $d_{i}\!$, and $W\!$ has dimension $d\!$. If we choose a basis $\{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}$ for each $V_{i}\!$ and a basis $\{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}$ for $W\!$ (using bold for vectors), then we can define a collection of scalars $A_{j_{1}\cdots j_{n}}^{k}$ by
$f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.$
Then the scalars $\{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}$ completely determine the multilinear function $f\!$. In particular, if
${\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!$
for $1\leq i\leq n\!$, then
$f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.$
Example
Let's take a trilinear function
$g\colon R^{2}\times R^{2}\times R^{2}\to R,$
where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.
A basis for each Vi is $\{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.$ Let
$g({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk},$
where $i,j,k\in \{1,2\}$. In other words, the constant $A_{ijk}$ is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three $V_{i}$), namely:
$\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.$
Each vector ${\textbf {v}}_{i}\in V_{i}=R^{2}$ can be expressed as a linear combination of the basis vectors
${\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1).$
The function value at an arbitrary collection of three vectors ${\textbf {v}}_{i}\in R^{2}$ can be expressed as
$g({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k}.$
Or, in expanded form as
${\begin{aligned}g((a,b),(c,d)&,(e,f))=ace\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+ade\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+bde\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}).\end{aligned}}$
Relation to tensor products
There is a natural one-to-one correspondence between multilinear maps
$f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}$
and linear maps
$F\colon V_{1}\otimes \cdots \otimes V_{n}\to W{\text{,}}$
where $V_{1}\otimes \cdots \otimes V_{n}\!$ denotes the tensor product of $V_{1},\ldots ,V_{n}$. The relation between the functions $f\!$ and $F\!$ is given by the formula
$f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}).$
Multilinear functions on n×n matrices
One can consider multilinear functions, on an n×n matrix over a commutative ring K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as
$D(A)=D(a_{1},\ldots ,a_{n}),$
satisfying
$D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).$
If we let ${\hat {e}}_{j}$ represent the jth row of the identity matrix, we can express each row ai as the sum
$a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}.$
Using the multilinearity of D we rewrite D(A) as
$D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).$
Continuing this substitution for each ai we get, for 1 ≤ i ≤ n,
$D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).$
Therefore, D(A) is uniquely determined by how D operates on ${\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}$.
Example
In the case of 2×2 matrices we get
$D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2})\,$
Where ${\hat {e}}_{1}=[1,0]$ and ${\hat {e}}_{2}=[0,1]$. If we restrict $D$ to be an alternating function then $D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0$ and $D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)$. Letting $D(I)=1$ we get the determinant function on 2×2 matrices:
$D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}.$
Properties
• A multilinear map has a value of zero whenever one of its arguments is zero.
See also
• Algebraic form
• Multilinear form
• Homogeneous polynomial
• Homogeneous function
• Tensors
References
1. Lang, Serge (2005) [2002]. "XIII. Matrices and Linear Maps §S Determinants". Algebra. Graduate Texts in Mathematics. Vol. 211 (3rd ed.). Springer. pp. 511–. ISBN 978-0-387-95385-4.
| Wikipedia |
Trilinear coordinates
In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z : x and vertices C and A.
In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a', b', c'), or equivalently in ratio form, ka' : kb' : kc' for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.
Notation
The ratio notation x : y : z for trilinear coordinates is different from the ordered triple notation (a', b', c' ) for actual directed distances. Here each of x, y, and z has no meaning by itself; its ratio to one of the others does have meaning. Thus "comma notation" for trilinear coordinates should be avoided, because the notation (x, y, z), which means an ordered triple, does not allow, for example, (x, y, z) = (2x, 2y, 2z), whereas the "colon notation" does allow x : y : z = 2x : 2y : 2z.
Examples
The trilinear coordinates of the incenter of a triangle △ABC are 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines BC, CA, AB are proportional to the actual distances denoted by (r, r, r), where r is the inradius of △ABC. Given side lengths a, b, c we have:
• A = 1 : 0 : 0
• B = 0 : 1 : 0
• C = 0 : 0 : 1
• incenter = 1 : 1 : 1
• centroid = bc : ca : ab = 1/a : 1/b : 1/c = csc A : csc B : csc C.
• circumcenter = cos A : cos B : cos C.
• orthocenter = sec A : sec B : sec C.
• nine-point center = cos(B − C) : cos(C − A) : cos(A − B).
• symmedian point = a : b : c = sin A : sin B : sin C.
• A-excenter = −1 : 1 : 1
• B-excenter = 1 : −1 : 1
• C-excenter = 1 : 1 : −1.
Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas of the triangles △BGC, △CGA, △AGB, where G = centroid.)
The midpoint of, for example, side BC has trilinear coordinates in actual sideline distances $(0,{\tfrac {\Delta }{b}},{\tfrac {\Delta }{c}})$ for triangle area Δ, which in arbitrarily specified relative distances simplifies to 0 : ca : ab. The coordinates in actual sideline distances of the foot of the altitude from A to BC are $(0,{\tfrac {2\Delta }{a}}\cos C,{\tfrac {2\Delta }{a}}\cos B),$ which in purely relative distances simplifies to 0 : cos C : cos B.[1]: p. 96
Formulas
Collinearities and concurrencies
Trilinear coordinates enable many algebraic methods in triangle geometry. For example, three points
${\begin{aligned}P&=p:q:r\\U&=u:v:w\\X&=x:y:z\\\end{aligned}}$
are collinear if and only if the determinant
$D={\begin{vmatrix}p&q&r\\u&v&w\\x&y&z\end{vmatrix}}$
equals zero. Thus if x : y : z is a variable point, the equation of a line through the points P and U is D = 0.[1]: p. 23 From this, every straight line has a linear equation homogeneous in x, y, z. Every equation of the form $lx+my+nz=0$ in real coefficients is a real straight line of finite points unless l : m : n is proportional to a : b : c, the side lengths, in which case we have the locus of points at infinity.[1]: p. 40
The dual of this proposition is that the lines
${\begin{aligned}p\alpha +q\beta +r\gamma &=0\\u\alpha +v\beta +w\gamma &=0\\x\alpha +y\beta +z\gamma &=0\end{aligned}}$
concur in a point (α, β, γ) if and only if D = 0.[1]: p. 28
Also, if the actual directed distances are used when evaluating the determinant of D, then the area of triangle △PUX is KD, where $K={\tfrac {-abc}{8\Delta ^{2}}}$ (and where Δ is the area of triangle △ABC, as above) if triangle △PUX has the same orientation (clockwise or counterclockwise) as △ABC, and $K={\tfrac {-abc}{8\Delta ^{2}}}$ otherwise.
Parallel lines
Two lines with trilinear equations $lx+my+nz=0$ and $l'x+m'y+n'z=0$ are parallel if and only if[1]: p. 98, #xi
${\begin{vmatrix}l&m&n\\l'&m'&n'\\a&b&c\end{vmatrix}}=0,$
where a, b, c are the side lengths.
Angle between two lines
The tangents of the angles between two lines with trilinear equations $lx+my+nz=0$ and $l'x+m'y+n'z=0$ are given by[1]: p.50
$\pm {\frac {(mn'-m'n)\sin A+(nl'-n'l)\sin B+(lm'-l'm)\sin C}{ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C}}.$
Perpendicular lines
Thus two lines with trilinear equations $lx+my+nz=0$ and $l'x+m'y+n'z=0$ are perpendicular if and only if
$ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0.$
Altitude
The equation of the altitude from vertex A to side BC is[1]: p.98, #x
$y\cos B-z\cos C=0.$
Line in terms of distances from vertices
The equation of a line with variable distances p, q, r from the vertices A, B, C whose opposite sides are a, b, c is[1]: p. 97, #viii
$apx+bqy+crz=0.$
Actual-distance trilinear coordinates
The trilinears with the coordinate values a', b', c' being the actual perpendicular distances to the sides satisfy[1]: p. 11
$aa'+bb'+cc'=2\Delta $
for triangle sides a, b, c and area Δ. This can be seen in the figure at the top of this article, with interior point P partitioning triangle △ABC into three triangles △PBC, △PCA, △PAB with respective areas ${\tfrac {1}{2}}aa',{\tfrac {1}{2}}bb',{\tfrac {1}{2}}cc'.$
Distance between two points
The distance d between two points with actual-distance trilinears ai : bi : ci is given by[1]: p. 46
$d^{2}\sin ^{2}C=(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}+2(a_{1}-a_{2})(b_{1}-b_{2})\cos C$
or in a more symmetric way
$d^{2}={\frac {abc}{4\Delta ^{2}}}\left(a(b_{1}-b_{2})(c_{2}-c_{1})+b(c_{1}-c_{2})(a_{2}-a_{1})+c(a_{1}-a_{2})(b_{2}-b_{1})\right).$
Distance from a point to a line
The distance d from a point a' : b' : c' , in trilinear coordinates of actual distances, to a straight line $lx+my+nz=0$ is[1]: p. 48
$d={\frac {la'+mb'+nc'}{\sqrt {l^{2}+m^{2}+n^{2}-2mn\cos A-2nl\cos B-2lm\cos C}}}.$
Quadratic curves
The equation of a conic section in the variable trilinear point x : y : z is[1]: p.118
$rx^{2}+sy^{2}+tz^{2}+2uyz+2vzx+2wxy=0.$
It has no linear terms and no constant term.
The equation of a circle of radius r having center at actual-distance coordinates (a', b', c' ) is[1]: p.287
$(x-a')^{2}\sin 2A+(y-b')^{2}\sin 2B+(z-c')^{2}\sin 2C=2r^{2}\sin A\sin B\sin C.$
Circumconics
The equation in trilinear coordinates x, y, z of any circumconic of a triangle is[1]: p. 192
$lyz+mzx+nxy=0.$
If the parameters l, m, n respectively equal the side lengths a, b, c (or the sines of the angles opposite them) then the equation gives the circumcircle.[1]: p. 199
Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center x' : y' : z' is[1]: p. 203
$yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0.$
Inconics
Every conic section inscribed in a triangle has an equation in trilinear coordinates:[1]: p. 208
$l^{2}x^{2}+m^{2}y^{2}+n^{2}z^{2}\pm 2mnyz\pm 2nlzx\pm 2lmxy=0,$
with exactly one or three of the unspecified signs being negative.
The equation of the incircle can be simplified to[1]: p. 210, p.214
$\pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0,$
while the equation for, for example, the excircle adjacent to the side segment opposite vertex A can be written as[1]: p. 215
$\pm {\sqrt {-x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0.$
Cubic curves
Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic Z(U, P), as the locus of a point X such that the P-isoconjugate of X is on the line UX is given by the determinant equation
${\begin{vmatrix}x&y&z\\qryz&rpzx&pqxy\\u&v&w\end{vmatrix}}=0.$
Among named cubics Z(U, P) are the following:
Thomson cubic: Z(X(2),X(1)), where X(2) = centroid, X(1) = incenter
Feuerbach cubic: Z(X(5),X(1)), where X(5) = Feuerbach point
Darboux cubic: Z(X(20),X(1)), where X(20) = De Longchamps point
Neuberg cubic: Z(X(30),X(1)), where X(30) = Euler infinity point.
Conversions
Between trilinear coordinates and distances from sidelines
For any choice of trilinear coordinates x : y : z to locate a point, the actual distances of the point from the sidelines are given by a' = kx, b' = ky, c' = kz where k can be determined by the formula $k={\tfrac {2\Delta }{ax+by+cz}}$ in which a, b, c are the respective sidelengths BC, CA, AB, and ∆ is the area of △ABC.
Between barycentric and trilinear coordinates
A point with trilinear coordinates x : y : z has barycentric coordinates ax : by : cz where a, b, c are the sidelengths of the triangle. Conversely, a point with barycentrics α : β : γ has trilinear coordinates ${\tfrac {\alpha }{a}}:{\tfrac {\beta }{b}}:{\tfrac {\gamma }{c}}.$
Between Cartesian and trilinear coordinates
Given a reference triangle △ABC, express the position of the vertex B in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector ${\vec {B}},$ using vertex C as the origin. Similarly define the position vector of vertex A as ${\vec {A}}.$ Then any point P associated with the reference triangle △ABC can be defined in a Cartesian system as a vector ${\vec {P}}=k_{1}{\vec {A}}+k_{2}{\vec {B}}.$ If this point P has trilinear coordinates x : y : z then the conversion formula from the coefficients k1 and k2 in the Cartesian representation to the trilinear coordinates is, for side lengths a, b, c opposite vertices A, B, C,
$x:y:z={\frac {k_{1}}{a}}:{\frac {k_{2}}{b}}:{\frac {1-k_{1}-k_{2}}{c}},$
and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is
$k_{1}={\frac {ax}{ax+by+cz}},\quad k_{2}={\frac {by}{ax+by+cz}}.$
More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors ${\vec {A}},{\vec {B}},{\vec {C}}$ and if the point P has trilinear coordinates x : y : z, then the Cartesian coordinates of ${\vec {P}}$ are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ax, by, cz as the weights. Hence the conversion formula from the trilinear coordinates x, y, z to the vector of Cartesian coordinates ${\vec {P}}$ of the point is given by
${\vec {P}}={\frac {ax}{ax+by+cz}}{\vec {A}}+{\frac {by}{ax+by+cz}}{\vec {B}}+{\frac {cz}{ax+by+cz}}{\vec {C}},$
where the side lengths are
${\begin{aligned}&|{\vec {C}}-{\vec {B}}|=a,\\&|{\vec {A}}-{\vec {C}}|=b,\\&|{\vec {B}}-{\vec {A}}|=c.\end{aligned}}$
See also
• Morley's trisector theorem#Morley's triangles, giving examples of numerous points expressed in trilinear coordinates
• Ternary plot
• Viviani's theorem
References
1. William Allen Whitworth (1866) Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise, link from Cornell University Historical Math Monographs.
External links
• Weisstein, Eric W. "Trilinear Coordinates". MathWorld.
• Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 7000 triangle centers
Authority control: National
• Israel
• United States
| Wikipedia |
Carroll diagram
A Carroll diagram, Lewis Carroll's square, biliteral diagram or a two-way table is a diagram used for grouping things in a yes/no fashion. Numbers or objects are either categorised as 'x' (having an attribute x) or 'not x' (not having an attribute 'x'). They are named after Lewis Carroll, the pseudonym of polymath Charles Lutwidge Dodgson.[1][2]
Usage
Although Carroll diagrams can be as simple as the first one above, the most well known types are those similar to the second one, where two attributes are shown. The 'universe' of a Carroll diagram is contained within the boxes in the diagram, as any number or object has to either have an attribute or not have it.
Carroll diagrams are often learnt by schoolchildren, but they can also be used outside the field of education, since they are a tidy way of categorising and displaying information.
See also
Wikimedia Commons has media related to Carroll diagrams.
• Diagram
• Karnaugh map
• Set theory
• Venn diagram
• The Game of Logic
References
1. Ameis, Jerry (2010). "Venn and Carroll Diagrams". Mathematical Tale Winds. Faculty of Education, University of Winnipeg, Winnipeg, Canada. Archived from the original on 2017-05-03. Retrieved 2010-09-10.
2. Shin, Sun-Joo; Lemon, Oliver; Mumma, John (2013-09-17) [2001-08-28]. "Diagrams". The Stanford Encyclopedia of Philosophy (SEP). Archived from the original on 1998-05-03. Retrieved 2015-05-03.
Further reading
• Mac Queen, Gailand (October 1967). The Logic Diagram (PDF) (Thesis). McMaster University. Archived from the original (PDF) on 2017-04-14. Retrieved 2017-04-14.
• Edwards, Anthony William Fairbank (2004). Cogwheels of the Mind: The Story of Venn Diagrams. Baltimore, Maryland, USA: Johns Hopkins University Press. ISBN 0-8018-7434-3.
External links
• Bogomolny, Alexander (2017) [1996]. "Lewis Carroll's Logic Game". Cut-the-knot. Archived from the original on 2017-05-03. Retrieved 2017-05-03.
• "Carroll Diagram (and Game of Logic) Interactive Demonstrator at lewiscarrollresources.net".
• Lewis Carroll: Logic, Internet Encyclopedia of Philosophy
| Wikipedia |
List of aperiodic sets of tiles
In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles).[1] A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single fundamental unit or primitive cell which repeats endlessly and regularly in two independent directions.[2] An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic.[3] The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".)
The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete.
Explanations
AbbreviationMeaningExplanation
E2Euclidean planenormal flat plane
H2hyperbolic planeplane, where the parallel postulate does not hold
E3Euclidean 3 spacespace defined by three perpendicular coordinate axes
MLDMutually locally derivabletwo tilings are said to be mutually locally derivable from each other, if one tiling can be obtained from the other by a simple local rule (such as deleting or inserting an edge)
List
ImageNameNumber of tilesSpacePublication DateRefs.Comments
Trilobite and cross tiles2E21999[4]Tilings MLD from the chair tilings.
Penrose P1 tiles6E21974[5][6]Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex".
Penrose P2 tiles2E21977[7][8]Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex".
Penrose P3 tiles2E21978[9][10]Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex".
Binary tiles2E21988[11][12]Although similar in shape to the P3 tiles, the tilings are not MLD from each other. Developed in an attempt to model the atomic arrangement in binary alloys.
Robinson tiles6E21971[13][14]Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Ammann A1 tiles6E21977[15][16]Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2 tiles2E21986[17][18]
Ammann A3 tiles3E21986[17][18]
Ammann A4 tiles2E21986[17][18][19]Tilings MLD with Ammann A5.
Ammann A5 tiles2E21982[20][21][22]Tilings MLD with Ammann A4.
No imagePenrose hexagon-triangle tiles3E21997[23][23][24]Uses mirror images of tiles for tiling.
No imagePegasus tiles2E22016[25][25][26]Variant of the Penrose hexagon-triangle tiles. Discovered in 2003 or earlier.
Golden triangle tiles10E22001[27][28]Date is for discovery of matching rules. Dual to Ammann A2.
Socolar tiles3E21989[29][30][31]Tilings MLD from the tilings by the Shield tiles.
Shield tiles4E21988[32][33][34]Tilings MLD from the tilings by the Socolar tiles.
Square triangle tiles5E21986[35][36]
Starfish, ivy leaf and hex tiles3E2[37][38][39]Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles.
Robinson triangle4E2[17]Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles6E21996[40][41]
Pinwheel tilesE21994[42][43][44][45]Date is for publication of matching rules.
Socolar–Taylor tile1E22010[46][47]Not a connected set. Aperiodic hierarchical tiling.
No imageWang tiles20426E21966[48]
No imageWang tiles104E22008[49]
No imageWang tiles52E21971[13][50]Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Wang tiles32E21986[51]Locally derivable from the Penrose tiles.
No imageWang tiles24E21986[51]Locally derivable from the A2 tiling.
Wang tiles16E21986[17][52]Derived from tiling A2 and its Ammann bars.
Wang tiles14E21996[53][54]
Wang tiles13E21996[55][56]
Wang tiles11E22015[57]Smallest aperiodic set of Wang tiles.
No imageDecagonal Sponge tile1E22002[58][59]Porous tile consisting of non-overlapping point sets.
No imageGoodman-Strauss strongly aperiodic tiles85H22005[60]
No imageGoodman-Strauss strongly aperiodic tiles26H22005[61]
Böröczky hyperbolic tile1Hn1974[62][63][61][64]Only weakly aperiodic.
No imageSchmitt tile1E31988[65]Screw-periodic.
Schmitt–Conway–Danzer tile1E3[65]Screw-periodic and convex.
Socolar–Taylor tile1E32010[46][47]Periodic in third dimension.
No imagePenrose rhombohedra2E31981[66][67][68][69][70][71][72][73]
Mackay–Amman rhombohedra4E31981[37]Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No imageWang cubes21E31996[74]
No imageWang cubes18E31999[75]
No imageDanzer tetrahedra4E31989[76][77]
I and L tiles2En for all n ≥ 31999[78]
Smith–Myers–Kaplan–Goodman-Strauss or "Hat" polytile1E22023[79]Mirrored monotiles, the first example of an "einstein".
Smith–Myers–Kaplan–Goodman-Strauss or "Spectre" polytile1E22023[80]"Strictly chiral" aperiodic monotile, the first example of a real "einstein".
TS1 2 E2 2014 [81]
References
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51. Lagae, Ares; Kari, Jarkko; Dutré, Phillip (2006), Aperiodic Sets of Square Tiles with Colored Corners, Report CW, vol. 460, KU Leuven, p. 15, CiteSeerX 10.1.1.89.1294
52. Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000), Pattern Formation in Biology, Vision and Dynamics, Singapore: World Scientific, ISBN 978-981-02-3792-9
53. Kari, Jarkko (1996), "A small aperiodic set of Wang tiles", Discrete Mathematics, 160 (1–3): 259–264, doi:10.1016/0012-365X(95)00120-L
54. Lagae, Ares (2007), Tile Based Methods in Computer Graphics (PDF) (PhD thesis), KU Leuven, p. 149, ISBN 978-90-5682-789-2, archived from the original (PDF) on 2011-07-20
55. Culik, Karel; Kari, Jarkko (1997), "On aperiodic sets of Wang tiles", Foundations of Computer Science, Lecture Notes in Computer Science, vol. 1337, pp. 153–162, doi:10.1007/BFb0052084, ISBN 978-3-540-63746-2
56. Culik, Karel (1996), "An aperiodic set of 13 Wang tiles", Discrete Mathematics, 160 (1–3): 245–251, CiteSeerX 10.1.1.53.5421, doi:10.1016/S0012-365X(96)00118-5
57. Jeandel, Emmanuel; Rao, Michaël (2021), "An aperiodic set of 11 Wang tiles", Advances in Combinatorics: Paper No. 1, 37, arXiv:1506.06492, doi:10.19086/aic.18614, MR 4210631, S2CID 13261182
58. Zhu, Feng (2002), The Search for a Universal Tile (PDF) (BA thesis), Williams College
59. Bailey, Duane A.; Zhu, Feng (2001), A Sponge-Like (Almost) Universal Tile (PDF), CiteSeerX 10.1.1.103.3739
60. Goodman-Strauss, Chaim (2010), "A hierarchical strongly aperiodic set of tiles in the hyperbolic plane" (PDF), Theoretical Computer Science, 411 (7–9): 1085–1093, doi:10.1016/j.tcs.2009.11.018
61. Goodman-Strauss, Chaim (2005), "A strongly aperiodic set of tiles in the hyperbolic plane", Invent. Math., 159 (1): 130–132, Bibcode:2004InMat.159..119G, CiteSeerX 10.1.1.477.1974, doi:10.1007/s00222-004-0384-1, S2CID 5348203
62. Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben I", Matematikai Lapok, 25: 265–306
63. Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben II", Matematikai Lapok, 26: 67–90
64. Dolbilin, Nikkolai; Frettlöh, Dirk (2010), "Properties of Böröczky tilings in high dimensional hyperbolic spaces" (PDF), European J. Combin., 31 (4): 1181–1195, arXiv:0705.0291, CiteSeerX 10.1.1.246.9821, doi:10.1016/j.ejc.2009.11.016, S2CID 13607905
65. Radin, Charles (1995), "Aperiodic tilings in higher dimensions" (PDF), Proceedings of the American Mathematical Society, American Mathematical Society, 123 (11): 3543–3548, doi:10.2307/2161105, JSTOR 2161105, retrieved 2013-09-25
66. Mackay, Alan L. (1981), "De Nive Quinquangula: On the pentagonal snowflake" (PDF), Sov. Phys. Crystallogr., 26 (5): 517–522, archived (PDF) from the original on 2011-07-21
67. Meisterernst, Götz, Experimente zur Wachstumskinetik Dekagonaler Quasikristalle (PDF) (Dissertation), Ludwig Maximilian University of Munich, pp. 18–19, archived (PDF) from the original on 2011-06-17
68. Jirong, Sun (1993), "Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field", Chinese Physics Letters, 10 (8): 449–452, Bibcode:1993ChPhL..10..449S, doi:10.1088/0256-307x/10/8/001, S2CID 250911962
69. Inchbald, Guy (2002), A 3-D Quasicrystal Structure
70. Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2001), "Quasicrystals: tiling versus clustering" (PDF), Philosophical Magazine A, 81 (11): 2645–2651, Bibcode:2001PMagA..81.2645L, CiteSeerX 10.1.1.487.2640, doi:10.1080/01418610108216660, S2CID 138403519, archived (PDF) from the original on 2011-07-21
71. Rudhart, Christoph Paul (June 1999), Zur numerischen Simulation des Bruchs von Quasikristallen (Thesis), University of Stuttgart, p. 11, doi:10.18419/opus-4639
72. Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000), "Tilings, coverings, clusters and quasicrystals" (PDF), Current Science, 78 (1): 64–72, archived (PDF) from the original on 2011-07-21
73. Katz, A. (1988), "Theory of Matching Rules for the 3-Dimensional Penrose Tilings", Communications in Mathematical Physics, 118 (2): 263–288, Bibcode:1988CMaPh.118..263K, doi:10.1007/BF01218580, S2CID 121086829
74. Culik, Karel; Kari, Jarkko (1995), "An aperiodic set of Wang cubes", Journal of Universal Computer Science, 1 (10), CiteSeerX 10.1.1.54.5897, doi:10.3217/jucs-001-10-0675
75. Walther. Gerd; Selter, Christoph, eds. (1999), Mathematikdidaktik als design science : Festschrift für Erich Christian Wittmann, Leipzig: Ernst Klett Grundschulverlag, ISBN 978-3-12-200060-8
76. Danzer, L. (1989), "Three-Dimensional Analogs of the Planar Penrose Tilings and Quasicrystals", Discrete Mathematics, 76 (1): 1–7, doi:10.1016/0012-365X(89)90282-3
77. Zerhusen, Aaron (1997), Danzer's three dimensional tiling, University of Kentucky
78. Goodman-Strauss, Chaim (1999), "An Aperiodic Pair of Tiles in En for all n ≥ 3", European J. Combin., 20 (5): 385–395, doi:10.1006/eujc.1998.0282 (preprint available)
79. Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "An aperiodic monotile". arXiv:2303.10798 [math.CO].
80. Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "A chiral aperiodic monotile". arXiv:2305.17743 [math.CO].
81. Mehta, Chirag (2021-04-03). "The art of what if". Journal of Mathematics and the Arts. 15 (2): 198–200. doi:10.1080/17513472.2021.1919977. ISSN 1751-3472.
External links
• Stephens P. W., Goldman A. I. The Structure of Quasicrystals
• Levine D., Steinhardt P. J. Quasicrystals I Definition and structure
• Tilings Encyclopedia
Tessellation
Periodic
• Pythagorean
• Rhombille
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regular
• 32.4.3.4
• V32.4.3.4
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Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
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• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
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• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Multimagic cube
In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k th powers for 1 ≤ k ≤ P. 2-multimagic cubes are called bimagic, 3-multimagic cubes are called trimagic, and 4-multimagic cubes tetramagic.[1] A P-multimagic cube is said to be semi-perfect if the k th power cubes are perfect for 1 ≤ k < P, and the P th power cube is semiperfect. If all P of the power cubes are perfect, the P-multimagic cube is said to be perfect.
The first known example of a bimagic cube was given by John Hendricks in 2000; it is a semiperfect cube of order 25 and magic constant 195325. In 2003, C. Bower discovered two semi-perfect bimagic cubes of order 16, and a perfect bimagic cube of order 32.[2]
MathWorld reports that only two trimagic cubes are known, discovered by C. Bower in 2003; a semiperfect cube of order 64 and a perfect cube of order 256.[3] It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.[4]
References
1. Weisstein, Eric W. "Multimagic cube". MathWorld.
2. Weisstein, Eric W. "Bimagic Cube". MathWorld.
3. Weisstein, Eric W. "Trimagic Cube". MathWorld.
4. Weisstein, Eric W. "Tetramagic Cube". MathWorld.
See also
• Magic square
• Multimagic square
| Wikipedia |
Trinity Mathematical Society
The Trinity Mathematical Society, abbreviated TMS, was founded in Trinity College, Cambridge in 1919 by G. H. Hardy to "promote the discussion of subjects of mathematical interest". It is the oldest mathematical university society in the United Kingdom and is believed to be the oldest existing subject society at any British university.
Today, the society is one of the largest societies in Trinity College, with nearly 600 members, and each year holds an extensive range of talks, together with social events including an annual cricket match against the Adams Society of St John's College, Cambridge.
The society has hosted a range of distinguished speakers, including: M.Atiyah, A.Baker; B.Birch; C.Birkar; B.Bollobás; M.Born; J.H.Conway; H.S.M.Coxeter; H.Davenport; P.Dirac; F.W.Dyson; O.R.Frisch; W.T.Gowers; G.H.Hardy; W.V.D.Hodge; P.Kaptiza; E.Landau; J.E.Littlewood; L.J.Mordell; R.Penrose; G.Polya; R.Rado; F.Ramsey; B.Russell; E.Rutherford; L.Susskind; P.Swinnerton-Dyer; J.J.Thomson; W.Thurston; F.Wilczek; L.Wittgenstein.[1]
The logo of the society is the minimal perfect squared square.
Significance of the apple
For historical reasons, the apple is very important symbolically to the society. An apple is dropped at the end of meetings to signify that the meeting is now social; the President bowls an apple as the first 'ball' at the annual cricket match; and, as outlined in the society's Standing Orders, an apple is part of the design of the Society tie.
See also
• Apple (symbolism)
• Ulam spiral
References
1. "TMS Meetings Archive".{{cite web}}: CS1 maint: url-status (link)
External links
• Official website
Trinity College, Cambridge
Founder
Henry VIII
Master
Professor Dame Sally Davies
List of Masters
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Student life
Choir
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Related
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Categories
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| Wikipedia |
k-noid
In differential geometry, a k-noid is a minimal surface with k catenoid openings. In particular, the 3-noid is often called trinoid. The first k-noid minimal surfaces were described by Jorge and Meeks in 1983.[1]
The term k-noid and trinoid is also sometimes used for constant mean curvature surfaces, especially branched versions of the unduloid ("triunduloids").[2]
k-noids are topologically equivalent to k-punctured spheres (spheres with k points removed). k-noids with symmetric openings can be generated using the Weierstrass–Enneper parameterization $f(z)=1/(z^{k}-1)^{2},g(z)=z^{k-1}\,\!$.[3] This produces the explicit formula
${\begin{aligned}X(z)={\frac {1}{2}}\Re {\Bigg \{}{\Big (}{\frac {-1}{kz(z^{k}-1)}}{\Big )}{\Big [}&(k-1)(z^{k}-1)_{2}F_{1}(1,-1/k;(k-1)/k;z^{k})\\&{}-(k-1)z^{2}(z^{k}-1)_{2}F_{1}(1,1/k;1+1/k;z^{k})\\&{}-kz^{k}+k+z^{2}-1{\Big ]}{\Bigg \}}\end{aligned}}$
${\begin{aligned}Y(z)={\frac {1}{2}}\Re {\Bigg \{}{\Big (}{\frac {i}{kz(z^{k}-1)}}{\Big )}{\Big [}&(k-1)(z^{k}-1)_{2}F_{1}(1,-1/k;(k-1)/k;z^{k})\\&{}+(k-1)z^{2}(z^{k}-1)_{2}F_{1}(1,1/k;1+1/k;z^{k})\\&{}-kz^{k}+k-z^{2}-1){\Big ]}{\Bigg \}}\end{aligned}}$
$Z(z)=\Re \left\{{\frac {1}{k-kz^{k}}}\right\}$
where $_{2}F_{1}(a,b;c;z)$ is the Gaussian hypergeometric function and $\Re \{z\}$ denotes the real part of $z$.
It is also possible to create k-noids with openings in different directions and sizes,[4] k-noids corresponding to the platonic solids and k-noids with handles.[5]
References
1. L. P. Jorge and W. H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983)
2. N Schmitt (2007). "Constant Mean Curvature n-noids with Platonic Symmetries". arXiv:math/0702469.
3. Matthias Weber (2001). "Classical Minimal Surfaces in Euclidean Space by Examples" (PDF). Indiana.edu. Retrieved 2012-10-05.
4. H. Karcher. "Construction of minimal surfaces, in "Surveys in Geometry", University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1-96" (PDF). Math.uni-bonn-de. Retrieved 2012-10-05.
5. Jorgen Berglund, Wayne Rossman (1995). "Minimal Surfaces with Catenoid Ends". Pacific J. Math. 171 (2): 353–371. arXiv:0804.4203. Bibcode:2008arXiv0804.4203B. doi:10.2140/pjm.1995.171.353. S2CID 11328539.
External links
• Indiana.edu
• Page.mi.fu-berlin.de
Minimal surfaces
• Associate family
• Bour's
• Catalan's
• Catenoid
• Chen–Gackstatter
• Costa's
• Enneper
• Gyroid
• Helicoid
• Henneberg
• k-noid
• Lidinoid
• Neovius
• Richmond
• Riemann's
• Saddle tower
• Scherk
• Schwarz
• Triply periodic
| Wikipedia |
Trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.[1]
Examples of trinomial expressions
1. $3x+5y+8z$ with $x,y,z$ variables
2. $3t+9s^{2}+3y^{3}$ with $t,s,y$ variables
3. $3ts+9t+5s$ with $t,s$ variables
4. $ax^{2}+bx+c$, the quadratic polynomial in standard form with $a,b,c$ variables.[note 1]
5. $Ax^{a}y^{b}z^{c}+Bt+Cs$ with $x,y,z,t,s$ variables, $a,b,c$ nonnegative integers and $A,B,C$ any constants.
6. $Px^{a}+Qx^{b}+Rx^{c}$ where $x$ is variable and constants $a,b,c$ are nonnegative integers and $P,Q,R$ any constants.
Trinomial equation
A trinomial equation is a polynomial equation involving three terms. An example is the equation $x=q+x^{m}$ studied by Johann Heinrich Lambert in the 18th century.[2]
Some notable trinomials
• The quadratic trinomial in standard form (as from above):
$ax^{2}+bx+c$
• sum or difference of two cubes:
$a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})$
• A special type of trinomial can be factored in a manner similar to quadratics since it can be viewed as a quadratic in a new variable (xn below). This form is factored as:
$x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),$
where
${\begin{aligned}a_{1}+a_{2}&=r\\a_{1}\cdot a_{2}&=s.\end{aligned}}$
For instance, the polynomial x2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a1 = −2 and a2 = −1 of the above system gives the trinomial factorization:
x2 + 3x + 2 = (x + a1)(x + a2) = (x + 2)(x + 1).
The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
See also
• Trinomial expansion
• Monomial
• Binomial
• Multinomial
• Simple expression
• Compound expression
• Sparse polynomial
Notes
1. Quadratic expressions are not always trinomials, the expressions' appearance can vary.
References
1. "Definition of Trinomial". Math Is Fun. Retrieved 16 April 2016.
2. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jerey, D. J.; Knuth, D. E. (1996). "On the Lambert W Function" (PDF). Advances in Computational Mathematics. 5 (1): 329–359. doi:10.1007/BF02124750.
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
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• Irreducible
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Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
| Wikipedia |
Trinomial tree
The trinomial tree is a lattice-based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the binomial options pricing model, and is conceptually similar. It can also be shown that the approach is equivalent to the explicit finite difference method for option pricing.[1] For fixed income and interest rate derivatives see Lattice model (finance)#Interest rate derivatives.
Formula
Under the trinomial method, the underlying stock price is modeled as a recombining tree, where, at each node the price has three possible paths: an up, down and stable or middle path.[2] These values are found by multiplying the value at the current node by the appropriate factor $u\,$, $d\,$ or $m\,$ where
$u=e^{\sigma {\sqrt {2\Delta t}}}$
$d=e^{-\sigma {\sqrt {2\Delta t}}}={\frac {1}{u}}\,$ (the structure is recombining)
$m=1\,$
and the corresponding probabilities are:
$p_{u}=\left({\frac {e^{(r-q)\Delta t/2}-e^{-\sigma {\sqrt {\Delta t/2}}}}{e^{\sigma {\sqrt {\Delta t/2}}}-e^{-\sigma {\sqrt {\Delta t/2}}}}}\right)^{2}\,$
$p_{d}=\left({\frac {e^{\sigma {\sqrt {\Delta t/2}}}-e^{(r-q)\Delta t/2}}{e^{\sigma {\sqrt {\Delta t/2}}}-e^{-\sigma {\sqrt {\Delta t/2}}}}}\right)^{2}\,$
$p_{m}=1-(p_{u}+p_{d})\,$.
In the above formulae: $\Delta t\,$ is the length of time per step in the tree and is simply time to maturity divided by the number of time steps; $r\,$ is the risk-free interest rate over this maturity; $\sigma \,$ is the corresponding volatility of the underlying; $q\,$ is its corresponding dividend yield.[3]
As with the binomial model, these factors and probabilities are specified so as to ensure that the price of the underlying evolves as a martingale, while the moments – considering node spacing and probabilities – are matched to those of the log-normal distribution[4] (and with increasing accuracy for smaller time-steps). Note that for $p_{u}$, $p_{d}$, and $p_{m}$ to be in the interval $(0,1)$ the following condition on $\Delta t$ has to be satisfied $\Delta t<2{\frac {\sigma ^{2}}{(r-q)^{2}}}$.
Once the tree of prices has been calculated, the option price is found at each node largely as for the binomial model, by working backwards from the final nodes to the present node ($t_{0}$). The difference being that the option value at each non-final node is determined based on the three – as opposed to two – later nodes and their corresponding probabilities.[5]
If the length of time-steps $\Delta t$ is taken as an exponentially distributed random variable and interpreted as the waiting time between two movements of the stock price then the resulting stochastic process is a birth–death process. The resulting model is soluble and there exist analytic pricing and hedging formulae for various options.
Application
The trinomial model is considered[6] to produce more accurate results than the binomial model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For vanilla options, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For exotic options the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.
See also
• Binomial options pricing model
• Valuation of options
• Option: Model implementation
• Korn–Kreer–Lenssen model
• Implied trinomial tree
References
1. Mark Rubinstein
2. Trinomial Tree, geometric Brownian motion Archived 2011-07-21 at the Wayback Machine
3. John Hull presents alternative formulae; see: Hull, John C. (2002). Options, Futures and Other Derivatives (5th ed.). Prentice Hall. ISBN 978-0-13-009056-0..
4. Pricing Options Using Trinomial Trees
5. Binomial and Trinomial Trees Versus Bjerksund and Stensland Approximations for American Options Pricing
6. On-Line Options Pricing & Probability Calculators
External links
• Phelim Boyle, 1986. "Option Valuation Using a Three-Jump Process", International Options Journal 3, 7–12.
• Rubinstein, M. (2000). "On the Relation Between Binomial and Trinomial Option Pricing Models". Journal of Derivatives. 8 (2): 47–50. CiteSeerX 10.1.1.43.5394. doi:10.3905/jod.2000.319149. Archived from the original on June 22, 2007.
• Paul Clifford et al. 2010. Pricing Options Using Trinomial Trees, University of Warwick
• Tero Haahtela, 2010. "Recombining Trinomial Tree for Real Option Valuation with Changing Volatility", Aalto University, Working Paper Series.
• Ralf Korn, Markus Kreer and Mark Lenssen, 1998. "Pricing of european options when the underlying stock price follows a linear birth-death process", Stochastic Models Vol. 14(3), pp 647 – 662
• Peter Hoadley. Trinomial Tree Option Calculator (Tree Visualized)
Derivatives market
Derivative (finance)
Options
Terms
• Delta neutral
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Strategies
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Valuation
• Bachelier
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• Commodity derivative
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| Wikipedia |
Trioctagonal tiling
In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.
Trioctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(3.8)2
Schläfli symbolr{8,3} or ${\begin{Bmatrix}8\\3\end{Bmatrix}}$
Wythoff symbol2 | 8 3|
3 3 | 4
Coxeter diagram or
Symmetry group[8,3], (*832)
[(4,3,3)], (*433)
DualOrder-8-3 rhombille tiling
PropertiesVertex-transitive edge-transitive
Symmetry
The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles, by Coxeter diagram .
Dual tiling
Related polyhedra and tilings
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
or
or
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
It can also be generated from the (4 3 3) hyperbolic tilings:
Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
h{8,3}
t0(4,3,3)
r{3,8}1/2
t0,1(4,3,3)
h{8,3}
t1(4,3,3)
h2{8,3}
t1,2(4,3,3)
{3,8}1/2
t2(4,3,3)
h2{8,3}
t0,2(4,3,3)
t{3,8}1/2
t0,1,2(4,3,3)
s{3,8}1/2
s(4,3,3)
Uniform duals
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4
The trioctagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:
Quasiregular tilings: (3.n)2
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Figure
Figure
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2
Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i}
Coxeter
Dual uniform figures
Dual
conf.
V(3.3)2
V(3.4)2
V(3.5)2
V(3.6)2
V(3.7)2
V(3.8)2
V(3.∞)2
Dimensional family of quasiregular polyhedra and tilings: (8.n)2
Symmetry
*8n2
[n,8]
Hyperbolic... Paracompact Noncompact
*832
[3,8]
*842
[4,8]
*852
[5,8]
*862
[6,8]
*872
[7,8]
*882
[8,8]...
*∞82
[∞,8]
[iπ/λ,8]
Coxeter
Quasiregular
figures
configuration
3.8.3.8
4.8.4.8
8.5.8.5
8.6.8.6
8.7.8.7
8.8.8.8
8.∞.8.∞
8.∞.8.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-8-3-8.
• Trihexagonal tiling - 3.6.3.6 tiling
• Rhombille tiling - dual V3.6.3.6 tiling
• Tilings of regular polygons
• List of uniform tilings
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Complex polytope
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.
A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on.
Precise definitions exist only for the regular complex polytopes, which are configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Coxeter.
Some complex polytopes which are not fully regular have also been described.
Definitions and introduction
The complex line $\mathbb {C} ^{1}$ has one dimension with real coordinates and another with imaginary coordinates. Applying real coordinates to both dimensions is said to give it two dimensions over the real numbers. A real plane, with the imaginary axis labelled as such, is called an Argand diagram. Because of this it is sometimes called the complex plane. Complex 2-space (also sometimes called the complex plane) is thus a four-dimensional space over the reals, and so on in higher dimensions.
A complex n-polytope in complex n-space is the analogue of a real n-polytope in real n-space.
There is no natural complex analogue of the ordering of points on a real line (or of the associated combinatorial properties). Because of this a complex polytope cannot be seen as a contiguous surface and it does not bound an interior in the way that a real polytope does.
In the case of regular polytopes, a precise definition can be made by using the notion of symmetry. For any regular polytope the symmetry group (here a complex reflection group, called a Shephard group) acts transitively on the flags, that is, on the nested sequences of a point contained in a line contained in a plane and so on.
More fully, say that a collection P of affine subspaces (or flats) of a complex unitary space V of dimension n is a regular complex polytope if it meets the following conditions:[1][2]
• for every −1 ≤ i < j < k ≤ n, if F is a flat in P of dimension i and H is a flat in P of dimension k such that F ⊂ H then there are at least two flats G in P of dimension j such that F ⊂ G ⊂ H;
• for every i, j such that −1 ≤ i < j − 2, j ≤ n, if F ⊂ G are flats of P of dimensions i, j, then the set of flats between F and G is connected, in the sense that one can get from any member of this set to any other by a sequence of containments; and
• the subset of unitary transformations of V that fix P are transitive on the flags F0 ⊂ F1 ⊂ … ⊂Fn of flats of P (with Fi of dimension i for all i).
(Here, a flat of dimension −1 is taken to mean the empty set.) Thus, by definition, regular complex polytopes are configurations in complex unitary space.
The regular complex polytopes were discovered by Shephard (1952), and the theory was further developed by Coxeter (1974).
Three views of regular complex polygon 4{4}2,
This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.[3] In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen.
A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image.
A complex polytope exists in the complex space of equivalent dimension. For example, the vertices of a complex polygon are points in the complex plane $\mathbb {C} ^{2}$, and the edges are complex lines $\mathbb {C} ^{1}$ existing as (affine) subspaces of the plane and intersecting at the vertices. Thus, an edge can be given a coordinate system consisting of a single complex number.
In a regular complex polytope the vertices incident on the edge are arranged symmetrically about their centroid, which is often used as the origin of the edge's coordinate system (in the real case the centroid is just the midpoint of the edge). The symmetry arises from a complex reflection about the centroid; this reflection will leave the magnitude of any vertex unchanged, but change its argument by a fixed amount, moving it to the coordinates of the next vertex in order. So we may assume (after a suitable choice of scale) that the vertices on the edge satisfy the equation $x^{p}-1=0$ where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the vertex points lie at the vertices of a regular polygon centered on the origin.
Three real projections of regular complex polygon 4{4}2 are illustrated above, with edges a, b, c, d, e, f, g, h. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices and each vertex lies on two edges, hence each edge meets four other edges. In the first diagram, each edge is represented by a square. The sides of the square are not parts of the polygon but are drawn purely to help visually relate the four vertices. The edges are laid out symmetrically. (Note that the diagram looks similar to the B4 Coxeter plane projection of the tesseract, but it is structurally different).
The middle diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a real line, and each meeting point of two lines is a vertex. The connectivity between the various edges is clear to see.
The last diagram gives a flavour of the structure projected into three dimensions: the two cubes of vertices are in fact the same size but are seen in perspective at different distances away in the fourth dimension.
Regular complex one-dimensional polytopes
A real 1-dimensional polytope exists as a closed segment in the real line $\mathbb {R} ^{1}$, defined by its two end points or vertices in the line. Its Schläfli symbol is {} .
Analogously, a complex 1-polytope exists as a set of p vertex points in the complex line $\mathbb {C} ^{1}$. These may be represented as a set of points in an Argand diagram (x,y)=x+iy. A regular complex 1-dimensional polytope p{} has p (p ≥ 2) vertex points arranged to form a convex regular polygon {p} in the Argand plane.[4]
Unlike points on the real line, points on the complex line have no natural ordering. Thus, unlike real polytopes, no interior can be defined.[5] Despite this, complex 1-polytopes are often drawn, as here, as a bounded regular polygon in the Argand plane.
A regular real 1-dimensional polytope is represented by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram . The dot or node of the Coxeter-Dynkin diagram itself represents a reflection generator while the circle around the node means the generator point is not on the reflection, so its reflective image is a distinct point from itself. By extension, a regular complex 1-dimensional polytope in $\mathbb {C} ^{1}$ has Coxeter-Dynkin diagram , for any positive integer p, 2 or greater, containing p vertices. p can be suppressed if it is 2. It can also be represented by an empty Schläfli symbol p{}, }p{, {}p, or p{2}1. The 1 is a notational placeholder, representing a nonexistent reflection, or a period 1 identity generator. (A 0-polytope, real or complex is a point, and is represented as } {, or 1{2}1.)
The symmetry is denoted by the Coxeter diagram , and can alternatively be described in Coxeter notation as p[], []p or ]p[, p[2]1 or p[1]p. The symmetry is isomorphic to the cyclic group, order p.[6] The subgroups of p[] are any whole divisor d, d[], where d≥2.
A unitary operator generator for is seen as a rotation by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is e2πi/p = cos(2π/p) + i sin(2π/p). When p = 2, the generator is eπi = –1, the same as a point reflection in the real plane.
In higher complex polytopes, 1-polytopes form p-edges. A 2-edge is similar to an ordinary real edge, in that it contains two vertices, but need not exist on a real line.
Regular complex polygons
While 1-polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons p{4}2, are limited to 5-edge (pentagonal edges) elements, and infinite regular apeirogons also include 6-edge (hexagonal edges) elements.
Shephard's modified Schläfli notation
Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.
The number of vertices V is then g/p2 and the number of edges E is g/p1.
The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
Coxeter's revised modified Schläfli notation
A more modern notation p1{q}p2 is due to Coxeter,[7] and is based on group theory. As a symmetry group, its symbol is p1[q]p2.
The symmetry group p1[q]p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q−1)/2R2 = (R1R2)(q−1)/2R1. When q is odd, p1=p2.
For 4[4]2 has R14 = R22 = I, (R2R1)2 = (R1R2)2.
For 3[5]3 has R13 = R23 = I, (R2R1)2R2 = (R1R2)2R1.
Coxeter-Dynkin diagrams
Coxeter also generalised the use of Coxeter-Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is represented by and the equivalent symmetry group, p[q]r, is a ringless diagram . The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is 2{q}2 or {q} or .
One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So and are ordinary, while is starry.
12 Irreducible Shephard groups
Coxeter enumerated this list of regular complex polygons in $\mathbb {C} ^{2}$. A regular complex polygon, p{q}r or , has p-edges, and r-gonal vertex figures. p{q}r is a finite polytope if (p+r)q>pr(q-2).
Its symmetry is written as p[q]r, called a Shephard group, analogous to a Coxeter group, while also allowing unitary reflections.
For nonstarry groups, the order of the group p[q]r can be computed as $g=8/q\cdot (1/p+2/q+1/r-1)^{-2}$.[9]
The Coxeter number for p[q]r is $h=2/(1/p+2/q+1/r-1)$, so the group order can also be computed as $g=2h^{2}/q$. A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.
The rank 2 solutions that generate complex polygons are:
Group G3=G(q,1,1)G2=G(p,1,2)G4G6G5G8G14G9G10G20G16G21G17G18
2[q]2, q=3,4...p[4]2, p=2,3...3[3]33[6]23[4]34[3]43[8]24[6]24[4]33[5]35[3]53[10]25[6]25[4]3
Order 2q2p22448729614419228836060072012001800
h q2p612243060
Excluded solutions with odd q and unequal p and r are: 6[3]2, 6[3]3, 9[3]3, 12[3]3, ..., 5[5]2, 6[5]2, 8[5]2, 9[5]2, 4[7]2, 9[5]2, 3[9]2, and 3[11]2.
Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and .
The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors.[10]
The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd.
Matrix generators
The group p[q]r, , can be represented by two matrices:[11]
NameR1
R2
Order p r
Matrix
$\left[{\begin{smallmatrix}e^{2\pi i/p}&0\\(e^{2\pi i/p}-1)k&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&(e^{2\pi i/r}-1)k\\0&e^{2\pi i/r}\\\end{smallmatrix}}\right]$
With
k=${\sqrt {\frac {cos({\frac {\pi }{p}}-{\frac {\pi }{r}})+cos({\frac {2\pi }{q}})}{2\sin {\frac {\pi }{p}}\sin {\frac {\pi }{r}}}}}$
Examples
NameR1
R2
Order p q
Matrix
$\left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&0\\0&e^{2\pi i/q}\\\end{smallmatrix}}\right]$
NameR1
R2
Order p 2
Matrix
$\left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]$
NameR1
R2
Order 3 3
Matrix
$\left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&{\frac {-3+{\sqrt {3}}i}{2}}\\0&{\frac {-1+{\sqrt {3}}i}{2}}\\\end{smallmatrix}}\right]$
NameR1
R2
Order 4 4
Matrix
$\left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&0\\0&i\\\end{smallmatrix}}\right]$
NameR1
R2
Order 4 2
Matrix
$\left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]$
NameR1
R2
Order 3 2
Matrix
$\left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&-2\\0&-1\\\end{smallmatrix}}\right]$
Enumeration of regular complex polygons
Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.[12]
GroupOrderCoxeter
number
PolygonVerticesEdgesNotes
G(q,q,2)
2[q]2 = [q]
q=2,3,4,...
2qq2{q}2qq{}Real regular polygons
Same as
Same as if q even
GroupOrderCoxeter
number
PolygonVerticesEdgesNotes
G(p,1,2)
p[4]2
p=2,3,4,...
2p22pp(2p2)2p{4}2
p22pp{}same as p{}×p{} or
$\mathbb {R} ^{4}$ representation as p-p duoprism
2(2p2)p2{4}p2pp2{}$\mathbb {R} ^{4}$ representation as p-p duopyramid
G(2,1,2)
2[4]2 = [4]
842{4}2 = {4}44{}same as {}×{} or
Real square
G(3,1,2)
3[4]2
1866(18)23{4}2963{}same as 3{}×3{} or
$\mathbb {R} ^{4}$ representation as 3-3 duoprism
2(18)32{4}369{}$\mathbb {R} ^{4}$ representation as 3-3 duopyramid
G(4,1,2)
4[4]2
3288(32)24{4}21684{}same as 4{}×4{} or
$\mathbb {R} ^{4}$ representation as 4-4 duoprism or {4,3,3}
2(32)42{4}4816{}$\mathbb {R} ^{4}$ representation as 4-4 duopyramid or {3,3,4}
G(5,1,2)
5[4]2
50255(50)25{4}225105{}same as 5{}×5{} or
$\mathbb {R} ^{4}$ representation as 5-5 duoprism
2(50)52{4}51025{}$\mathbb {R} ^{4}$ representation as 5-5 duopyramid
G(6,1,2)
6[4]2
72366(72)26{4}236126{}same as 6{}×6{} or
$\mathbb {R} ^{4}$ representation as 6-6 duoprism
2(72)62{4}61236{}$\mathbb {R} ^{4}$ representation as 6-6 duopyramid
G4=G(1,1,2)
3[3]3
<2,3,3>
2463(24)33{3}3883{}Möbius–Kantor configuration
self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,3,4}
G6
3[6]2
48123(48)23{6}224163{}same as
3{3}2starry polygon
2(48)32{6}31624{}
2{3}3starry polygon
G5
3[4]3
72123(72)33{4}324243{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,4,3}
G8
4[3]4
96124(96)44{3}424244{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,4,3}
G14
3[8]2
144243(144)23{8}272483{}same as
3{8/3}2starry polygon, same as
2(144)32{8}34872{}
2{8/3}3starry polygon
G9
4[6]2
192244(192)24{6}296484{}same as
2(192)42{6}44896{}
4{3}29648{}starry polygon
2{3}44896{}starry polygon
G10
4[4]3
288244(288)34{4}396724{}
124{8/3}3starry polygon
243(288)43{4}472963{}
123{8/3}4starry polygon
G20
3[5]3
360303(360)33{5}31201203{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,3,5}
3{5/2}3self-dual, starry polygon
G16
5[3]5
600305(600)55{3}51201205{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,3,5}
105{5/2}5self-dual, starry polygon
G21
3[10]2
720603(720)23{10}23602403{}same as
3{5}2starry polygon
3{10/3}2starry polygon, same as
3{5/2}2starry polygon
2(720)32{10}3240360{}
2{5}3starry polygon
2{10/3}3starry polygon
2{5/2}3starry polygon
G17
5[6]2
1200605(1200)25{6}26002405{}same as
205{5}2starry polygon
205{10/3}2starry polygon
605{3}2starry polygon
602(1200)52{6}5240600{}
202{5}5starry polygon
202{10/3}5starry polygon
602{3}5starry polygon
G18
5[4]3
1800605(1800)35{4}36003605{}
155{10/3}3starry polygon
305{3}3starry polygon
305{5/2}3starry polygon
603(1800)53{4}53606003{}
153{10/3}5starry polygon
303{3}5starry polygon
303{5/2}5starry polygon
Visualizations of regular complex polygons
Polygons of the form p{2r}q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.
2D orthogonal projections of complex polygons 2{r}q
Polygons of the form 2{4}q are called generalized orthoplexes. They share vertices with the 4D q-q duopyramids, vertices connected by 2-edges.
• 2{4}2, , with 4 vertices, and 4 edges
• 2{4}3, , with 6 vertices, and 9 edges[13]
• 2{4}4, , with 8 vertices, and 16 edges
• 2{4}5, , with 10 vertices, and 25 edges
• 2{4}6, , with 12 vertices, and 36 edges
• 2{4}7, , with 14 vertices, and 49 edges
• 2{4}8, , with 16 vertices, and 64 edges
• 2{4}9, , with 18 vertices, and 81 edges
• 2{4}10, , with 20 vertices, and 100 edges
Complex polygons p{4}2
Polygons of the form p{4}2 are called generalized hypercubes (squares for polygons). They share vertices with the 4D p-p duoprisms, vertices connected by p-edges. Vertices are drawn in green, and p-edges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.
• 2{4}2, or , with 4 vertices, and 4 2-edges
• 3{4}2, or , with 9 vertices, and 6 (triangular) 3-edges[13]
• 4{4}2, or , with 16 vertices, and 8 (square) 4-edges
• 5{4}2, or , with 25 vertices, and 10 (pentagonal) 5-edges
• 6{4}2, or , with 36 vertices, and 12 (hexagonal) 6-edges
• 7{4}2, or , with 49 vertices, and 14 (heptagonal)7-edges
• 8{4}2, or , with 64 vertices, and 16 (octagonal) 8-edges
• 9{4}2, or , with 81 vertices, and 18 (enneagonal) 9-edges
• 10{4}2, or , with 100 vertices, and 20 (decagonal) 10-edges
3D perspective projections of complex polygons p{4}2. The duals 2{4}p
are seen by adding vertices inside the edges, and adding edges in place of vertices.
• 3{4}2, or with 9 vertices, 6 3-edges in 2 sets of colors
• 2{4}3, with 6 vertices, 9 edges in 3 sets
• 4{4}2, or with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges
• 5{4}2, or with 25 vertices, 10 5-edges in 2 sets of colors
Other Complex polygons p{r}2
• 3{6}2, or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue[14]
• 3{8}2, or , with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue[15]
2D orthogonal projections of complex polygons, p{r}p
Polygons of the form p{r}p have equal number of vertices and edges. They are also self-dual.
• 3{3}3, or , with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue[16]
• 3{4}3, or , with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled[17]
• 4{3}4, or , with 24 vertices and 24 4-edges shown in 4 sets of colors[17]
• 3{5}3, or , with 120 vertices and 120 3-edges[18]
• 5{3}5, or , with 120 vertices and 120 5-edges[19]
Regular complex polytopes
In general, a regular complex polytope is represented by Coxeter as p{z1}q{z2}r{z3}s… or Coxeter diagram …, having symmetry p[z1]q[z2]r[z3]s… or ….[20]
There are infinite families of regular complex polytopes that occur in all dimensions, generalizing the hypercubes and cross polytopes in real space. Shephard's "generalized orthotope" generalizes the hypercube; it has symbol given by γp
n
= p{4}2{3}2…2{3}2 and diagram …. Its symmetry group has diagram p[4]2[3]2…2[3]2; in the Shephard–Todd classification, this is the group G(p, 1, n) generalizing the signed permutation matrices. Its dual regular polytope, the "generalized cross polytope", is represented by the symbol βp
n
= 2{3}2{3}2…2{4}p and diagram ….[21]
A 1-dimensional regular complex polytope in $\mathbb {C} ^{1}$ is represented as , having p vertices, with its real representation a regular polygon, {p}. Coxeter also gives it symbol γp
1
or βp
1
as 1-dimensional generalized hypercube or cross polytope. Its symmetry is p[] or , a cyclic group of order p. In a higher polytope, p{} or represents a p-edge element, with a 2-edge, {} or , representing an ordinary real edge between two vertices.[21]
A dual complex polytope is constructed by exchanging k and (n-1-k)-elements of an n-polytope. For example, a dual complex polygon has vertices centered on each edge, and new edges are centered at the old vertices. A v-valence vertex creates a new v-edge, and e-edges become e-valence vertices.[22] The dual of a regular complex polytope has a reversed symbol. Regular complex polytopes with symmetric symbols, i.e. p{q}p, p{q}r{q}p, p{q}r{s}r{q}p, etc. are self dual.
Enumeration of regular complex polyhedra
Coxeter enumerated this list of nonstarry regular complex polyhedra in $\mathbb {C} ^{3}$, including the 5 platonic solids in $\mathbb {R} ^{3}$.[23]
A regular complex polyhedron, p{n1}q{n2}r or , has faces, edges, and vertex figures.
A complex regular polyhedron p{n1}q{n2}r requires both g1 = order(p[n1]q) and g2 = order(q[n2]r) be finite.
Given g = order(p[n1]q[n2]r), the number of vertices is g/g2, and the number of faces is g/g1. The number of edges is g/pr.
SpaceGroupOrderCoxeter numberPolygonVerticesEdgesFacesVertex
figure
Van Oss
polygon
Notes
$\mathbb {R} ^{3}$G(1,1,3)
2[3]2[3]2
= [3,3]
244α3 = 2{3}2{3}2
= {3,3}
46{}4{3}{3}noneReal tetrahedron
Same as
$\mathbb {R} ^{3}$G23
2[3]2[5]2
= [3,5]
120102{3}2{5}2 = {3,5}1230{}20{3}{5}noneReal icosahedron
2{5}2{3}2 = {5,3}2030{}12{5}{3}noneReal dodecahedron
$\mathbb {R} ^{3}$G(2,1,3)
2[3]2[4]2
= [3,4]
486β2
3
= β3 = {3,4}
612{}8{3}{4}{4}Real octahedron
Same as {}+{}+{}, order 8
Same as , order 24
$\mathbb {R} ^{3}$γ2
3
= γ3 = {4,3}
812{}6{4}{3}noneReal cube
Same as {}×{}×{} or
$\mathbb {C} ^{3}$G(p,1,3)
2[3]2[4]p
p=2,3,4,...
6p33pβp
3
= 2{3}2{4}p
3p3p2{}p3{3}2{4}p2{4}pGeneralized octahedron
Same as p{}+p{}+p{}, order p3
Same as , order 6p2
$\mathbb {C} ^{3}$γp
3
= p{4}2{3}2
p33p2p{}3pp{4}2{3}noneGeneralized cube
Same as p{}×p{}×p{} or
$\mathbb {C} ^{3}$G(3,1,3)
2[3]2[4]3
1629β3
3
= 2{3}2{4}3
927{}27{3}2{4}32{4}3Same as 3{}+3{}+3{}, order 27
Same as , order 54
$\mathbb {C} ^{3}$γ3
3
= 3{4}2{3}2
27273{}93{4}2{3}noneSame as 3{}×3{}×3{} or
$\mathbb {C} ^{3}$G(4,1,3)
2[3]2[4]4
38412β4
3
= 2{3}2{4}4
1248{}64{3}2{4}42{4}4Same as 4{}+4{}+4{}, order 64
Same as , order 96
$\mathbb {C} ^{3}$γ4
3
= 4{4}2{3}2
64484{}124{4}2{3}noneSame as 4{}×4{}×4{} or
$\mathbb {C} ^{3}$G(5,1,3)
2[3]2[4]5
75015β5
3
= 2{3}2{4}5
1575{}125{3}2{4}52{4}5Same as 5{}+5{}+5{}, order 125
Same as , order 150
$\mathbb {C} ^{3}$γ5
3
= 5{4}2{3}2
125755{}155{4}2{3}noneSame as 5{}×5{}×5{} or
$\mathbb {C} ^{3}$G(6,1,3)
2[3]2[4]6
129618β6
3
= 2{3}2{4}6
36108{}216{3}2{4}62{4}6Same as 6{}+6{}+6{}, order 216
Same as , order 216
$\mathbb {C} ^{3}$γ6
3
= 6{4}2{3}2
2161086{}186{4}2{3}noneSame as 6{}×6{}×6{} or
$\mathbb {C} ^{3}$G25
3[3]3[3]3
64893{3}3{3}327723{}273{3}33{3}33{4}2Same as .
$\mathbb {R} ^{6}$ representation as 221
Hessian polyhedron
G26
2[4]3[3]3
1296182{4}3{3}354216{}722{4}33{3}3{6}
3{3}3{4}2722163{}543{3}33{4}23{4}3Same as [24]
$\mathbb {R} ^{6}$ representation as 122
Visualizations of regular complex polyhedra
2D orthogonal projections of complex polyhedra, p{s}t{r}r
• Real {3,3}, or has 4 vertices, 6 edges, and 4 faces
• 3{3}3{3}3, or , has 27 vertices, 72 3-edges, and 27 faces, with one face highlighted blue.[25]
• 2{4}3{3}3, has 54 vertices, 216 simple edges, and 72 faces, with one face highlighted blue.[26]
• 3{3}3{4}2, or , has 72 vertices, 216 3-edges, and 54 vertices, with one face highlighted blue.[27]
Generalized octahedra
Generalized octahedra have a regular construction as and quasiregular form as . All elements are simplexes.
• Real {3,4}, or , with 6 vertices, 12 edges, and 8 faces
• 2{3}2{4}3, or , with 9 vertices, 27 edges, and 27 faces
• 2{3}2{4}4, or , with 12 vertices, 48 edges, and 64 faces
• 2{3}2{4}5, or , with 15 vertices, 75 edges, and 125 faces
• 2{3}2{4}6, or , with 18 vertices, 108 edges, and 216 faces
• 2{3}2{4}7, or , with 21 vertices, 147 edges, and 343 faces
• 2{3}2{4}8, or , with 24 vertices, 192 edges, and 512 faces
• 2{3}2{4}9, or , with 27 vertices, 243 edges, and 729 faces
• 2{3}2{4}10, or , with 30 vertices, 300 edges, and 1000 faces
Generalized cubes
Generalized cubes have a regular construction as and prismatic construction as , a product of three p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.
• Real {4,3}, or has 8 vertices, 12 edges, and 6 faces
• 3{4}2{3}2, or has 27 vertices, 27 3-edges, and 9 faces[25]
• 4{4}2{3}2, or , with 64 vertices, 48 edges, and 12 faces
• 5{4}2{3}2, or , with 125 vertices, 75 edges, and 15 faces
• 6{4}2{3}2, or , with 216 vertices, 108 edges, and 18 faces
• 7{4}2{3}2, or , with 343 vertices, 147 edges, and 21 faces
• 8{4}2{3}2, or , with 512 vertices, 192 edges, and 24 faces
• 9{4}2{3}2, or , with 729 vertices, 243 edges, and 27 faces
• 10{4}2{3}2, or , with 1000 vertices, 300 edges, and 30 faces
Enumeration of regular complex 4-polytopes
Coxeter enumerated this list of nonstarry regular complex 4-polytopes in $\mathbb {C} ^{4}$, including the 6 convex regular 4-polytopes in $\mathbb {R} ^{4}$.[23]
SpaceGroupOrderCoxeter
number
PolytopeVerticesEdgesFacesCellsVan Oss
polygon
Notes
$\mathbb {R} ^{4}$G(1,1,4)
2[3]2[3]2[3]2
= [3,3,3]
1205α4 = 2{3}2{3}2{3}2
= {3,3,3}
510
{}
10
{3}
5
{3,3}
noneReal 5-cell (simplex)
$\mathbb {R} ^{4}$G28
2[3]2[4]2[3]2
= [3,4,3]
1152122{3}2{4}2{3}2 = {3,4,3}
2496
{}
96
{3}
24
{3,4}
{6}Real 24-cell
G30
2[3]2[3]2[5]2
= [3,3,5]
14400302{3}2{3}2{5}2 = {3,3,5}
120720
{}
1200
{3}
600
{3,3}
{10}Real 600-cell
2{5}2{3}2{3}2 = {5,3,3}
6001200
{}
720
{5}
120
{5,3}
Real 120-cell
$\mathbb {R} ^{4}$G(2,1,4)
2[3]2[3]2[4]p
=[3,3,4]
3848β2
4
= β4 = {3,3,4}
824
{}
32
{3}
16
{3,3}
{4}Real 16-cell
Same as , order 192
$\mathbb {R} ^{4}$γ2
4
= γ4 = {4,3,3}
1632
{}
24
{4}
8
{4,3}
noneReal tesseract
Same as {}4 or , order 16
$\mathbb {C} ^{4}$G(p,1,4)
2[3]2[3]2[4]p
p=2,3,4,...
24p44pβp
4
= 2{3}2{3}2{4}p
4p6p2
{}
4p3
{3}
p4
{3,3}
2{4}pGeneralized 4-orthoplex
Same as , order 24p3
$\mathbb {C} ^{4}$γp
4
= p{4}2{3}2{3}2
p44p3
p{}
6p2
p{4}2
4p
p{4}2{3}2
noneGeneralized tesseract
Same as p{}4 or , order p4
$\mathbb {C} ^{4}$G(3,1,4)
2[3]2[3]2[4]3
194412β3
4
= 2{3}2{3}2{4}3
1254
{}
108
{3}
81
{3,3}
2{4}3Generalized 4-orthoplex
Same as , order 648
$\mathbb {C} ^{4}$γ3
4
= 3{4}2{3}2{3}2
81108
3{}
54
3{4}2
12
3{4}2{3}2
noneSame as 3{}4 or , order 81
$\mathbb {C} ^{4}$G(4,1,4)
2[3]2[3]2[4]4
614416β4
4
= 2{3}2{3}2{4}4
1696
{}
256
{3}
64
{3,3}
2{4}4Same as , order 1536
$\mathbb {C} ^{4}$γ4
4
= 4{4}2{3}2{3}2
256256
4{}
96
4{4}2
16
4{4}2{3}2
noneSame as 4{}4 or , order 256
$\mathbb {C} ^{4}$G(5,1,4)
2[3]2[3]2[4]5
1500020β5
4
= 2{3}2{3}2{4}5
20150
{}
500
{3}
625
{3,3}
2{4}5Same as , order 3000
$\mathbb {C} ^{4}$γ5
4
= 5{4}2{3}2{3}2
625500
5{}
150
5{4}2
20
5{4}2{3}2
noneSame as 5{}4 or , order 625
$\mathbb {C} ^{4}$G(6,1,4)
2[3]2[3]2[4]6
3110424β6
4
= 2{3}2{3}2{4}6
24216
{}
864
{3}
1296
{3,3}
2{4}6Same as , order 5184
$\mathbb {C} ^{4}$γ6
4
= 6{4}2{3}2{3}2
1296864
6{}
216
6{4}2
24
6{4}2{3}2
noneSame as 6{}4 or , order 1296
$\mathbb {C} ^{4}$G32
3[3]3[3]3[3]3
155520303{3}3{3}3{3}3
2402160
3{}
2160
3{3}3
240
3{3}3{3}3
3{4}3Witting polytope
$\mathbb {R} ^{8}$ representation as 421
Visualizations of regular complex 4-polytopes
• Real {3,3,3}, , had 5 vertices, 10 edges, 10 {3} faces, and 5 {3,3} cells
• Real {3,4,3}, , had 24 vertices, 96 edges, 96 {3} faces, and 24 {3,4} cells
• Real {5,3,3}, , had 600 vertices, 1200 edges, 720 {5} faces, and 120 {5,3} cells
• Real {3,3,5}, , had 120 vertices, 720 edges, 1200 {3} faces, and 600 {3,3} cells
• Witting polytope, , has 240 vertices, 2160 3-edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells
Generalized 4-orthoplexes
Generalized 4-orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.
• Real {3,3,4}, or , with 8 vertices, 24 edges, 32 faces, and 16 cells
• 2{3}2{3}2{4}3, or , with 12 vertices, 54 edges, 108 faces, and 81 cells
• 2{3}2{3}2{4}4, or , with 16 vertices, 96 edges, 256 faces, and 256 cells
• 2{3}2{3}2{4}5, or , with 20 vertices, 150 edges, 500 faces, and 625 cells
• 2{3}2{3}2{4}6, or , with 24 vertices, 216 edges, 864 faces, and 1296 cells
• 2{3}2{3}2{4}7, or , with 28 vertices, 294 edges, 1372 faces, and 2401 cells
• 2{3}2{3}2{4}8, or , with 32 vertices, 384 edges, 2048 faces, and 4096 cells
• 2{3}2{3}2{4}9, or , with 36 vertices, 486 edges, 2916 faces, and 6561 cells
• 2{3}2{3}2{4}10, or , with 40 vertices, 600 edges, 4000 faces, and 10000 cells
Generalized 4-cubes
Generalized tesseracts have a regular construction as and prismatic construction as , a product of four p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.
• Real {4,3,3}, or , with 16 vertices, 32 edges, 24 faces, and 8 cells
• 3{4}2{3}2{3}2, or , with 81 vertices, 108 edges, 54 faces, and 12 cells
• 4{4}2{3}2{3}2, or , with 256 vertices, 96 edges, 96 faces, and 16 cells
• 5{4}2{3}2{3}2, or , with 625 vertices, 500 edges, 150 faces, and 20 cells
• 6{4}2{3}2{3}2, or , with 1296 vertices, 864 edges, 216 faces, and 24 cells
• 7{4}2{3}2{3}2, or , with 2401 vertices, 1372 edges, 294 faces, and 28 cells
• 8{4}2{3}2{3}2, or , with 4096 vertices, 2048 edges, 384 faces, and 32 cells
• 9{4}2{3}2{3}2, or , with 6561 vertices, 2916 edges, 486 faces, and 36 cells
• 10{4}2{3}2{3}2, or , with 10000 vertices, 4000 edges, 600 faces, and 40 cells
Enumeration of regular complex 5-polytopes
Regular complex 5-polytopes in $\mathbb {C} ^{5}$ or higher exist in three families, the real simplexes and the generalized hypercube, and orthoplex.
SpaceGroupOrderPolytopeVerticesEdgesFacesCells4-facesVan Oss
polygon
Notes
$\mathbb {R} ^{5}$G(1,1,5)
= [3,3,3,3]
720α5 = {3,3,3,3}
615
{}
20
{3}
15
{3,3}
6
{3,3,3}
noneReal 5-simplex
$\mathbb {R} ^{5}$G(2,1,5)
=[3,3,3,4]
3840β2
5
= β5 = {3,3,3,4}
1040
{}
80
{3}
80
{3,3}
32
{3,3,3}
{4}Real 5-orthoplex
Same as , order 1920
$\mathbb {R} ^{5}$γ2
5
= γ5 = {4,3,3,3}
3280
{}
80
{4}
40
{4,3}
10
{4,3,3}
noneReal 5-cube
Same as {}5 or , order 32
$\mathbb {C} ^{5}$G(p,1,5)
2[3]2[3]2[3]2[4]p
120p5βp
5
= 2{3}2{3}2{3}2{4}p
5p10p2
{}
10p3
{3}
5p4
{3,3}
p5
{3,3,3}
2{4}pGeneralized 5-orthoplex
Same as , order 120p4
$\mathbb {C} ^{5}$γp
5
= p{4}2{3}2{3}2{3}2
p55p4
p{}
10p3
p{4}2
10p2
p{4}2{3}2
5p
p{4}2{3}2{3}2
noneGeneralized 5-cube
Same as p{}5 or , order p5
$\mathbb {C} ^{5}$G(3,1,5)
2[3]2[3]2[3]2[4]3
29160β3
5
= 2{3}2{3}2{3}2{4}3
1590
{}
270
{3}
405
{3,3}
243
{3,3,3}
2{4}3Same as , order 9720
$\mathbb {C} ^{5}$γ3
5
= 3{4}2{3}2{3}2{3}2
243405
3{}
270
3{4}2
90
3{4}2{3}2
15
3{4}2{3}2{3}2
noneSame as 3{}5 or , order 243
$\mathbb {C} ^{5}$G(4,1,5)
2[3]2[3]2[3]2[4]4
122880β4
5
= 2{3}2{3}2{3}2{4}4
20160
{}
640
{3}
1280
{3,3}
1024
{3,3,3}
2{4}4Same as , order 30720
$\mathbb {C} ^{5}$γ4
5
= 4{4}2{3}2{3}2{3}2
10241280
4{}
640
4{4}2
160
4{4}2{3}2
20
4{4}2{3}2{3}2
noneSame as 4{}5 or , order 1024
$\mathbb {C} ^{5}$G(5,1,5)
2[3]2[3]2[3]2[4]5
375000β5
5
= 2{3}2{3}2{3}2{5}5
25250
{}
1250
{3}
3125
{3,3}
3125
{3,3,3}
2{5}5Same as , order 75000
$\mathbb {C} ^{5}$γ5
5
= 5{4}2{3}2{3}2{3}2
31253125
5{}
1250
5{5}2
250
5{5}2{3}2
25
5{4}2{3}2{3}2
noneSame as 5{}5 or , order 3125
$\mathbb {C} ^{5}$G(6,1,5)
2[3]2[3]2[3]2[4]6
933210β6
5
= 2{3}2{3}2{3}2{4}6
30360
{}
2160
{3}
6480
{3,3}
7776
{3,3,3}
2{4}6Same as , order 155520
$\mathbb {C} ^{5}$γ6
5
= 6{4}2{3}2{3}2{3}2
77766480
6{}
2160
6{4}2
360
6{4}2{3}2
30
6{4}2{3}2{3}2
noneSame as 6{}5 or , order 7776
Visualizations of regular complex 5-polytopes
Generalized 5-orthoplexes
Generalized 5-orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.
• Real {3,3,3,4}, , with 10 vertices, 40 edges, 80 faces, 80 cells, and 32 4-faces
• 2{3}2{3}2{3}2{4}3, , with 15 vertices, 90 edges, 270 faces, 405 cells, and 243 4-faces
• 2{3}2{3}2{3}2{4}4, , with 20 vertices, 160 edges, 640 faces, 1280 cells, and 1024 4-faces
• 2{3}2{3}2{3}2{4}5, , with 25 vertices, 250 edges, 1250 faces, 3125 cells, and 3125 4-faces
• 2{3}2{3}2{3}2{4}6, , with 30 vertices, 360 edges, 2160 faces, 6480 cells, 7776 4-faces
• 2{3}2{3}2{3}2{4}7, , with 35 vertices, 490 edges, 3430 faces, 12005 cells, 16807 4-faces
• 2{3}2{3}2{3}2{4}8, , with 40 vertices, 640 edges, 5120 faces, 20480 cells, 32768 4-faces
• 2{3}2{3}2{3}2{4}9, , with 45 vertices, 810 edges, 7290 faces, 32805 cells, 59049 4-faces
• 2{3}2{3}2{3}2{4}10, , with 50 vertices, 1000 edges, 10000 faces, 50000 cells, 100000 4-faces
Generalized 5-cubes
Generalized 5-cubes have a regular construction as and prismatic construction as , a product of five p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.
• Real {4,3,3,3}, , with 32 vertices, 80 edges, 80 faces, 40 cells, and 10 4-faces
• 3{4}2{3}2{3}2{3}2, , with 243 vertices, 405 edges, 270 faces, 90 cells, and 15 4-faces
• 4{4}2{3}2{3}2{3}2, , with 1024 vertices, 1280 edges, 640 faces, 160 cells, and 20 4-faces
• 5{4}2{3}2{3}2{3}2, , with 3125 vertices, 3125 edges, 1250 faces, 250 cells, and 25 4-faces
• 6{4}2{3}2{3}2{3}2, , with 7776 vertices, 6480 edges, 2160 faces, 360 cells, and 30 4-faces
Enumeration of regular complex 6-polytopes
SpaceGroupOrderPolytopeVerticesEdgesFacesCells4-faces5-facesVan Oss
polygon
Notes
$\mathbb {R} ^{6}$G(1,1,6)
= [3,3,3,3,3]
720α6 = {3,3,3,3,3}
721
{}
35
{3}
35
{3,3}
21
{3,3,3}
7
{3,3,3,3}
noneReal 6-simplex
$\mathbb {R} ^{6}$G(2,1,6)
[3,3,3,4]
46080β2
6
= β6 = {3,3,3,4}
1260
{}
160
{3}
240
{3,3}
192
{3,3,3}
64
{3,3,3,3}
{4}Real 6-orthoplex
Same as , order 23040
$\mathbb {R} ^{6}$γ2
6
= γ6 = {4,3,3,3}
64192
{}
240
{4}
160
{4,3}
60
{4,3,3}
12
{4,3,3,3}
noneReal 6-cube
Same as {}6 or , order 64
$\mathbb {C} ^{6}$G(p,1,6)
2[3]2[3]2[3]2[4]p
720p6βp
6
= 2{3}2{3}2{3}2{4}p
6p15p2
{}
20p3
{3}
15p4
{3,3}
6p5
{3,3,3}
p6
{3,3,3,3}
2{4}pGeneralized 6-orthoplex
Same as , order 720p5
$\mathbb {C} ^{6}$γp
6
= p{4}2{3}2{3}2{3}2
p66p5
p{}
15p4
p{4}2
20p3
p{4}2{3}2
15p2
p{4}2{3}2{3}2
6p
p{4}2{3}2{3}2{3}2
noneGeneralized 6-cube
Same as p{}6 or , order p6
Visualizations of regular complex 6-polytopes
Generalized 6-orthoplexes
Generalized 6-orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.
• Real {3,3,3,3,4}, , with 12 vertices, 60 edges, 160 faces, 240 cells, 192 4-faces, and 64 5-faces
• 2{3}2{3}2{3}2{3}2{4}3, , with 18 vertices, 135 edges, 540 faces, 1215 cells, 1458 4-faces, and 729 5-faces
• 2{3}2{3}2{3}2{3}2{4}4, , with 24 vertices, 240 edges, 1280 faces, 3840 cells, 6144 4-faces, and 4096 5-faces
• 2{3}2{3}2{3}2{3}2{4}5, , with 30 vertices, 375 edges, 2500 faces, 9375 cells, 18750 4-faces, and 15625 5-faces
• 2{3}2{3}2{3}2{3}2{4}6, , with 36 vertices, 540 edges, 4320 faces, 19440 cells, 46656 4-faces, and 46656 5-faces
• 2{3}2{3}2{3}2{3}2{4}7, , with 42 vertices, 735 edges, 6860 faces, 36015 cells, 100842 4-faces, 117649 5-faces
• 2{3}2{3}2{3}2{3}2{4}8, , with 48 vertices, 960 edges, 10240 faces, 61440 cells, 196608 4-faces, 262144 5-faces
• 2{3}2{3}2{3}2{3}2{4}9, , with 54 vertices, 1215 edges, 14580 faces, 98415 cells, 354294 4-faces, 531441 5-faces
• 2{3}2{3}2{3}2{3}2{4}10, , with 60 vertices, 1500 edges, 20000 faces, 150000 cells, 600000 4-faces, 1000000 5-faces
Generalized 6-cubes
Generalized 6-cubes have a regular construction as and prismatic construction as , a product of six p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.
• Real {3,3,3,3,3,4}, , with 64 vertices, 192 edges, 240 faces, 160 cells, 60 4-faces, and 12 5-faces
• 3{4}2{3}2{3}2{3}2{3}2, , with 729 vertices, 1458 edges, 1215 faces, 540 cells, 135 4-faces, and 18 5-faces
• 4{4}2{3}2{3}2{3}2{3}2, , with 4096 vertices, 6144 edges, 3840 faces, 1280 cells, 240 4-faces, and 24 5-faces
• 5{4}2{3}2{3}2{3}2{3}2, , with 15625 vertices, 18750 edges, 9375 faces, 2500 cells, 375 4-faces, and 30 5-faces
Enumeration of regular complex apeirotopes
Coxeter enumerated this list of nonstarry regular complex apeirotopes or honeycombs.[28]
For each dimension there are 12 apeirotopes symbolized as δp,r
n+1
exists in any dimensions $\mathbb {C} ^{n}$, or $\mathbb {R} ^{n}$ if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2.[29]
Each has proportional element counts given as:
k-faces = ${n \choose k}p^{n-k}r^{k}$, where ${n \choose m}={\frac {n!}{m!\,(n-m)!}}$ and n! denotes the factorial of n.
Regular complex 1-polytopes
The only regular complex 1-polytope is ∞{}, or . Its real representation is an apeirogon, {∞}, or .
Regular complex apeirogons
Rank 2 complex apeirogons have symmetry p[q]r, where 1/p + 2/q + 1/r = 1. Coxeter expresses them as δp,r
2
where q is constrained to satisfy q = 2/(1 – (p + r)/pr).[30]
There are 8 solutions:
2[∞]23[12]24[8]26[6]23[6]36[4]34[4]46[3]6
There are two excluded solutions odd q and unequal p and r: 10[5]2 and 12[3]4, or and .
A regular complex apeirogon p{q}r has p-edges and r-gonal vertex figures. The dual apeirogon of p{q}r is r{q}p. An apeirogon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular apeirogon is the same as quasiregular .[31]
Apeirogons can be represented on the Argand plane share four different vertex arrangements. Apeirogons of the form 2{q}r have a vertex arrangement as {q/2,p}. The form p{q}2 have vertex arrangement as r{p,q/2}. Apeirogons of the form p{4}r have vertex arrangements {p,r}.
Including affine nodes, and $\mathbb {C} ^{2}$, there are 3 more infinite solutions: ∞[2]∞, ∞[4]2, ∞[3]3, and , , and . The first is an index 2 subgroup of the second. The vertices of these apeirogons exist in $\mathbb {C} ^{1}$.
Rank 2
SpaceGroupApeirogonEdge$\mathbb {R} ^{2}$ rep.[32]PictureNotes
$\mathbb {R} ^{1}$2[∞]2 = [∞]δ2,2
2
= {∞}
{}Real apeirogon
Same as
$\mathbb {C} ^{2}$ / $\mathbb {C} ^{1}$∞[4]2∞{4}2∞{}{4,4}Same as
$\mathbb {C} ^{1}$∞[3]3∞{3}3∞{}{3,6}Same as
$\mathbb {C} ^{1}$p[q]rδp,r
2
= p{q}r
p{}
$\mathbb {C} ^{1}$3[12]2δ3,2
2
= 3{12}2
3{}r{3,6}Same as
δ2,3
2
= 2{12}3
{}{6,3}
$\mathbb {C} ^{1}$3[6]3δ3,3
2
= 3{6}3
3{}{3,6}Same as
$\mathbb {C} ^{1}$4[8]2δ4,2
2
= 4{8}2
4{}{4,4}Same as
δ2,4
2
= 2{8}4
{}{4,4}
$\mathbb {C} ^{1}$4[4]4δ4,4
2
= 4{4}4
4{}{4,4}Same as
$\mathbb {C} ^{1}$6[6]2δ6,2
2
= 6{6}2
6{}r{3,6}Same as
δ2,6
2
= 2{6}6
{}{3,6}
$\mathbb {C} ^{1}$6[4]3δ6,3
2
= 6{4}3
6{}{6,3}
δ3,6
2
= 3{4}6
3{}{3,6}
$\mathbb {C} ^{1}$6[3]6δ6,6
2
= 6{3}6
6{}{3,6}Same as
Regular complex apeirohedra
There are 22 regular complex apeirohedra, of the form p{a}q{b}r. 8 are self-dual (p=r and a=b), while 14 exist as dual polytope pairs. Three are entirely real (p=q=r=2).
Coxeter symbolizes 12 of them as δp,r
3
or p{4}2{4}r is the regular form of the product apeirotope δp,r
2
× δp,r
2
or p{q}r × p{q}r, where q is determined from p and r.
is the same as , as well as , for p,r=2,3,4,6. Also = .[33]
Rank 3
SpaceGroupApeirohedronVertexEdgeFacevan Oss
apeirogon
Notes
$\mathbb {C} ^{3}$2[3]2[4]∞∞{4}2{3}2∞{}∞{4}2Same as ∞{}×∞{}×∞{} or
Real representation {4,3,4}
$\mathbb {C} ^{2}$p[4]2[4]rp{4}2{4}r
p22prp{}r2p{4}22{q}rSame as , p,r=2,3,4,6
$\mathbb {R} ^{2}$[4,4]δ2,2
3
= {4,4}
48{}4{4}{∞}Real square tiling
Same as or or
$\mathbb {C} ^{2}$ 3[4]2[4]2
3[4]2[4]3
4[4]2[4]2
4[4]2[4]4
6[4]2[4]2
6[4]2[4]3
6[4]2[4]6
3{4}2{4}2
2{4}2{4}3
3{4}2{4}3
4{4}2{4}2
2{4}2{4}4
4{4}2{4}4
6{4}2{4}2
2{4}2{4}6
6{4}2{4}3
3{4}2{4}6
6{4}2{4}6
9
4
9
16
4
16
36
4
36
9
36
12
12
18
16
16
32
24
24
36
36
72
3{}
{}
3{}
4{}
{}
4{}
6{}
{}
6{}
3{}
6{}
4
9
9
4
16
16
4
36
9
36
36
3{4}2
{4}
3{4}2
4{4}2
{4}
4{4}2
6{4}2
{4}
6{4}2
3{4}2
6{4}2
p{q}r Same as or or
Same as
Same as
Same as or or
Same as
Same as
Same as or or
Same as
Same as
Same as
Same as
SpaceGroupApeirohedronVertexEdgeFacevan Oss
apeirogon
Notes
$\mathbb {C} ^{2}$2[4]r[4]22{4}r{4}2
2{}2p{4}2'2{4}rSame as and , r=2,3,4,6
$\mathbb {R} ^{2}$[4,4]{4,4}24{}2{4}{∞}Same as and
$\mathbb {C} ^{2}$ 2[4]3[4]2
2[4]4[4]2
2[4]6[4]2
2{4}3{4}2
2{4}4{4}2
2{4}6{4}2
29
16
36
{}2 2{4}3
2{4}4
2{4}6
2{q}r Same as and
Same as and
Same as and [34]
SpaceGroupApeirohedronVertexEdgeFacevan Oss
apeirogon
Notes
$\mathbb {R} ^{2}$ 2[6]2[3]2
= [6,3]
{3,6}
13{}2{3}{∞}Real triangular tiling
{6,3}23{}1{6}noneReal hexagonal tiling
$\mathbb {C} ^{2}$ 3[4]3[3]33{3}3{4}3183{}33{3}33{4}6Same as
3{4}3{3}3383{}13{4}33{12}2
$\mathbb {C} ^{2}$ 4[3]4[3]44{3}4{3}4164{}14{3}44{4}4Self-dual, same as
$\mathbb {C} ^{2}$ 4[3]4[4]24{3}4{4}21124{}34{3}42{8}4Same as
2{4}4{3}4312{}12{4}44{4}4
Regular complex 3-apeirotopes
There are 16 regular complex apeirotopes in $\mathbb {C} ^{3}$. Coxeter expresses 12 of them by δp,r
3
where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: = . The first case is the $\mathbb {R} ^{3}$ cubic honeycomb.
Rank 4
SpaceGroup3-apeirotopeVertexEdgeFaceCellvan Oss
apeirogon
Notes
$\mathbb {C} ^{3}$p[4]2[3]2[4]rδp,r
3
= p{4}2{3}2{4}r
p{}p{4}2p{4}2{3}2p{q}rSame as
$\mathbb {R} ^{3}$2[4]2[3]2[4]2
=[4,3,4]
δ2,2
3
= 2{4}2{3}2{4}2
{}{4}{4,3}Cubic honeycomb
Same as or or
$\mathbb {C} ^{3}$3[4]2[3]2[4]2δ3,2
3
= 3{4}2{3}2{4}2
3{}3{4}23{4}2{3}2Same as or or
δ2,3
3
= 2{4}2{3}2{4}3
{}{4}{4,3}Same as
$\mathbb {C} ^{3}$3[4]2[3]2[4]3δ3,3
3
= 3{4}2{3}2{4}3
3{}3{4}23{4}2{3}2Same as
$\mathbb {C} ^{3}$4[4]2[3]2[4]2δ4,2
3
= 4{4}2{3}2{4}2
4{}4{4}24{4}2{3}2Same as or or
δ2,4
3
= 2{4}2{3}2{4}4
{}{4}{4,3}Same as
$\mathbb {C} ^{3}$4[4]2[3]2[4]4δ4,4
3
= 4{4}2{3}2{4}4
4{}4{4}24{4}2{3}2Same as
$\mathbb {C} ^{3}$6[4]2[3]2[4]2δ6,2
3
= 6{4}2{3}2{4}2
6{}6{4}26{4}2{3}2Same as or or
δ2,6
3
= 2{4}2{3}2{4}6
{}{4}{4,3}Same as
$\mathbb {C} ^{3}$6[4]2[3]2[4]3δ6,3
3
= 6{4}2{3}2{4}3
6{}6{4}26{4}2{3}2Same as
δ3,6
3
= 3{4}2{3}2{4}6
3{}3{4}23{4}2{3}2Same as
$\mathbb {C} ^{3}$6[4]2[3]2[4]6δ6,6
3
= 6{4}2{3}2{4}6
6{}6{4}26{4}2{3}2Same as
Rank 4, exceptional cases
SpaceGroup3-apeirotopeVertexEdgeFaceCellvan Oss
apeirogon
Notes
$\mathbb {C} ^{3}$2[4]3[3]3[3]33{3}3{3}3{4}2
124 3{}27 3{3}32 3{3}3{3}33{4}6Same as
2{4}3{3}3{3}3
227 {}24 2{4}31 2{4}3{3}32{12}3
$\mathbb {C} ^{3}$2[3]2[4]3[3]32{3}2{4}3{3}3
127 {}72 2{3}28 2{3}2{4}32{6}6
3{3}3{4}2{3}2
872 3{}27 3{3}31 3{3}3{4}23{6}3Same as or
Regular complex 4-apeirotopes
There are 15 regular complex apeirotopes in $\mathbb {C} ^{4}$. Coxeter expresses 12 of them by δp,r
4
where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: = . The first case is the $\mathbb {R} ^{4}$ tesseractic honeycomb. The 16-cell honeycomb and 24-cell honeycomb are real solutions. The last solution is generated has Witting polytope elements.
Rank 5
SpaceGroup4-apeirotopeVertexEdgeFaceCell4-facevan Oss
apeirogon
Notes
$\mathbb {C} ^{4}$p[4]2[3]2[3]2[4]rδp,r
4
= p{4}2{3}2{3}2{4}r
p{}p{4}2p{4}2{3}2p{4}2{3}2{3}2p{q}rSame as
$\mathbb {R} ^{4}$2[4]2[3]2[3]2[4]2δ2,2
4
= {4,3,3,3}
{}{4}{4,3}{4,3,3}{∞}Tesseractic honeycomb
Same as
$\mathbb {R} ^{4}$2[3]2[4]2[3]2[3]2
=[3,4,3,3]
{3,3,4,3}
112 {}32 {3}24 {3,3}3 {3,3,4}Real 16-cell honeycomb
Same as
{3,4,3,3}
324 {}32 {3}12 {3,4}1 {3,4,3}Real 24-cell honeycomb
Same as or
$\mathbb {C} ^{4}$3[3]3[3]3[3]3[3]33{3}3{3}3{3}3{3}3
180 3{}270 3{3}380 3{3}3{3}31 3{3}3{3}3{3}33{4}6$\mathbb {R} ^{8}$ representation 521
Regular complex 5-apeirotopes and higher
There are only 12 regular complex apeirotopes in $\mathbb {C} ^{5}$ or higher,[35] expressed δp,r
n
where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed a product of n apeirogons: ... = ... . The first case is the real $\mathbb {R} ^{n}$ hypercube honeycomb.
Rank 6
SpaceGroup5-apeirotopesVerticesEdgeFaceCell4-face5-facevan Oss
apeirogon
Notes
$\mathbb {C} ^{5}$p[4]2[3]2[3]2[3]2[4]rδp,r
5
= p{4}2{3}2{3}2{3}2{4}r
p{}p{4}2p{4}2{3}2p{4}2{3}2{3}2p{4}2{3}2{3}2{3}2p{q}rSame as
$\mathbb {R} ^{5}$2[4]2[3]2[3]2[3]2[4]2
=[4,3,3,3,4]
δ2,2
5
= {4,3,3,3,4}
{}{4}{4,3}{4,3,3}{4,3,3,3}{∞}5-cubic honeycomb
Same as
van Oss polygon
A van Oss polygon is a regular polygon in the plane (real plane $\mathbb {R} ^{2}$, or unitary plane $\mathbb {C} ^{2}$) in which both an edge and the centroid of a regular polytope lie, and formed of elements of the polytope. Not all regular polytopes have Van Oss polygons.
For example, the van Oss polygons of a real octahedron are the three squares whose planes pass through its center. In contrast a cube does not have a van Oss polygon because the edge-to-center plane cuts diagonally across two square faces and the two edges of the cube which lie in the plane do not form a polygon.
Infinite honeycombs also have van Oss apeirogons. For example, the real square tiling and triangular tiling have apeirogons {∞} van Oss apeirogons.[36]
If it exists, the van Oss polygon of regular complex polytope of the form p{q}r{s}t... has p-edges.
Non-regular complex polytopes
Product complex polytopes
Example product complex polytope
Complex product polygon or {}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges, with its real representation as a 3-dimensional pentagonal prism.
The dual polygon,{}+5{} has 7 vertices centered on the edges of the original, connected by 10 edges. Its real representation is a pentagonal bipyramid.
Some complex polytopes can be represented as Cartesian products. These product polytopes are not strictly regular since they'll have more than one facet type, but some can represent lower symmetry of regular forms if all the orthogonal polytopes are identical. For example, the product p{}×p{} or of two 1-dimensional polytopes is the same as the regular p{4}2 or . More general products, like p{}×q{} have real representations as the 4-dimensional p-q duoprisms. The dual of a product polytope can be written as a sum p{}+q{} and have real representations as the 4-dimensional p-q duopyramid. The p{}+p{} can have its symmetry doubled as a regular complex polytope 2{4}p or .
Similarly, a $\mathbb {C} ^{3}$ complex polyhedron can be constructed as a triple product: p{}×p{}×p{} or is the same as the regular generalized cube, p{4}2{3}2 or , as well as product p{4}2×p{} or .[37]
Quasiregular polygons
A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons and . The quasiregular polygon has p vertices on the p-edges of the regular form.
Example quasiregular polygons
p[q]r2[4]23[4]24[4]25[4]26[4]27[4]28[4]23[3]33[4]3
Regular
4 2-edges
9 3-edges
16 4-edges
25 5-edges
36 6-edges
49 8-edges
64 8-edges
Quasiregular
=
4+4 2-edges
6 2-edges
9 3-edges
8 2-edges
16 4-edges
10 2-edges
25 5-edges
12 2-edges
36 6-edges
14 2-edges
49 7-edges
16 2-edges
64 8-edges
=
=
Regular
4 2-edges
6 2-edges
8 2-edges
10 2-edges
12 2-edges
14 2-edges
16 2-edges
Quasiregular apeirogons
There are 7 quasiregular complex apeirogons which alternate edges of a regular apeirogon and its regular dual. The vertex arrangements of these apeirogon have real representations with the regular and uniform tilings of the Euclidean plane. The last column for the 6{3}6 apeirogon is not only self-dual, but the dual coincides with itself with overlapping hexagonal edges, thus their quasiregular form also has overlapping hexagonal edges, so it can't be drawn with two alternating colors like the others. The symmetry of the self-dual families can be doubled, so creating an identical geometry as the regular forms: =
p[q]r4[8]24[4]46[6]26[4]33[12]23[6]36[3]6
Regular
or p{q}r
Quasiregular
=
=
=
Regular dual
or r{q}p
Quasiregular polyhedra
Like real polytopes, a complex quasiregular polyhedron can be constructed as a rectification (a complete truncation) of a regular polyhedron. Vertices are created mid-edge of the regular polyhedron and faces of the regular polyhedron and its dual are positioned alternating across common edges.
For example, a p-generalized cube, , has p3 vertices, 3p2 edges, and 3p p-generalized square faces, while the p-generalized octahedron, , has 3p vertices, 3p2 edges and p3 triangular faces. The middle quasiregular form p-generalized cuboctahedron, , has 3p2 vertices, 3p3 edges, and 3p+p3 faces.
Also the rectification of the Hessian polyhedron , is , a quasiregular form sharing the geometry of the regular complex polyhedron .
Quasiregular examples
Generalized cube/octahedraHessian polyhedron
p=2 (real)p=3p=4p=5p=6
Generalized
cubes
(regular)
Cube
, 8 vertices, 12 2-edges, and 6 faces.
, 27 vertices, 27 3-edges, and 9 faces, with one face blue and red
, 64 vertices, 48 4-edges, and 12 faces.
, 125 vertices, 75 5-edges, and 15 faces.
, 216 vertices, 108 6-edges, and 18 faces.
, 27 vertices, 72 6-edges, and 27 faces.
Generalized
cuboctahedra
(quasiregular)
Cuboctahedron
, 12 vertices, 24 2-edges, and 6+8 faces.
, 27 vertices, 81 2-edges, and 9+27 faces, with one face blue
, 48 vertices, 192 2-edges, and 12+64 faces, with one face blue
, 75 vertices, 375 2-edges, and 15+125 faces.
, 108 vertices, 648 2-edges, and 18+216 faces.
= , 72 vertices, 216 3-edges, and 54 faces.
Generalized
octahedra
(regular)
Octahedron
, 6 vertices, 12 2-edges, and 8 {3} faces.
, 9 vertices, 27 2-edges, and 27 {3} faces.
, 12 vertices, 48 2-edges, and 64 {3} faces.
, 15 vertices, 75 2-edges, and 125 {3} faces.
, 18 vertices, 108 2-edges, and 216 {3} faces.
, 27 vertices, 72 6-edges, and 27 faces.
Other complex polytopes with unitary reflections of period two
Other nonregular complex polytopes can be constructed within unitary reflection groups that don't make linear Coxeter graphs. In Coxeter diagrams with loops Coxeter marks a special period interior, like or symbol (11 1 1)3, and group [1 1 1]3.[38][39] These complex polytopes have not been systematically explored beyond a few cases.
The group is defined by 3 unitary reflections, R1, R2, R3, all order 2: R12 = R12 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = (R1R2R3R1)p = 1. The period p can be seen as a double rotation in real $\mathbb {R} ^{4}$.
As with all Wythoff constructions, polytopes generated by reflections, the number of vertices of a single-ringed Coxeter diagram polytope is equal to the order of the group divided by the order of the subgroup where the ringed node is removed. For example, a real cube has Coxeter diagram , with octahedral symmetry order 48, and subgroup dihedral symmetry order 6, so the number of vertices of a cube is 48/6=8. Facets are constructed by removing one node furthest from the ringed node, for example for the cube. Vertex figures are generated by removing a ringed node and ringing one or more connected nodes, and for the cube.
Coxeter represents these groups by the following symbols. Some groups have the same order, but a different structure, defining the same vertex arrangement in complex polytopes, but different edges and higher elements, like and with p≠3.[40]
Groups generated by unitary reflections
Coxeter diagramOrderSymbol or Position in Table VII of Shephard and Todd (1954)
, ( and ), , ...
pn − 1 n!, p ≥ 3G(p, p, n), [p], [1 1 1]p, [1 1 (n−2)p]3
, 72·6!, 108·9!Nos. 33, 34, [1 2 2]3, [1 2 3]3
, ( and ), ( and )14·4!, 3·6!, 64·5!Nos. 24, 27, 29
Coxeter calls some of these complex polyhedra almost regular because they have regular facets and vertex figures. The first is a lower symmetry form of the generalized cross-polytope in $\mathbb {C} ^{3}$. The second is a fractional generalized cube, reducing p-edges into single vertices leaving ordinary 2-edges. Three of them are related to the finite regular skew polyhedron in $\mathbb {R} ^{4}$.
Some almost regular complex polyhedra[41]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacesVertex
figure
Notes
$\mathbb {C} ^{3}$[1 1 1p]3
p=2,3,4...
6p2(1 1 11p)3
3p3p2{3}{2p}Shephard symbol (1 1; 11)p
same as βp
3
=
(11 1 1p)3
p2{3}{6}Shephard symbol (11 1; 1)p
1/p γp
3
$\mathbb {R} ^{3}$[1 1 12]3
24(1 1 112)3
6128 {3}{4}Same as β2
3
= = real octahedron
(11 1 12)3
464 {3}{3}1/2 γ2
3
= = α3 = real tetrahedron
$\mathbb {C} ^{3}$[1 1 1]3
54(1 1 11)3
927{3}{6}Shephard symbol (1 1; 11)3
same as β3
3
=
(11 1 1)3
927{3}{6}Shephard symbol (11 1; 1)3
1/3 γ3
3
= β3
3
$\mathbb {C} ^{3}$[1 1 14]3
96(1 1 114)3
1248{3}{8}Shephard symbol (1 1; 11)4
same as β4
3
=
(11 1 14)3
16{3}{6}Shephard symbol (11 1; 1)4
1/4 γ4
3
$\mathbb {C} ^{3}$[1 1 15]3
150(1 1 115)3
1575{3}{10}Shephard symbol (1 1; 11)5
same as β5
3
=
(11 1 15)3
25{3}{6}Shephard symbol (11 1; 1)5
1/5 γ5
3
$\mathbb {C} ^{3}$[1 1 16]3
216(1 1 116)3
18216{3}{12}Shephard symbol (1 1; 11)6
same as β6
3
=
(11 1 16)3
36{3}{6}Shephard symbol (11 1; 1)6
1/6 γ6
3
$\mathbb {C} ^{3}$[1 1 14]4
336(1 1 114)4
42168112 {3}{8}$\mathbb {R} ^{4}$ representation {3,8|,4} = {3,8}8
(11 1 14)4
56{3}{6}
$\mathbb {C} ^{3}$[1 1 15]4
2160(1 1 115)4
2161080720 {3}{10}$\mathbb {R} ^{4}$ representation {3,10|,4} = {3,10}8
(11 1 15)4
360{3}{6}
$\mathbb {C} ^{3}$[1 1 14]5
(1 1 114)5
2701080720 {3}{8}$\mathbb {R} ^{4}$ representation {3,8|,5} = {3,8}10
(11 1 14)5
360{3}{6}
Coxeter defines other groups with anti-unitary constructions, for example these three. The first was discovered and drawn by Peter McMullen in 1966.[42]
More almost regular complex polyhedra[41]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacesVertex
figure
Notes
$\mathbb {C} ^{3}$[1 14 14](3)
336(11 14 14)(3)
5616884 {4}{6}$\mathbb {R} ^{4}$ representation {4,6|,3} = {4,6}6
$\mathbb {C} ^{3}$[15 14 14](3)
2160(115 14 14)(3)
2161080540 {4}{10}$\mathbb {R} ^{4}$ representation {4,10|,3} = {4,10}6
$\mathbb {C} ^{3}$[14 15 15](3)
(114 15 15)(3)
2701080432 {5}{8}$\mathbb {R} ^{4}$ representation {5,8|,3} = {5,8}6
Some complex 4-polytopes[41]
SpaceGroupOrderCoxeter
symbols
VerticesOther
elements
CellsVertex
figure
Notes
$\mathbb {C} ^{4}$[1 1 2p]3
p=2,3,4...
24p3(1 1 22p)3
4pShephard (22 1; 1)p
same as βp
4
=
(11 1 2p )3
p3
Shephard (2 1; 11)p
1/p γp
4
$\mathbb {R} ^{4}$[1 1 22]3
=[31,1,1]
192(1 1 222)3
824 edges
32 faces
16 β2
4
= , real 16-cell
(11 1 22 )3
1/2 γ2
4
= = β2
4
, real 16-cell
$\mathbb {C} ^{4}$[1 1 2]3
648(1 1 22)3
12Shephard (22 1; 1)3
same as β3
4
=
(11 1 23)3
27
Shephard (2 1; 11)3
1/3 γ3
4
$\mathbb {C} ^{4}$[1 1 24]3
1536(1 1 224)3
16Shephard (22 1; 1)4
same as β4
4
=
(11 1 24 )3
64
Shephard (2 1; 11)4
1/4 γ4
4
$\mathbb {C} ^{4}$[14 1 2]3
7680(22 14 1)3
80Shephard (22 1; 1)4
(114 1 2)3
160
Shephard (2 1; 11)4
(11 14 2)3
320
Shephard (2 11; 1)4
$\mathbb {C} ^{4}$[1 1 2]4
(1 1 22)4
80640 edges
1280 triangles
640
(11 1 2)4
320
Some complex 5-polytopes[41]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacetsVertex
figure
Notes
$\mathbb {C} ^{5}$[1 1 3p]3
p=2,3,4...
120p4(1 1 33p)3
5pShephard (33 1; 1)p
same as βp
5
=
(11 1 3p)3
p4
Shephard (3 1; 11)p
1/p γp
5
$\mathbb {C} ^{5}$[2 2 1]3
51840(2 1 22)3
80
Shephard (2 1; 22)3
(2 11 2)3
432Shephard (2 11; 2)3
Some complex 6-polytopes[41]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacetsVertex
figure
Notes
$\mathbb {C} ^{6}$[1 1 4p]3
p=2,3,4...
720p5(1 1 44p)3
6pShephard (44 1; 1)p
same as βp
6
=
(11 1 4p)3
p5
Shephard (4 1; 11)p
1/p γp
6
$\mathbb {C} ^{6}$[1 2 3]3
39191040(2 1 33)3
756
Shephard (2 1; 33)3
(22 1 3)3
4032
Shephard (22 1; 3)3
(2 11 3)3
54432
Shephard (2 11; 3)3
Visualizations
• (1 1 114)4, has 42 vertices, 168 edges and 112 triangular faces, seen in this 14-gonal projection.
• (14 14 11)(3), has 56 vertices, 168 edges and 84 square faces, seen in this 14-gonal projection.
• (1 1 22)4, has 80 vertices, 640 edges, 1280 triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection.[43]
See also
• Quaternionic polytope
Notes
1. Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische Annalen. March 2002, Volume 322, Issue 3, pp 477–492. DOI:10.1007/s002080200001
2. Coxeter, Regular Complex Polytopes, p. 115
3. Coxeter, Regular Complex Polytopes, 11.3 Petrie Polygon, a simple h-gon formed by the orbit of the flag (O0,O0O1) for the product of the two generating reflections of any nonstarry regular complex polygon, p1{q}p2.
4. Complex Regular Polytopes,11.1 Regular complex polygons p.103
5. Shephard, 1952; "It is from considerations such as these that we derive the notion of the interior of a polytope, and it will be seen that in unitary space where the numbers cannot be so ordered such a concept of interior is impossible. [Para break] Hence ... we have to consider unitary polytopes as configurations."
6. Coxeter, Regular Complex polytopes, p. 96
7. Coxeter, Regular Complex Polytopes, p. xiv
8. Coxeter, Complex Regular Polytopes, p. 177, Table III
9. Lehrer & Taylor 2009, p. 87
10. Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179
11. Complex Polytopes, 8.9 The Two-Dimensional Case, p. 88
12. Regular Complex Polytopes, Coxeter, pp. 177-179
13. Coxeter, Regular Complex Polytopes, p. 108
14. Coxeter, Regular Complex Polytopes, p. 109
15. Coxeter, Regular Complex Polytopes, p. 111
16. Coxeter, Regular Complex Polytopes, p. 30 diagram and p. 47 indices for 8 3-edges
17. Coxeter, Regular Complex Polytopes, p. 110
18. Coxeter, Regular Complex Polytopes, p. 48
19. Coxeter, Regular Complex Polytopes, p. 49
20. Coxeter, Regular Complex Polytopes, pp. 116–140.
21. Coxeter, Regular Complex Polytopes, pp. 118–119.
22. Complex Regular Polytopes, p.29
23. Coxeter, Regular Complex Polytopes, Table V. The nonstarry regular polyhedra and 4-polytopes. p. 180.
24. Coxeter, Kaleidoscopes — Selected Writings of H.S.M. Coxeter, Paper 25 Surprising relationships among unitary reflection groups, p. 431.
25. Coxeter, Regular Complex Polytopes, p. 131
26. Coxeter, Regular Complex Polytopes, p. 126
27. Coxeter, Regular Complex Polytopes, p. 125
28. Coxeter, Regular Complex Polytopes, Table VI. The regular honeycombs. p. 180.
29. Complex regular polytope, p.174
30. Coxeter, Regular Complex Polytopes, Table VI. The regular honeycombs. p. 111, 136.
31. Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179
32. Coxeter, Regular Complex Polytopes, 11.6 Apeirogons, pp. 111-112
33. Coxeter, Complex Regular Polytopes, p.140
34. Coxeter, Regular Complex Polytopes, pp. 139-140
35. Complex Regular Polytopes, p.146
36. Complex Regular Polytopes, p.141
37. Coxeter, Regular Complex Polytopes, pp. 118–119, 138.
38. Coxeter, Regular Complex Polytopes, Chapter 14, Almost regular polytopes, pp. 156–174.
39. Coxeter, Groups Generated by Unitary Reflections of Period Two, 1956
40. Coxeter, Finite Groups Generated by Unitary Reflections, 1966, 4. The Graphical Notation, Table of n-dimensional groups generated by n Unitary Reflections. pp. 422-423
41. Coxeter, Groups generated by Unitary Reflections of Period Two (1956), Table III: Some Complex Polytopes, p.413
42. Coxeter, Complex Regular Polytopes, (1991), 14.6 McMullen's two polyhedral with 84 square faces, pp.166-171
43. Coxeter, Complex Regular Polytopes, pp.172-173
References
• Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
• Coxeter, H.S.M. (1991), Regular Complex Polytopes, Cambridge University Press, ISBN 0-521-39490-2
• Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,
• Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.
• G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canadian Journal of Mathematics. 6(1954), 274-304
• Gustav I. Lehrer and Donald E. Taylor, Unitary Reflection Groups, Cambridge University Press 2009
Further reading
Wikimedia Commons has media related to Complex polytopes.
• F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.S.M. Coxeter., Paper 25, Finite groups generated by unitary reflections, p 415-425, John Wiley, 1995, ISBN 0-471-01003-0
• McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0 Chapter 9 Unitary Groups and Hermitian Forms, pp. 289–298
| Wikipedia |
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.
Uniform triangular prism
TypePrismatic uniform polyhedron
ElementsF = 5, E = 9
V = 6 (χ = 2)
Faces by sides3{4}+2{3}
Schläfli symbolt{2,3} or {3}×{}
Wythoff symbol2 3 | 2
Coxeter diagram
Symmetry groupD3h, [3,2], (*322), order 12
Rotation groupD3, [3,2]+, (322), order 6
ReferencesU76(a)
DualTriangular dipyramid
Propertiesconvex
Vertex figure
4.4.3
Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.
As a semiregular (or uniform) polyhedron
A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t{2,3}. Alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product, The dual of a triangular prism is a triangular bipyramid.
The symmetry group of a right 3-sided prism with triangular base is D3h of order 12. The rotation group is D3 of order 6. The symmetry group does not contain inversion.
Volume
The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to compute the area of the triangle and multiply this by the length of the prism:
$V={\frac {bhl}{2}},$
where b is the length of one side of the triangle, h is the length of an altitude drawn to that side, and l is the distance between the triangular faces.
Truncated triangular prism
A truncated right triangular prism has one triangular face truncated (planed) at an oblique angle.[1]
The volume of a truncated triangular prism with base area A and the three heights h1, h2, and h3 is determined by
$V={\frac {A(h_{1}+h_{2}+h_{3})}{3}}.$
Facetings
There are two full D3h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C3v symmetry facetings have one base triangle, 3 lateral crossed square faces, and 3 isosceles triangle lateral faces.
Convex Facetings
D3h symmetry C3v symmetry
2 {3}
3 {4}
3 {4}
6 ( ) v { }
2 {3}
6 ( ) v { }
1 {3}
3 t'{2}
6 ( ) v { }
1 {3}
3 t'{2}
3 ( ) v { }
Related polyhedra and tilings
Family of uniform n-gonal prisms
Prism name Digonal prism (Trigonal)
Triangular prism
(Tetragonal)
Square prism
Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism Enneagonal prism Decagonal prism Hendecagonal prism Dodecagonal prism ... Apeirogonal prism
Polyhedron image ...
Spherical tiling image Plane tiling image
Vertex config. 2.4.43.4.44.4.45.4.46.4.47.4.48.4.49.4.410.4.411.4.412.4.4...∞.4.4
Coxeter diagram ...
Family of convex cupolae
n2345678
Schläfli symbol{2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6} {7} || t{7} {8} || t{8}
Cupola
Digonal cupola
Triangular cupola
Square cupola
Pentagonal cupola
Hexagonal cupola
(Flat)
Heptagonal cupola
(Non-regular face)
Octagonal cupola
(Non-regular face)
Related
uniform
polyhedra
Rhombohedron
Cuboctahedron
Rhombicuboctahedron
Rhombicosidodecahedron
Rhombitrihexagonal tiling
Rhombitriheptagonal tiling
Rhombitrioctagonal tiling
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated tilings: t{n,3}
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3}
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞
This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paracomp.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4
Compounds
There are 4 uniform compounds of triangular prisms:
Compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms.
Honeycombs
There are 9 uniform honeycombs that include triangular prism cells:
Gyroelongated alternated cubic honeycomb, elongated alternated cubic honeycomb, gyrated triangular prismatic honeycomb, snub square prismatic honeycomb, triangular prismatic honeycomb, triangular-hexagonal prismatic honeycomb, truncated hexagonal prismatic honeycomb, rhombitriangular-hexagonal prismatic honeycomb, snub triangular-hexagonal prismatic honeycomb, elongated triangular prismatic honeycomb
Related polytopes
The triangular prism is first in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (equilateral triangles and squares in the case of the triangular prism). In Coxeter's notation the triangular prism is given the symbol −121.
k21 figures in n dimensions
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde {E}}_{8}$ = E8+ E10 = ${\bar {T}}_{8}$ = E8++
Coxeter
diagram
Symmetry [3−1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 51,840 2,903,040 696,729,600 ∞
Graph - -
Name −121 021 121 221 321 421 521 621
Four dimensional space
The triangular prism exists as cells of a number of four-dimensional uniform 4-polytopes, including:
Four dimensional polytopes with triangular prisms
Tetrahedral prism
Octahedral prism
Cuboctahedral prism
Icosahedral prism
Icosidodecahedral prism
Truncated dodecahedral prism
Rhomb-icosidodecahedral prism
Rhombi-cuboctahedral prism
Truncated cubic prism
Snub dodecahedral prism
n-gonal antiprismatic prism
Cantellated 5-cell
Cantitruncated 5-cell
Runcinated 5-cell
Runcitruncated 5-cell
Cantellated tesseract
Cantitruncated tesseract
Runcinated tesseract
Runcitruncated tesseract
Cantellated 24-cell
Cantitruncated 24-cell
Runcinated 24-cell
Runcitruncated 24-cell
Cantellated 120-cell
Cantitruncated 120-cell
Runcinated 120-cell
Runcitruncated 120-cell
See also
• Wedge (geometry)
References
1. Kern, William F.; Bland, James R. (1938). Solid Mensuration with proofs. p. 81. OCLC 1035479.
• Weisstein, Eric W. "Triangular prism". MathWorld.
• Interactive Polyhedron: Triangular Prism
• Surface area and volume of a triangular prism
| Wikipedia |
3-dimensional matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph).
3-dimensional matching, often abbreviated as 3DM, is also the name of a well-known computational problem: finding a largest 3-dimensional matching in a given hypergraph. 3DM is one of the first problems that were proved to be NP-hard.
Definition
Let X, Y, and Z be finite sets, and let T be a subset of X × Y × Z. That is, T consists of triples (x, y, z) such that x ∈ X, y ∈ Y, and z ∈ Z. Now M ⊆ T is a 3-dimensional matching if the following holds: for any two distinct triples (x1, y1, z1) ∈ M and (x2, y2, z2) ∈ M, we have x1 ≠ x2, y1 ≠ y2, and z1 ≠ z2.
Example
The figure on the right illustrates 3-dimensional matchings. The set X is marked with red dots, Y is marked with blue dots, and Z is marked with green dots. Figure (a) shows the set T (gray areas). Figure (b) shows a 3-dimensional matching M with |M| = 2, and Figure (c) shows a 3-dimensional matching M with |M| = 3.
The matching M illustrated in Figure (c) is a maximum 3-dimensional matching, i.e., it maximises |M|. The matching illustrated in Figures (b)–(c) are maximal 3-dimensional matchings, i.e., they cannot be extended by adding more elements from T.
Here is example interactive visualisation in javascript
Comparison with bipartite matching
A 2-dimensional matching can be defined in a completely analogous manner. Let X and Y be finite sets, and let T be a subset of X × Y. Now M ⊆ T is a 2-dimensional matching if the following holds: for any two distinct pairs (x1, y1) ∈ M and (x2, y2) ∈ M, we have x1 ≠ x2 and y1 ≠ y2.
In the case of 2-dimensional matching, the set T can be interpreted as the set of edges in a bipartite graph G = (X, Y, T); each edge in T connects a vertex in X to a vertex in Y. A 2-dimensional matching is then a matching in the graph G, that is, a set of pairwise non-adjacent edges.
Hence 3-dimensional matchings can be interpreted as a generalization of matchings to hypergraphs: the sets X, Y, and Z contain the vertices, each element of T is a hyperedge, and the set M consists of pairwise non-adjacent edges (edges that do not have a common vertex). In case of 2-dimensional matching, we have Y = Z.
Comparison with set packing
A 3-dimensional matching is a special case of a set packing: we can interpret each element (x, y, z) of T as a subset {x, y, z} of X ∪ Y ∪ Z; then a 3-dimensional matching M consists of pairwise disjoint subsets.
Decision problem
In computational complexity theory, 3-dimensional matching (3DM) is the name of the following decision problem: given a set T and an integer k, decide whether there exists a 3-dimensional matching M ⊆ T with |M| ≥ k.
This decision problem is known to be NP-complete; it is one of Karp's 21 NP-complete problems.[1] It is NP-complete even in the special case that k = |X| = |Y| = |Z| and when each element is contained in exactly 3 sets, i.e., when we want a perfect matching in a 3-regular hypergraph.[1][2][3] In this case, a 3-dimensional matching is not only a set packing, but also an exact cover: the set M covers each element of X, Y, and Z exactly once.[4] The proof is by reduction from 3SAT. Given a 3SAT instance, we construct a 3DM instance as follows:[2][5]
• For each variable xi, there is a "variable gadget" shaped like a wheel. It is made of overlapping triplets. The number of triplets is twice the number of occurrences of xi in the formula. There are exactly two ways to cover all the vertices in the gadget: one is to choose all even-indexed triplets, and one is to choose all odd-indexed triplets. These two ways correspond to setting xi to "true" or "false". The "true" selection leaves uncovered exactly one vertex in every odd-indexed triplet, and the "false" selection leaves uncovered exactly one vertex in every even-indexed triplet.
• For each clause xi u xj u xk, there is a "clause gadget" shaped like a rose. It is made of three overlapping triplets, one for each variable in the clause. It can be covered iff at least one of the nodes is left uncovered by the selection of the variable gadgets.
• Since it is possible that two or more nodes are left uncovered, we also need a "garbage collection gadget". It is shaped like a larger rose. It is made of several overlapping triplets, one for each vertex that can be left uncovered in the variable gadget. The number of such gadgets is determined so that they can be covered exactly if and only if there is a satisfying assignment.
Special cases
There exist polynomial time algorithms for solving 3DM in dense hypergraphs.[6][7]
Optimization problem
A maximum 3-dimensional matching is a largest 3-dimensional matching. In computational complexity theory, this is also the name of the following optimization problem: given a set T, find a 3-dimensional matching M ⊆ T that maximizes |M|.
Since the decision problem described above is NP-complete, this optimization problem is NP-hard, and hence it seems that there is no polynomial-time algorithm for finding a maximum 3-dimensional matching. However, there are efficient polynomial-time algorithms for finding a maximum bipartite matching (maximum 2-dimensional matching), for example, the Hopcroft–Karp algorithm.
Approximation algorithms
There is a very simple polynomial-time 3-approximation algorithm for 3-dimensional matching: find any maximal 3-dimensional matching.[8] Just like a maximal matching is within factor 2 of a maximum matching,[9] a maximal 3-dimensional matching is within factor 3 of a maximum 3-dimensional matching.
For any constant ε > 0 there is a polynomial-time (4/3 + ε)-approximation algorithm for 3-dimensional matching.[10]
However, attaining better approximation factors is probably hard: the problem is APX-complete, that is, it is hard to approximate within some constant.[11][12][8]
It is NP-hard to achieve an approximation factor of 95/94 for maximum 3-d matching, and an approximation factor of 48/47 for maximum 4-d matching. The hardness remains even when restricted to instances with exactly two occurrences of each element.[13]
Parallel algorithms
There are various algorithms for 3-d matching in the Massively parallel computation model.[14]
See also
• List of NP-complete problems
• Rainbow-independent set – a problem that can be reduced from 3-dimensional matching.
• Numerical 3-dimensional matching
Notes
1. Karp (1972).
2. Garey, Michael R.; Johnson, David S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, ISBN 0-7167-1045-5, Section 3.1 and problem SP1 in Appendix A.3.1.
3. Korte, Bernhard; Vygen, Jens (2006), Combinatorial Optimization: Theory and Algorithms (3rd ed.), Springer, Section 15.5.
4. Papadimitriou & Steiglitz (1998), Section 15.7.
5. Demaine, Erik (2016). "16. Complexity: P, NP, NP-completeness, Reductions". YouTube.
6. Karpinski, Rucinski & Szymanska (2009)
7. Keevash, Knox & Mycroft (2013)
8. Kann (1991)
9. Matching (graph theory)#Properties.
10. Cygan, Marek (2013). "Improved Approximation for 3-Dimensional Matching via Bounded Pathwidth Local Search". 2013 IEEE 54th Annual Symposium on Foundations of Computer Science. pp. 509–518. arXiv:1304.1424. Bibcode:2013arXiv1304.1424C. doi:10.1109/FOCS.2013.61. ISBN 978-0-7695-5135-7. S2CID 14160646.
11. Crescenzi et al. (2000).
12. Ausiello et al. (2003), problem SP1 in Appendix B.
13. Chlebík, Miroslav; Chlebíková, Janka (2006-04-04). "Complexity of approximating bounded variants of optimization problems". Theoretical Computer Science. Foundations of Computation Theory (FCT 2003). 354 (3): 320–338. doi:10.1016/j.tcs.2005.11.029. ISSN 0304-3975.
14. Hanguir, Oussama; Stein, Clifford (2020-09-21). "Distributed Algorithms for Matching in Hypergraphs". arXiv:2009.09605 [cs.DS].
References
• Ausiello, Giorgio; Crescenzi, Pierluigi; Gambosi, Giorgio; Kann, Viggo; Marchetti-Spaccamela, Alberto; Protasi, Marco (2003), Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, Springer.
• Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (2000), "Maximum 3-dimensional matching", A Compendium of NP Optimization Problems.
• Kann, Viggo (1991), "Maximum bounded 3-dimensional matching is MAX SNP-complete", Information Processing Letters, 37 (1): 27–35, doi:10.1016/0020-0190(91)90246-E.
• Karp, Richard M. (1972), "Reducibility among combinatorial problems", in Miller, Raymond E.; Thatcher, James W. (eds.), Complexity of Computer Computations, Plenum, pp. 85–103.
• Karpinski, Marek; Rucinski, Andrzej; Szymanska, Edyta (2009), "The Complexity of Perfect Matching Problems on Dense Hypergraphs", ISAAC '09 Proceedings of the 20th International Symposium on Algorithms, Lecture Notes in Computer Science, 5878: 626–636, doi:10.1007/978-3-642-10631-6_64, ISBN 978-3-642-10630-9.
• Keevash, Peter; Knox, Fiachra; Mycroft, Richard (2013), "Polynomial-time perfect matchings in dense hypergraphs", Proceedings of the forty-fifth annual ACM symposium on Theory of Computing, pp. 311–320, arXiv:1307.2608, Bibcode:2013arXiv1307.2608K, doi:10.1145/2488608.2488647, ISBN 9781450320290, S2CID 5393523{{citation}}: CS1 maint: date and year (link).
• Papadimitriou, Christos H.; Steiglitz, Kenneth (1998), Combinatorial Optimization: Algorithms and Complexity, Dover Publications.
| Wikipedia |
Triple helix
In the fields of geometry and biochemistry, a triple helix (PL: triple helices) is a set of three congruent geometrical helices with the same axis, differing by a translation along the axis. This means that each of the helices keeps the same distance from the central axis. As with a single helix, a triple helix may be characterized by its pitch, diameter, and handedness. Examples of triple helices include triplex DNA,[1] triplex RNA,[2] the collagen helix,[3] and collagen-like proteins.
Structure
A triple helix is named such because it is made up of three separate helices. Each of these helices shares the same axis, but they do not take up the same space because each helix is translated angularly around the axis. Generally, the identity of a triple helix depends on the type of helices that make it up. For example: a triple helix made of three strands of collagen protein is a collagen triple helix, and a triple helix made of three strands of DNA is a DNA triple helix.
As with other types of helices, triple helices have handedness: right-handed or left-handed. A right-handed helix moves around its axis in a clockwise direction from beginning to end. A left-handed helix is the right-handed helix's mirror image, and it moves around the axis in a counterclockwise direction from beginning to end.[4] The beginning and end of a helical molecule are defined based on certain markers in the molecule that do not change easily. For example: the beginning of a helical protein is its N terminus, and the beginning of a single strand of DNA is its 5' end.[4]
The collagen triple helix is made of three collagen peptides, each of which forms its own left-handed polyproline helix.[5] When the three chains combine, the triple helix adopts a right-handed orientation. The collagen peptide is composed of repeats of Gly-X-Y, with the second residue (X) usually being Pro and the third (Y) being hydroxyproline.[6][5]
A DNA triple helix is made up of three separate DNA strands, each oriented with the sugar/phosphate backbone on the outside of the helix and the bases on the inside of the helix. The bases are the part of the molecule closest to the triple helix's axis, and the backbone is the part of the molecule farthest away from the axis. The third strand occupies the major groove of relatively normal duplex DNA.[7] The bases in triplex DNA are arranged to match up according to a Hoogsteen base pairing scheme.[8] Similarly, RNA triple helices are formed as a result of a single stranded RNA forming hydrogen bonds with an RNA duplex; the duplex consists of Watson-Crick base pairing while the third strand binds via Hoogsteen base pairing.[9]
Stabilizing factors
The collagen triple helix has several characteristics that increase its stability. When proline is incorporated into the Y position of the Gly-X-Y sequence, it is post-translationally modified to hydroxyproline.[10] The hydroxyproline can enter into favorable interactions with water, which stabilizes the triple helix because the Y residues are solvent-accessible in the triple helix structure. The individual helices are also held together by an extensive network of amide-amide hydrogen bonds formed between the strands, each of which contributes approximately -2 kcal/mol to the overall free energy of the triple helix.[5] The formation of the superhelix not only protects the critical glycine residues on the interior of the helix, but also protects the overall protein from proteolysis.[6]
Triple helix DNA and RNA are stabilized by many of the same forces that stabilize double-stranded DNA helices. With nucleotide bases oriented to the inside of the helix, closer to its axis, bases engage in hydrogen bonding with other bases. The bonded bases in the center exclude water, so the hydrophobic effect is particularly important in the stabilization of DNA triple helices.[4]
Biological role
Proteins
Members of the collagen superfamily are major contributors to the extracellular matrix. The triple helical structure provides strength and stability to collagen fibers by providing great resistance to tensile stress. The rigidity of the collagen fibers is an important factor that can withstand most mechanical stress, making it an ideal protein for macromolecular transport and overall structural support throughout the body.[6]
DNA
There are some oligonucleotide sequences, called triplet-forming oligonucleotides (TFOs) that can bind to form a triplex with a longer molecule of double-stranded DNA; TFOs can inactivate a gene or help to induce mutations.[7] TFOs can only bind to certain sites in a larger molecule, so researchers must first determine whether a TFO can bind to the gene of interest. Twisted intercalating nucleic acid is sometimes used to improve this process.
RNA
In recent years, the biological function of triplex RNA has become more studied. Some roles include increasing stability, translation, influencing ligand binding, and catalysis. One example of ligand binding being influenced by a triple helix is in the SAM-II riboswitch where the triple helix creates a binding site that will uniquely accept S-adenosylmethionine (SAM).[9] The ribonucleoprotein complex telomerase, responsible for replicating the tail-ends of DNA (telomeres) also contains triplex RNA believed to be necessary for proper telomerase functioning.[9][11] The triple helix at the 3' end of the PAN and MALAT1 long-noncoding RNAs serves to stabilize the RNA by protecting the Poly(A) tail from deadenylation, which subsequently affect their functions in viral pathogenesis and multiple human cancers.[9][12] Additionally, RNA triple helices can stabilize mRNAs by formation of a poly(A) tail 3'-end binding pocket.[13]
References
1. Bernués J, Azorín F (1995). "Triple-stranded DNA.". Nucleic Acids and Molecular Biology. Vol. 9. Berlin, Heidelberg: Springer. pp. 1–21. doi:10.1007/978-3-642-79488-9_1. ISBN 978-3-642-79490-2.
2. Buske FA, Mattick JS, Bailey TL (May 2011). "Potential in vivo roles of nucleic acid triple-helices". RNA Biology. 8 (3): 427–39. doi:10.4161/rna.8.3.14999. PMC 3218511. PMID 21525785.
3. Bächinger HP (2005-05-03). Collagen: Primer in Structure, Processing and Assembly. Springer Science & Business Media. ISBN 9783540232728.
4. John, Kuriyan (2012-07-25). The molecules of life : physical and chemical principles. Konforti, Boyana,, Wemmer, David. New York. ISBN 9780815341888. OCLC 779577263.{{cite book}}: CS1 maint: location missing publisher (link)
5. Shoulders MD, Raines RT (2009). "Collagen structure and stability". Annual Review of Biochemistry. 78: 929–58. doi:10.1146/annurev.biochem.77.032207.120833. PMC 2846778. PMID 19344236.
6. Fidler AL, Boudko SP, Rokas A, Hudson BG (April 2018). "The triple helix of collagens - an ancient protein structure that enabled animal multicellularity and tissue evolution". Journal of Cell Science. 131 (7): jcs203950. doi:10.1242/jcs.203950. PMC 5963836. PMID 29632050.
7. Jain A, Wang G, Vasquez KM (August 2008). "DNA triple helices: biological consequences and therapeutic potential". Biochimie. 90 (8): 1117–30. doi:10.1016/j.biochi.2008.02.011. PMC 2586808. PMID 18331847.
8. Duca M, Vekhoff P, Oussedik K, Halby L, Arimondo PB (September 2008). "The triple helix: 50 years later, the outcome". Nucleic Acids Research. 36 (16): 5123–38. doi:10.1093/nar/gkn493. PMC 2532714. PMID 18676453.
9. Conrad NK (2014). "The emerging role of triple helices in RNA biology". Wiley Interdisciplinary Reviews: RNA. 5 (1): 15–29. doi:10.1002/wrna.1194. PMC 4721660. PMID 24115594.
10. Brodsky B, Persikov AV (2005-01-01). "Molecular structure of the collagen triple helix". Advances in Protein Chemistry. 70: 301–39. doi:10.1016/S0065-3233(05)70009-7. ISBN 9780120342709. PMID 15837519.
11. Theimer CA, Blois CA, Feigon J (March 2005). "Structure of the human telomerase RNA pseudoknot reveals conserved tertiary interactions essential for function". Molecular Cell. 17 (5): 671–82. doi:10.1016/j.molcel.2005.01.017. PMID 15749017.
12. Brown JA, Bulkley D, Wang J, Valenstein ML, Yario TA, Steitz TA, Steitz JA (July 2014). "Structural insights into the stabilization of MALAT1 noncoding RNA by a bipartite triple helix". Nature Structural & Molecular Biology. 21 (7): 633–40. doi:10.1038/nsmb.2844. PMC 4096706. PMID 24952594.
13. Torabi, Seyed-Fakhreddin; Vaidya, Anand T.; Tycowski, Kazimierz T.; DeGregorio, Suzanne J.; Wang, Jimin; Shu, Mei-Di; Steitz, Thomas A.; Steitz, Joan A. (2021-01-07). "RNA stabilization by a poly(A) tail 3ʹ-end binding pocket and other modes of poly(A)-RNA interaction". Science. 371 (6529): eabe6523. doi:10.1126/science.abe6523. ISSN 0036-8075. PMC 9491362. PMID 33414189. S2CID 231195473.
Spirals, curves and helices
Curves
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| Wikipedia |
Triple correlation
The triple correlation of an ordinary function on the real line is the integral of the product of that function with two independently shifted copies of itself:
$\int _{-\infty }^{\infty }f^{*}(x)f(x+s_{1})f(x+s_{2})dx.$
The Fourier transform of triple correlation is the bispectrum. The triple correlation extends the concept of autocorrelation, which correlates a function with a single shifted copy of itself and thereby enhances its latent periodicities.
History
The theory of the triple correlation was first investigated by statisticians examining the cumulant structure of non-Gaussian random processes. It was also independently studied by physicists as a tool for spectroscopy of laser beams. Hideya Gamo in 1963 described an apparatus for measuring the triple correlation of a laser beam, and also showed how phase information can be recovered from the real part of the bispectrum—up to sign reversal and linear offset. However, Gamo's method implicitly requires the Fourier transform to never be zero at any frequency. This requirement was relaxed, and the class of functions which are known to be uniquely identified by their triple (and higher-order) correlations was considerably expanded, by the study of Yellott and Iverson (1992). Yellott & Iverson also pointed out the connection between triple correlations and the visual texture discrimination theory proposed by Bela Julesz.
Applications
Triple correlation methods are frequently used in signal processing for treating signals that are corrupted by additive white Gaussian noise; in particular, triple correlation techniques are suitable when multiple observations of the signal are available and the signal may be translating in between the observations, e.g.,a sequence of images of an object translating on a noisy background. What makes the triple correlation particularly useful for such tasks are three properties: (1) it is invariant under translation of the underlying signal; (2) it is unbiased in additive Gaussian noise; and (3) it retains nearly all of the relevant phase information in the underlying signal. Properties (1)-(3) of the triple correlation extend in many cases to functions on an arbitrary locally compact group, in particular to the groups of rotations and rigid motions of euclidean space that arise in computer vision and signal processing.
Extension to groups
The triple correlation may be defined for any locally compact group by using the group's left-invariant Haar measure. It is easily shown that the resulting object is invariant under left translation of the underlying function and unbiased in additive Gaussian noise. What is more interesting is the question of uniqueness : when two functions have the same triple correlation, how are the functions related? For many cases of practical interest, the triple correlation of a function on an abstract group uniquely identifies that function up to a single unknown group action. This uniqueness is a mathematical result that relies on the Pontryagin duality theorem, the Tannaka–Krein duality theorem, and related results of Iwahori-Sugiura, and Tatsuuma. Algorithms exist for recovering bandlimited functions from their triple correlation on Euclidean space, as well as rotation groups in two and three dimensions. There is also an interesting link with Wiener's tauberian theorem: any function whose translates are dense in $L_{1}(G)$, where $G$ is a locally compact Abelian group, is also uniquely identified by its triple correlation.
References
• K. Hasselman, W. Munk, and G. MacDonald (1963), "Bispectra of ocean waves", in Time Series Analysis, M. Rosenblatt, Ed., New York: Wiley, 125-139.
• Gamo, H. (1963). "Triple Correlator of Photoelectric Fluctuations as a Spectroscopic Tool". Journal of Applied Physics. 34 (4): 875–876. Bibcode:1963JAP....34..875G. doi:10.1063/1.1729553.
• Yellott, J.; Iverson, G. J. (1992). "Uniqueness properties of higher-order autocorrelation functions". Journal of the Optical Society of America A. 9 (3): 388. Bibcode:1992JOSAA...9..388Y. doi:10.1364/JOSAA.9.000388.
• R. Kakarala (1992) Triple correlation on groups, Ph.D. Thesis, Department of Mathematics, University of California, Irvine.
• R. Kondor (2007), "A complete set of rotationally and translationally invariant features for images", arXiv:cs/0701127
| Wikipedia |
Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity:
$\prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+{\frac {x^{2m-1}}{y^{2}}}\right)=\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n},$
for complex numbers x and y, with |x| < 1 and y ≠ 0.
It was introduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum.
The Jacobi triple product identity is the Macdonald identity for the affine root system of type A1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra.
Properties
The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity.
Let $x=q{\sqrt {q}}$ and $y^{2}=-{\sqrt {q}}$. Then we have
$\phi (q)=\prod _{m=1}^{\infty }\left(1-q^{m}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {3n^{2}-n}{2}}.$
The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:
Let $x=e^{i\pi \tau }$ and $y=e^{i\pi z}.$
Then the Jacobi theta function
$\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }e^{\pi {\rm {i}}n^{2}\tau +2\pi {\rm {i}}nz}$
can be written in the form
$\sum _{n=-\infty }^{\infty }y^{2n}x^{n^{2}}.$
Using the Jacobi Triple Product Identity we can then write the theta function as the product
$\vartheta (z;\tau )=\prod _{m=1}^{\infty }\left(1-e^{2m\pi {\rm {i}}\tau }\right)\left[1+e^{(2m-1)\pi {\rm {i}}\tau +2\pi {\rm {i}}z}\right]\left[1+e^{(2m-1)\pi {\rm {i}}\tau -2\pi {\rm {i}}z}\right].$
There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of q-Pochhammer symbols:
$\sum _{n=-\infty }^{\infty }q^{\frac {n(n+1)}{2}}z^{n}=(q;q)_{\infty }\;\left(-{\tfrac {1}{z}};q\right)_{\infty }\;(-zq;q)_{\infty },$
where $(a;q)_{\infty }$ is the infinite q-Pochhammer symbol.
It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For $|ab|<1$ it can be written as
$\sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}\;b^{\frac {n(n-1)}{2}}=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.$
Proof
Let $f_{x}(y)=\prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+x^{2m-1}y^{-2}\right)$
Substituting xy for y and multiplying the new terms out gives
$f_{x}(xy)={\frac {1+x^{-1}y^{-2}}{1+xy^{2}}}f_{x}(y)=x^{-1}y^{-2}f_{x}(y)$
Since $f_{x}$ is meromorphic for $|y|>0$, it has a Laurent series
$f_{x}(y)=\sum _{n=-\infty }^{\infty }c_{n}(x)y^{2n}$
which satisfies
$\sum _{n=-\infty }^{\infty }c_{n}(x)x^{2n+1}y^{2n}=xf_{x}(xy)=y^{-2}f_{x}(y)=\sum _{n=-\infty }^{\infty }c_{n+1}(x)y^{2n}$
so that
$c_{n+1}(x)=c_{n}(x)x^{2n+1}=\dots =c_{0}(x)x^{(n+1)^{2}}$
and hence
$f_{x}(y)=c_{0}(x)\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n}$
Evaluating c0(x)
Showing that $c_{0}(x)=1$ is technical. One way is to set $y=1$ and show both the numerator and the denominator of
${\frac {1}{c_{0}(e^{2i\pi z})}}={\frac {\sum \limits _{n=-\infty }^{\infty }e^{2i\pi n^{2}z}}{\prod \limits _{m=1}^{\infty }(1-e^{2i\pi mz})(1+e^{2i\pi (2m-1)z})^{2}}}$
are weight 1/2 modular under $z\mapsto -{\frac {1}{4z}}$, since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that $c_{0}(x)=c_{0}(0)=1$.
Other proofs
A different proof is given by G. E. Andrews based on two identities of Euler.[1]
For the analytic case, see Apostol.[2]
References
1. Andrews, George E. (1965-02-01). "A simple proof of Jacobi's triple product identity". Proceedings of the American Mathematical Society. 16 (2): 333. doi:10.1090/S0002-9939-1965-0171725-X. ISSN 0002-9939.
2. Chapter 14, theorem 14.6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press, ISBN 0-521-45761-0
• Jacobi, C. G. J. (1829), Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012
• Carlitz, L (1962), A note on the Jacobi theta formula, American Mathematical Society
• Wright, E. M. (1965), "An Enumerative Proof of An Identity of Jacobi", Journal of the London Mathematical Society, London Mathematical Society: 55–57, doi:10.1112/jlms/s1-40.1.55
| Wikipedia |
Triple product rule
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem, and is given by
$\left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1,$
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where each factor is a partial derivative of the variable in the numerator, considered to be a function of the other two.
The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,
$\left({\frac {\partial x}{\partial y}}\right)=-{\frac {\left({\frac {\partial z}{\partial y}}\right)}{\left({\frac {\partial z}{\partial x}}\right)}}$
Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}.
Derivation
An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total differential dz is
$dz=\left({\frac {\partial z}{\partial x}}\right)dx+\left({\frac {\partial z}{\partial y}}\right)dy$
Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve
$dy=\left({\frac {\partial y}{\partial x}}\right)dx$
Therefore, the equation for dz = 0 becomes
$0=\left({\frac {\partial z}{\partial x}}\right)\,dx+\left({\frac {\partial z}{\partial y}}\right)\left({\frac {\partial y}{\partial x}}\right)\,dx$
Since this must be true for all dx, rearranging terms gives
$\left({\frac {\partial z}{\partial x}}\right)=-\left({\frac {\partial z}{\partial y}}\right)\left({\frac {\partial y}{\partial x}}\right)$
Dividing by the derivatives on the right hand side gives the triple product rule
$\left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1$
Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the exact differential dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities.
Alternative derivation
Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables
$dx=\left({\frac {\partial x}{\partial y}}\right)dy+\left({\frac {\partial x}{\partial z}}\right)dz$
$dy=\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz$
Substitute dy into dx
$dx=\left({\frac {\partial x}{\partial y}}\right)\left[\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz\right]+\left({\frac {\partial x}{\partial z}}\right)dz$
By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero
$\left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)+\left({\frac {\partial x}{\partial z}}\right)=0$
Subtracting the second term and multiplying by its inverse gives the triple product rule
$\left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1.$
Applications
Example: Ideal Gas Law
The ideal gas law relates the state variables of pressure (P), volume (V), and temperature (T) via
$PV=nRT$
which can be written as
$f(P,V,T)=PV-nRT=0$
so each state variable can be written as an implicit function of the other state variables:
${\begin{aligned}P&=P(V,T)={\frac {nRT}{V}}\\[1em]V&=V(P,T)={\frac {nRT}{P}}\\[1em]T&=T(P,V)={\frac {PV}{nR}}\end{aligned}}$
From the above expressions, we have
${\begin{aligned}-1&=\left({\frac {\partial P}{\partial V}}\right)\left({\frac {\partial V}{\partial T}}\right)\left({\frac {\partial T}{\partial P}}\right)\\[1em]&=\left(-{\frac {nRT}{V^{2}}}\right)\left({\frac {nR}{P}}\right)\left({\frac {V}{nR}}\right)\\[1em]&=\left(-{\frac {nRT}{PV}}\right)\\[1em]&=-{\frac {P}{P}}=-1\end{aligned}}$
Geometric Realization
A geometric realization of the triple product rule can be found in its close ties to the velocity of a traveling wave
$\phi (x,t)=A\cos(kx-\omega t)$
shown on the right at time t (solid blue line) and at a short time later t+Δt (dashed). The wave maintains its shape as it propagates, so that a point at position x at time t will correspond to a point at position x+Δx at time t+Δt,
$A\cos(kx-\omega t)=A\cos(k(x+\Delta x)-\omega (t+\Delta t)).$
This equation can only be satisfied for all x and t if k Δx − ω Δt = 0, resulting in the formula for the phase velocity
$v={\frac {\Delta x}{\Delta t}}={\frac {\omega }{k}}.$
To elucidate the connection with the triple product rule, consider the point p1 at time t and its corresponding point (with the same height) p̄1 at t+Δt. Define p2 as the point at time t whose x-coordinate matches that of p̄1, and define p̄2 to be the corresponding point of p2 as shown in the figure on the right. The distance Δx between p1 and p̄1 is the same as the distance between p2 and p̄2 (green lines), and dividing this distance by Δt yields the speed of the wave.
To compute Δx, consider the two partial derivatives computed at p2,
$\left({\frac {\partial \phi }{\partial t}}\right)\Delta t={\text{rise from }}p_{2}{\text{ to }}{\bar {p}}_{1}{\text{ in time }}\Delta t{\text{ (gold line)}}$
$\left({\frac {\partial \phi }{\partial x}}\right)={\text{slope of the wave (red line) at time }}t.$
Dividing these two partial derivatives and using the definition of the slope (rise divided by run) gives us the desired formula for
$\Delta x=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)\Delta t}{\left({\frac {\partial \phi }{\partial x}}\right)}},$
where the negative sign accounts for the fact that p1 lies behind p2 relative to the wave's motion. Thus, the wave's velocity is given by
$v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)}{\left({\frac {\partial \phi }{\partial x}}\right)}}.$
For infinitesimal Δt, ${\frac {\Delta x}{\Delta t}}=\left({\frac {\partial x}{\partial t}}\right)$ and we recover the triple product rule
$v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)}{\left({\frac {\partial \phi }{\partial x}}\right)}}.$
See also
• Differentiation rules – Rules for computing derivatives of functions
• Exact differential – type of infinitesimal in calculusPages displaying wikidata descriptions as a fallback (has another derivation of the triple product rule)
• Product rule – Formula for the derivative of a product
• Total derivative – Type of derivative in mathematics
• Triple product – Ternary operation on vectors and scalars.
References
• Elliott, J. R.; Lira, C. T. (1999). Introductory Chemical Engineering Thermodynamics (1st ed.). Prentice Hall. p. 184. ISBN 0-13-011386-7.
• Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Prentice Hall. p. 392. ISBN 0-13-779208-5.
| Wikipedia |
Triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
This article is about ternary operations on vectors. For the identity in number theory, see Jacobi triple product. For the calculus chain rule for three interdependent variables, see Triple product rule. For the product in nuclear fusion, see Lawson criterion. For the abstract algebra identity satisfied in some groups, see Triple product property.
Scalar triple product
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
Geometric interpretation
Geometrically, the scalar triple product
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )$
is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined.
Properties
• The scalar triple product is unchanged under a circular shift of its three operands (a, b, c):
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )$
• Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product:
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} $
• Swapping any two of the three operands negates the triple product. This follows from the circular-shift property and the anticommutativity of the cross product:
${\begin{aligned}\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )&=-\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\&=-\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\&=-\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}$
• The scalar triple product can also be understood as the determinant of the 3×3 matrix that has the three vectors either as its rows or its columns (a matrix has the same determinant as its transpose):
$\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}=\det {\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{bmatrix}}=\det {\begin{bmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{bmatrix}}.$
• If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume.
• If any two vectors in the scalar triple product are equal, then its value is zero:
$\mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {b} )=\mathbf {b} \cdot (\mathbf {a} \times \mathbf {a} )=0$
• Also:
$(\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\,\mathbf {a} =(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )$
• The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products:[1]
$((\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} )\;((\mathbf {d} \times \mathbf {e} )\cdot \mathbf {f} )=\det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}$
This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is a Gram determinant.
• The ratio of the triple product and the product of the three vector norms is known as a polar sine:
${\frac {\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}{\|{\mathbf {a} }\|\|{\mathbf {b} }\|\|{\mathbf {c} }\|}}=\operatorname {psin} (\mathbf {a} ,\mathbf {b} ,\mathbf {c} )$
which ranges between -1 and 1.
Scalar or pseudoscalar
Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the orientation of the frame or the parity of the permutation of the vectors. This means the product is negated if the orientation is reversed, for example by a parity transformation, and so is more properly described as a pseudoscalar if the orientation can change.
This also relates to the handedness of the cross product; the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector. The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product must be pseudoscalar-valued.
If T is a rotation operator, then
$\mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ),$
but if T is an improper rotation, then
$\mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=-\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ).$
As an exterior product
In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element.
Given vectors a, b and c, the product
$\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} $
is a trivector with magnitude equal to the scalar triple product, i.e.
$|\mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} |=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|$,
and is the Hodge dual of the scalar triple product. As the exterior product is associative brackets are not needed as it does not matter which of a ∧ b or b ∧ c is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector a ∧ b ∧ c corresponds to the parallelepiped spanned by a, b, and c, with bivectors a ∧ b, b ∧ c and a ∧ c matching the parallelogram faces of the parallelepiped.
As a trilinear function
The triple product is identical to the volume form of the Euclidean 3-space applied to the vectors via interior product. It also can be expressed as a contraction of vectors with a rank-3 tensor equivalent to the form (or a pseudotensor equivalent to the volume pseudoform); see below.
Vector triple product
The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds:
$\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} $.
This is known as triple product expansion, or Lagrange's formula,[2][3] although the latter name is also used for several other formulas. Its right hand side can be remembered by using the mnemonic "ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is provided below. Some textbooks write the identity as $\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )$ such that a more familiar mnemonic "BAC − CAB" is obtained, as in “back of the cab”.
Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as:
$(\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} $
From Lagrange's formula it follows that the vector triple product satisfies:
$\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=\mathbf {0} $
which is the Jacobi identity for the cross product. Another useful formula follows:
$(\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )$
These formulas are very useful in simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product identity:[4]
${\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {A} )-({\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }})\mathbf {A} $
This can be also regarded as a special case of the more general Laplace–de Rham operator $\Delta =d\delta +\delta d$.
Proof
The $x$ component of $\mathbf {u} \times (\mathbf {v} \times \mathbf {w} )$ is given by:
${\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\mathbf {w} _{y}-\mathbf {v} _{y}\mathbf {w} _{x})-\mathbf {u} _{z}(\mathbf {v} _{z}\mathbf {w} _{x}-\mathbf {v} _{x}\mathbf {w} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})+(\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x}-\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x})\\&=\mathbf {v} _{x}(\mathbf {u} _{x}\mathbf {w} _{x}+\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{x}\mathbf {v} _{x}+\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{x}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{x}\end{aligned}}$
Similarly, the $y$ and $z$ components of $\mathbf {u} \times (\mathbf {v} \times \mathbf {w} )$ are given by:
${\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{y}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{y}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{y}\\(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{z}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{z}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{z}\end{aligned}}$
By combining these three components we obtain:
$\mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\ \mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\ \mathbf {w} $[5]
Using geometric algebra
If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product b∧c, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction[6] can be used, so the formula becomes[7]
${\begin{aligned}-\mathbf {a} \;{\big \lrcorner }\;(\mathbf {b} \wedge \mathbf {c} )&=\mathbf {b} \wedge (\mathbf {a} \;{\big \lrcorner }\;\mathbf {c} )-(\mathbf {a} \;{\big \lrcorner }\;\mathbf {b} )\wedge \mathbf {c} \\&=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} \end{aligned}}$
The proof follows from the properties of the contraction.[6] The result is the same vector as calculated using a × (b × c).
Interpretations
Tensor calculus
In tensor notation, the triple product is expressed using the Levi-Civita symbol:[8]
$\mathbf {a} \cdot [\mathbf {b} \times \mathbf {c} ]=\varepsilon _{ijk}a^{i}b^{j}c^{k}$
and
$(\mathbf {a} \times [\mathbf {b} \times \mathbf {c} ])_{i}=\varepsilon _{ijk}a^{j}\varepsilon ^{k\ell m}b_{\ell }c_{m}=\varepsilon _{ijk}\varepsilon ^{k\ell m}a^{j}b_{\ell }c_{m},$
referring to the $i$-th component of the resulting vector. This can be simplified by performing a contraction on the Levi-Civita symbols, $\varepsilon _{ijk}\varepsilon ^{k\ell m}=\delta _{ij}^{\ell m}=\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell }\,,$ where $\delta _{j}^{i}$ is the Kronecker delta function ($\delta _{j}^{i}=0$ when $i\neq j$ and $\delta _{j}^{i}=1$ when $i=j$) and $\delta _{ij}^{\ell m}$ is the generalized Kronecker delta function. We can reason out this identity by recognizing that the index $k$ will be summed out leaving only $i$ and $j$. In the first term, we fix $i=l$ and thus $j=m$. Likewise, in the second term, we fix $i=m$ and thus $l=j$.
Returning to the triple cross product,
$(\mathbf {a} \times [\mathbf {b} \times \mathbf {c} ])_{i}=(\delta _{i}^{\ell }\delta _{j}^{m}-\delta _{i}^{m}\delta _{j}^{\ell })a^{j}b_{\ell }c_{m}=a^{j}b_{i}c_{j}-a^{j}b_{j}c_{i}=b_{i}(\mathbf {a} \cdot \mathbf {c} )-c_{i}(\mathbf {a} \cdot \mathbf {b} )\,.$
Vector calculus
Consider the flux integral of the vector field $\mathbf {F} $ across the parametrically-defined surface $S=\mathbf {r} (u,v)$: $ \iint _{S}\mathbf {F} \cdot {\hat {\mathbf {n} }}\,dS$. The unit normal vector ${\hat {\mathbf {n} }}$ to the surface is given by $ {\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}$, so the integrand $ \mathbf {F} \cdot {\frac {(\mathbf {r} _{u}\times \mathbf {r} _{v})}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}$ is a scalar triple product.
See also
• Quadruple product
• Vector algebra relations
Notes
1. Wong, Chun Wa (2013). Introduction to Mathematical Physics: Methods & Concepts. Oxford University Press. p. 215. ISBN 9780199641390.
2. Joseph Louis Lagrange did not develop the cross product as an algebraic product on vectors, but did use an equivalent form of it in components: see Lagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires". Oeuvres. Vol. 3. He may have written a formula similar to the triple product expansion in component form. See also Lagrange's identity and Kiyosi Itô (1987). Encyclopedic Dictionary of Mathematics. MIT Press. p. 1679. ISBN 0-262-59020-4.
3. Kiyosi Itô (1993). "§C: Vector product". Encyclopedic dictionary of mathematics (2nd ed.). MIT Press. p. 1679. ISBN 0-262-59020-4.
4. Pengzhi Lin (2008). Numerical Modelling of Water Waves: An Introduction to Engineers and Scientists. Routledge. p. 13. ISBN 978-0-415-41578-1.
5. J. Heading (1970). Mathematical Methods in Science and Engineering. American Elsevier Publishing Company, Inc. pp. 262–263.
6. Pertti Lounesto (2001). Clifford algebras and spinors (2nd ed.). Cambridge University Press. p. 46. ISBN 0-521-00551-5.
7. Janne Pesonen. "Geometric Algebra of One and Many Multivector Variables" (PDF). p. 37.
8. "Permutation Tensor". Wolfram. Retrieved 21 May 2014.
References
• Lass, Harry (1950). Vector and Tensor Analysis. McGraw-Hill Book Company, Inc. pp. 23–25.
External links
• Khan Academy video of the proof of the triple product expansion
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| Wikipedia |
Triple system
In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map
$(\cdot ,\cdot ,\cdot )\colon V\times V\times V\to V.$
For an account of that concept in combinatorics, see Steiner triple system and block design.
The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals).
Lie triple systems
A triple system is said to be a Lie triple system if the trilinear map, denoted $[\cdot ,\cdot ,\cdot ]$, satisfies the following identities:
$[u,v,w]=-[v,u,w]$
$[u,v,w]+[w,u,v]+[v,w,u]=0$
$[u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].$
The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map Lu,v: V → V, defined by Lu,v(w) = [u, v, w], is a derivation of the triple product. The identity also shows that the space k = span {Lu,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.
Writing m in place of V, it follows that
${\mathfrak {g}}:=k\oplus {\mathfrak {m}}$
can be made into a $\mathbb {Z} _{2}$-graded Lie algebra, the standard embedding of m, with bracket
$[(L,u),(M,v)]=([L,M]+L_{u,v},L(v)-M(u)).$
The decomposition of g is clearly a symmetric decomposition for this Lie bracket, and hence if G is a connected Lie group with Lie algebra g and K is a subgroup with Lie algebra k, then G/K is a symmetric space.
Conversely, given a Lie algebra g with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket [[u, v], w] makes m into a Lie triple system.
Jordan triple systems
A triple system is said to be a Jordan triple system if the trilinear map, denoted {.,.,.}, satisfies the following identities:
$\{u,v,w\}=\{u,w,v\}$
$\{u,v,\{w,x,y\}\}=\{w,x,\{u,v,y\}\}+\{w,\{u,v,x\},y\}-\{\{v,u,w\},x,y\}.$
The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if Lu,v:V→V is defined by Lu,v(y) = {u, v, y} then
$[L_{u,v},L_{w,x}]:=L_{u,v}\circ L_{w,x}-L_{w,x}\circ L_{u,v}=L_{w,\{u,v,x\}}-L_{\{v,u,w\},x}$
so that the space of linear maps span {Lu,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra g0.
Any Jordan triple system is a Lie triple system with respect to the product
$[u,v,w]=\{u,v,w\}-\{v,u,w\}.$
A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of Lu,v is positive definite (resp. nondegenerate). In either case, there is an identification of V with its dual space, and a corresponding involution on g0. They induce an involution of
$V\oplus {\mathfrak {g}}_{0}\oplus V^{*}$
which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space. It has a noncompact dual given by replacing the Cartan involution by its composite with the involution equal to +1 on g0 and −1 on V and V*. A special case of this construction arises when g0 preserves a complex structure on V. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type (the latter being bounded symmetric domains).
Jordan pair
A Jordan pair is a generalization of a Jordan triple system involving two vector spaces V+ and V−. The trilinear map is then replaced by a pair of trilinear maps
$\{\cdot ,\cdot ,\cdot \}_{+}\colon V_{-}\times S^{2}V_{+}\to V_{+}$
$\{\cdot ,\cdot ,\cdot \}_{-}\colon V_{+}\times S^{2}V_{-}\to V_{-}$
which are often viewed as quadratic maps V+ → Hom(V−, V+) and V− → Hom(V+, V−). The other Jordan axiom (apart from symmetry) is likewise replaced by two axioms, one being
$\{u,v,\{w,x,y\}_{+}\}_{+}=\{w,x,\{u,v,y\}_{+}\}_{+}+\{w,\{u,v,x\}_{+},y\}_{+}-\{\{v,u,w\}_{-},x,y\}_{+}$
and the other being the analogue with + and − subscripts exchanged.
As in the case of Jordan triple systems, one can define, for u in V− and v in V+, a linear map
$L_{u,v}^{+}:V_{+}\to V_{+}\quad {\text{by}}\quad L_{u,v}^{+}(y)=\{u,v,y\}_{+}$
and similarly L−. The Jordan axioms (apart from symmetry) may then be written
$[L_{u,v}^{\pm },L_{w,x}^{\pm }]=L_{w,\{u,v,x\}_{\pm }}^{\pm }-L_{\{v,u,w\}_{\mp },x}^{\pm }$
which imply that the images of L+ and L− are closed under commutator brackets in End(V+) and End(V−). Together they determine a linear map
$V_{+}\otimes V_{-}\to {\mathfrak {gl}}(V_{+})\oplus {\mathfrak {gl}}(V_{-})$
whose image is a Lie subalgebra ${\mathfrak {g}}_{0}$, and the Jordan identities become Jacobi identities for a graded Lie bracket on
$V_{+}\oplus {\mathfrak {g}}_{0}\oplus V_{-},$
so that conversely, if
${\mathfrak {g}}={\mathfrak {g}}_{+1}\oplus {\mathfrak {g}}_{0}\oplus {\mathfrak {g}}_{-1}$
is a graded Lie algebra, then the pair $({\mathfrak {g}}_{+1},{\mathfrak {g}}_{-1})$ is a Jordan pair, with brackets
$\{X_{\mp },Y_{\pm },Z_{\pm }\}_{\pm }:=[[X_{\mp },Y_{\pm }],Z_{\pm }].$
Jordan triple systems are Jordan pairs with V+ = V− and equal trilinear maps. Another important case occurs when V+ and V− are dual to one another, with dual trilinear maps determined by an element of
$\mathrm {End} (S^{2}V_{+})\cong S^{2}V_{+}^{*}\otimes S^{2}V_{-}^{*}\cong \mathrm {End} (S^{2}V_{-}).$
These arise in particular when ${\mathfrak {g}}$ above is semisimple, when the Killing form provides a duality between ${\mathfrak {g}}_{+1}$ and ${\mathfrak {g}}_{-1}$.
See also
• Associator
• Quadratic Jordan algebra
References
• Bertram, Wolfgang (2000), The geometry of Jordan and Lie structures, Lecture Notes in Mathematics, vol. 1754, Springer, ISBN 978-3-540-41426-1
• Helgason, Sigurdur (2001) [1978], Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, ISBN 978-0-8218-2848-9
• Jacobson, Nathan (1949), "Lie and Jordan triple systems", American Journal of Mathematics, 71 (1): 149–170, doi:10.2307/2372102, JSTOR 2372102
• Kamiya, Noriaki (2001) [1994], "Lie triple system", Encyclopedia of Mathematics, EMS Press.
• Kamiya, Noriaki (2001) [1994], "Jordan triple system", Encyclopedia of Mathematics, EMS Press.
• Koecher, M. (1969), An elementary approach to bounded symmetric domains, Lecture Notes, Rice University
• Loos, Ottmar (1969), General Theory, Symmetric spaces, vol. 1, W. A. Benjamin, OCLC 681278693
• Loos, Ottmar (1969), Compact Spaces and Classification, Symmetric spaces, vol. 2, W. A. Benjamin
• Loos, Ottmar (1971), "Jordan triple systems, R-spaces, and bounded symmetric domains", Bulletin of the American Mathematical Society, 77 (4): 558–561, doi:10.1090/s0002-9904-1971-12753-2
• Loos, Ottmar (2006) [1975], Jordan pairs, Lecture Notes in Mathematics, vol. 460, Springer, ISBN 978-3-540-37499-2
• Loos, Ottmar (1977), Bounded symmetric domains and Jordan pairs (PDF), Mathematical lectures, University of California, Irvine, archived from the original (PDF) on 2016-03-03
• Meyberg, K. (1972), Lectures on algebras and triple systems (PDF), University of Virginia
• Rosenfeld, Boris (1997), Geometry of Lie groups, Mathematics and its Applications, vol. 393, Kluwer, p. 92, ISBN 978-0792343905, Zbl 0867.53002
• Tevelev, E. (2002), "Moore-Penrose inverse, parabolic subgroups, and Jordan pairs", Journal of Lie Theory, 12: 461–481, arXiv:math/0101107, Bibcode:2001math......1107T
| Wikipedia |
Triplet loss
Triplet loss is a loss function for machine learning algorithms where a reference input (called anchor) is compared to a matching input (called positive) and a non-matching input (called negative). The distance from the anchor to the positive is minimized, and the distance from the anchor to the negative input is maximized.[1][2] An early formulation equivalent to triplet loss was introduced (without the idea of using anchors) for metric learning from relative comparisons by M. Schultze and T. Joachims in 2003.[3]
By enforcing the order of distances, triplet loss models embed in the way that a pair of samples with same labels are smaller in distance than those with different labels. Unlike t-SNE which preserves embedding orders via probability distributions, triplet loss works directly on embedded distances. Therefore, in its common implementation, it needs soft margin treatment with a slack variable $\alpha $ in its hinge loss-style formulation. It is often used for learning similarity for the purpose of learning embeddings, such as learning to rank, word embeddings, thought vectors, and metric learning.[4]
Consider the task of training a neural network to recognize faces (e.g. for admission to a high security zone). A classifier trained to classify an instance would have to be retrained every time a new person is added to the face database. This can be avoided by posing the problem as a similarity learning problem instead of a classification problem. Here the network is trained (using a contrastive loss) to output a distance which is small if the image belongs to a known person and large if the image belongs to an unknown person. However, if we want to output the closest images to a given image, we want to learn a ranking and not just a similarity. A triplet loss is used in this case.
The loss function can be described by means of the Euclidean distance function
${\mathcal {L}}\left(A,P,N\right)=\operatorname {max} \left({\|\operatorname {f} \left(A\right)-\operatorname {f} \left(P\right)\|}_{2}-{\|\operatorname {f} \left(A\right)-\operatorname {f} \left(N\right)\|}_{2}+\alpha ,0\right)$
where $A$ is an anchor input, $P$ is a positive input of the same class as $A$, $N$ is a negative input of a different class from $A$, $\alpha $ is a margin between positive and negative pairs, and $\operatorname {f} $ is an embedding.
This can then be used in a cost function, that is the sum of all losses, which can then be used for minimization of the posed optimization problem
${\mathcal {J}}=\sum _{i=1}^{{}M}{\mathcal {L}}\left(A^{(i)},P^{(i)},N^{(i)}\right)$
The indices are for individual input vectors given as a triplet. The triplet is formed by drawing an anchor input, a positive input that describes the same entity as the anchor entity, and a negative input that does not describe the same entity as the anchor entity. These inputs are then run through the network, and the outputs are used in the loss function.
Comparison and Extensions
In computer vision tasks such as re-identification, a prevailing belief has been that the triplet loss is inferior to using surrogate losses (i.e., typical classification losses) followed by separate metric learning steps. Recent work showed that for models trained from scratch, as well as pretrained models, a special version of triplet loss doing end-to-end deep metric learning outperforms most other published methods as of 2017.[5]
Additionally, triplet loss has been extended to simultaneously maintain a series of distance orders by optimizing a continuous relevance degree with a chain (i.e., ladder) of distance inequalities. This leads to the Ladder Loss, which has been demonstrated to offer performance enhancements of visual-semantic embedding in learning to rank tasks.[6]
In Natural Language Processing, triplet loss is one of the loss functions considered for BERT fine-tuning in the SBERT architecture.[7]
Other extensions involve specifying multiple negatives (multiple negatives ranking loss).
See also
• Siamese neural network
• t-distributed stochastic neighbor embedding
• Learning to rank
• Similarity learning
References
1. Chechik, G.; Sharma, V.; Shalit, U.; Bengio, S. (2010). "Large Scale Online Learning of Image Similarity Through Ranking" (PDF). Journal of Machine Learning Research. 11: 1109–1135.
2. Schroff, F.; Kalenichenko, D.; Philbin, J. (June 2015). "FaceNet: A unified embedding for face recognition and clustering". 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp. 815–823. arXiv:1503.03832. doi:10.1109/CVPR.2015.7298682. ISBN 978-1-4673-6964-0. S2CID 206592766.
3. Schultz, M.; Joachims, T. (2004). "Learning a distance metric from relative comparisons" (PDF). Advances in Neural Information Processing Systems. 16: 41–48.
4. Ailon, Nir; Hoffer, Elad (2014-12-20). "Deep metric learning using Triplet network". arXiv:1412.6622. Bibcode:2014arXiv1412.6622H. {{cite journal}}: Cite journal requires |journal= (help)
5. Hermans, Alexander; Beyer, Lucas; Leibe, Bastian (2017-03-22). "In Defense of the Triplet Loss for Person Re-Identification". arXiv:1703.07737 [cs.CV].
6. Zhou, Mo; Niu, Zhenxing; Wang, Le; Gao, Zhanning; Zhang, Qilin; Hua, Gang (2020-04-03). "Ladder Loss for Coherent Visual-Semantic Embedding" (PDF). Proceedings of the AAAI Conference on Artificial Intelligence. 34 (7): 13050–13057. doi:10.1609/aaai.v34i07.7006. ISSN 2374-3468. S2CID 208139521.
7. Reimers, Nils; Gurevych, Iryna (2019-08-27). "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks". arXiv:1908.10084 [cs.CL].
| Wikipedia |
Triply periodic minimal surface
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations.
These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.[1]
TPMS are of relevance in natural science. TPMS have been observed as biological membranes,[2] as block copolymers,[3] equipotential surfaces in crystals[4] etc. They have also been of interest in architecture, design and art.
Properties
Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).[5]
All connected TPMS have genus ≥ 3,[6] and in every lattice there exist orientable embedded TPMS of every genus ≥3.[7]
Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths). If they are congruent the surface is said to be a balance surface.[8]
History
The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student E. R. Neovius in 1883.[9][10]
In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells.[11] [12] While Schoen's surfaces became popular in natural science the construction did not lend itself to a mathematical existence proof and remained largely unknown in mathematics, until H. Karcher proved their existence in 1989.[13]
Using conjugate surfaces many more surfaces were found. While Weierstrass representations are known for the simpler examples, they are not known for many surfaces. Instead methods from Discrete differential geometry are often used.[5]
Families
The classification of TPMS is an open problem.
TPMS often come in families that can be continuously deformed into each other. Meeks found an explicit 5-parameter family for genus 3 TPMS that contained all then known examples of genus 3 surfaces except the gyroid.[6] Members of this family can be continuously deformed into each other, remaining embedded in the process (although the lattice may change). The gyroid and lidinoid are each inside a separate 1-parameter family.[14]
Another approach to classifying TPMS is to examine their space groups. For surfaces containing lines the possible boundary polygons can be enumerated, providing a classification.[8][15]
Generalisations
Periodic minimal surfaces can be constructed in S3[16] and H3.[17]
It is possible to generalise the division of space into labyrinths to find triply periodic (but possibly branched) minimal surfaces that divide space into more than two sub-volumes.[18]
Quasiperiodic minimal surfaces have been constructed in ℝ2×S1.[19] It has been suggested but not been proven that minimal surfaces with a quasicrystalline order in ℝ3 exist.[20]
External galleries of images
• TPMS at the Minimal Surface Archive
• Periodic minimal surfaces gallery
References
1. "Mathematics of the EPINET Project".
2. Deng, Yuru; Mieczkowski, Mark (1998). "Three-dimensional periodic cubic membrane structure in the mitochondria of amoebae Chaos carolinensis". Protoplasma. Springer Science and Business Media LLC. 203 (1–2): 16–25. doi:10.1007/bf01280583. ISSN 0033-183X. S2CID 25569139.
3. Jiang, Shimei; Göpfert, Astrid; Abetz, Volker (2003). "Novel Morphologies of Block Copolymer Blends via Hydrogen Bonding". Macromolecules. American Chemical Society (ACS). 36 (16): 6171–6177. Bibcode:2003MaMol..36.6171J. doi:10.1021/ma0342933. ISSN 0024-9297.
4. Mackay, Alan L. (1985). "Periodic minimal surfaces". Physica B+C. Elsevier BV. 131 (1–3): 300–305. Bibcode:1985PhyBC.131..300M. doi:10.1016/0378-4363(85)90163-9. ISSN 0378-4363. S2CID 4267918.
5. Karcher, Hermann; Polthier, Konrad (1996-09-16). "Construction of triply periodic minimal surfaces" (PDF). Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. The Royal Society. 354 (1715): 2077–2104. arXiv:1002.4805. Bibcode:1996RSPTA.354.2077K. doi:10.1098/rsta.1996.0093. ISSN 1364-503X. S2CID 15540887.
6. William H. Meeks, III. The Geometry and the Conformal Structure of Triply Periodic Minimal Surfaces in R3. PhD thesis, University of California, Berkeley, 1975.
7. Traizet, M. (2008). "On the genus of triply periodic minimal surfaces" (PDF). Journal of Differential Geometry. International Press of Boston. 79 (2): 243–275. doi:10.4310/jdg/1211512641. ISSN 0022-040X.
8. "Without self-intersections". Archived from the original on 2007-02-22.
9. H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.
10. E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimal Flachen", Akad. Abhandlungen, Helsingfors, 1883.
11. Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)"Infinite periodic minimal surfaces without self-intersections by Alan H. Schoen" (PDF). Archived (PDF) from the original on 2018-04-13. Retrieved 2019-04-12.
12. "Triply-periodic minimal surfaces by Alan H. Schoen". Archived from the original on 2018-10-22. Retrieved 2019-04-12.
13. Karcher, Hermann (1989-03-05). "The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions". Manuscripta Mathematica. 64 (3): 291–357. doi:10.1007/BF01165824. S2CID 119894224.
14. Adam G. Weyhaupt. New families of embedded triply periodic minimal surfaces of genus three in euclidean space. PhD thesis, Indiana University, 2006
15. Fischer, W.; Koch, E. (1996-09-16). "Spanning minimal surfaces". Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. The Royal Society. 354 (1715): 2105–2142. Bibcode:1996RSPTA.354.2105F. doi:10.1098/rsta.1996.0094. ISSN 1364-503X. S2CID 118170498.
16. Karcher, H.; Pinkall, U.; Sterling, I. (1988). "New minimal surfaces in S3". Journal of Differential Geometry. International Press of Boston. 28 (2): 169–185. doi:10.4310/jdg/1214442276. ISSN 0022-040X.
17. K. Polthier. New periodic minimal surfaces in h3. In G. Dziuk, G. Huisken, and J. Hutchinson, editors, Theoretical and Numerical Aspects of Geometric Variational Problems, volume 26, pages 201–210. CMA Canberra, 1991.
18. Góźdź, Wojciech T.; Hołyst, Robert (1996-11-01). "Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions". Physical Review E. American Physical Society (APS). 54 (5): 5012–5027. Bibcode:1996PhRvE..54.5012G. doi:10.1103/physreve.54.5012. ISSN 1063-651X. PMID 9965680.
19. Laurent Mazet, Martin Traizet, A quasi-periodic minimal surface, Commentarii Mathematici Helvetici, pp. 573–601, 2008
20. Sheng, Qing; Elser, Veit (1994-04-01). "Quasicrystalline minimal surfaces". Physical Review B. American Physical Society (APS). 49 (14): 9977–9980. Bibcode:1994PhRvB..49.9977S. doi:10.1103/physrevb.49.9977. ISSN 0163-1829. PMID 10009804.
Minimal surfaces
• Associate family
• Bour's
• Catalan's
• Catenoid
• Chen–Gackstatter
• Costa's
• Enneper
• Gyroid
• Helicoid
• Henneberg
• k-noid
• Lidinoid
• Neovius
• Richmond
• Riemann's
• Saddle tower
• Scherk
• Schwarz
• Triply periodic
| Wikipedia |
Tripod packing
In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axis-aligned rays with a shared apex.[1]
Unsolved problem in mathematics:
How many tripods can have their apexes packed into a given cube?
(more unsolved problems in mathematics)
Several problems of tiling and packing tripods and related shapes were formulated in 1967 by Sherman K. Stein.[2] Stein originally called the tripods of this problem "semicrosses", and they were also called Stein corners by Solomon W. Golomb.[3] A collection of disjoint tripods can be represented compactly as a monotonic matrix, a square matrix whose nonzero entries increase along each row and column and whose equal nonzero entries are placed in a monotonic sequence of cells,[4] and the problem can also be formulated in terms of finding sets of triples satisfying a compatibility condition called "2-comparability", or of finding compatible sets of triangles in a convex polygon.[1]
The best lower bound known for the number of tripods that can have their apexes packed into an $n\times n\times n$ grid is $\Omega (n^{1.546})$, and the best upper bound is $n^{2}/\exp \Omega (\log ^{*}n)$, both expressed in big Omega notation.[1][5]
Equivalent problems
The coordinates $(x_{i},y_{i},z_{i})$ of the apexes of a solution to the tripod problem form a 2-comparable sets of triples, where two triples are defined as being 2-comparable if there are either at least two coordinates where one triple is smaller than the other, or at least two coordinates where one triple is larger than the other. This condition ensures that the tripods defined from these triples do not have intersecting rays.[1]
Another equivalent two-dimensional version of the question asks how many cells of an $n\times n$ array of square cells (indexed from $1$ to $n$) can be filled in by the numbers from $1$ to $n$ in such a way that the non-empty cells of each row and each column of the array form strictly increasing sequences of numbers, and the positions holding each value $i$ form a monotonic chain within the array. An array with these properties is called a monotonic matrix. A collection of disjoint tripods with apexes $(x_{i},y_{i},z_{i})$ can be transformed into a monotonic matrix by placing the number $z_{i}$ in array cell $(x_{i},y_{i})$ and vice versa.[1]
The problem is also equivalent to finding as many triangles as possible among the vertices of a convex polygon, such that no two triangles that share a vertex have nested angles at that vertex. This triangle-counting problem was posed by Peter Braß[6] and its equivalence to tripod packing was observed by Aronov et al.[1]
Lower bounds
It is straightforward to find a solution to the tripod packing problem with $\Omega (n^{3/2})$ tripods.[1] For instance, for $k=\lfloor {\sqrt {n}}\rfloor $, the $\Omega (n^{3/2})$ triples
${\bigl \{}(ak+b+1,bk+c+1,ak+c+1){\big |}a,b,c\in [0,k-1]{\bigr \}}$
are 2-comparable.
After several earlier improvements to this naïve bound,[7][8][9] Gowers and Long found solutions to the tripod problem of cardinality $\Omega (n^{1.546})$.[5]
Upper bounds
From any solution to the tripod packing problem, one can derive a balanced tripartite graph whose vertices are three copies of the numbers from $0$ to $n-1$ (one for each of the three coordinates) with a triangle of edges connecting the three vertices corresponding to the coordinates of the apex of each tripod. There are no other triangles in these graphs (they are locally linear graphs) because any other triangle would lead to a violation of 2-comparability. Therefore, by the known upper bounds to the Ruzsa–Szemerédi problem (one version of which is to find the maximum density of edges in a balanced tripartite locally linear graph), the maximum number of disjoint tripods that can be packed in an $n\times n\times n$ grid is $o(n^{2})$, and more precisely $n^{2}/\exp \Omega (\log ^{*}n)$.[1][5][9][10] Although Tiskin writes that "tighter analysis of the parameters" can produce a bound that is less than quadratic by a polylogarithmic factor,[9] he does not supply details and his proof that the number is $o(n^{2})$ uses only the same techniques that are known for the Ruzsa–Szemerédi problem, so this stronger claim appears to be a mistake.[1]
An argument of Dean Hickerson shows that, because tripods cannot pack space with constant density, the same is true for analogous problems in higher dimensions.[4]
Small instances
For small instances of the tripod problem, the exact solution is known. The numbers of tripods that can be packed into an $n\times n\times n$ cube, for $n\leq 11$, are:[9][11][12][13]
1, 2, 5, 8, 11, 14, 19, 23, 28, 32, 38, ...
For instance, the figure shows the 11 tripods that can be packed into a $5\times 5\times 5$ cube.
The number of distinct monotonic matrices of order $n$, for $n=1,2,3,\dots $ is[14]
2, 19, 712, 87685, 31102080, 28757840751, ...
References
1. Aronov, Boris; Dujmović, Vida; Morin, Pat; Ooms, Aurélien; Schultz Xavier da Silveira, Luís Fernando (2019), "More Turán-type theorems for triangles in convex point sets", Electronic Journal of Combinatorics, 26 (1): P1.8
2. Stein, S. K. (1967), "Factoring by subsets", Pacific Journal of Mathematics, 22: 523–541, doi:10.2140/pjm.1967.22.523, MR 0219435
3. Golomb, S. W. (1969), "A general formulation of error metrics", IEEE Transactions on Information Theory, IT-15: 425–426, doi:10.1109/tit.1969.1054308, MR 0243902
4. Stein, Sherman K.; Szabó, Sándor (1994), Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Washington, DC: Mathematical Association of America, p. 97, ISBN 0-88385-028-1, MR 1311249
5. Gowers, W. T.; Long, J. (January 2021), "The length of an $s$-increasing sequence of $r$-tuples", Combinatorics, Probability and Computing: 1–36, arXiv:1609.08688, doi:10.1017/s0963548320000371
6. Braß, Peter (2004), "Turán-type extremal problems for convex geometric hypergraphs", in Pach, János (ed.), Towards a theory of geometric graphs, Contemporary Mathematics, vol. 342, Providence, Rhode Island: American Mathematical Society, pp. 25–33, doi:10.1090/conm/342/06128, MR 2065250
7. Hamaker, William; Stein, Sherman (1984), "Combinatorial packing of $\mathbf {R} ^{3}$ by certain error spheres", IEEE Transactions on Information Theory, 30 (2, part 2): 364–368, doi:10.1109/TIT.1984.1056868, MR 0754867
8. Stein, Sherman K. (March 1995), "Packing tripods", Mathematical entertainments, The Mathematical Intelligencer, 17 (2): 37–39, doi:10.1007/bf03024896. Reprinted in Gale, David (1998), Tracking the Automatic ANT, Springer-Verlag, pp. 131–136, doi:10.1007/978-1-4612-2192-0, ISBN 0-387-98272-8, MR 1661863
9. Tiskin, Alexander (2007), "Packing tripods: narrowing the density gap", Discrete Mathematics, 307 (16): 1973–1981, doi:10.1016/j.disc.2004.12.028, MR 2326159
10. Loh, Po-Shen (2015), Directed paths: from Ramsey to Ruzsa and Szemerédi, arXiv:1505.07312
11. Szabó, Sándor (2013), "Monotonic matrices and clique search in graphs", Ann. Univ. Sci. Budapest Sect. Comput., 41: 307–322, doi:10.1080/00455091.2001.10717569, MR 3129145
12. Östergård, Patric R. J.; Pöllänen, Antti (2019), "New results on tripod packings" (PDF), Discrete & Computational Geometry, 61 (2): 271–284, doi:10.1007/s00454-018-0012-2, MR 3903789
13. Sloane, N. J. A. (ed.), "Sequence A070214", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
14. Sloane, N. J. A. (ed.), "Sequence A086976", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
| Wikipedia |
Proprism
In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for product prism. The dimension of the space of a proprism equals the sum of the dimensions of all its product elements. Proprisms are often seen as k-face elements of uniform polytopes.[1]
Properties
The number of vertices in a proprism is equal to the product of the number of vertices in all the polytopes in the product.
The minimum symmetry order of a proprism is the product of the symmetry orders of all the polytopes. A higher symmetry order is possible if polytopes in the product are identical.
A proprism is convex if all its product polytopes are convex.
f-vectors
An f-vector is a number of k-face elements in a polytope from k=0 (points) to k=n-1 (facets). An extended f-vector can also include k=-1 (nullitope), or k=n (body). Prism products include the body element. (The dual to prism products includes the nullitope, while pyramid products include both.)
The f-vector of prism product, A×B, can be computed as (fA,1)*(fB,1), like polynomial multiplication polynomial coefficients.
For example for product of a triangle, f=(3,3), and dion, f=(2) makes a triangular prism with 6 vertices, 9 edges, and 5 faces:
fA(x) = (3,3,1) = 3 + 3x + x2 (triangle)
fB(x) = (2,1) = 2 + x (dion)
fA∨B(x) = fA(x) * fB(x)
= (3 + 3x + x2) * (2 + x)
= 6 + 9x + 5x2 + x3
= (6,9,5,1)
Hypercube f-vectors can be computed as Cartesian products of n dions, { }n. Each { } has f=(2), extended to f=(2,1).
For example, an 8-cube will have extended f-vector power product: f=(2,1)8 = (4,4,1)4 = (16,32,24,8,1)2 = (256,1024,1792,1792,1120,448,112,16,1). If equal lengths, this doubling represents { }8, a square tetra-prism {4}4, a tesseract duo-prism {4,3,3}2, and regular 8-cube {4,3,3,3,3,3,3}.
Double products or duoprisms
Further information: Duoprism
In geometry of 4 dimensions or higher, duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope is an (a+b)-polytope, where a and b are 2-polytopes (polygon) or higher.
Most commonly this refers to the product of two polygons in 4-dimensions. In the context of a product of polygons, Henry P. Manning's 1910 work explaining the fourth dimension called these double prisms.[2]
The Cartesian product of two polygons is the set of points:
$P_{1}\times P_{2}=\{(x,y,u,v)|(x,y)\in P_{1},(u,v)\in P_{2}\}$
where P1 and P2 are the sets of the points contained in the respective polygons.
The smallest is a 3-3 duoprism, made as the product of 2 triangles. If the triangles are regular it can be written as a product of Schläfli symbols, {3} × {3}, and is composed of 9 vertices.
The tesseract, can be constructed as the duoprism {4} × {4}, the product of two equal-size orthogonal squares, composed of 16 vertices. The 5-cube can be constructed as a duoprism {4} × {4,3}, the product of a square and cube, while the 6-cube can be constructed as the product of two cubes, {4,3} × {4,3}.
Triple products
In geometry of 6 dimensions or higher, a triple product is a polytope resulting from the Cartesian product of three polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope, and a c-polytope is an (a + b + c)-polytope, where a, b and c are 2-polytopes (polygon) or higher.
The lowest-dimensional forms are 6-polytopes being the Cartesian product of three polygons. The smallest can be written as {3} × {3} × {3} in Schläfli symbols if they are regular, and contains 27 vertices. This is the product of three equilateral triangles and is a uniform polytope. The f-vectors can be computed by (3,3,1)3 = (27,81,108,81,36,9,1).
The 6-cube, can be constructed as a triple product {4} × {4} × {4}. The f-vectors can be computed by (4,4,1)3 = (64,192,240,160,60,12,1).
References
1. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, p. 391 "proprism")
2. The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms (double prisms) and duocylinders (double cylinders). Googlebook
| Wikipedia |
1 32 polytope
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.
321
231
132
Rectified 321
birectified 321
Rectified 231
Rectified 132
Orthogonal projections in E7 Coxeter plane
Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.
The rectified 132 is constructed by points at the mid-edges of the 132.
These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
1_32 polytope
132
TypeUniform 7-polytope
Family1k2 polytope
Schläfli symbol{3,33,2}
Coxeter symbol132
Coxeter diagram
6-faces182:
56 122
126 131
5-faces4284:
756 121
1512 121
2016 {34}
4-faces23688:
4032 {33}
7560 111
12096 {33}
Cells50400:
20160 {32}
30240 {32}
Faces40320 {3}
Edges10080
Vertices576
Vertex figuret2{35}
Petrie polygonOctadecagon
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex
This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E7* lattice.[1]
Alternate names
• Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.[2]
• Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
• Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)[3]
Images
Coxeter plane projections
E7 E6 / F4 B7 / A6
[18]
[12]
[7x2]
A5 D7 / B6 D6 / B5
[6]
[12/2]
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
[8]
[6]
[4]
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram,
Removing the node on the end of the 2-length branch leaves the 6-demicube, 131,
Removing the node on the end of the 3-length branch leaves the 122,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 032,
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[4]
E7k-facefkf0f1f2f3f4f5f6k-figuresnotes
A6( ) f0 5763521014021035105105214221772r{3,3,3,3,3}E7/A6 = 72*8!/7! = 576
A3A2A1{ } f1 21008012121841212612343{3,3}x{3}E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A2A1{3} f2 33403202316336132{ }∨{3}E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A3A2{3,3} f3 46420160*13033031{3}∨( )E7/A3A2 = 72*8!/4!/3! = 20160
A3A1A1 464*3024002214122Phyllic disphenoidE7/A3A1A1 = 72*8!/4!/2/2 = 30240
A4A2{3,3,3} f4 51010504032**30030{3}E7/A4A2 = 72*8!/5!/3! = 4032
D4A1{3,3,4} 8243288*7560*12021{ }∨( )E7/D4A1 = 72*8!/8/4!/2 = 7560
A4A1{3,3,3} 5101005**1209602112E7/A4A1 = 72*8!/5!/2 = 12096
D5A1h{4,3,3,3} f5 1680160804016100756**20{ }E7/D5A1 = 72*8!/16/5!/2 = 756
D5 1680160408001016*1512*11E7/D5 = 72*8!/16/5! = 1512
A5A1{3,3,3,3,3} 61520015006**201602E7/A5A1 = 72*8!/6!/2 = 2016
E6{3,32,2} f6 727202160108010802162702162727056*( )E7/E6 = 72*8!/72/6! = 56
D6h{4,3,3,3,3} 3224064016048006019201232*126E7/D6 = 72*8!/32/6! = 126
Related polytopes and honeycombs
The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
13k dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\tilde {E}}_{7}$=E7+ ${\bar {T}}_{8}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name 13,-1 130 131 132 133 134
1k2 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde {E}}_{8}$ = E8+ E10 = ${\bar {T}}_{8}$ = E8++
Coxeter
diagram
Symmetry
(order)
[3−1,2,1] [30,2,1] [31,2,1] [[32,2,1]] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 103,680 2,903,040 696,729,600 ∞
Graph - -
Name 1−1,2 102 112 122 132 142 152 162
Rectified 1_32 polytope
Rectified 132
TypeUniform 7-polytope
Schläfli symbolt1{3,33,2}
Coxeter symbol0321
Coxeter-Dynkin diagram
6-faces758
5-faces12348
4-faces72072
Cells191520
Faces241920
Edges120960
Vertices10080
Vertex figure{3,3}×{3}×{}
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex
The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.
Alternate names
• Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)[5]
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).
Removing the node on the end of the 3-length branch leaves the rectified 122 polytope,
Removing the node on the end of the 2-length branch leaves the demihexeract, 131,
Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[6]
E7k-facefkf0f1f2f3f4f5f6k-figuresnotes
A3A2A1( ) f0 10080242412368123618244121824126681263423{3,3}x{3}x{ }E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A1A1{ } f1 21209602131263313663133621312( )v{3}v{ }E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A2A201 f2 3380640**1130013330033310311{3}v( )v( )E7/A2A2 = 72*8!/3!/3! = 80640
A2A2A1 33*40320*0203010603030601302{3}v{ }E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A2A1A1 33**1209600021201242112421212{ }v{ }v( )E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A3A202 f3 4640020160****13000033000310{3}v( )E7/A3A2 = 72*8!/4!/3! = 20160
011 612440*20160***10300030300301
A3A1 612404**60480**01120012210211SphenoidE7/A3A1 = 72*8!/4!/2 = 60480
A3A1A1 612044***30240*00202010401202{ }v{ }E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A3A102 46004****6048000021101221112SphenoidE7/A3A1 = 72*8!/4!/2 = 60480
A4A2021 f4 103020100550004032*****30000300{3}E7/A4A2 = 72*8!/5!/3! = 4032
A4A1 10302001050500*12096****12000210{ }v()E7/A4A1 = 72*8!/5!/2 = 12096
D4A10111 249632323208880**7560***10200201E7/D4A1 = 72*8!/8/4!/2 = 7560
A4021 10301002000505***24192**01110111( )v( )v( )E7/A4 = 72*8!/5! = 34192
A4A1 10300102000055****12096*00201102{ }v()E7/A4A1 = 72*8!/5!/2 = 12096
03 510001000005*****1209600021012
D5A10211 f5 80480320160160808080400161610000756****200{ }E7/D5A1 = 72*8!/16/5!/2 = 756
A5022 20906006015030015060600*4032***110E7/A5 = 72*8!/6! = 4032
D50211 80480160160320040808080001016160**1512**101E7/D5 = 72*8!/16/5! = 1512
A5031 1560200600015030000606***4032*011E7/A5 = 72*8!/6! = 4032
A5A1 1560020600001530000066****2016002E7/A5A1 = 72*8!/6!/2 = 2016
E60221 f6 72064804320216043201080108021601080108021643227043221602772270056**( )E7/E6 = 72*8!/72/6! = 56
A6032 352101400210350105010502104202107070*576*E7/A6 = 72*8!/7! = 576
D60311 240192064064019200160480480960006019219219200123232**126E7/D6 = 72*8!/32/6! = 126
Images
Coxeter plane projections
E7 E6 / F4 B7 / A6
[18]
[12]
[14]
A5 D7 / B6 D6 / B5
[6]
[12/2]
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
[8]
[6]
[4]
See also
• List of E7 polytopes
Notes
1. The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin
2. Elte, 1912
3. Klitzing, (o3o3o3x *c3o3o3o - lin)
4. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
5. Klitzing, (o3o3x3o *c3o3o3o - rolin)
6. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
References
• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
• H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.[1]
421
142
241
Rectified 421
Rectified 142
Rectified 241
Birectified 421
Trirectified 421
Orthogonal projections in E6 Coxeter plane
Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .
The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421.
These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: .
421 polytope
421
TypeUniform 8-polytope
Familyk21 polytope
Schläfli symbol{3,3,3,3,32,1}
Coxeter symbol421
Coxeter diagrams
=
7-faces19440 total:
2160 411
17280 {36}
6-faces207360:
138240 {35}
69120 {35}
5-faces483840 {34}
4-faces483840 {33}
Cells241920 {3,3}
Faces60480 {3}
Edges6720
Vertices240
Vertex figure321 polytope
Petrie polygon30-gon
Coxeter groupE8, [34,2,1], order 696729600
Propertiesconvex
The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 321 polytope. As its vertices represent the root vectors of the simple Lie group E8, this polytope is sometimes referred to as the E8 root polytope.
The vertices of this polytope can also be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop.
For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
Alternate names
• This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure.[1] It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.
• E. L. Elte named it V240 (for its 240 vertices) in his 1912 listing of semiregular polytopes.[2]
• H.S.M. Coxeter called it 421 because its Coxeter-Dynkin diagram has three branches of length 4, 2, and 1, with a single node on the terminal node of the 4 branch.
• Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton (Acronym Fy) - 2160-17280 facetted polyzetton (Jonathan Bowers)[3]
Coordinates
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The 240 vertices of the 421 polytope can be constructed in two sets: 112 (22×8C2) with coordinates obtained from $(\pm 2,\pm 2,0,0,0,0,0,0)\,$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from $(\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,$ by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be a multiple of 4).
Each vertex has 56 nearest neighbors; for example, the nearest neighbors of the vertex $(1,1,1,1,1,1,1,1)$ are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of $(2,2,0,0,0,0,0,0)\,$ and the 28 obtained by permuting the coordinates of $(1,1,1,1,1,1,-1,-1)$. These 56 points are the vertices of a 321 polytope in 7 dimensions.
Each vertex has 126 second nearest neighbors: for example, the nearest neighbors of the vertex $(1,1,1,1,1,1,1,1)$ are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of $(2,-2,0,0,0,0,0,0)\,$ and the 70 obtained by permuting the coordinates of $(1,1,1,1,-1,-1,-1,-1)$. These 126 points are the vertices of a 231 polytope in 7 dimensions.
Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, and one antipodal vertex, for a total of $1+56+126+56+1=240$ vertices.
Another construction is by taking signed combination of 14 codewords of 8-bit Extended Hamming code(8,4) that give 14 × 24 = 224 vertices and adding trivial signed axis $(\pm 2,0,0,0,0,0,0,0)$ for last 16 vertices. In this case, vertices are distance of ${\sqrt {4}}$ from origin rather than ${\sqrt {8}}$.
Hamming 8-bit Code
0 0 0 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 ⇒ ± ± ± ± 0 0 0 0
2 1 1 0 0 1 1 0 0 ⇒ ± ± 0 0 ± ± 0 0
3 0 0 1 1 1 1 0 0 ⇒ 0 0 ± ± ± ± 0 0
4 1 0 1 0 1 0 1 0 ⇒ ± 0 ± 0 ± 0 ± 0 ±2 0 0 0 0 0 0 0
5 0 1 0 1 1 0 1 0 ⇒ 0 ± 0 ± ± 0 ± 0 0 ±2 0 0 0 0 0 0
6 0 1 1 0 0 1 1 0 ⇒ 0 ± ± 0 0 ± ± 0 0 0 ±2 0 0 0 0 0
7 1 0 0 1 0 1 1 0 ⇒ ± 0 0 ± 0 ± ± 0 0 0 0 ±2 0 0 0 0
8 0 1 1 0 1 0 0 1 ⇒ 0 ± ± 0 ± 0 0 ± 0 0 0 0 ±2 0 0 0
9 1 0 0 1 1 0 0 1 ⇒ ± 0 0 ± ± 0 0 ± 0 0 0 0 0 ±2 0 0
A 1 0 1 0 0 1 0 1 ⇒ ± 0 ± 0 0 ± 0 ± 0 0 0 0 0 0 ±2 0
B 0 1 0 1 0 1 0 1 ⇒ 0 ± 0 ± 0 ± 0 ± 0 0 0 0 0 0 0 ±2
C 1 1 0 0 0 0 1 1 ⇒ ± ± 0 0 0 0 ± ±
D 0 0 1 1 0 0 1 1 ⇒ 0 0 ± ± 0 0 ± ±
E 0 0 0 0 1 1 1 1 ⇒ 0 0 0 0 ± ± ± ±
F 1 1 1 1 1 1 1 1
( 224 vertices + 16 vertices )
Another decomposition gives the 240 points in 9-dimensions as an expanded 8-simplex, and two opposite birectified 8-simplexes, and .
(3,-3,0,0,0,0,0,0,0) : 72 vertices
(-2,-2,-2,1,1,1,1,1,1) : 84 vertices
(2,2,2,-1,-1,-1,-1,-1,-1) : 84 vertices
This arises similarly to the relation of the A8 lattice and E8 lattice, sharing 8 mirrors of A8: .
A7 Coxeter plane projections
Name 421
expanded 8-simplex
birectified 8-simplex
birectified 8-simplex
Vertices240728484
Image
Tessellations
This polytope is the vertex figure for a uniform tessellation of 8-dimensional space, represented by symbol 521 and Coxeter-Dynkin diagram:
Construction and faces
The facet information of this polytope can be extracted from its Coxeter-Dynkin diagram:
Removing the node on the short branch leaves the 7-simplex:
Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form (411):
Every 7-simplex facet touches only 7-orthoplex facets, while alternate facets of an orthoplex facet touch either a simplex or another orthoplex. There are 17,280 simplex facets and 2160 orthoplex facets.
Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, the 421 polytope has 120,960 (7×17,280) 6-simplex faces that are facets of 7-simplexes. Since every 7-orthoplex has 128 (27) 6-simplex facets, half of which are not incident to 7-simplexes, the 421 polytope has 138,240 (26×2160) 6-simplex faces that are not facets of 7-simplexes. The 421 polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope. The total number of 6-simplex faces is 259200 (120,960+138,240).
The vertex figure of a single-ring polytope is obtained by removing the ringed node and ringing its neighbor(s). This makes the 321 polytope.
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[4]
Configuration matrix
E8k-facefkf0f1f2f3f4f5f6f7k-figurenotes
E7( ) f0 2405675640321008012096403220165761263_21 polytopeE8/E7 = 192×10!/(72×8!) = 240
A1E6{ } f1 2672027216720108043221672272_21 polytopeE8/A1E6 = 192×10!/(2×72×6!) = 6720
A2D5{3} f2 33604801680160804016105-demicubeE8/A2D5 = 192×10!/(6×24×5!) = 60480
A3A4{3,3} f3 4642419201030201055Rectified 5-cellE8/A3A4 = 192×10!/(4!×5!) = 241920
A4A2A1{3,3,3} f4 51010548384066323Triangular prismE8/A4A2A1 = 192×10!/(5!×3!×2) = 483840
A5A1{3,3,3,3} f5 615201564838402112Isosceles triangleE8/A5A1 = 192×10!/(6!×2) = 483840
A6{3,3,3,3,3} f6 7213535217138240*11{ }E8/A6 = 192×(10!×7!) = 138240
A6A1 7213535217*6912002E8/A6A1 = 192×10!/(7!×2) = 69120
A7{3,3,3,3,3,3} f7 828567056288017280*( )E8/A7 = 192×10!/8! = 17280
D7{3,3,3,3,3,4} 14842805606724486464*2160E8/D7 = 192×10!/(26×7!) = 2160
Projections
The 421 graph created as string art.
E8 Coxeter plane projection
3D
Mathematical representation of the physical Zome model isomorphic (?) to E8. This is constructed from VisibLie_E8 pictured with all 3360 edges of length √2(√5-1) from two concentric 600-cells (at the golden ratio) with orthogonal projections to perspective 3-space
The actual split real even E8 421 polytope projected into perspective 3-space pictured with all 6720 edges of length √2 [5]
E8 rotated to H4+H4φ, projected to 3D, converted to STL, and printed in nylon plastic. Projection basis used:
x = {1, φ, 0, -1, φ, 0,0,0}
y = {φ, 0, 1, φ, 0, -1,0,0}
z = {0, 1, φ, 0, -1, φ,0,0}
2D
These graphs represent orthographic projections in the E8, E7, E6, and B8, D8, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
Orthogonal projections
E8 / H4
[30]
[20] [24]
(Colors: 1)
(Colors: 1)
(Colors: 1)
E7
[18]
E6 / F4
[12]
[6]
(Colors: 1,3,6)
(Colors: 1,8,24)
(Colors: 1,2,3)
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
(Colors: 1,12,32,60)
(Colors: 1,27,72)
(Colors: 1,8,24)
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
(Colors: 1,5,10,20)
(Colors: 1,3,9,12)
(Colors: 1,2,3)
B8
[16/2]
A5
[6]
A7
[8]
(Colors: 1)
(Colors: 3,8,24,30)
(Colors: 1,2,4,8)
k21 family
The 421 polytope is last in a family called the k21 polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).
Geometric folding
The 421 is related to the 600-cell by a geometric folding of the Coxeter-Dynkin diagrams. This can be seen in the E8/H4 Coxeter plane projections. The 240 vertices of the 421 polytope are projected into 4-space as two copies of the 120 vertices of the 600-cell, one copy smaller (scaled by the golden ratio) than the other with the same orientation. Seen as a 2D orthographic projection in the E8/H4 Coxeter plane, the 120 vertices of the 600-cell are projected in the same four rings as seen in the 421. The other 4 rings of the 421 graph also match a smaller copy of the four rings of the 600-cell.
E8/H4 Coxeter plane foldings
E8 H4
421
600-cell
[20] symmetry planes
421
600-cell
Related polytopes
In 4-dimensional complex geometry, the regular complex polytope 3{3}3{3}3{3}3, and Coxeter diagram exists with the same vertex arrangement as the 421 polytope. It is self-dual. Coxeter called it the Witting polytope, after Alexander Witting. Coxeter expresses its Shephard group symmetry by 3[3]3[3]3[3]3.[7]
The 421 is sixth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.
k21 figures in n dimensions
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde {E}}_{8}$ = E8+ E10 = ${\bar {T}}_{8}$ = E8++
Coxeter
diagram
Symmetry [3−1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 51,840 2,903,040 696,729,600 ∞
Graph - -
Name −121 021 121 221 321 421 521 621
Rectified 4_21 polytope
Rectified 421
TypeUniform 8-polytope
Schläfli symbolt1{3,3,3,3,32,1}
Coxeter symbolt1(421)
Coxeter diagram
7-faces19680 total:
240 321
17280 t1{36}
2160 t1{35,4}
6-faces375840
5-faces1935360
4-faces3386880
Cells2661120
Faces1028160
Edges181440
Vertices6720
Vertex figure221 prism
Coxeter groupE8, [34,2,1]
Propertiesconvex
The rectified 421 can be seen as a rectification of the 421 polytope, creating new vertices on the center of edges of the 421.
Alternative names
• Rectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for rectified 2160-17280 polyzetton (Acronym riffy) (Jonathan Bowers)[8]
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a rectification of the 421. Vertices are positioned at the midpoint of all the edges of 421, and new edges connecting them.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the rectified 7-simplex:
Removing the node on the end of the 2-length branch leaves the rectified 7-orthoplex in its alternated form:
Removing the node on the end of the 4-length branch leaves the 321:
The vertex figure is determined by removing the ringed node and adding a ring to the neighboring node. This makes a 221 prism.
Coordinates
The Cartesian coordinates of the 6720 vertices of the rectified 421 is given by all permutations of coordinates from three other uniform polytope:
• hexic 8-cube - odd negatives: ½(±1,±1,±1,±1,±1,±1,±3,±3) - 3584 vertices[9]
• birectified 8-cube - (0,0,±1,±1,±1,±1,±1,±1) - 1792 vertices[10]
• cantellated 8-orthoplex - (0,0,0,0,0,0,±1,±1,±2) - 1344 vertices[11]
D8 Coxeter plane projections
Name Rectified 421
birectified 8-cube
=
hexic 8-cube
=
cantellated 8-orthoplex
=
Vertices6720179235841344
Image
2D
These graphs represent orthographic projections in the E8, E7, E6, and B8, D8, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
Orthogonal projections
E8 / H4
[30]
[20] [24]
E7
[18]
E6 / F4
[12]
[6]
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
B8
[16/2]
A5
[6]
A7
[8]
Birectified 4_21 polytope
Birectified 421 polytope
TypeUniform 8-polytope
Schläfli symbolt2{3,3,3,3,32,1}
Coxeter symbolt2(421)
Coxeter diagram
7-faces19680 total:
17280 t2{36}
2160 t2{35,4}
240 t1(321)
6-faces382560
5-faces2600640
4-faces7741440
Cells9918720
Faces5806080
Edges1451520
Vertices60480
Vertex figure5-demicube-triangular duoprism
Coxeter groupE8, [34,2,1]
Propertiesconvex
The birectified 421 can be seen as a second rectification of the uniform 421 polytope. Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 421.
Alternative names
• Birectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for birectified 2160-17280 polyzetton (acronym borfy) (Jonathan Bowers)[12]
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a birectification of the 421. Vertices are positioned at the center of all the triangle faces of 421.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the birectified 7-simplex. There are 17280 of these facets.
Removing the node on the end of the 2-length branch leaves the birectified 7-orthoplex in its alternated form. There are 2160 of these facets.
Removing the node on the end of the 4-length branch leaves the rectified 321. There are 240 of these facets.
The vertex figure is determined by removing the ringed node and adding rings to the neighboring nodes. This makes a 5-demicube-triangular duoprism.
2D
These graphs represent orthographic projections in the E8, E7, E6, and B8, D8, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. Edges are not drawn. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green, etc.
Orthogonal projections
E8 / H4
[30]
[20] [24]
E7
[18]
E6 / F4
[12]
[6]
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
B8
[16/2]
A5
[6]
A7
[8]
Trirectified 4_21 polytope
Trirectified 421 polytope
TypeUniform 8-polytope
Schläfli symbolt3{3,3,3,3,32,1}
Coxeter symbolt3(421)
Coxeter diagram
7-faces19680
6-faces382560
5-faces2661120
4-faces9313920
Cells16934400
Faces14515200
Edges4838400
Vertices241920
Vertex figuretetrahedron-rectified 5-cell duoprism
Coxeter groupE8, [34,2,1]
Propertiesconvex
Alternative names
• Trirectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for trirectified 2160-17280 polyzetton (acronym torfy) (Jonathan Bowers)[13]
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a birectification of the 421. Vertices are positioned at the center of all the triangle faces of 421.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the trirectified 7-simplex:
Removing the node on the end of the 2-length branch leaves the trirectified 7-orthoplex in its alternated form:
Removing the node on the end of the 4-length branch leaves the birectified 321:
The vertex figure is determined by removing the ringed node and ring the neighbor nodes. This makes a tetrahedron-rectified 5-cell duoprism.
2D
These graphs represent orthographic projections in the E7, E6, B8, D8, D7, D6, D5, D4, D3, A7, and A5 Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
(E8 and B8 were too large to display)
Orthogonal projections
E7
[18]
E6 / F4
[12]
D4 - E6
[6]
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
A5
[6]
A7
[8]
See also
• List of E8 polytopes
Notes
1. Gosset, 1900
2. Elte, 1912
3. Klitzing, (o3o3o3o *c3o3o3o3x - fy)
4. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
5. e8Flyer.nb
6. David Richter: Gosset's Figure in 8 Dimensions, A Zome Model
7. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
8. Klitzing, (o3o3o3o *c3o3o3x3o - riffy)
9. "Sotho".
10. "Bro".
11. "Srek".
12. Klitzing, (o3o3o3o *c3o3x3o3o - borfy)
13. Klitzing, (o3o3o3o *c3x3o3o3o - torfy)
References
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
• Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, (1974).
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p347 (figure 3.8c) by Peter McMullen: (30-gonal node-edge graph of 421)
• Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3o3o3o *c3o3o3o3x - fy, o3o3o3o *c3o3o3x3o - riffy, o3o3o3o *c3o3x3o3o - borfy, o3o3o3o *c3x3o3o3o - torfy
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
6-cubic honeycomb
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.
6-cubic honeycomb
(no image)
TypeRegular 6-honeycomb
Uniform 6-honeycomb
FamilyHypercube honeycomb
Schläfli symbol{4,34,4}
{4,33,31,1}
Coxeter-Dynkin diagrams
6-face type{4,34}
5-face type{4,33}
4-face type{4,3,3}
Cell type{4,3}
Face type{4}
Face figure{4,3}
(octahedron)
Edge figure8 {4,3,3}
(16-cell)
Vertex figure64 {4,34}
(6-orthoplex)
Coxeter group${\tilde {C}}_{6}$, [4,34,4]
${\tilde {B}}_{6}$, [4,33,31,1]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,34,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,33,31,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(6).
Related honeycombs
The [4,34,4], , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.
The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.
Trirectified 6-cubic honeycomb
A trirectified 6-cubic honeycomb, , contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{6}$×2, [[4,34,4]] symmetry, alternately colored from ${\tilde {C}}_{6}$, [4,34,4] symmetry, three colors from ${\tilde {B}}_{6}$, [4,33,31,1] symmetry, and 4 colors from ${\tilde {D}}_{6}$, [31,1,3,3,31,1] symmetry.
See also
• List of regular polytopes
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
Rectified 7-orthoplexes
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
7-orthoplex
Rectified 7-orthoplex
Birectified 7-orthoplex
Trirectified 7-orthoplex
Birectified 7-cube
Rectified 7-cube
7-cube
Orthogonal projections in B7 Coxeter plane
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.
Rectified 7-orthoplex
Rectified 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolr{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces142
5-faces1344
4-faces3360
Cells3920
Faces2520
Edges840
Vertices84
Vertex figure5-orthoplex prism
Coxeter groupsB7, [3,3,3,3,3,4]
D7, [34,1,1]
Propertiesconvex
The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.
or
Alternate names
• rectified heptacross
• rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon[1]
Images
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Construction
There are two Coxeter groups associated with the rectified heptacross, one with the C7 or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or [34,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length ${\sqrt {2}}\ $ are all permutations of:
(±1,±1,0,0,0,0,0)
Root vectors
Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.
Birectified 7-orthoplex
Birectified 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol2r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces142
5-faces1428
4-faces6048
Cells10640
Faces8960
Edges3360
Vertices280
Vertex figure{3}×{3,3,4}
Coxeter groupsB7, [3,3,3,3,3,4]
D7, [34,1,1]
Propertiesconvex
Alternate names
• Birectified heptacross
• Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexon[2]
Images
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length ${\sqrt {2}}\ $ are all permutations of:
(±1,±1,±1,0,0,0,0)
Trirectified 7-orthoplex
A trirectified 7-orthoplex is the same as a trirectified 7-cube.
Notes
1. Klitzing, (o3o3x3o3o3o4o - rez)
2. Klitzing, (o3o3x3o3o3o4o - barz)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz
External links
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Rectified 8-orthoplexes
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
8-orthoplex
Rectified 8-orthoplex
Birectified 8-orthoplex
Trirectified 8-orthoplex
Trirectified 8-cube
Birectified 8-cube
Rectified 8-cube
8-cube
Orthogonal projections in A8 Coxeter plane
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.
Rectified 8-orthoplex
Rectified 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces272
6-faces3072
5-faces8960
4-faces12544
Cells10080
Faces4928
Edges1344
Vertices112
Vertex figure6-orthoplex prism
Petrie polygonhexakaidecagon
Coxeter groupsC8, [4,36]
D8, [35,1,1]
Propertiesconvex
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.
Related polytopes
The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
or
Alternate names
• rectified octacross
• rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)[1]
Construction
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
(±1,±1,0,0,0,0,0,0)
Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]
Birectified 8-orthoplex
Birectified 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces272
6-faces3184
5-faces16128
4-faces34048
Cells36960
Faces22400
Edges6720
Vertices448
Vertex figure{3,3,3,4}x{3}
Coxeter groupsC8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex
Alternate names
• birectified octacross
• birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
(±1,±1,±1,0,0,0,0,0)
Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]
Trirectified 8-orthoplex
Trirectified 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces16+256
6-faces1024 + 2048 + 112
5-faces1792 + 7168 + 7168 + 448
4-faces1792 + 10752 + 21504 + 14336
Cells8960 + 126880 + 35840
Faces17920 + 35840
Edges17920
Vertices1120
Vertex figure{3,3,4}x{3,3}
Coxeter groupsC8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.
Alternate names
• trirectified octacross
• trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)[3]
Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
(±1,±1,±1,±1,0,0,0,0)
Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]
Notes
1. Klitzing, (o3x3o3o3o3o3o4o - rek)
2. Klitzing, (o3o3x3o3o3o3o4o - bark)
3. Klitzing, (o3o3o3x3o3o3o4o - tark)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark
External links
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Trisected perimeter point
In geometry, given a triangle ABC, there exist unique points A´, B´, and C´ on the sides BC, CA, AB respectively, such that:[1]
• A´, B´, and C´ partition the perimeter of the triangle into three equal-length pieces. That is,
C´B + BA´ = B´A + AC´ = A´C + CB´.
• The three lines AA´, BB´, and CC´ meet in a point, the trisected perimeter point.
This is point X369 in Clark Kimberling's Encyclopedia of Triangle Centers.[2] Uniqueness and a formula for the trilinear coordinates of X369 were shown by Peter Yff late in the twentieth century. The formula involves the unique real root of a cubic equation.[2]
See also
• Bisected perimeter point
References
1. Weisstein, Eric W. "Trisected Perimeter Point". MathWorld.
2. Kimberling, C. Encyclopedia of Triangle Centers. X(369) = 1st TRISECTED PERIMETER POINT.
| Wikipedia |
Angle trisection
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle (that is, to construct an angle of 30 degrees).
It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries.
Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.[1]
Background and problem statement
Using only an unmarked straightedge and a compass, Greek mathematicians found means to divide a line into an arbitrary set of equal segments, to draw parallel lines, to bisect angles, to construct many polygons, and to construct squares of equal or twice the area of a given polygon.
Three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads:
Construct an angle equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools:
1. an unmarked straightedge, and
2. a compass.
Proof of impossibility
Pierre Wantzel published a proof of the impossibility of classically trisecting an arbitrary angle in 1837.[2] Wantzel's proof, restated in modern terminology, uses the concept of field extensions, a topic now typically combined with Galois theory. However, Wantzel published these results earlier than Évariste Galois (whose work, written in 1830, was published only in 1846) and did not use the concepts introduced by Galois.[3]
The problem of constructing an angle of a given measure θ is equivalent to constructing two segments such that the ratio of their length is cos θ. From a solution to one of these two problems, one may pass to a solution of the other by a compass and straightedge construction. The triple-angle formula gives an expression relating the cosines of the original angle and its trisection: cos θ = 4 cos3 θ/3 − 3 cos θ/3.
It follows that, given a segment that is defined to have unit length, the problem of angle trisection is equivalent to constructing a segment whose length is the root of a cubic polynomial. This equivalence reduces the original geometric problem to a purely algebraic problem.
Every rational number is constructible. Every irrational number that is constructible in a single step from some given numbers is a root of a polynomial of degree 2 with coefficients in the field generated by these numbers. Therefore, any number that is constructible by a sequence of steps is a root of a minimal polynomial whose degree is a power of two. The angle π/3 radians (60 degrees, written 60°) is constructible. The argument below shows that it is impossible to construct a 20° angle. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected.
Denote the set of rational numbers by Q. If 60° could be trisected, the degree of a minimal polynomial of cos 20° over Q would be a power of two. Now let x = cos 20°. Note that cos 60° = cos π/3 = 1/2. Then by the triple-angle formula, cos π/3 = 4x3 − 3x and so 4x3 − 3x = 1/2. Thus 8x3 − 6x − 1 = 0. Define p(t) to be the polynomial p(t) = 8t3 − 6t − 1.
Since x = cos 20° is a root of p(t), the minimal polynomial for cos 20° is a factor of p(t). Because p(t) has degree 3, if it is reducible over by Q then it has a rational root. By the rational root theorem, this root must be ±1, ±1/2, ±1/4 or ±1/8, but none of these is a root. Therefore, p(t) is irreducible over by Q, and the minimal polynomial for cos 20° is of degree 3.
So an angle of measure 60° cannot be trisected.
Angles which can be trisected
However, some angles can be trisected. For example, for any constructible angle θ, an angle of measure 3θ can be trivially trisected by ignoring the given angle and directly constructing an angle of measure θ. There are angles that are not constructible but are trisectible (despite the one-third angle itself being non-constructible). For example, 3π/7 is such an angle: five angles of measure 3π/7 combine to make an angle of measure 15π/7, which is a full circle plus the desired π/7.
For a positive integer N, an angle of measure 2π/N is trisectible if and only if 3 does not divide N.[4][5] In contrast, 2π/N is constructible if and only if N is a power of 2 or the product of a power of 2 with the product of one or more distinct Fermat primes.
Algebraic characterization
Again, denote the set of rational numbers by Q.
Theorem: An angle of measure θ may be trisected if and only if q(t) = 4t3 − 3t − cos(θ) is reducible over the field extension Q(cos(θ)).
The proof is a relatively straightforward generalization of the proof given above that a 60° angle is not trisectible.[6]
Other numbers of parts
For any nonzero integer N, an angle of measure 2π⁄N radians can be divided into n equal parts with straightedge and compass if and only if n is either a power of 2 or is a power of 2 multiplied by the product of one or more distinct Fermat primes, none of which divides N. In the case of trisection (n = 3, which is a Fermat prime), this condition becomes the above-mentioned requirement that N not be divisible by 3.[5]
Other methods
The general problem of angle trisection is solvable by using additional tools, and thus going outside of the original Greek framework of compass and straightedge.
Many incorrect methods of trisecting the general angle have been proposed. Some of these methods provide reasonable approximations; others (some of which are mentioned below) involve tools not permitted in the classical problem. The mathematician Underwood Dudley has detailed some of these failed attempts in his book The Trisectors.[1]
Approximation by successive bisections
Trisection can be approximated by repetition of the compass and straightedge method for bisecting an angle. The geometric series 1/3 = 1/4 + 1/16 + 1/64 + 1/256 + ⋯ or 1/3 = 1/2 − 1/4 + 1/8 − 1/16 + ⋯ can be used as a basis for the bisections. An approximation to any degree of accuracy can be obtained in a finite number of steps.[7]
Using origami
Main article: Mathematics of origami § Trisecting an angle
Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the operations of paper folding, or origami. Huzita's axioms (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots).
Using a linkage
There are a number of simple linkages which can be used to make an instrument to trisect angles including Kempe's Trisector and Sylvester's Link Fan or Isoklinostat.[8]
With a right triangular ruler
In 1932, Ludwig Bieberbach published in Journal für die reine und angewandte Mathematik his work Zur Lehre von den kubischen Konstruktionen.[9] He states therein (free translation):
"As is known ... every cubic construction can be traced back to the trisection of the angle and to the multiplication of the cube, that is, the extraction of the third root. I need only to show how these two classical tasks can be solved by means of the right angle hook."
The construction begins with drawing a circle passing through the vertex P of the angle to be trisected, centered at A on an edge of this angle, and having B as its second intersection with the edge. A circle centered at P and of the same radius intersects the line supporting the edge in A and O.
Now the right triangular ruler is placed on the drawing in the following manner: one leg of its right angle passes through O; the vertex of its right angle is placed at a point S on the line PC in such a way that the second leg of the ruler is tangent at E to the circle centered at A. It follows that the original angle is trisected by the line PE, and the line PD perpendicular to SE and passing through P. This line can be drawn either by using again the right triangular ruler, or by using a traditional straightedge and compass construction. With a similar construction, one can improve the location of E, by using that it is the intersection of the line SE and its perpendicular passing through A.
Proof: One has to prove the angle equalities ${\widehat {EPD}}={\widehat {DPS}}$ and ${\widehat {BPE}}={\widehat {EPD}}.$ The three lines OS, PD, and AE are parallel. As the line segments OP and PA are equal, these three parallel lines delimit two equal segments on every other secant line, and in particular on their common perpendicular SE. Thus SD' = D'E, where D' is the intersection of the lines PD and SE. It follows that the right triangles PD'S and PD'E are congruent, and thus that ${\widehat {EPD}}={\widehat {DPS}},$ the first desired equality. On the other hand, the triangle PAE is isosceles, since all radiuses of a circle are equal; this implies that ${\widehat {APE}}={\widehat {AEP}}.$ One has also ${\widehat {AEP}}={\widehat {EPD}},$ since these two angles are alternate angles of a transversal to two parallel lines. This proves the second desired equality, and thus the correctness of the construction.
With an auxiliary curve
• Trisection using the Archimedean spiral
• Trisection using the Maclaurin trisectrix
There are certain curves called trisectrices which, if drawn on the plane using other methods, can be used to trisect arbitrary angles.[10] Examples include the trisectrix of Colin Maclaurin, given in Cartesian coordinates by the implicit equation
$2x(x^{2}+y^{2})=a(3x^{2}-y^{2}),$
and the Archimedean spiral. The spiral can, in fact, be used to divide an angle into any number of equal parts. Archimedes described how to trisect an angle using the Archimedean spiral in On Spirals around 225 BC.
With a marked ruler
Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to Archimedes, called a Neusis construction, i.e., that uses tools other than an un-marked straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees.
This requires three facts from geometry (at right):
1. Any full set of angles on a straight line add to 180°,
2. The sum of angles of any triangle is 180°, and,
3. Any two equal sides of an isosceles triangle will meet the third side at the same angle.
Let l be the horizontal line in the adjacent diagram. Angle a (left of point B) is the subject of trisection. First, a point A is drawn at an angle's ray, one unit apart from B. A circle of radius AB is drawn. Then, the markedness of the ruler comes into play: one mark of the ruler is placed at A and the other at B. While keeping the ruler (but not the mark) touching A, the ruler is slid and rotated until one mark is on the circle and the other is on the line l. The mark on the circle is labeled C and the mark on the line is labeled D. This ensures that CD = AB. A radius BC is drawn to make it obvious that line segments AB, BC, and CD all have equal length. Now, triangles ABC and BCD are isosceles, thus (by Fact 3 above) each has two equal angles.
Hypothesis: Given AD is a straight line, and AB, BC, and CD all have equal length,
Conclusion: angle b = a/3.
Proof:
1. From Fact 1) above, $e+c=180$°.
2. Looking at triangle BCD, from Fact 2) $e+2b=180$°.
3. From the last two equations, $c=2b$.
4. From Fact 2), $d+2c=180$°, thus $d=180$°$-2c$, so from last, $d=180$°$-4b$.
5. From Fact 1) above, $a+d+b=180$°, thus $a+(180$°$-4b)+b=180$°.
Clearing, a − 3b = 0, or a = 3b, and the theorem is proved.
Again, this construction stepped outside the framework of allowed constructions by using a marked straightedge.
With a string
Thomas Hutcheson published an article in the Mathematics Teacher[11] that used a string instead of a compass and straight edge. A string can be used as either a straight edge (by stretching it) or a compass (by fixing one point and identifying another), but can also wrap around a cylinder, the key to Hutcheson's solution.
Hutcheson constructed a cylinder from the angle to be trisected by drawing an arc across the angle, completing it as a circle, and constructing from that circle a cylinder on which a, say, equilateral triangle was inscribed (a 360-degree angle divided in three). This was then "mapped" onto the angle to be trisected, with a simple proof of similar triangles.
With a "tomahawk"
Main article: Tomahawk (geometry)
A "tomahawk" is a geometric shape consisting of a semicircle and two orthogonal line segments, such that the length of the shorter segment is equal to the circle radius. Trisection is executed by leaning the end of the tomahawk's shorter segment on one ray, the circle's edge on the other, so that the "handle" (longer segment) crosses the angle's vertex; the trisection line runs between the vertex and the center of the semicircle.
While a tomahawk is constructible with compass and straightedge, it is not generally possible to construct a tomahawk in any desired position. Thus, the above construction does not contradict the nontrisectibility of angles with ruler and compass alone.
As a tomahawk can be used as a set square, it can be also used for trisection angles by the method described in § With a right triangular ruler.
The tomahawk produces the same geometric effect as the paper-folding method: the distance between circle center and the tip of the shorter segment is twice the distance of the radius, which is guaranteed to contact the angle. It is also equivalent to the use of an architects L-Ruler (Carpenter's Square).
With interconnected compasses
An angle can be trisected with a device that is essentially a four-pronged version of a compass, with linkages between the prongs designed to keep the three angles between adjacent prongs equal.[12]
Uses of angle trisection
A cubic equation with real coefficients can be solved geometrically with compass, straightedge, and an angle trisector if and only if it has three real roots.[13]: Thm. 1
A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if $n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},$ where r, s, k ≥ 0 and where the pi are distinct primes greater than 3 of the form $2^{t}3^{u}+1$ (i.e. Pierpont primes greater than 3).[13]: Thm. 2
See also
• Bisection
• Constructible number
• Constructible polygon
• Morley's trisector theorem
• Trisectrix
References
1. Dudley, Underwood (1994), The trisectors, Mathematical Association of America, ISBN 978-0-88385-514-0
2. Wantzel, P M L (1837). "Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas" (PDF). Journal de Mathématiques Pures et Appliquées. 1. 2: 366–372. Archived (PDF) from the original on 2022-10-09. Retrieved 3 March 2014.
3. For the historical basis of Wantzel's proof in the earlier work of Ruffini and Abel, and its timing vis-a-vis Galois, see Smorynski, Craig (2007), History of Mathematics: A Supplement, Springer, p. 130, ISBN 9780387754802.
4. MacHale, Desmond. "Constructing integer angles", Mathematical Gazette 66, June 1982, 144–145.
5. McLean, K. Robin (July 2008). "Trisecting angles with ruler and compasses". Mathematical Gazette. 92: 320–323. doi:10.1017/S0025557200183317. S2CID 126351853. See also Feedback on this article in vol. 93, March 2009, p. 156.
6. Stewart, Ian (1989). Galois Theory. Chapman and Hall Mathematics. pp. g. 58. ISBN 978-0-412-34550-0.
7. Jim Loy (2003) [1997]. "Trisection of an Angle". Archived from the original on February 25, 2012. Retrieved 30 March 2012.
8. Yates, Robert C (1942). The Trisection Problem (PDF). The National Council of Teachers of Mathematics. pp. 39–42. Archived (PDF) from the original on 2022-10-09.
9. Ludwig Bieberbach (1932) "Zur Lehre von den kubischen Konstruktionen", Journal für die reine und angewandte Mathematik, H. Hasse und L. Schlesinger, Band 167 Berlin, p. 142–146 online-copie (GDZ). Retrieved on June 2, 2017.
10. Jim Loy "Trisection of an Angle". Archived from the original on November 4, 2013. Retrieved 2013-11-04.
11. Hutcheson, Thomas W. (May 2001). "Dividing Any Angle into Any Number of Equal Parts". Mathematics Teacher. 94 (5): 400–405. doi:10.5951/MT.94.5.0400.
12. Isaac, Rufus, "Two mathematical papers without words", Mathematics Magazine 48, 1975, p. 198. Reprinted in Mathematics Magazine 78, April 2005, p. 111.
13. Gleason, Andrew Mattei (March 1988). "Angle trisection, the heptagon, and the triskaidecagon" (PDF). The American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. JSTOR 2323624. Archived from the original (PDF) on November 5, 2014.
Further reading
• Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: an elementary approach to ideas and methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3.
External links
• MathWorld site
• Geometric problems of antiquity, including angle trisection
• Some history
• One link of marked ruler construction
• Another, mentioning Archimedes
• A long article with many approximations & means going outside the Greek framework
• Geometry site
Other means of trisection
• Approximate angle trisection as an animation, max. error of the angle ≈ ±4E-8°
• Trisecting via (Archived 2009-10-25) the limacon of Pascal; see also Trisectrix
• Trisecting via an Archimedean Spiral
• Trisecting via the Conchoid of Nicomedes
• sciencenews.org site on using origami
• Hyperbolic trisection and the spectrum of regular polygons
Ancient Greek mathematics
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