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Trisectrix
In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:
• Limaçon trisectrix (some sources refer to this curve as simply the trisectrix.)
• Trisectrix of Maclaurin
• Equilateral trefoil (a.k.a. Longchamps' Trisectrix)
• Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic)
• Durer's folium
• Cubic parabola
• Hyperbola with eccentricity 2
• Rose curve specified by a sinusoid with angular frequency of one-third.
• Parabola
A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. Examples include:
• Archimedean Spiral
• Quadratrix of Hippias
• Sectrix of Maclaurin
• Sectrix of Ceva
• Sectrix of Delanges
See also
• Doubling the cube
• Neusis construction
• Quadratrix
References
• Loy, Jim "Trisection of an Angle", Part VI
• Weisstein, Eric W. "Trisectrix". MathWorld.
• "Sectrix curve" at Encyclopédie des Formes Mathématiques Remarquables (In French)
• This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Trisectrix". Encyclopædia Britannica. Vol. 27 (11th ed.). Cambridge University Press.
Wikimedia Commons has media related to Trisectrix.
| Wikipedia |
Trisectrix of Maclaurin
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.
Equations
Let two lines rotate about the points $P=(0,0)$ and $P_{1}=(a,0)$ so that when the line rotating about $P$ has angle $\theta $ with the x axis, the rotating about $P_{1}$ has angle $3\theta $. Let $Q$ be the point of intersection, then the angle formed by the lines at $Q$ is $2\theta $. By the law of sines,
${r \over \sin 3\theta }={a \over \sin 2\theta }\!$
so the equation in polar coordinates is (up to translation and rotation)
$r=a{\frac {\sin 3\theta }{\sin 2\theta }}={a \over 2}{\frac {4\cos ^{2}\theta -1}{\cos \theta }}={a \over 2}(4\cos \theta -\sec \theta )\!$.
The curve is therefore a member of the Conchoid of de Sluze family.
In Cartesian coordinates the equation of this curve is
$2x(x^{2}+y^{2})=a(3x^{2}-y^{2})\!$.
If the origin is moved to (a, 0) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes
$r=2a\cos {\theta \over 3}\!$
making it an example of a limacon with a loop.
The trisection property
Given an angle $\phi $, draw a ray from $(a,0)$ whose angle with the $x$-axis is $\phi $. Draw a ray from the origin to the point where the first ray intersects the curve. Then, by the construction of the curve, the angle between the second ray and the $x$-axis is $\phi /3$.
Notable points and features
The curve has an x-intercept at $3a \over 2$ and a double point at the origin. The vertical line $x={-{a \over 2}}$ is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at $(a,{\pm {1 \over {\sqrt {3}}}a})$. As a nodal cubic, it is of genus zero.
Relationship to other curves
The trisectrix of Maclaurin can be defined from conic sections in three ways. Specifically:
• It is the inverse with respect to the unit circle of the hyperbola
$2x=a(3x^{2}-y^{2})$.
• It is cissoid of the circle
$(x+a)^{2}+y^{2}=a^{2}$
and the line $x={a \over 2}$ relative to the origin.
• It is the pedal with respect to the origin of the parabola
$y^{2}=2a(x-{\tfrac {3}{2}}a)$.
In addition:
• The inverse with respect to the point $(a,0)$ is the Limaçon trisectrix.
• The trisectrix of Maclaurin is related to the Folium of Descartes by affine transformation.
References
• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 95, 104–106. ISBN 0-486-60288-5.
• Weisstein, Eric W. "Maclaurin Trisectrix". MathWorld.
• "Trisectrix of Maclaurin" at MacTutor's Famous Curves Index
• Maclaurin Trisectrix at mathcurve.com
• "Trisectrix of Maclaurin" at Visual Dictionary Of Special Plane Curves
External links
Wikimedia Commons has media related to Maclaurin's Trisectrix.
• Loy, Jim "Trisection of an Angle", Part VI
| Wikipedia |
Bispectrum
In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.
Definitions
The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum.
The Fourier transform of C3(t1, t2) (third-order cumulant-generating function) is called the bispectrum or bispectral density.
Calculation
Applying the convolution theorem allows fast calculation of the bispectrum: $B(f_{1},f_{2})=F(f_{1})\cdot F(f_{2})\cdot F^{*}(f_{1}+f_{2})$, where $F$ denotes the Fourier transform of the signal, and $F^{*}$ its conjugate.
Applications
Bispectrum and bicoherence may be applied to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension.[1]
Bispectral measurements have been carried out for EEG signals monitoring.[2] It was also shown that bispectra characterize differences between families of musical instruments.[3]
In seismology, signals rarely have adequate duration for making sensible bispectral estimates from time averages.
Bispectral analysis describes observations made at two wavelengths. It is often used by scientists to analyze elemental makeup of a planetary atmosphere by analyzing the amount of light reflected and received through various color filters. By combining and removing two filters, much can be gleaned from only two filters. Through modern computerized interpolation, a third virtual filter can be created to recreate true color photographs that, while not particularly useful for scientific analysis, are popular for public display in textbooks and fund raising campaigns.
Bispectral analysis can also be used to analyze interactions between wave patterns and tides on Earth.[4]
A form of bispectral analysis called the bispectral index is applied to EEG waveforms to monitor depth of anesthesia.[5]
Biphase (phase of polyspectrum) can be used for detection of phase couplings,[6] noise reduction of polharmonic (particularly, speech [7]) signal analysis.
Generalizations
Bispectra fall in the category of higher-order spectra, or polyspectra and provide supplementary information to the power spectrum. The third order polyspectrum (bispectrum) is the easiest to compute, and hence the most popular.
A statistic defined analogously is the bispectral coherency or bicoherence.
Trispectrum
The Fourier transform of C4 (t1, t2, t3) (fourth-order cumulant-generating function) is called the trispectrum or trispectral density.
The trispectrum T(f1,f2,f3) falls into the category of higher-order spectra, or polyspectra, and provides supplementary information to the power spectrum. The trispectrum is a three-dimensional construct. The symmetries of the trispectrum allow a much reduced support set to be defined, contained within the following vertices, where 1 is the Nyquist frequency. (0,0,0) (1/2,1/2,-1/2) (1/3,1/3,0) (1/2,0,0) (1/4,1/4,1/4). The plane containing the points (1/6,1/6,1/6) (1/4,1/4,0) (1/2,0,0) divides this volume into an inner and an outer region. A stationary signal will have zero strength (statistically) in the outer region. The trispectrum support is divided into regions by the plane identified above and by the (f1,f2) plane. Each region has different requirements in terms of the bandwidth of signal required for non-zero values.
In the same way that the bispectrum identifies contributions to a signal's skewness as a function of frequency triples, the trispectrum identifies contributions to a signal's kurtosis as a function of frequency quadruplets.
The trispectrum has been used to investigate the domains of applicability of maximum kurtosis phase estimation used in the deconvolution of seismic data to find layer structure.
References
1. Greb U, Rusbridge MG (1988). "The interpretation of the bispectrum and bicoherence for non-linear interactions of continuous spectra". Plasma Phys. Control. Fusion. 30 (5): 537–49. Bibcode:1988PPCF...30..537G. doi:10.1088/0741-3335/30/5/005. S2CID 250741815.
2. Johansen JW, Sebel PS (November 2000). "Development and clinical application of electroencephalographic bispectrum monitoring". Anesthesiology. 93 (5): 1336–44. doi:10.1097/00000542-200011000-00029. PMID 11046224. S2CID 379085.
3. Dubnov S, Tishby N and Cohen D. (1997). "Polyspectra as Measures of Sound Texture and Timbre". Journal of New Music Research. 26 (4): 277–314. doi:10.1080/09298219708570732.
4. Kamalabadi, F.; Forbes, J. M.; Makarov, N. M.; Portnyagin, Yu. I. (27 February 1997). "Evidence for nonlinear coupling of planetary waves and tides in the Antarctic mesopause". Journal of Geophysical Research: Atmospheres. 102 (D4): 4437–4446. Bibcode:1997JGR...102.4437K. doi:10.1029/96JD01996.
5. Mathur, Surbhi; Patel, Jashvin; Goldstein, Sheldon; Jain, Ankit (2021), "Bispectral Index", StatPearls, Treasure Island (FL): StatPearls Publishing, PMID 30969631, retrieved 2021-04-08
6. Fackrell, Justin W. A. (September 1996). "Bispectral analysis of speech signals" (Document). Edinburgh: The University of Edinburgh.
7. Nemer, Elias J. (1999). "Speech analysis and quality enhancement using higher order cumulants" (Document). Ottawa: Ottawa-Carleton Institute for Electrical and Computer Engineering.
Further reading
• Mendel JM (1991). "Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications". Proc. IEEE. 79 (3): 278–305. doi:10.1109/5.75086.
• HOSA - Higher Order Spectral Analysis Toolbox: A MATLAB toolbox for spectral and polyspectral analysis, and time-frequency distributions. The documentation explains polyspectra in great detail.
| Wikipedia |
Tristan Rivière
Tristan Rivière (born 26 November 1967, Brest) is a French mathematician, working on partial differential equations and the calculus of variations.
Tristan Rivière
Tristan Rivière at Ushant Island, 2011
Born26 November 1967 (1967-11-26) (age 55)
Brest, France
NationalityFrench
Alma materPierre and Marie Curie University (Ph.D., 1993)
AwardsBronze Medal of the CNRS (1996)
Stampacchia Medal (2003)
Scientific career
FieldsCalculus of variations, Partial differential equations
InstitutionsETH Zurich
Doctoral advisorFabrice Bethuel
Biography
Rivière studied at the École Polytechnique and obtained his PhD in 1993 at the Pierre and Marie Curie University, under the supervision of Fabrice Bethuel, with a thesis on harmonic maps between manifolds. In 1992 he was appointed chargé de recherche at CNRS. In 1997 he received his habilitation at the University of Paris-Sud in Orsay. From 1999 to 2000 he was a visiting associate professor at the Courant Institute of Mathematical Sciences (New York University). Since 2003 he is full professor at ETH Zurich and since 2009 he is the Director of the Institute for Mathematical Research at ETH.
Research activity
His research interests include partial differential equations in physics (liquid crystals, Bose–Einstein condensates, micromagnetics, Ginzburg–Landau theory of superconductivity, gauge theory) and differential geometry (harmonic maps between manifolds, geometric flows, minimal surfaces, the Willmore functional and Yang–Mills fields). His work focuses in particular on non-linear phenomena, formation of vortices, energy quantization and regularity issues.
Awards and recognition
In 1996 he received the Bronze Medal of the CNRS, while in 2003 he was awarded the first Stampacchia Medal. In 2002 he was an invited speaker at the International Congress of Mathematicians in Beijing, where he gave a talk on bubbling, quantization and regularity issues in geometric non-linear analysis.
Selected publications
• "Everywhere discontinuous Harmonic Maps into Spheres. Acta Mathematica, 175 (1995), 197-226
• with F. Pacard: Linear and Nonlinear Aspects of Vortices. Birkhäuser 2000
• "Conservation laws for conformally invariant variational problems". Inventiones Math., 168 (2007), 1-22
• with R. Hardt: "Connecting rational homotopy type singularities of maps between manifolds". Acta Mathematica, 200 (2008), 15-83
• "Analysis Aspects of Willmore Surfaces". Inventiones Math., 174 (2008), no. 1, 1-45
• with G. Tian: "The singular set of 1-1 Integral currents". Annals of Mathematics, 169 (2009), no. 3, 741-794
• with Y. Bernard: "Energy Quantization for Willmore Surfaces and Applications". Annals of Mathematics, 180 (2014), no. 1, 87-136
• "A viscosity method in the min-max theory of minimal surfaces". Publications mathématiques de l'IHÉS, 126 (2017), no. 1, 177-246
External links
• Homepage, ETH Zürich
• Tristan Rivière at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
Academics
• CiNii
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
| Wikipedia |
Alternated octagonal tiling
In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)} or h{8,3}.
Alternated octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(3.4)3
Schläfli symbol(4,3,3)
s(4,4,4)
Wythoff symbol3 | 3 4
Coxeter diagram
Symmetry group[(4,3,3)], (*433)
[(4,4,4)]+, (444)
DualAlternated octagonal tiling#Dual tiling
PropertiesVertex-transitive
Geometry
Although a sequence of edges seem to represent straight lines (projected into curves), careful attention will show they are not straight, as can be seen by looking at it from different projective centers.
Triangle-centered
hyperbolic straight edges
Edge-centered
projective straight edges
Point-centered
projective straight edges
Dual tiling
In art
Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came". White curves within the figure, through the middle of each line of fish, divide the plane into squares and triangles in the pattern of the tritetragonal tiling. However, in the tritetragonal tiling, the corresponding curves are chains of hyperbolic line segments, with a slight angle at each vertex, while in Escher's woodcut they appear to be smooth hypercycles.
Related polyhedra and tiling
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
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3
See also
• Circle Limit III
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
Wikimedia Commons has media related to Uniform tiling 3-4-3-4-3-4.
• Douglas Dunham Department of Computer Science University of Minnesota, Duluth
• Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation, 2008:A “Circle Limit III” Backbone Arc Formula
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery Archived 2013-03-24 at the Wayback Machine
• 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 |
Cantic order-4 hexagonal tiling
In geometry, the cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}.
Cantic order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.8.4.8
Schläfli symbolt0,1(4,4,3)
Wythoff symbol4 4 | 3
Coxeter diagram
Symmetry group[(4,4,3)], (*443)
DualOrder-4-4-3 t01 dual tiling
PropertiesVertex-transitive
Related polyhedra and tiling
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-8-4-8.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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 |
Cantic octagonal tiling
In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2(4,3,3). It can also be named as a cantic octagonal tiling, h2{8,3}.
Cantic octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.6.4.6
Schläfli symbolh2{8,3}
Wythoff symbol4 3 | 3
Coxeter diagram =
Symmetry group[(4,3,3)], (*433)
DualOrder-4-3-3 t12 dual tiling
PropertiesVertex-transitive
Dual tiling
Related polyhedra and tiling
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
*n33 orbifold symmetries of cantic tilings: 3.6.n.6
Symmetry
*n32
[1+,2n,3]
= [(n,3,3)]
Spherical Euclidean Compact Hyperbolic Paracompact
*233
[1+,4,3]
= [3,3]
*333
[1+,6,3]
= [(3,3,3)]
*433
[1+,8,3]
= [(4,3,3)]
*533
[1+,10,3]
= [(5,3,3)]
*633...
[1+,12,3]
= [(6,3,3)]
*∞33
[1+,∞,3]
= [(∞,3,3)]
Coxeter
Schläfli
=
h2{4,3}
=
h2{6,3}
=
h2{8,3}
=
h2{10,3}
=
h2{12,3}
=
h2{∞,3}
Cantic
figure
Vertex 3.6.2.6 3.6.3.6 3.6.4.6 3.6.5.6 3.6.6.6 3.6.∞.6
Domain
Wythoff 2 3 | 3 3 3 | 3 4 3 | 3 5 3 | 3 6 3 | 3 ∞ 3 | 3
Dual
figure
Face V3.6.2.6 V3.6.3.6 V3.6.4.6 V3.6.5.6 V3.6.6.6 V3.6.∞.6
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.
See also
Wikimedia Commons has media related to Uniform tiling 3-6-4-6.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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 |
Truncated 6-cubes
In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.
6-cube
Truncated 6-cube
Bitruncated 6-cube
Tritruncated 6-cube
6-orthoplex
Truncated 6-orthoplex
Bitruncated 6-orthoplex
Orthogonal projections in B6 Coxeter plane
There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.
Truncated 6-cube
Truncated 6-cube
Typeuniform 6-polytope
ClassB6 polytope
Schläfli symbolt{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces76
4-faces464
Cells1120
Faces1520
Edges1152
Vertices384
Vertex figure
( )v{3,3,3}
Coxeter groupsB6, [3,3,3,3,4]
Propertiesconvex
Alternate names
• Truncated hexeract (Acronym: tox) (Jonathan Bowers)[1]
Construction and coordinates
The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at $1/({\sqrt {2}}+2)$ of the edge length. A regular 5-simplex replaces each original vertex.
The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:
$\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)$
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
The truncated 6-cube, is fifth in a sequence of truncated hypercubes:
Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}
Bitruncated 6-cube
Bitruncated 6-cube
Typeuniform 6-polytope
ClassB6 polytope
Schläfli symbol2t{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
{ }v{3,3}
Coxeter groupsB6, [3,3,3,3,4]
Propertiesconvex
Alternate names
• Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)[2]
Construction and coordinates
The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:
$\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2,\ \pm 2\right)$
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:
Bitruncated hypercubes
Image ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v{ }
{ }v{ }
{ }v{3}
{ }v{3,3}
{ }v{3,3,3} { }v{3,3,3,3}
Tritruncated 6-cube
Tritruncated 6-cube
Typeuniform 6-polytope
ClassB6 polytope
Schläfli symbol3t{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
{3}v{4}[3]
Coxeter groupsB6, [3,3,3,3,4]
Propertiesconvex
Alternate names
• Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)[4]
Construction and coordinates
The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:
$\left(0,\ 0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)$
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8 n
Name t{4} r{4,3} 2t{4,3,3} 2r{4,3,3,3} 3t{4,3,3,3,3} 3r{4,3,3,3,3,3} 4t{4,3,3,3,3,3,3} ...
Coxeter
diagram
Images
Facets {3}
{4}
t{3,3}
t{3,4}
r{3,3,3}
r{3,3,4}
2t{3,3,3,3}
2t{3,3,3,4}
2r{3,3,3,3,3}
2r{3,3,3,3,4}
3t{3,3,3,3,3,3}
3t{3,3,3,3,3,4}
Vertex
figure
( )v( )
{ }×{ }
{ }v{ }
{3}×{4}
{3}v{4}
{3,3}×{3,4} {3,3}v{3,4}
Related polytopes
These polytopes are from a set of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes
β6
t1β6
t2β6
t2γ6
t1γ6
γ6
t0,1β6
t0,2β6
t1,2β6
t0,3β6
t1,3β6
t2,3γ6
t0,4β6
t1,4γ6
t1,3γ6
t1,2γ6
t0,5γ6
t0,4γ6
t0,3γ6
t0,2γ6
t0,1γ6
t0,1,2β6
t0,1,3β6
t0,2,3β6
t1,2,3β6
t0,1,4β6
t0,2,4β6
t1,2,4β6
t0,3,4β6
t1,2,4γ6
t1,2,3γ6
t0,1,5β6
t0,2,5β6
t0,3,4γ6
t0,2,5γ6
t0,2,4γ6
t0,2,3γ6
t0,1,5γ6
t0,1,4γ6
t0,1,3γ6
t0,1,2γ6
t0,1,2,3β6
t0,1,2,4β6
t0,1,3,4β6
t0,2,3,4β6
t1,2,3,4γ6
t0,1,2,5β6
t0,1,3,5β6
t0,2,3,5γ6
t0,2,3,4γ6
t0,1,4,5γ6
t0,1,3,5γ6
t0,1,3,4γ6
t0,1,2,5γ6
t0,1,2,4γ6
t0,1,2,3γ6
t0,1,2,3,4β6
t0,1,2,3,5β6
t0,1,2,4,5β6
t0,1,2,4,5γ6
t0,1,2,3,5γ6
t0,1,2,3,4γ6
t0,1,2,3,4,5γ6
Notes
1. Klitzing, (o3o3o3o3x4x - tox)
2. Klitzing, (o3o3o3x3x4o - botox)
3. https://bendwavy.org/klitzing/incmats/squete.htm
4. Klitzing, (o3o3x3x3o4o - xog)
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. "6D uniform polytopes (polypeta)". o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• 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 |
Truncated 7-cubes
In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.
7-cube
Truncated 7-cube
Bitruncated 7-cube
Tritruncated 7-cube
7-orthoplex
Truncated 7-orthoplex
Bitruncated 7-orthoplex
Tritruncated 7-orthoplex
Orthogonal projections in B7 Coxeter plane
There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.
Truncated 7-cube
Truncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges3136
Vertices896
Vertex figureElongated 5-simplex pyramid
Coxeter groupsB7, [35,4]
Propertiesconvex
Alternate names
• Truncated hepteract (Jonathan Bowers)[1]
Coordinates
Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of
(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√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]
Related polytopes
The truncated 7-cube, is sixth in a sequence of truncated hypercubes:
Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}
Bitruncated 7-cube
Bitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol2t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges9408
Vertices2688
Vertex figure{ }v{3,3,3}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex
Alternate names
• Bitruncated hepteract (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±2,±2,±1,0)
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]
Related polytopes
The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:
Bitruncated hypercubes
Image ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v{ }
{ }v{ }
{ }v{3}
{ }v{3,3}
{ }v{3,3,3} { }v{3,3,3,3}
Tritruncated 7-cube
Tritruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol3t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges13440
Vertices3360
Vertex figure{4}v{3,3}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex
Alternate names
• Tritruncated hepteract (Jonathan Bowers)[3]
Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±2,±1,0,0)
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]
Notes
1. Klitizing (x3x3o3o3o3o4o - taz)
2. Klitizing (o3x3x3o3o3o4o - botaz)
3. Klitizing (o3o3x3x3o3o4o - totaz)
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)". o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz
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 |
Truncated 7-orthoplexes
In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.
7-orthoplex
Truncated 7-orthoplex
Bitruncated 7-orthoplex
Tritruncated 7-orthoplex
7-cube
Truncated 7-cube
Bitruncated 7-cube
Tritruncated 7-cube
Orthogonal projections in B7 Coxeter plane
There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.
Truncated 7-orthoplex
Truncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells3920
Faces2520
Edges924
Vertices168
Vertex figure( )v{3,3,4}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex
Alternate names
• Truncated heptacross
• Truncated hecatonicosoctaexon (Jonathan Bowers)[1]
Coordinates
Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of
(±2,±1,0,0,0,0,0)
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 truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.
Bitruncated 7-orthoplex
Bitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol2t{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges4200
Vertices840
Vertex figure{ }v{3,3,4}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex
Alternate names
• Bitruncated heptacross
• Bitruncated hecatonicosoctaexon (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±1,0,0,0,0)
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]
Tritruncated 7-orthoplex
The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.
Tritruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbol3t{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges10080
Vertices2240
Vertex figure{3}v{3,4}
Coxeter groupsB7, [35,4]
D7, [34,1,1]
Propertiesconvex
Alternate names
• Tritruncated heptacross
• Tritruncated hecatonicosoctaexon (Jonathan Bowers)[3]
Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±1,0,0,0)
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]
Notes
1. Klitzing, (x3x3o3o3o3o4o - tez)
2. Klitzing, (o3x3x3o3o3o4o - botaz)
3. Klitzing, (o3o3x3x3o3o4o - totaz)
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)". x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz
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 |
Cubic graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.
Not to be confused with graphs of cubic functions, hypercube graph, cube graph, cubical graph.
A bicubic graph is a cubic bipartite graph.
Symmetry
In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.[1] Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number s such that each two oriented paths of length s can be mapped to each other by exactly one symmetry of the graph. He showed that s is at most 5, and provided examples of graphs with each possible value of s from 1 to 5.[2]
Semi-symmetric cubic graphs include the Gray graph (the smallest semi-symmetric cubic graph), the Ljubljana graph, and the Tutte 12-cage.
The Frucht graph is one of the five smallest cubic graphs without any symmetries:[3] it possesses only a single graph automorphism, the identity automorphism.[4]
Coloring and independent sets
According to Brooks' theorem every connected cubic graph other than the complete graph K4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
According to Vizing's theorem every cubic graph needs either three or four colors for an edge coloring. A 3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings. By Kőnig's line coloring theorem every bicubic graph has a Tait coloring.
The bridgeless cubic graphs that do not have a Tait coloring are known as snarks. They include the Petersen graph, Tietze's graph, the Blanuša snarks, the flower snark, the double-star snark, the Szekeres snark and the Watkins snark. There is an infinite number of distinct snarks.[5]
Topology and geometry
Cubic graphs arise naturally in topology in several ways. For example, if one considers a graph to be a 1-dimensional CW complex, cubic graphs are generic in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are also formed as the graphs of simple polyhedra in three dimensions, polyhedra such as the regular dodecahedron with the property that three faces meet at every vertex.
An arbitrary graph embedding on a two-dimensional surface may be represented as a cubic graph structure known as a graph-encoded map. In this structure, each vertex of a cubic graph represents a flag of the embedding, a mutually incident triple of a vertex, edge, and face of the surface. The three neighbors of each flag are the three flags that may be obtained from it by changing one of the members of this mutually incident triple and leaving the other two members unchanged.[6]
Hamiltonicity
There has been much research on Hamiltonicity of cubic graphs. In 1880, P.G. Tait conjectured that every cubic polyhedral graph has a Hamiltonian circuit. William Thomas Tutte provided a counter-example to Tait's conjecture, the 46-vertex Tutte graph, in 1946. In 1971, Tutte conjectured that all bicubic graphs are Hamiltonian. However, Joseph Horton provided a counterexample on 96 vertices, the Horton graph.[7] Later, Mark Ellingham constructed two more counterexamples: the Ellingham–Horton graphs.[8][9] Barnette's conjecture, a still-open combination of Tait's and Tutte's conjecture, states that every bicubic polyhedral graph is Hamiltonian. When a cubic graph is Hamiltonian, LCF notation allows it to be represented concisely.
If a cubic graph is chosen uniformly at random among all n-vertex cubic graphs, then it is very likely to be Hamiltonian: the proportion of the n-vertex cubic graphs that are Hamiltonian tends to one in the limit as n goes to infinity.[10]
David Eppstein conjectured that every n-vertex cubic graph has at most 2n/3 (approximately 1.260n) distinct Hamiltonian cycles, and provided examples of cubic graphs with that many cycles.[11] The best proven estimate for the number of distinct Hamiltonian cycles is $O({1.276}^{n})$.[12]
Other properties
Unsolved problem in mathematics:
What is the largest possible pathwidth of an $n$-vertex cubic graph?
(more unsolved problems in mathematics)
The pathwidth of any n-vertex cubic graph is at most n/6. The best known lower bound on the pathwidth of cubic graphs is 0.082n. It is not known how to reduce this gap between this lower bound and the n/6 upper bound.[13]
It follows from the handshaking lemma, proven by Leonhard Euler in 1736 as part of the first paper on graph theory, that every cubic graph has an even number of vertices.
Petersen's theorem states that every cubic bridgeless graph has a perfect matching.[14] Lovász and Plummer conjectured that every cubic bridgeless graph has an exponential number of perfect matchings. The conjecture was recently proved, showing that every cubic bridgeless graph with n vertices has at least 2n/3656 perfect matchings.[15]
Algorithms and complexity
Several researchers have studied the complexity of exponential time algorithms restricted to cubic graphs. For instance, by applying dynamic programming to a path decomposition of the graph, Fomin and Høie showed how to find their maximum independent sets in time 2n/6 + o(n).[13] The travelling salesman problem in cubic graphs can be solved in time O(1.2312n) and polynomial space.[16][17]
Several important graph optimization problems are APX hard, meaning that, although they have approximation algorithms whose approximation ratio is bounded by a constant, they do not have polynomial time approximation schemes whose approximation ratio tends to 1 unless P=NP. These include the problems of finding a minimum vertex cover, maximum independent set, minimum dominating set, and maximum cut.[18] The crossing number (the minimum number of edges which cross in any graph drawing) of a cubic graph is also NP-hard for cubic graphs but may be approximated.[19] The Travelling Salesman Problem on cubic graphs has been proven to be NP-hard to approximate to within any factor less than 1153/1152.[20]
See also
Wikimedia Commons has media related to 3-regular graphs.
• Table of simple cubic graphs
References
1. Foster, R. M. (1932), "Geometrical Circuits of Electrical Networks", Transactions of the American Institute of Electrical Engineers, 51 (2): 309–317, doi:10.1109/T-AIEE.1932.5056068, S2CID 51638449.
2. Tutte, W. T. (1959), "On the symmetry of cubic graphs", Can. J. Math., 11: 621–624, doi:10.4153/CJM-1959-057-2, S2CID 124273238.
3. Bussemaker, F. C.; Cobeljic, S.; Cvetkovic, D. M.; Seidel, J. J. (1976), Computer investigation of cubic graphs, EUT report, vol. 76-WSK-01, Dept. of Mathematics and Computing Science, Eindhoven University of Technology
4. Frucht, R. (1949), "Graphs of degree three with a given abstract group", Canadian Journal of Mathematics, 1 (4): 365–378, doi:10.4153/CJM-1949-033-6, ISSN 0008-414X, MR 0032987, S2CID 124723321.
5. Isaacs, R. (1975), "Infinite families of nontrivial trivalent graphs which are not Tait colorable", American Mathematical Monthly, 82 (3): 221–239, doi:10.2307/2319844, JSTOR 2319844.
6. Bonnington, C. Paul; Little, Charles H. C. (1995), The Foundations of Topological Graph Theory, Springer-Verlag.
7. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976.
8. Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs."Research Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981.
9. Ellingham, M. N.; Horton, J. D. (1983), "Non-Hamiltonian 3-connected cubic bipartite graphs", Journal of Combinatorial Theory, Series B, 34 (3): 350–353, doi:10.1016/0095-8956(83)90046-1.
10. Robinson, R.W.; Wormald, N.C. (1994), "Almost all regular graphs are Hamiltonian", Random Structures and Algorithms, 5 (2): 363–374, doi:10.1002/rsa.3240050209.
11. Eppstein, David (2007), "The traveling salesman problem for cubic graphs" (PDF), Journal of Graph Algorithms and Applications, 11 (1): 61–81, arXiv:cs.DS/0302030, doi:10.7155/jgaa.00137.
12. Gebauer, H. (2008), "On the number of Hamilton cycles in bounded degree graphs", Proc. 4th Workshop on Analytic Algorithmics and Combinatorics (ANALCO '08), pp. 241–248, doi:10.1137/1.9781611972986.8, ISBN 9781611972986.
13. Fomin, Fedor V.; Høie, Kjartan (2006), "Pathwidth of cubic graphs and exact algorithms", Information Processing Letters, 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012.
14. Petersen, Julius Peter Christian (1891), "Die Theorie der regulären Graphs (The theory of regular graphs)", Acta Mathematica, 15 (15): 193–220, doi:10.1007/BF02392606, S2CID 123779343.
15. Esperet, Louis; Kardoš, František; King, Andrew D.; Kráľ, Daniel; Norine, Serguei (2011), "Exponentially many perfect matchings in cubic graphs", Advances in Mathematics, 227 (4): 1646–1664, arXiv:1012.2878, doi:10.1016/j.aim.2011.03.015, S2CID 4401537.
16. Xiao, Mingyu; Nagamochi, Hiroshi (2013), "An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure", Theory and Applications of Models of Computation, Lecture Notes in Computer Science, vol. 7876, Springer-Verlag, pp. 96–107, arXiv:1212.6831, doi:10.1007/978-3-642-38236-9_10, ISBN 978-3-642-38236-9.
17. Xiao, Mingyu; Nagamochi, Hiroshi (2012), "An Exact Algorithm for TSP in Degree-3 Graphs Via Circuit Procedure and Amortization on Connectivity Structure", Algorithmica, 74 (2): 713–741, arXiv:1212.6831, Bibcode:2012arXiv1212.6831X, doi:10.1007/s00453-015-9970-4, S2CID 7654681.
18. Alimonti, Paola; Kann, Viggo (2000), "Some APX-completeness results for cubic graphs", Theoretical Computer Science, 237 (1–2): 123–134, doi:10.1016/S0304-3975(98)00158-3.
19. Hliněný, Petr (2006), "Crossing number is hard for cubic graphs", Journal of Combinatorial Theory, Series B, 96 (4): 455–471, doi:10.1016/j.jctb.2005.09.009.
20. Karpinski, Marek; Schmied, Richard (2013), Approximation Hardness of Graphic TSP on Cubic Graphs, arXiv:1304.6800, Bibcode:2013arXiv1304.6800K.
External links
• Royle, Gordon. "Cubic symmetric graphs (The Foster Census)". Archived from the original on 2011-10-23.
• Weisstein, Eric W. "Bicubic Graph". MathWorld.
• Weisstein, Eric W. "Cubic Graph". MathWorld.
| Wikipedia |
Unlink
[1]In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.
Unlink
2-component unlink
Common nameCircle
Crossing no.0
Linking no.0
Stick no.6
Unknotting no.0
Conway notation-
A–B notation02
1
Dowker notation-
NextL2a1
Other
, tricolorable (if n>1)
Look up unlink in Wiktionary, the free dictionary.
Properties
• An n-component link L ⊂ S3 is an unlink if and only if there exists n disjointly embedded discs Di ⊂ S3 such that L = ∪i∂Di.
• A link with one component is an unlink if and only if it is the unknot.
• The link group of an n-component unlink is the free group on n generators, and is used in classifying Brunnian links.
Examples
• The Hopf link is a simple example of a link with two components that is not an unlink.
• The Borromean rings form a link with three components that is not an unlink; however, any two of the rings considered on their own do form a two-component unlink.
• Taizo Kanenobu has shown that for all n > 1 there exists a hyperbolic link of n components such that any proper sublink is an unlink (a Brunnian link). The Whitehead link and Borromean rings are such examples for n = 2, 3.[1]
See also
• Linking number
References
1. Kanenobu, Taizo (1986), "Hyperbolic links with Brunnian properties", Journal of the Mathematical Society of Japan, 38 (2): 295–308, doi:10.2969/jmsj/03820295, MR 0833204
Further reading
• Kawauchi, A. A Survey of Knot Theory. Birkhauser.
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
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
Trivial cylinder
In geometry and topology, trivial cylinders are certain pseudoholomorphic curves appearing in certain cylindrical manifolds.
In Floer homology and its variants, chain complexes or differential graded algebras are generated by certain combinations of closed orbits of vector fields. In symplectic Floer homology, one considers the Hamiltonian vector field of a Hamiltonian function on a symplectic manifold; in symplectic field theory, contact homology, and their variants, one considers the Reeb vector field associated to a contact form, or more generally a stable Hamiltonian structure.
The differentials all count some flavor of pseudoholomorphic curves in a manifold with a cylindrical almost-complex structure whose ends at negative infinity are the given collection of closed orbits. For instance, in symplectic Floer homology, one considers the product of the mapping torus of a symplectomorphism with the real numbers; in symplectic field theory, one considers the symplectization of a contact manifold.
The product of a given embedded closed orbit with R is always a pseudoholomorphic curve, and such a curve is called a trivial cylinder. Trivial cylinders do not generally contribute to the aforementioned differentials, but they may appear as components of more complicated curves which do.
| Wikipedia |
Trivial measure
In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.[1]
Properties of the trivial measure
Let μ denote the trivial measure on some measurable space (X, Σ).
• A measure ν is the trivial measure μ if and only if ν(X) = 0.
• μ is an invariant measure (and hence a quasi-invariant measure) for any measurable function f : X → X.
Suppose that X is a topological space and that Σ is the Borel σ-algebra on X.
• μ trivially satisfies the condition to be a regular measure.
• μ is never a strictly positive measure, regardless of (X, Σ), since every measurable set has zero measure.
• Since μ(X) = 0, μ is always a finite measure, and hence a locally finite measure.
• If X is a Hausdorff topological space with its Borel σ-algebra, then μ trivially satisfies the condition to be a tight measure. Hence, μ is also a Radon measure. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on X.
• If X is an infinite-dimensional Banach space with its Borel σ-algebra, then μ is the only measure on (X, Σ) that is locally finite and invariant under all translations of X. See the article There is no infinite-dimensional Lebesgue measure.
• If X is n-dimensional Euclidean space Rn with its usual σ-algebra and n-dimensional Lebesgue measure λn, μ is a singular measure with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.
References
1. Porter, Christopher P. (2015-04-01). "Trivial Measures are not so Trivial". Theory of Computing Systems. 56 (3): 487–512. arXiv:1503.06332. doi:10.1007/s00224-015-9614-8. ISSN 1433-0490.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation (V, φ) of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector.
For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations.
Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentations is the whole topic of invariant theory.
The trivial character is the character that takes the value of one for all group elements.
References
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..
| Wikipedia |
Zero ring
In ring theory, a branch of mathematics, the zero ring[1][2][3][4][5] or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which xy = 0 for all x and y. This article refers to the one-element ring.)
Algebraic structure → Ring theory
Ring theory
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• Module
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• Field
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• Zero ring $\mathbb {Z} _{1}$
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• Base-p circle ring $\mathbb {T} $
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• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
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In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object.
Definition
The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and · defined such that 0 + 0 = 0 and 0 · 0 = 0.
Properties
• The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide.[6][7] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. The proof of the last equality is found here.)
• The zero ring is commutative.
• The element 0 in the zero ring is a unit, serving as its own multiplicative inverse.
• The unit group of the zero ring is the trivial group {0}.
• The element 0 in the zero ring is not a zero divisor.
• The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither maximal nor prime.
• The zero ring is generally excluded from fields, while occasionally called as the trivial field. Excluding it agrees with the fact that its zero ideal is not maximal. (When mathematicians speak of the "field with one element", they are referring to a non-existent object, and their intention is to define the category that would be the category of schemes over this object if it existed.)
• The zero ring is generally excluded from integral domains.[8] Whether the zero ring is considered to be a domain at all is a matter of convention, but there are two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor (in particular, 0 is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer n, the ring Z/nZ is a domain if and only if n is prime, but 1 is not prime.
• For each ring A, there is a unique ring homomorphism from A to the zero ring. Thus the zero ring is a terminal object in the category of rings.[9]
• If A is a nonzero ring, then there is no ring homomorphism from the zero ring to A. In particular, the zero ring is not a subring of any nonzero ring.[10]
• The zero ring is the unique ring of characteristic 1.
• The only module for the zero ring is the zero module. It is free of rank א for any cardinal number א.
• The zero ring is not a local ring. It is, however, a semilocal ring.
• The zero ring is Artinian and (therefore) Noetherian.
• The spectrum of the zero ring is the empty scheme.[11]
• The Krull dimension of the zero ring is −∞.
• The zero ring is semisimple but not simple.
• The zero ring is not a central simple algebra over any field.
• The total quotient ring of the zero ring is itself.
Constructions
• For any ring A and ideal I of A, the quotient A/I is the zero ring if and only if I = A, i.e. if and only if I is the unit ideal.
• For any commutative ring A and multiplicative set S in A, the localization S−1A is the zero ring if and only if S contains 0.
• If A is any ring, then the ring M0(A) of 0 × 0 matrices over A is the zero ring.
• The direct product of an empty collection of rings is the zero ring.
• The endomorphism ring of the trivial group is the zero ring.
• The ring of continuous real-valued functions on the empty topological space is the zero ring.
Notes
1. Artin, p. 347.
2. Atiyah and Macdonald, p. 1.
3. Bosch, p. 10.
4. Bourbaki, p. 101.
5. Lam, p. 1.
6. Artin, p. 347.
7. Lang, p. 83.
8. Lam, p. 3.
9. Hartshorne, p. 80.
10. Hartshorne, p. 80.
11. Hartshorne, p. 80.
References
• Michael Artin, Algebra, Prentice-Hall, 1991.
• Siegfried Bosch, Algebraic geometry and commutative algebra, Springer, 2012.
• M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969.
• N. Bourbaki, Algebra I, Chapters 1-3.
• Robin Hartshorne, Algebraic geometry, Springer, 1977.
• T. Y. Lam, Exercises in classical ring theory, Springer, 2003.
• Serge Lang, Algebra 3rd ed., Springer, 2002.
| Wikipedia |
Subrepresentation
In representation theory, a subrepresentation of a representation $(\pi ,V)$ of a group G is a representation $(\pi |_{W},W)$ such that W is a vector subspace of V and $\pi |_{W}(g)=\pi (g)|_{W}$.
A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations.
If $(\pi ,V)$ is a representation of G, then there is the trivial subrepresentation:
$V^{G}=\{v\in V\mid \pi (g)v=v,\,g\in G\}.$
References
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
| Wikipedia |
Ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") $P$ is a certain subset of $P,$ namely a maximal filter on $P;$ that is, a proper filter on $P$ that cannot be enlarged to a bigger proper filter on $P.$
This article is about the mathematical concept in order theory. For ultrafilters on sets specifically, see Ultrafilter (set theory). For the physical device, see ultrafiltration.
If $X$ is an arbitrary set, its power set $\wp (X),$ ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on $\wp (X)$ are usually called ultrafilter on the set $X$.[note 1] An ultrafilter on a set $X$ may be considered as a finitely additive measure on $X$. In this view, every subset of $X$ is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.
Ultrafilters have many applications in set theory, model theory, topology[1]: 186 and combinatorics.[2]
Ultrafilters on partial orders
In order theory, an ultrafilter is a subset of a partially ordered set that is maximal among all proper filters. This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset.
Formally, if $P$ is a set, partially ordered by $\,\leq \,$ then
• a subset $F\subseteq P$ is called a filter on $P$ if
• $F$ is nonempty,
• for every $x,y\in F,$ there exists some element $z\in F$ such that $z\leq x$ and $z\leq y,$ and
• for every $x\in F$ and $y\in P,$ $x\leq y$ implies that $y$ is in $F$ too;
• a proper subset $U$ of $P$ is called an ultrafilter on $P$ if
• $U$ is a filter on $P,$ and
• there is no proper filter $F$ on $P$ that properly extends $U$ (that is, such that $U$ is a proper subset of $F$).
Types and existence of ultrafilters
Every ultrafilter falls into exactly one of two categories: principal or free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form $F_{a}=\{x:a\leq x\}$ for some (but not all) elements $a$ of the given poset. In this case $a$ is called the principal element of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter.
For ultrafilters on a powerset $\wp (X),$ a principal ultrafilter consists of all subsets of $X$ that contain a given element $x\in X.$ Each ultrafilter on $\wp (X)$ that is also a principal filter is of this form.[1]: 187 Therefore, an ultrafilter $U$ on $\wp (X)$ is principal if and only if it contains a finite set.[note 2] If $X$ is infinite, an ultrafilter $U$ on $\wp (X)$ is hence non-principal if and only if it contains the Fréchet filter of cofinite subsets of $X.$[note 3] If $X$ is finite, every ultrafilter is principal.[1]: 187 If $X$ is infinite then the Fréchet filter is not an ultrafilter on the power set of $X$ but it is an ultrafilter on the finite–cofinite algebra of $X.$
Every filter on a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice (AC) in the form of Zorn's lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo–Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). In general, proofs involving the axiom of choice do not produce explicit examples of free ultrafilters, though it is possible to find explicit examples in some models of ZFC; for example, Gödel showed that this can be done in the constructible universe where one can write down an explicit global choice function. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.[3]
Ultrafilter on a Boolean algebra
An important special case of the concept occurs if the considered poset is a Boolean algebra. In this case, ultrafilters are characterized by containing, for each element $a$ of the Boolean algebra, exactly one of the elements $a$ and $\lnot a$ (the latter being the Boolean complement of $a$):
If $P$ is a Boolean algebra and $F$ is a proper filter on $P,$ then the following statements are equivalent:
1. $F$ is an ultrafilter on $P,$
2. $F$ is a prime filter on $P,$
3. for each $a\in P,$ either $a\in F$ or ($\lnot a$) $\in F.$[1]: 186
A proof that 1. and 2. are equivalent is also given in (Burris, Sankappanavar, 2012, Corollary 3.13, p.133).[4]
Moreover, ultrafilters on a Boolean algebra can be related to maximal ideals and homomorphisms to the 2-element Boolean algebra {true, false} (also known as 2-valued morphisms) as follows:
• Given a homomorphism of a Boolean algebra onto {true, false}, the inverse image of "true" is an ultrafilter, and the inverse image of "false" is a maximal ideal.
• Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".
• Given an ultrafilter on a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".
Ultrafilter on the power set of a set
Main article: Ultrafilter (set theory)
Given an arbitrary set $X,$ its power set $\wp (X),$ ordered by set inclusion, is always a Boolean algebra; hence the results of the above section apply. An (ultra)filter on $\wp (X)$ is often called just an "(ultra)filter on $X$".[note 1] Given an arbitrary set $X,$ an ultrafilter on $\wp (X)$ is a set ${\mathcal {U}}$ consisting of subsets of $X$ such that:
1. The empty set is not an element of ${\mathcal {U}}$.
2. If $A$ is an element of ${\mathcal {U}}$ then so is every superset $B\supset A$.
3. If $A$ and $B$ are elements of ${\mathcal {U}}$ then so is the intersection $A\cap B$.
4. If $A$ is a subset of $X,$ then either[note 4] $A$ or its complement $X\setminus A$ is an element of ${\mathcal {U}}$.
An equivalent form of a given ${\mathcal {U}}$ is a 2-valued morphism, a function $m$ on $\wp (X)$ defined as $m(A)=1$ if $A$ is an element of ${\mathcal {U}}$ and $m(A)=0$ otherwise. Then $m$ is finitely additive, and hence a content on $\wp (X),$ and every property of elements of $X$ is either true almost everywhere or false almost everywhere. However, $m$ is usually not countably additive, and hence does not define a measure in the usual sense.
For a filter ${\mathcal {F}}$ that is not an ultrafilter, one can define $m(A)=1$ if $A\in {\mathcal {F}}$ and $m(A)=0$ if $X\setminus A\in {\mathcal {F}},$ leaving $m$ undefined elsewhere.[5]
Applications
Ultrafilters on power sets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on Boolean algebras play a central role in Stone's representation theorem. In set theory ultrafilters are used to show that the axiom of constructibility is incompatible with the existence of a measurable cardinal κ. This is proved by taking the ultrapower of the set theoretical universe modulo a κ-complete, non-principal ultrafilter.[6]
The set $G$ of all ultrafilters of a poset $P$ can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element $a$ of $P$, let $D_{a}=\left\{U\in G:a\in U\right\}.$ This is most useful when $P$ is again a Boolean algebra, since in this situation the set of all $D_{a}$ is a base for a compact Hausdorff topology on $G$. Especially, when considering the ultrafilters on a powerset $\wp (S),$ the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality $|S|.$
The ultraproduct construction in model theory uses ultrafilters to produce a new model starting from a sequence of $X$-indexed models; for example, the compactness theorem can be proved this way. In the special case of ultrapowers, one gets elementary extensions of structures. For example, in nonstandard analysis, the hyperreal numbers can be constructed as an ultraproduct of the real numbers, extending the domain of discourse from real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead the functions and relations are defined "pointwise modulo" $U$, where $U$ is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If $U$ is nonprincipal, then the extension thereby obtained is nontrivial.
In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This construction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces.
Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.
In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for aggregating the preferences of infinitely many individuals. Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (for example, Kirman and Sondermann, 1972).[7] Mihara (1997,[8] 1999)[9] shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.
See also
• Filter (mathematics) – In mathematics, a special subset of a partially ordered set
• Filter (set theory) – Family of sets representing "large" sets
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
• The ultrafilter lemma – Maximal proper filterPages displaying short descriptions of redirect targets
• Universal net – A generalization of a sequence of pointsPages displaying short descriptions of redirect targets
Notes
1. If $X$ happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on $\wp (X)$ or an (ultra)filter just on $X$ is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter of a partial ordered set" vs. "on an arbitrary set"; i.e. they write "(ultra)filter on $X$" to abbreviate "(ultra)filter of $\wp (X)$".
2. To see the "if" direction: If $\left\{x_{1},\ldots ,x_{n}\right\}\in U,$ then $\left\{x_{1}\right\}\in U,{\text{ or }}\ldots {\text{ or }}\left\{x_{n}\right\}\in U,$ by the characterization Nr.7 from Ultrafilter (set theory)#Characterizations. That is, some $\left\{x_{i}\right\}$ is the principal element of $U.$
3. $U$ is non-principal if and only if it contains no finite set, that is, (by Nr.3 of the above characterization theorem) if and only if it contains every cofinite set, that is, every member of the Fréchet filter.
4. Properties 1 and 3 imply that $A$ and $X\setminus A$ cannot both be elements of $U.$
References
1. Davey, B. A.; Priestley, H. A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press.
2. Goldbring, Isaac (2021). Marta Maggioni, Sophia Jahns. "Ultrafilter methods in combinatorics". Snapshots of Modern Mathematics from Oberwolfach. doi:10.14760/SNAP-2021-006-EN.
3. Halbeisen, L. J. (2012). Combinatorial Set Theory. Springer Monographs in Mathematics. Springer.
4. Burris, Stanley N.; Sankappanavar, H. P. (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
5. "Notes on Ultrafilters" (PDF).
6. Kanamori, The Higher infinite, p. 49.
7. Kirman, A.; Sondermann, D. (1972). "Arrow's theorem, many agents, and invisible dictators". Journal of Economic Theory. 5 (2): 267–277. doi:10.1016/0022-0531(72)90106-8.
8. Mihara, H. R. (1997). "Arrow's Theorem and Turing computability" (PDF). Economic Theory. 10 (2): 257–276. CiteSeerX 10.1.1.200.520. doi:10.1007/s001990050157. S2CID 15398169. Archived from the original (PDF) on 2011-08-12Reprinted in K. V. Velupillai, S. Zambelli, and S. Kinsella, ed., Computable Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.{{cite journal}}: CS1 maint: postscript (link)
9. Mihara, H. R. (1999). "Arrow's theorem, countably many agents, and more visible invisible dictators". Journal of Mathematical Economics. 32 (3): 267–277. CiteSeerX 10.1.1.199.1970. doi:10.1016/S0304-4068(98)00061-5.
Bibliography
• Arkhangel'skii, Alexander Vladimirovich; Ponomarev, V.I. (1984). Fundamentals of General Topology: Problems and Exercises. Mathematics and Its Applications. Vol. 13. Dordrecht Boston: D. Reidel. ISBN 978-90-277-1355-1. OCLC 9944489.
• Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
• Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
• Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
• Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
• Jech, Thomas (2006). Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-44085-7. OCLC 50422939.
• Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753.
Further reading
• Comfort, W. W. (1977). "Ultrafilters: some old and some new results". Bulletin of the American Mathematical Society. 83 (4): 417–455. doi:10.1090/S0002-9904-1977-14316-4. ISSN 0002-9904. MR 0454893.
• Comfort, W. W.; Negrepontis, S. (1974), The theory of ultrafilters, Berlin, New York: Springer-Verlag, MR 0396267
• Ultrafilter at the nLab
• "Mathematical Logic 15, The Ultrafilter Theorem" on YouTube
| Wikipedia |
Trochoid
In geometry, a trochoid (from Greek trochos 'wheel') is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line.[1] If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" was coined by Gilles de Roberval.
Basic description
As a circle of radius a rolls without slipping along a line L, the center C moves parallel to L, and every other point P in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let CP = b. Parametric equations of the trochoid for which L is the x-axis are
${\begin{aligned}&x=a\theta -b\sin \theta \\&y=a-b\cos \theta \end{aligned}}$
where θ is the variable angle through which the circle rolls.
Curtate, common, prolate
If P lies inside the circle (b < a), on its circumference (b = a), or outside (b > a), the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively.[2] A curtate trochoid is traced by a pedal (relative to the ground) when a normally geared bicycle is pedaled along a straight line.[3] A prolate trochoid is traced by the tip of a paddle (relative to the water's surface) when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a cycloid, has cusps at the points where P touches the line L.
General description
A more general approach would define a trochoid as the locus of a point $(x,y)$ orbiting at a constant rate around an axis located at $(x',y')$,
$x=x'+r_{1}\cos(\omega _{1}t+\phi _{1}),\ y=y'+r_{1}\sin(\omega _{1}t+\phi _{1}),\ r_{1}>0,$
which axis is being translated in the x-y-plane at a constant rate in either a straight line,
${\begin{array}{lcl}x'=x_{0}+v_{2x}t,\ y'=y_{0}+v_{2y}t\\\therefore x=x_{0}+r_{1}\cos(\omega _{1}t+\phi _{1})+v_{2x}t,\ y=y_{0}+r_{1}\sin(\omega _{1}t+\phi _{1})+v_{2y}t,\\\end{array}}$
or a circular path (another orbit) around $(x_{0},y_{0})$ (the hypotrochoid/epitrochoid case),
${\begin{array}{lcl}x'=x_{0}+r_{2}\cos(\omega _{2}t+\phi _{2}),\ y'=y_{0}+r_{2}\sin(\omega _{2}t+\phi _{2}),\ r_{2}\geq 0\\\therefore x=x_{0}+r_{1}\cos(\omega _{1}t+\phi _{1})+r_{2}\cos(\omega _{2}t+\phi _{2}),\ y=y_{0}+r_{1}\sin(\omega _{1}t+\phi _{1})+r_{2}\sin(\omega _{2}t+\phi _{2}),\\\end{array}}$
The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, $\omega _{1}/\omega _{2}$, is a rational number, say $p/q$, where $p$ & $q$ are coprime, in which case, one period consists of $p$ orbits around the moving axis and $q$ orbits of the moving axis around the point $(x_{0},y_{0})$. The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius $r_{1}$ while it is rolled on the perimeter of a stationary circle of radius $R$, have the following properties:
${\begin{array}{lcl}{\text{epicycloid: }}&\omega _{1}/\omega _{2}&=p/q=r_{2}/r_{1}=R/r_{1}+1,\ |p-q|{\text{ cusps}}\\{\text{hypocycloid: }}&\omega _{1}/\omega _{2}&=p/q=-r_{2}/r_{1}=-(R/r_{1}-1),\ |p-q|=|p|+|q|{\text{ cusps}}\end{array}}$
where $r_{2}$ is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".
See also
• Aristotle's wheel paradox
• Brachistochrone
• Cyclogon
• Cycloid
• Epitrochoid
• Hypotrochoid
• List of periodic functions
• Roulette (curve)
• Spirograph
• Trochoidal wave
References
1. Weisstein, Eric W. "Trochoid". MathWorld.
2. "Trochoid". Xah Math. Retrieved October 4, 2014.
3. The Bicycle Pulling Puzzle. YouTube. Archived from the original on 2021-12-11.
External links
• Online experiments with the Trochoid using JSXGraph
| Wikipedia |
Trombi–Varadarajan theorem
In mathematics, the Trombi–Varadarajan theorem, introduced by Trombi and Varadarjan (1971), gives an isomorphism between a certain space of spherical functions on a semisimple Lie group, and a certain space of holomorphic functions defined on a tubular neighborhood of the dual of a Cartan subalgebra.
References
• Trombi, P. C.; Varadarajan, V. S. (1971), "Spherical transforms of semisimple Lie groups", Annals of Mathematics, Second Series, 94: 246–303, doi:10.2307/1970861, JSTOR 1970861, MR 0289725.
| Wikipedia |
Tromino
A tromino or triomino is a polyomino of size 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.[1]
Symmetry and enumeration
When rotations and reflections are not considered to be distinct shapes, there are only two different free trominoes: "I" and "L" (the "L" shape is also called "V").
Since both free trominoes have reflection symmetry, they are also the only two one-sided trominoes (trominoes with reflections considered distinct). When rotations are also considered distinct, there are six fixed trominoes: two I and four L shapes. They can be obtained by rotating the above forms by 90°, 180° and 270°.[2][3]
Rep-tiling and Golomb's tromino theorem
Both types of tromino can be dissected into n2 smaller trominos of the same type, for any integer n > 1. That is, they are rep-tiles.[4] Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, the L-tromino is called a chair, and its tiling by recursive subdivision into four smaller L-trominos is called the chair tiling.[5]
Motivated by the mutilated chessboard problem, Solomon W. Golomb used this tiling as the basis for what has become known as Golomb's tromino theorem: if any square is removed from a 2n × 2n chessboard, the remaining board can be completely covered with L-trominoes. To prove this by mathematical induction, partition the board into a quarter-board of size 2n−1 × 2n−1 that contains the removed square, and a large tromino formed by the other three quarter-boards. The tromino can be recursively dissected into unit trominoes, and a dissection of the quarter-board with one square removed follows by the induction hypothesis. In contrast, when a chessboard of this size has one square removed, it is not always possible to cover the remaining squares by I-trominoes.[6]
See also
Previous and next orders
• Domino
• Tetromino
References
1. Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
2. Weisstein, Eric W. "Triomino". MathWorld.
3. Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36: 191–203. doi:10.1016/0012-365X(81)90237-5.
4. Nițică, Viorel (2003), "Rep-tiles revisited", MASS selecta, Providence, RI: American Mathematical Society, pp. 205–217, MR 2027179.
5. Robinson, E. Arthur, Jr. (1999). "On the table and the chair". Indagationes Mathematicae. 10 (4): 581–599. doi:10.1016/S0019-3577(00)87911-2. MR 1820555.{{cite journal}}: CS1 maint: multiple names: authors list (link).
6. Golomb, S. W. (1954). "Checker boards and polyominoes". American Mathematical Monthly. 61: 675–682. doi:10.2307/2307321. MR 0067055..
External links
• Golomb's inductive proof of a tromino theorem at cut-the-knot
• Tromino Puzzle at cut-the-knot
• Interactive Tromino Puzzle at Amherst College
Polyforms
Polyominoes
• Domino
• Tromino
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Higher dimensions
• Polyominoid
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| Wikipedia |
Trope (mathematics)
In geometry, trope is an archaic term for a singular (meaning special) tangent space of a variety, often a quartic surface. The term may have been introduced by Cayley (1869, p. 202), who defined it as "the reciprocal term to node". It is not easy to give a precise definition, because the term is used mainly in older books and papers on algebraic geometry, whose definitions are vague and different, and use archaic terminology. The term trope is used in the theory of quartic surfaces in projective space, where it is sometimes defined as a tangent space meeting the quartic surface in a conic; for example Kummer's surface has 16 tropes.
Hudson (1990, p. 14), describes a trope as a tangent plane where the envelope of nearby tangent planes forms a conic, rather than a plane pencil which we would expect for a generic point. The tangent plane would be tangent to the quartic along the conic, implying that the Gauss map would have a singular point. (Dolgachev 2012, p. 437)
See also
• Glossary of classical algebraic geometry
References
• Cayley, Arthur (1869), "A Memoir on the Theory of Reciprocal Surfaces", Philosophical Transactions of the Royal Society of London, The Royal Society, 159: 201–229, doi:10.1098/rstl.1869.0009, ISSN 0080-4614, JSTOR 108996 See page 202 for an early use of the term "trope".
• Hudson, R. W. H. T. (1990), Kummer's quartic surface, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-39790-2, MR 1097176
• Jessop, Charles Minshall (1916), Quartic surfaces with singular points, Cambridge University Press, ISBN 978-1-112-28262-1
• Dolgachev, Igor V. (2012), Classical Algebraic Geometry: A Modern View, Cambridge University Press, ISBN 978-1107017658
| Wikipedia |
Johannes Tropfke
Johannes Tropfke (14 October 1866 – 10 November 1939) was a German mathematician and teacher, who is best remembered for his influential work on the history of mathematics Geschichte der Elementarmathematik, which consists of seven volumes.
Life
Tropfke was born in Berlin at Marienstraße 14 as the older of two sons of the cabinet maker Franz Tropfke. The house in which Tropfke was born was built by his grandfather Franz Joseph Tropfke around 1830 and is one of the few houses in the area that was not destroyed during World War II. Tropfke grew up in Berlin and after his graduation from the Friedrichs-Gymnasium (high school) in 1884 he attended the university in Berlin to study sciences and mathematics. In 1889 he was awarded a degree to teach math and sciences at gymnasiums (high schools). Later he earned a PhD in mathematics from the University of Halle for a thesis on elliptic integrals (Zur Darstellung des elliptischen Integrales erster Gattung), his advisor was Lazarus Fuchs.[1][2]
Tropfke first worked as teacher at the Friedrichs-Realgymnasium and at the Realgymnasium of Dorotheenstadt and in 1913 he became the principal of the newly founded Kirschner-Oberrealschule in Moabit. Tropfke stayed on in this position until his retirement in 1932. In 1907 he was awarded the title of a professor. He was one of the first members of the International Academy of the History of Science and in 1939 he was awarded the Leibniz medal by the Prussian Academy of Sciences.[1]
Aside from his work in education and mathematics Tropfke also had a career in local politics. He was a member of the German People's Party and served as council member for the city of Berlin from 1907 to 1920.[1]
Tropfke married Frida Thyssen. The couple had a son Erich, who perished in World War I, and a daughter Elisabeth. Tropfke died on 10 November 1939 in the very same house in which he was born.[1]
Work
Tropfke's most important contributions were in the history of mathematics. His seminal work Geschichte der Elementarmathematik originally consisted of two volumes, when it was first published in 1902 and 1903. Later it got expanded into seven volumes for its second edition (1921-1924). For this second edition Tropfke was supported by the mathematicians and historians Gustaf Eneström, Julius Ruska and Heinrich Wieleitner. To incorporate the latest research Tropfke published revised third editions of the first three volumes in the 1930s. After his death the mathematician Kurt Vogel completed the third edition of the fourth volume in 1940. The structure and focus of Tropfke's work differed somewhat from most work in the history of mathematics at the time. Instead of structuring the material chronologically with a focus on the biography of mathematicians, Tropfke selected to organize it by mathematical fields and then focused on the development of concepts and terminology in those fields rather than on biographical aspects. In particular with its second edition Tropfke's Geschichte der Elementarmathematik was also one of the most extensive compilations on the history of mathematics, which led to it becoming a well known and influential reference work.[3][4][5][1]
The publication of a fourth revised edition under the direction of Kurt Vogel, Karin Reich and Helmuth Gericke began from 1980 onwards, more than 40 years after Tropfke's death.[6]
In 1930 Tropfke received the Ackermann–Teubner Memorial Award for his second edition of Geschichte der Elementarmathematik.[7]
References
1. Menso Folkerts: Johannes Tropfke (1866-1939) at the websites of the Berliner Mathematische Gesellschaft (Berlin mathematical society), retrieved 2019-01-25 (German)
2. Johannes Tropfke at the Mathematics Genealogy Project (retrieved 2019-01-25)
3. Solomon Gandz: Geschichte der Elementarmathematik - dritter Band, dritte Auflage (review) . Isis, vol. 29, no. 1, 1938, pp. 167–169 (JSTOR)
4. David Eugene Smith: Geschichte der Elementarmathematik - erster Band, dritte Auflage (review). The American Mathematical Monthly, vol. 38, no. 6, 1931, pp. 331–334 (JSTOR)
5. R. B. McClenon: Review: Johannes Tropfke, Geschichte der Elementarmathematik. Bull. Amer. Math. Soc., Volume 31, Number 8 (1925), pp. 461-462 (online copy)
6. Michael S. Mahoney: Geschichte der Elementarmathematik - erster Band, vierte Auflage (review). Isis, vol. 72, no. 1, 1981, pp. 115–116. (JSTOR)
7. Ackermann-Teubner memorial prize (Leipzig 1914-1941) (Html), Ackermann-Teubner-Preis mit Originalzitaten (DMV / AMS 1914-1941) (pdf) at www.weiss-leipzig.de (retrieved 2019-01-25)
External links
• Menso Folkerts: Johannes Tropfke (1866-1939) at the websites of the Berliner Mathematische Gesellschaft (German)
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| Wikipedia |
Tropical compactification
In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev.[1][2] Given an algebraic torus and a connected closed subvariety of that torus, a compactification of the subvariety is defined as a closure of it in a toric variety of the original torus. The concept of a tropical compactification arises when trying to make compactifications as "nice" as possible. For a torus $T$, a toric variety $\mathbb {P} $, the compactification ${\bar {X}}$ is tropical when the map
$\Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx$
is faithfully flat and ${\bar {X}}$ is proper.
See also
• Tropical geometry
• GIT quotient
• Chow quotient
• Toroidal embedding
References
1. Tevelev, Jenia (2007-08-07). "Compactifications of subvarieties of tori". American Journal of Mathematics. 129 (4): 1087–1104. arXiv:math/0412329. doi:10.1353/ajm.2007.0029. ISSN 1080-6377.
2. Brugallé, Erwan; Shaw, Kristin (2014). "A Bit of Tropical Geometry". The American Mathematical Monthly. 121 (7): 563–589. arXiv:1311.2360. doi:10.4169/amer.math.monthly.121.07.563. JSTOR 10.4169/amer.math.monthly.121.07.563.
• Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of $\mathbb {P} ^{1}$". Selecta Mathematica. 23: 1027–1060. arXiv:1410.2837. Bibcode:2014arXiv1410.2837C.
| Wikipedia |
Tropical geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:
$x\oplus y=\min\{x,y\},$
$x\otimes y=x+y.$
So for example, the classical polynomial $x^{3}+2xy+y^{4}$ would become $\min\{x+x+x,\;2+x+y,\;y+y+y+y\}$. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains.
Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from algebraic geometry, such as the Brill–Noether theorem, using the tools of tropical geometry.[1]
History
The basic ideas of tropical analysis were developed independently using the same notation by mathematicians working in various fields.[2] The central ideas of tropical geometry appeared in different forms in a number of earlier works. For example, Victor Pavlovich Maslov introduced a tropical version of the process of integration. He also noticed that the Legendre transformation and solutions of the Hamilton–Jacobi equation are linear operations in the tropical sense.[3] However, only since the late 1990s has an effort been made to consolidate the basic definitions of the theory. This was motivated by its application to enumerative algebraic geometry, with ideas from Maxim Kontsevich[4] and works by Grigory Mikhalkin[5] among others.
The adjective tropical was coined by French mathematicians in honor of the Hungarian-born Brazilian computer scientist Imre Simon, who wrote on the field. Jean-Éric Pin attributes the coinage to Dominique Perrin,[6] whereas Simon himself attributes the word to Christian Choffrut.[7]
Algebra background
Further information: Tropical semiring
Tropical geometry is based on the tropical semiring. This is defined in two ways, depending on max or min convention.
The min tropical semiring is the semiring $(\mathbb {R} \cup \{+\infty \},\oplus ,\otimes )$, with the operations:
$x\oplus y=\min\{x,y\},$
$x\otimes y=x+y.$
The operations $\oplus $ and $\otimes $ are referred to as tropical addition and tropical multiplication respectively. The identity element for $\oplus $ is $+\infty $, and the identity element for $\otimes $ is 0.
Similarly, the max tropical semiring is the semiring $(\mathbb {R} \cup \{-\infty \},\oplus ,\otimes )$, with operations:
$x\oplus y=\max\{x,y\},$
$x\otimes y=x+y.$
The identity element for $\oplus $ is $-\infty $, and the identity element for $\otimes $ is 0.
These semirings are isomorphic, under negation $x\mapsto -x$, and generally one of these is chosen and referred to simply as the tropical semiring. Conventions differ between authors and subfields: some use the min convention, some use the max convention.
The tropical semiring operations model how valuations behave under addition and multiplication in a valued field.
Some common valued fields encountered in tropical geometry (with min convention) are:
• $\mathbb {Q} $ or $\mathbb {C} $ with the trivial valuation, $v(a)=0$ for all $a\neq 0$.
• $\mathbb {Q} $ or its extensions with the p-adic valuation, $v_{p}(p^{n}a/b)=n$ for a and b coprime to p.
• The field of Laurent series $\mathbb {C} (\!(t)\!)$ (integer powers), or the field of (complex) Puiseux series $\mathbb {C} \{\!\{t\}\!\}$, with valuation returning the smallest exponent of t appearing in the series.
Tropical polynomials
A tropical polynomial is a function $F\colon \mathbb {R} ^{n}\to \mathbb {R} $ that can be expressed as the tropical sum of a finite number of monomial terms. A monomial term is a tropical product (and/or quotient) of a constant and variables from $X_{1},\ldots ,X_{n}$. Thus a tropical polynomial F is the minimum of a finite collection of affine-linear functions in which the variables have integer coefficients, so it is concave, continuous, and piecewise linear.[8]
${\begin{aligned}F(X_{1},\ldots ,X_{n})&=\left(C_{1}\otimes X_{1}^{\otimes a_{11}}\otimes \cdots \otimes X_{n}^{\otimes a_{n1}}\right)\oplus \cdots \oplus \left(C_{s}\otimes X_{1}^{\otimes a_{1s}}\otimes \cdots \otimes X_{n}^{\otimes a_{ns}}\right)\\&=\min\{C_{1}+a_{11}X_{1}+\cdots +a_{n1}X_{n},\;\ldots ,\;C_{s}+a_{1s}X_{1}+\cdots +a_{ns}X_{n}\}.\end{aligned}}$
Given a polynomial f in the Laurent polynomial ring $K[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}]$ where K is a valued field, the tropicalization of f, denoted $\operatorname {Trop} (f)$, is the tropical polynomial obtained from f by replacing multiplication and addition by their tropical counterparts and each constant in K by its valuation. That is, if
$f=\sum _{i=1}^{s}c_{i}x^{A_{i}}\quad {\text{ with }}A_{1},\ldots ,A_{s}\in \mathbb {Z} ^{n},$
then
$\operatorname {Trop} (f)=\bigoplus _{i=1}^{s}v(c_{i})\otimes X^{\otimes A_{i}}.$
The set of points where a tropical polynomial F is non-differentiable is called its associated tropical hypersurface, denoted $\mathrm {V} (F)$ (in analogy to the vanishing set of a polynomial). Equivalently, $\mathrm {V} (F)$ is the set of points where the minimum among the terms of F is achieved at least twice. When $F=\operatorname {Trop} (f)$ for a Laurent polynomial f, this latter characterization of $\mathrm {V} (F)$ reflects the fact that at any solution to $f=0$, the minimum valuation of the terms of f must be achieved at least twice in order for them all to cancel.[9]
Tropical varieties
Definitions
For X an algebraic variety in the algebraic torus $(K^{\times })^{n}$, the tropical variety of X or tropicalization of X, denoted $\operatorname {Trop} (X)$, is a subset of $\mathbb {R} ^{n}$ that can be defined in several ways. The equivalence of these definitions is referred to as the Fundamental Theorem of Tropical Geometry.[9]
Intersection of tropical hypersurfaces
Let $\mathrm {I} (X)$ be the ideal of Laurent polynomials that vanish on X in $K[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}]$. Define
$\operatorname {Trop} (X)=\bigcap _{f\in \mathrm {I} (X)}\mathrm {V} (\operatorname {Trop} (f))\subseteq \mathbb {R} ^{n}.$
When X is a hypersurface, its vanishing ideal $\mathrm {I} (X)$ is a principal ideal generated by a Laurent polynomial f, and the tropical variety $\operatorname {Trop} (X)$ is precisely the tropical hypersurface $\mathrm {V} (\operatorname {Trop} (f))$.
Every tropical variety is the intersection of a finite number of tropical hypersurfaces. A finite set of polynomials $\{f_{1},\ldots ,f_{r}\}\subseteq \mathrm {I} (X)$ is called a tropical basis for X if $\operatorname {Trop} (X)$ is the intersection of the tropical hypersurfaces of $\operatorname {Trop} (f_{1}),\ldots ,\operatorname {Trop} (f_{r})$. In general, a generating set of $\mathrm {I} (X)$ is not sufficient to form a tropical basis. The intersection of a finite number of a tropical hypersurfaces is called a tropical prevariety and in general is not a tropical variety.[9]
Initial ideals
Choosing a vector $\mathbf {w} $ in $\mathbb {R} ^{n}$ defines a map from the monomial terms of $K[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}]$ to $\mathbb {R} $ by sending the term m to $\operatorname {Trop} (m)(\mathbf {w} )$. For a Laurent polynomial $f=m_{1}+\cdots +m_{s}$, define the initial form of f to be the sum of the terms $m_{i}$ of f for which $\operatorname {Trop} (m_{i})(\mathbf {w} )$ is minimal. For the ideal $\mathrm {I} (X)$, define its initial ideal with respect to $\mathbf {w} $ to be
$\operatorname {in} _{\mathbf {w} }\mathrm {I} (X)=(\operatorname {in} _{\mathbf {w} }(f):f\in \mathrm {I} (X)).$
Then define
$\operatorname {Trop} (X)=\{\mathbf {w} \in \mathbb {R} ^{n}:\operatorname {in} _{\mathbf {w} }\mathrm {I} (X)\neq (1)\}.$
Since we are working in the Laurent ring, this is the same as the set of weight vectors for which $\operatorname {in} _{\mathbf {w} }\mathrm {I} (X)$ does not contain a monomial.
When K has trivial valuation, $\operatorname {in} _{\mathbf {w} }\mathrm {I} (X)$ is precisely the initial ideal of $\mathrm {I} (X)$ with respect to the monomial order given by a weight vector $\mathbf {w} $. It follows that $\operatorname {Trop} (X)$ is a subfan of the Gröbner fan of $\mathrm {I} (X)$.
Image of the valuation map
Suppose that X is a variety over a field K with valuation v whose image is dense in $\mathbb {R} $ (for example a field of Puiseux series). By acting coordinate-wise, v defines a map from the algebraic torus $(K^{\times })^{n}$ to $\mathbb {R} ^{n}$. Then define
$\operatorname {Trop} (X)={\overline {\{(v(x_{1}),\ldots ,v(x_{n})):(x_{1},\ldots ,x_{n})\in X\}}},$
where the overline indicates the closure in the Euclidean topology. If the valuation of K is not dense in $\mathbb {R} $, then the above definition can be adapted by extending scalars to larger field which does have a dense valuation.
This definition shows that $\operatorname {Trop} (X)$ is the non-Archimedean amoeba over an algebraically closed non-Archimedean field K.[10]
If X is a variety over $\mathbb {C} $, $\operatorname {Trop} (X)$ can be considered as the limiting object of the amoeba $\operatorname {Log} _{t}(X)$ as the base t of the logarithm map goes to infinity.[11]
Polyhedral complex
The following characterization describes tropical varieties intrinsically without reference to algebraic varieties and tropicalization. A set V in $\mathbb {R} ^{n}$ is an irreducible tropical variety if it is the support of a weighted polyhedral complex of pure dimension d that satisfies the zero-tension condition and is connected in codimension one. When d is one, the zero-tension condition means that around each vertex, the weighted-sum of the out-going directions of edges equals zero. For higher dimension, sums are taken instead around each cell of dimension $d-1$ after quotienting out the affine span of the cell.[8] The property that V is connected in codimension one means for any two points lying on dimension d cells, there is a path connecting them that does not pass through any cells of dimension less than $d-1$.[12]
Tropical curves
The study of tropical curves (tropical varieties of dimension one) is particularly well developed and is strongly related to graph theory. For instance, the theory of divisors of tropical curves are related to chip-firing games on graphs associated to the tropical curves.[13]
Many classical theorems of algebraic geometry have counterparts in tropical geometry, including:
• Pappus's hexagon theorem.[14]
• Bézout's theorem.
• The degree-genus formula.
• The Riemann–Roch theorem.[15]
• The group law of the cubics.[16]
Oleg Viro used tropical curves to classify real curves of degree 7 in the plane up to isotopy. His method of patchworking gives a procedure to build a real curve of a given isotopy class from its tropical curve.
Applications
A tropical line appeared in Paul Klemperer's design of auctions used by the Bank of England during the financial crisis in 2007.[17] Yoshinori Shiozawa defined subtropical algebra as max-times or min-times semiring (instead of max-plus and min-plus). He found that Ricardian trade theory (international trade without input trade) can be interpreted as subtropical convex algebra.[18] Tropical geometry has also been used for analyzing the complexity of feedforward neural networks with ReLU activation.[19]
Moreover, several optimization problems arising for instance in job scheduling, location analysis, transportation networks, decision making and discrete event dynamical systems can be formulated and solved in the framework of tropical geometry.[20] A tropical counterpart of the Abel–Jacobi map can be applied to a crystal design.[21] The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry can show self-organized criticality.[22]
See also
• Tropical analysis
• Tropical compactification
Notes
1. Hartnett, Kevin (5 September 2018). "Tinkertoy Models Produce New Geometric Insights". Quanta Magazine. Retrieved 12 December 2018.
2. See Cuninghame-Green, Raymond A. (1979). Minimax algebra. Lecture Notes in Economics and Mathematical Sciences. Vol. 166. Springer. ISBN 978-3-540-09113-4 and references therein.
3. Maslov, Victor (1987). "On a new superposition principle for optimization problems". Russian Mathematical Surveys. 42 (3): 43–54. Bibcode:1987RuMaS..42...43M. doi:10.1070/RM1987v042n03ABEH001439. S2CID 250889913.
4. Kontsevich, Maxim; Soibelman, Yan (7 November 2000). "Homological mirror symmetry and torus fibrations". arXiv:math/0011041.
5. Mikhalkin, Grigory (2005). "Enumerative tropical algebraic geometry in R2" (PDF). Journal of the American Mathematical Society. 18 (2): 313–377. arXiv:math/0312530. doi:10.1090/S0894-0347-05-00477-7.
6. Pin, Jean-Eric (1998). "Tropical semirings" (PDF). In Gunawardena, J. (ed.). Idempotency. Publications of the Newton Institute. Vol. 11. Cambridge University Press. pp. 50–69. doi:10.1017/CBO9780511662508.004. ISBN 9780511662508.
7. Simon, Imre (1988). "Recognizable sets with multiplicities in the tropical semiring". Mathematical Foundations of Computer Science 1988. Lecture Notes in Computer Science. Vol. 324. pp. 107–120. doi:10.1007/BFb0017135. ISBN 978-3-540-50110-7.
8. Speyer, David; Sturmfels, Bernd (2009), "Tropical mathematics" (PDF), Mathematics Magazine, 82 (3): 163–173, doi:10.1080/0025570X.2009.11953615, S2CID 15278805
9. Maclagan, Diane; Sturmfels, Bernd (2015). Introduction to Tropical Geometry. American Mathematical Society. ISBN 9780821851982.
10. Mikhalkin, Grigory (2004). "Amoebas of algebraic varieties and tropical geometry". In Donaldson, Simon; Eliashberg, Yakov; Gromov, Mikhael (eds.). Different faces of geometry. International Mathematical Series. Vol. 3. New York, NY: Kluwer Academic/Plenum Publishers. pp. 257–300. ISBN 978-0-306-48657-9. Zbl 1072.14013.
11. Katz, Eric (2017), "What is Tropical Geometry?" (PDF), Notices of the American Mathematical Society, 64 (4): 380–382, doi:10.1090/noti1507
12. Cartwright, Dustin; Payne, Sam (2012), "Connectivity of tropicalizations", Mathematical Research Letters, 19 (5): 1089–1095, arXiv:1204.6589, Bibcode:2012arXiv1204.6589C, doi:10.4310/MRL.2012.v19.n5.a10, S2CID 51767353
13. Hladký, Jan; Králʼ, Daniel; Norine, Serguei (1 September 2013). "Rank of divisors on tropical curves". Journal of Combinatorial Theory, Series A. 120 (7): 1521–1538. arXiv:0709.4485. doi:10.1016/j.jcta.2013.05.002. ISSN 0097-3165. S2CID 3045053.
14. Tabera, Luis Felipe (1 January 2005). "Tropical constructive Pappus' theorem". International Mathematics Research Notices. 2005 (39): 2373–2389. arXiv:math/0409126. doi:10.1155/IMRN.2005.2373. ISSN 1073-7928.
15. Kerber, Michael; Gathmann, Andreas (1 May 2008). "A Riemann–Roch theorem in tropical geometry". Mathematische Zeitschrift. 259 (1): 217–230. arXiv:math/0612129. doi:10.1007/s00209-007-0222-4. ISSN 1432-1823. S2CID 15239772.
16. Chan, Melody; Sturmfels, Bernd (2013). "Elliptic curves in honeycomb form". In Brugallé, Erwan (ed.). Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011. Contemporary Mathematics. Vol. 589. Providence, RI: American Mathematical Society. pp. 87–107. arXiv:1203.2356. Bibcode:2012arXiv1203.2356C. ISBN 978-0-8218-9146-9. Zbl 1312.14142.
17. "How geometry came to the rescue during the banking crisis". Department of Economics, University of Oxford. Retrieved 24 March 2014.
18. Shiozawa, Yoshinori (2015). "International trade theory and exotic algebras". Evolutionary and Institutional Economics Review. 12: 177–212. doi:10.1007/s40844-015-0012-3. S2CID 155827635. This is a digest of Y. Shiozawa, "Subtropical Convex Geometry as the Ricardian Theory of International Trade" draft paper.
19. Zhang, Liwen; Naitzat, Gregory; Lim, Lek-Heng (2018). "Tropical Geometry of Deep Neural Networks". Proceedings of the 35th International Conference on Machine Learning. 35th International Conference on Machine Learning. pp. 5824–5832.
20. Krivulin, Nikolai (2014). "Tropical optimization problems". In Leon A. Petrosyan; David W. K. Yeung; Joseph V. Romanovsky (eds.). Advances in Economics and Optimization: Collected Scientific Studies Dedicated to the Memory of L. V. Kantorovich. New York: Nova Science Publishers. pp. 195–214. arXiv:1408.0313. ISBN 978-1-63117-073-7.
21. Sunada, T. (2012). Topological Crystallography: With a View Towards Discrete Geometric Analysis. Surveys and Tutorials in the Applied Mathematical Sciences. Vol. 6. Springer Japan. ISBN 9784431541769.
22. Kalinin, N.; Guzmán-Sáenz, A.; Prieto, Y.; Shkolnikov, M.; Kalinina, V.; Lupercio, E. (15 August 2018). "Self-organized criticality and pattern emergence through the lens of tropical geometry". Proceedings of the National Academy of Sciences of the United States of America. 115 (35): E8135–E8142. arXiv:1806.09153. Bibcode:2018PNAS..115E8135K. doi:10.1073/pnas.1805847115. ISSN 0027-8424. PMC 6126730. PMID 30111541.
References
• Maslov, Victor (1986). "New superposition principle for optimization problems", Séminaire sur les Équations aux Dérivées Partielles 1985/6, Centre de Mathématiques de l’École Polytechnique, Palaiseau, exposé 24.
• Maslov, Victor (1987). "Méthodes Opératorielles". Moscou, Mir, 707 p. (See Chapter 8, Théorie linéaire sur semi moduli, pp. 652–701).
• Bogart, Tristram; Jensen, Anders; Speyer, David; Sturmfels, Bernd; Thomas, Rekha (2005). "Computing Tropical Varieties". Journal of Symbolic Computation. 42 (1–2): 54–73. arXiv:math/0507563. Bibcode:2005math......7563B. doi:10.1016/j.jsc.2006.02.004. S2CID 24788157.
• Einsiedler, Manfred; Kapranov, Mikhail; Lind, Douglas (2006). "Non-archimedean amoebas and tropical varieties". J. Reine Angew. Math. 601: 139–157. arXiv:math/0408311. Bibcode:2004math......8311E.
• Gathmann, Andreas (2006). "Tropical algebraic geometry". arXiv:math/0601322v1.
• Gross, Mark (2010). Tropical geometry and mirror symmetry. Providence, R.I.: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation. ISBN 9780821852323.
• Itenberg, Illia; Grigory Mikhalkin; Eugenii Shustin (2009). Tropical algebraic geometry (2nd ed.). Basel: Birkhäuser Basel. ISBN 9783034600484. Zbl 1165.14002.
• Maclagan, Diane; Sturmfels, Bernd (2015). Introduction to tropical geometry. American Mathematical Soc. ISBN 9780821851982.
• Mikhalkin, Grigory (2006). "Tropical Geometry and its applications". arXiv:math/0601041v2.
• Mikhalkin, Grigory (2004). "Enumerative tropical algebraic geometry in R2". arXiv:math/0312530v4.
• Mikhalkin, Grigory (2004). "Amoebas of algebraic varieties and tropical geometry". arXiv:math/0403015v1.
• Pachter, Lior; Sturmfels, Bernd (2004). "Tropical geometry of statistical models". Proceedings of the National Academy of Sciences of the United States of America. 101 (46): 16132–16137. arXiv:q-bio/0311009. Bibcode:2004PNAS..10116132P. doi:10.1073/pnas.0406010101. PMC 528960. PMID 15534224. Zbl 1135.62302.
• Speyer, David E. (2003). "The Tropical Grassmannian". arXiv:math/0304218v3.
• Speyer, David; Sturmfels, Bernd (2009) [2004]. "Tropical Mathematics". Mathematics Magazine. 82 (3): 163–173. arXiv:math/0408099. doi:10.4169/193009809x468760. S2CID 119142649. Zbl 1227.14051.
• Theobald, Thorsten (2003). "First steps in tropical geometry". arXiv:math/0306366v2.
Further reading
• Amini, Omid; Baker, Matthew; Faber, Xander, eds. (2013). Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011. Contemporary Mathematics. Vol. 605. Providence, RI: American Mathematical Society. ISBN 978-1-4704-1021-6. Zbl 1281.14002.
• Tropical geometry and mirror symmetry
External links
• Tropical Geometry, I
| Wikipedia |
Tropical projective space
In tropical geometry, a tropical projective space is the tropical analog of the classic projective space.
Definition
Given a module M over the tropical semiring T, its projectivization is the usual projective space of a module: the quotient space of the module (omitting the additive identity 0) under scalar multiplication, omitting multiplication by the scalar additive identity 0:[lower-alpha 1]
$\mathbf {T} (M):=(M\setminus \mathbf {0} )/(\mathbf {T} \setminus 0).$
In the tropical setting, tropical multiplication is classical addition, with unit real number 0 (not 1); tropical addition is minimum or maximum (depending on convention), with unit extended real number ∞ (not 0),[lower-alpha 2] so it is clearer to write this using the extended real numbers, rather than the abstract algebraic units:
$\mathbf {T} (M):=(M\setminus {\boldsymbol {\infty }})/(\mathbf {T} \setminus \infty ).$
Just as in the classical case, the standard n-dimensional tropical projective space is defined as the quotient of the standard (n+1)-dimensional coordinate space by scalar multiplication, with all operations defined coordinate-wise:[1]
$\mathbf {TP} ^{n}:=(\mathbf {T} ^{n+1}\setminus {\boldsymbol {\infty }})/(\mathbf {T} \setminus \infty ).$
Tropical multiplication corresponds to classical addition, so tropical scalar multiplication by c corresponds to adding c to all coordinates. Thus two elements of $\mathbf {T} ^{n+1}\setminus {\boldsymbol {\infty }}$ are identified if their coordinates differ by the same additive amount c:
$(x_{0},\dots ,x_{n})\sim (y_{0},\dots ,y_{n})\iff (x_{0}+c,\dots ,x_{n}+c)=(y_{0},\dots ,y_{n}).$
Notes
1. As usual, scalar multiplication of any vector by 0 yields the identity for vector addition 0, so these must be omitted or all vectors will be identified.
2. ∞ can be interpreted as either positive or negative infinity, depending on convention.
References
1. Mikhalkin 2006, p. 6, example 3.10.
• Richter-Gebert, Jürgen; Sturmfels, Bernd; Theobald, Thorsten (2003). "First steps in tropical geometry". arXiv:math/0306366.
• Mikhalkin, Grigory (2006). "Tropical Geometry and its applications". arXiv:math/0601041.
| Wikipedia |
Tropical analysis
In the mathematical discipline of idempotent analysis, tropical analysis is the study of the tropical semiring.
Applications
The max tropical semiring can be used appropriately to determine marking times within a given Petri net and a vector filled with marking state at the beginning: $-\infty $ (unit for max, tropical addition) means "never before", while 0 (unit for addition, tropical multiplication) is "no additional time".
Tropical cryptography is cryptography based on the tropical semiring.
Tropical geometry is an analog to algebraic geometry, using the tropical semiring.
References
• Litvinov, G. L. (2005). "The Maslov dequantization, idempotent and tropical mathematics: A brief introduction". arXiv:math/0507014v1.
Further reading
• Butkovič, Peter (2010), Max-linear Systems: Theory and Algorithms, Springer Monographs in Mathematics, Springer-Verlag, doi:10.1007/978-1-84996-299-5, ISBN 978-1-84996-298-8
• Bernd Heidergott; Geert Jan Olsder; Jacob van der Woude (2005). Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications. p. 224. ISBN 978-0-69111763-8.
See also
• Lunar arithmetic
External links
• MaxPlus algebra
• Max Plus working group, INRIA Rocquencourt
| Wikipedia |
Bitangents of a quartic
In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane.
An explicit quartic with twenty-eight real bitangents was first given by Plücker (1839)[1] As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines.[2] Shioda (1995) gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane.
Example
The Trott curve, another curve with 28 real bitangents, is the set of points (x,y) satisfying the degree four polynomial equation
$\displaystyle 144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0.$
These points form a nonsingular quartic curve that has genus three and that has twenty-eight real bitangents.[3]
Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate the two ovals, and two that do not. Additionally, each oval bounds a nonconvex region of the plane and has one bitangent spanning the nonconvex portion of its boundary.
Connections to other structures
The dual curve to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the primal curve.
The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form
${\begin{bmatrix}a&b&c\\d&e&f\\\end{bmatrix}}$
where a, b, c, d, e, f are all zero or one and where
$ad+be+cf=1\ (\operatorname {mod} \ 2).$[4]
There are 64 choices for a, b, c, d, e, f, but only 28 of these choices produce an odd sum. One may also interpret a, b, c as the homogeneous coordinates of a point of the Fano plane and d, e, f as the coordinates of a line in the same finite projective plane; the condition that the sum is odd is equivalent to requiring that the point and the line do not touch each other, and there are 28 different pairs of a point and a line that do not touch.
The points and lines of the Fano plane that are disjoint from a non-incident point-line pair form a triangle, and the bitangents of a quartic have been considered as being in correspondence with the 28 triangles of the Fano plane.[5] The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the Coxeter graph.[6]
The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2 del Pezzo surface,[5] and to the 28 odd theta characteristics.
The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonic sextic curve of genus 4, form a "trinity" in the sense of Vladimir Arnold, specifically a form of McKay correspondence,[7][8][9] and can be related to many further objects, including E7 and E8, as discussed at trinities.
Notes
1. See e.g. Gray (1982).
2. Blum & Guinand (1964).
3. Trott (1997).
4. Riemann (1876); Cayley (1879).
5. Manivel (2006).
6. Dejter, Italo J. (2011), "From the Coxeter graph to the Klein graph", Journal of Graph Theory, 70: 1–9, arXiv:1002.1960, doi:10.1002/jgt.20597, S2CID 754481.
7. le Bruyn, Lieven (17 June 2008), Arnold's trinities, archived from the original on 2011-04-11
8. Arnold 1997, p. 13 – Arnold, Vladimir, 1997, Toronto Lectures, Lecture 2: Symplectization, Complexification and Mathematical Trinities, June 1997 (last updated August, 1998). TeX, PostScript, PDF
9. (McKay & Sebbar 2007, p. 11)
References
• Blum, R.; Guinand, A. P. (1964). "A quartic with 28 real bitangents". Canadian Mathematical Bulletin. 7 (3): 399–404. doi:10.4153/cmb-1964-038-6.
• Cayley, Arthur (1879), "On the bitangents of a quartic", Salmon's Higher Plane Curves, pp. 387–389. In The collected mathematical papers of Arthur Cayley, Andrew Russell Forsyth, ed., The University Press, 1896, vol. 11, pp. 221–223.
• Gray, Jeremy (1982), "From the history of a simple group", The Mathematical Intelligencer, 4 (2): 59–67, doi:10.1007/BF03023483, MR 0672918, S2CID 14602496. Reprinted in Levy, Silvio, ed. (1999), The Eightfold Way, MSRI Publications, vol. 35, Cambridge University Press, pp. 115–131, ISBN 0-521-66066-1, MR 1722415.
• Manivel, L. (2006), "Configurations of lines and models of Lie algebras", Journal of Algebra, 304 (1): 457–486, arXiv:math/0507118, doi:10.1016/j.jalgebra.2006.04.029, S2CID 17374533.
• McKay, John; Sebbar, Abdellah (2007). "Replicable Functions: An Introduction". Frontiers in Number Theory, Physics, and Geometry II: 373–386. doi:10.1007/978-3-540-30308-4_10. ISBN 978-3-540-30307-7.
• Plücker, J. (1839), Theorie der algebraischen Curven: gegrundet auf eine neue Behandlungsweise der analytischen Geometrie, Berlin: Adolph Marcus.
• Riemann, G. F. B. (1876), "Zur Theorie der Abel'schen Funktionen für den Fall p = 3", Ges. Werke, Leipzig, pp. 456–472{{citation}}: CS1 maint: location missing publisher (link). As cited by Cayley.
• Shioda, Tetsuji (1995), "Weierstrass transformations and cubic surfaces" (PDF), Commentarii Mathematici Universitatis Sancti Pauli, 44 (1): 109–128, MR 1336422
• Trott, Michael (1997), "Applying GroebnerBasis to Three Problems in Geometry", Mathematica in Education and Research, 6 (1): 15–28.
| Wikipedia |
Truchet tiles
In information visualization and graphic design, Truchet tiles are square tiles decorated with patterns that are not rotationally symmetric. When placed in a square tiling of the plane, they can form varied patterns, and the orientation of each tile can be used to visualize information associated with the tile's position within the tiling.[1]
Truchet tiles were first described in a 1704 memoir by Sébastien Truchet entitled "Mémoire sur les combinaisons", and were popularized in 1987 by Cyril Stanley Smith.[1][2]
Variations
Contrasting triangles
The tile originally studied by Truchet is split along the diagonal into two triangles of contrasting colors. The tile has four possible orientations.
Some examples of surface filling made tiling such a pattern.
With a scheme:
With random placement:
Quarter-circles
A second common form of the Truchet tiles, due to Smith (1987), decorates each tile with two quarter-circles connecting the midpoints of adjacent sides. Each such tile has two possible orientations.
The Truchet tile
Inverse of the Truchet tile, created by any 90° rotation or orthogonal flip
We have such a tiling:
This type of tile has also been used in abstract strategy games Trax and the Black Path Game, prior to Smith's work.[1]
Diagonal
A labyrinth can be generated by tiles in the form of a white square with a black diagonal. As with the quarter-circle tiles, each such tile has two orientations.[3] The connectivity of the resulting labyrinth can be analyzed mathematically using percolation theory as bond percolation at the critical point of a diagonally-oriented grid. Nick Montfort considers the single line of Commodore 64 BASIC required to generate such patterns - 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 - to be "a concrete poem, a found poem".[3]
See also
Wikimedia Commons has media related to Truchet tiles.
• Girih tiles
• Wallpaper group
• Wang tiles
References
1. Browne, Cameron (2008), "Truchet curves and surfaces", Computers & Graphics, 32 (2): 268–281, doi:10.1016/j.cag.2007.10.001.
2. Smith, Cyril Stanley (1987), "The tiling patterns of Sebastian Truchet and the topology of structural hierarchy", Leonardo, 20 (4): 373–385, doi:10.2307/1578535. With a translation of Truchet's text by Pauline Boucher.
3. Montfort, Nick (2012). 10 PRINT CHR$(205.5+RND(1)); : GOTO 10. MIT Press.
External links
• Weisstein, Eric W. "Truchet Tiling". MathWorld.
• Online Truchet Pattern Generator: https://truchetpatterns.netlify.app/
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 |
True-range multilateration
True-range multilateration (also termed range-range multilateration and spherical multilateration) is a method to determine the location of a movable vehicle or stationary point in space using multiple ranges (distances) between the vehicle/point and multiple spatially-separated known locations (often termed "stations"). [1][2] Energy waves may be involved in determining range, but are not required.
True-range multilateration is both a mathematical topic and an applied technique used in several fields. A practical application involving a fixed location occurs in surveying.[3][4] Applications involving vehicle location are termed navigation when on-board persons/equipment are informed of its location, and are termed surveillance when off-vehicle entities are informed of the vehicle's location.
Two slant ranges from two known locations can be used to locate a third point in a two-dimensional Cartesian space (plane), which is a frequently applied technique (e.g., in surveying). Similarly, two spherical ranges can be used to locate a point on a sphere, which is a fundamental concept of the ancient discipline of celestial navigation — termed the altitude intercept problem. Moreover, if more than the minimum number of ranges are available, it is good practice to utilize those as well. This article addresses the general issue of position determination using multiple ranges.
In two-dimensional geometry, it is known that if a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two – one of which is the desired solution and the other is an ambiguous solution. Additional information often narrow the possibilities down to a unique location. In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii also provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line).
True-range multilateration can be contrasted to the more frequently encountered pseudo-range multilateration, which employs range differences to locate a (typically, movable) point. Pseudo range multilateration is almost always implemented by measuring times-of-arrival (TOAs) of energy waves. True-range multilateration can also be contrasted to triangulation, which involves the measurement of angles.
Terminology
There is no accepted or widely-used general term for what is termed true-range multilateration here . That name is selected because it: (a) is an accurate description and partially familiar terminology (multilateration is often used in this context); (b) avoids specifying the number of ranges involved (as does, e.g., range-range; (c) avoids implying an application (as do, e.g., DME/DME navigation or trilateration) and (d) and avoids confusion with the more common pseudo-range multilateration.
Obtaining ranges
For similar ranges and measurement errors, a navigation and surveillance system based on true-range multilateration provide service to a significantly larger 2-D area or 3-D volume than systems based on pseudo-range multilateration. However, it is often more difficult or costly to measure true=ranges than it is to measure pseudo ranges. For distances up to a few miles and fixed locations, true-range can be measured manually. This has been done in surveying for several thousand years – e.g., using ropes and chains.
For longer distances and/or moving vehicles, a radio/radar system is generally needed. This technology was first developed circa 1940 in conjunction with radar. Since then, three methods have been employed:
• Two-way range measurement, one party active – This is the method used by traditional radars (sometimes termed primary radars) to determine the range of a non-cooperative target, and now used by laser rangefinders. Its major limitations are that: (a) the target does not identify itself, and in a multiple target situation, mis-assignment of a return can occur; (b) the return signal is attenuated (relative to the transmitted signal) by the fourth power of the vehicle-station range (thus, for distances of tens of miles or more, stations generally require high-power transmitters and/or large/sensitive antennas); and (c) many systems utilize line-of-sight propagation, which limits their ranges to less than 20 miles when both parties are at similar heights above sea level.
• Two-way range measurement, both parties active – This method was reportedly first used for navigation by the Y-Gerät aircraft guidance system fielded in 1941 by the Luftwaffe. It is now used globally in air traffic control – e.g., secondary radar surveillance and DME/DME navigation. It requires that both parties have both transmitters and receivers, and may require that interference issues be addressed.
• One-way range measurement – The time of flight (TOF) of electromagnetic energy between multiple stations and the vehicle is measured based on transmission by one party and reception by the other. This is the most recently developed method, and was enabled by the development of atomic clocks; it requires that the vehicle (user) and stations having synchronized clocks. It has been successfully demonstrated (experimentally) with Loran-C and GPS.[2][5]
Solution methods
True-range multilateration algorithms may be partitioned based on
• problem space dimension (generally, two or three),
• problem space geometry (generally, Cartesian or spherical) and
• presence of redundant measurements (more than the problem space dimension).
Any pseudo-range multilateration algorithm can be specialized for use with true-range multilateration.
Two Cartesian dimensions, two measured slant ranges (trilateration)
An analytic solution has likely been known for over 1,000 years, and is given in several texts.[6] Moreover, one can easily adapt algorithms for a three dimensional Cartesian space.
The simplest algorithm employs analytic geometry and a station-based coordinate frame. Thus, consider the circle centers (or stations) C1 and C2 in Fig. 1 which have known coordinates (e.g., have already been surveyed) and thus whose separation $U$ is known. The figure 'page' contains C1 and C2. If a third 'point of interest' P (e.g., a vehicle or another point to be surveyed) is at unknown point $(x,y)$, then Pythagoras's theorem yields
${\begin{aligned}r_{1}^{2}&=x^{2}+y^{2}\\[4pt]r_{2}^{2}&=(U-x)^{2}+y^{2}\end{aligned}}$
Thus,
${\begin{aligned}x&={\frac {r_{1}^{2}-r_{2}^{2}+U^{2}}{2U}}\\[4pt]y&=\pm {\sqrt {r_{1}^{2}-x^{2}}}\end{aligned}}$
(1)
Note that $y$ has two values (i.e., solution is ambiguous); this is usually not a problem.
While there are many enhancements, Equation 1 is the most fundamental true-range multilateration relationship. Aircraft DME/DME navigation and the trilateration method of surveying are examples of its application. During World War II Oboe and during the Korean War SHORAN used the same principle to guide aircraft based on measured ranges to two ground stations. SHORAN was later used for off-shore oil exploration and for aerial surveying. The Australian Aerodist aerial survey system utilized 2-D Cartesian true-range multilateration.[7] This 2-D scenario is sufficiently important that the term trilateration is often applied to all applications involving a known baseline and two range measurements.
The baseline containing the centers of the circles is a line of symmetry. The correct and ambiguous solutions are perpendicular to and equally distant from (on opposite sides of) the baseline. Usually, the ambiguous solution is easily identified. For example, if P is a vehicle, any motion toward or away from the baseline will be opposite that of the ambiguous solution; thus, a crude measurement of vehicle heading is sufficient. A second example: surveyors are well aware of which side of the baseline that P lies. A third example: in applications where P is an aircraft and C1 and C2 are on the ground, the ambiguous solution is usually below ground.
If needed, the interior angles of triangle C1-C2-P can be found using the trigonometric law of cosines. Also, if needed, the coordinates of P can be expressed in a second, better-known coordinate system—e.g., the Universal Transverse Mercator (UTM) system—provided the coordinates of C1 and C2 are known in that second system. Both are often done in surveying when the trilateration method is employed.[8] Once the coordinates of P are established, lines C1-P and C2-P can be used as new baselines, and additional points surveyed. Thus, large areas or distances can be surveyed based on multiple, smaller triangles—termed a traverse.
An implied assumption for the above equation to be true is that $r_{1}$ and $r_{2}$ relate to the same position of P. When P is a vehicle, then typically $r_{1}$ and $r_{2}$ must be measured within a synchronization tolerance that depends on the vehicle speed and the allowable vehicle position error. Alternatively, vehicle motion between range measurements may be accounted for, often by dead reckoning.
A trigonometric solution is also possible (side-side-side case). Also, a solution employing graphics is possible. A graphical solution is sometimes employed during real-time navigation, as an overlay on a map.
Three Cartesian dimensions, three measured slant ranges
There are multiple algorithms that solve the 3-D Cartesian true-range multilateration problem directly (i.e., in closed-form) – e.g., Fang.[9] Moreover, one can adopt closed-form algorithms developed for pseudo range multilateration.[10][6] Bancroft's algorithm[11] (adapted) employs vectors, which is an advantage in some situations.
The simplest algorithm corresponds to the sphere centers in Fig. 2. The figure 'page' is the plane containing C1, C2 and C3. If P is a 'point of interest' (e.g., vehicle) at $(x,y,z)$, then Pythagoras's theorem yields the slant ranges between P and the sphere centers:
${\begin{aligned}r_{1}^{2}&=x^{2}+y^{2}+z^{2}\\[4pt]r_{2}^{2}&=(x-U)^{2}+y^{2}+z^{2}\\[4pt]r_{3}^{2}&=(x-V_{x})^{2}+(y-V_{y})^{2}+z^{2}\end{aligned}}$
Thus, the coordinates of P are:
${\begin{aligned}x&={\frac {r_{1}^{2}-r_{2}^{2}+U^{2}}{2U}}\\[4pt]y&={\frac {r_{1}^{2}-r_{3}^{2}+V_{x}^{2}+V_{y}^{2}-2V_{x}x}{2V_{y}}}\\[4pt]z&=\pm {\sqrt {r_{1}^{2}-x^{2}-y^{2}}}\end{aligned}}$
(2)
The plane containing the sphere centers is a plane of symmetry. The correct and ambiguous solutions are perpendicular to it and equally distant from it, on opposite sides.
Many applications of 3-D true-range multilateration involve short ranges—e.g., precision manufacturing.[12] Integrating range measurement from three or more radars (e.g., FAA's ERAM) is a 3-D aircraft surveillance application. 3-D true-range multilateration has been used on an experimental basis with GPS satellites for aircraft navigation.[5] The requirement that an aircraft be equipped with an atomic clock precludes its general use. However, GPS receiver clock aiding is an area of active research, including aiding over a network. Thus, conclusions may change.[13] 3-D true-range multilateration was evaluated by the International Civil Aviation Organization as an aircraft landing system, but another technique was found to be more efficient.[14] Accurately measuring the altitude of aircraft during approach and landing requires many ground stations along the flight path.
Two spherical dimensions, two or more measured spherical ranges
This is a classic celestial (or astronomical) navigation problem, termed the altitude intercept problem (Fig. 3). It's the spherical geometry equivalent of the trilateration method of surveying (although the distances involved are generally much larger). A solution at sea (not necessarily involving the sun and moon) was made possible by the marine chronometer (introduced in 1761) and the discovery of the 'line of position' (LOP) in 1837. The solution method now most taught at universities (e.g., U.S. Naval Academy) employs spherical trigonometry to solve an oblique spherical triangle based on sextant measurements of the 'altitude' of two heavenly bodies.[15][16] This problem can also be addressed using vector analysis.[17] Historically, graphical techniques – e.g., the intercept method – were employed. These can accommodate more than two measured 'altitudes'. Owing to the difficulty of making measurements at sea, 3 to 5 'altitudes' are often recommended.
As the earth is better modeled as an ellipsoid of revolution than a sphere, iterative techniques may be used in modern implementations.[18] In high-altitude aircraft and missiles, a celestial navigation subsystem is often integrated with an inertial navigation subsystem to perform automated navigation—e.g., U.S. Air Force SR-71 Blackbird and B-2 Spirit.
While intended as a 'spherical' pseudo range multilateration system, Loran-C has also been used as a 'spherical' true-range multilateration system by well-equipped users (e.g., Canadian Hydrographic Service).[2] This enabled the coverage area of a Loran-C station triad to be extended significantly (e.g., doubled or tripled) and the minimum number of available transmitters to be reduced from three to two. In modern aviation, slant ranges rather than spherical ranges are more often measured; however, when aircraft altitude is known, slant ranges are readily converted to spherical ranges.[6]
Redundant range measurements
When there are more range measurements available than there are problem dimensions, either from the same C1 and C2 (or C1, C2 and C3) stations, or from additional stations, at least these benefits accrue:
• 'Bad' measurements can be identified and rejected
• Ambiguous solutions can be identified automatically (i.e., without human involvement) -- requires an additional station
• Errors in 'good' measurements can be averaged, reducing their effect.
The iterative Gauss–Newton algorithm for solving non-linear least squares (NLLS) problems is generally preferred when there are more 'good' measurements than the minimum necessary. An important advantage of the Gauss–Newton method over many closed-form algorithms is that it treats range errors linearly, which is often their nature, thereby reducing the effect of range errors by averaging.[10] The Gauss–Newton method may also be used with the minimum number of measured ranges. Since it is iterative, the Gauss–Newton method requires an initial solution estimate.
In 3-D Cartesian space, a fourth sphere eliminates the ambiguous solution that occurs with three ranges, provided its center is not co-planar with the first three. In 2-D Cartesian or spherical space, a third circle eliminates the ambiguous solution that occurs with two ranges, provided its center is not co-linear with the first two.
One-time application versus repetitive application
This article largely describes 'one-time' application of the true-range multilateration technique, which is the most basic use of the technique. With reference to Fig. 1, the characteristic of 'one-time' situations is that point P and at least one of C1 and C2 change from one application of the true-range multilateration technique to the next. This is appropriate for surveying, celestial navigation using manual sightings, and some aircraft DME/DME navigation.
However, in other situations, the true-range multilateration technique is applied repetitively (essentially continuously). In those situations, C1 and C2 (and perhaps Cn, n = 3,4,...) remain constant and P is the same vehicle. Example applications (and selected intervals between measurements) are: multiple radar aircraft surveillance (5 and 12 seconds, depending upon radar coverage range), aerial surveying, Loran-C navigation with a high-accuracy user clock (roughly 0.1 seconds), and some aircraft DME/DME navigation (roughly 0.1 seconds). Generally, implementations for repetitive use: (a) employ a 'tracker' algorithm[19] (in addition to the multilateration solution algorithm), which enables measurements collected at different times to be compared and averaged in some manner; and (b) utilize an iterative solution algorithm, as they (b1) admit varying numbers of measurements (including redundant measurements) and (b2) inherently have an initial guess each time the solution algorithm is invoked.
Hybrid multilateration systems
Hybrid multilateration systems – those that are neither true-range nor pseudo range systems – are also possible. For example, in Fig. 1, if the circle centers are shifted to the left so that C1 is at $x_{1}^{\prime }=-{\tfrac {1}{2}}U,y_{1}^{\prime }=0$ and C2 is at $x_{2}^{\prime }={\tfrac {1}{2}}U,y_{2}^{\prime }=0$ then the point of interest P is at
${\begin{aligned}x^{\prime }&={\frac {(r_{1}^{\prime }+r_{2}^{\prime })(r_{1}^{\prime }-r_{2}^{\prime })}{2U}}\\[4pt]y^{\prime }&=\pm {\frac {{\sqrt {(r_{1}^{\prime }+r_{2}^{\prime })^{2}-U^{2}}}{\sqrt {U^{2}-(r_{1}^{\prime }-r_{2}^{\prime })^{2}}}}{2U}}\end{aligned}}$
This form of the solution explicitly depends on the sum and difference of $r_{1}^{\prime }$ and $r_{2}^{\prime }$ and does not require 'chaining' from the $x^{\prime }$-solution to the $y^{\prime }$-solution. It could be implemented as a true-range multilateration system by measuring $r_{1}^{\prime }$ and $r_{2}^{\prime }$.
However, it could also be implemented as a hybrid multilateration system by measuring $r_{1}^{\prime }+r_{2}^{\prime }$ and $r_{1}^{\prime }-r_{2}^{\prime }$ using different equipment – e.g., for surveillance by a multistatic radar with one transmitter and two receivers (rather than two monostatic radars). While eliminating one transmitter is a benefit, there is a countervailing 'cost': the synchronization tolerance for the two stations becomes dependent on the propagation speed (typically, the speed of light) rather that the speed of point P, in order to accurately measure both $r_{1}^{\prime }\pm r_{2}^{\prime }$.
While not implemented operationally, hybrid multilateration systems have been investigated for aircraft surveillance near airports and as a GPS navigation backup system for aviation.[20]
Preliminary and final computations
The position accuracy of a true-range multilateration system—e.g., accuracy of the $(x,y)$ coordinates of point P in Fig. 1 -- depends upon two factors: (1) the range measurement accuracy, and (2) the geometric relationship of P to the system's stations C1 and C2. This can be understood from Fig. 4. The two stations are shown as dots, and BLU denotes baseline units. (The measurement pattern is symmetric about both the baseline and the perpendicular bisector of the baseline, and is truncated in the figure.) As is commonly done, individual range measurement errors are taken to be independent of range, statistically independent and identically distributed. This reasonable assumption separates the effects of user-station geometry and range measurement errors on the error in the calculated $(x,y)$ coordinates of P. Here, the measurement geometry is simply the angle at which two circles cross—or equivalently, the angle between lines P-C1 and P-C2. When point P- is not on a circle, the error in its position is approximately proportional to the area bounded by the nearest two blue and nearest two magenta circles.
Without redundant measurements, a true-range multilateration system can be no more accurate than the range measurements, but can be significantly less accurate if the measurement geometry is not chosen properly. Accordingly, some applications place restrictions on the location of point P. For a 2-D Cartesian (trilateration) situation, these restrictions take one of two equivalent forms:
• The allowable interior angle at P between lines P-C1 and P-C2: The ideal is a right angle, which occurs at distances from the baseline of one-half or less of the baseline length; maximum allowable deviations from the ideal 90 degrees may be specified.
• The horizontal dilution of precision (HDOP), which multiplies the range error in determining the position error: For two dimensions, the ideal (minimum) HDOP is the square root of 2 (${\sqrt {2}}\approx 1.414$), which occurs when the angle between P-C1 and P-C2 is 90 degrees; a maximum allowable HDOP value may be specified. (Here, equal HDOPs are simply the locus of points in Fig. 4 having the same crossing angle.)
Planning a true-range multilateration navigation or surveillance system often involves a dilution of precision (DOP) analysis to inform decisions on the number and location of the stations and the system's service area (two dimensions) or service volume (three dimensions).[21][22] Fig. 5 shows horizontal DOPs (HDOPs) for a 2-D, two-station true-range multilateration system. HDOP is infinite along the baseline and its extensions, as only one of the two dimensions is actually measured. A user of such a system should be roughly broadside of the baseline and within an application-dependent range band. For example, for DME/DME navigation fixes by aircraft, the maximum HDOP permitted by the U.S. FAA is twice the minimum possible value, or 2.828,[23] which limits the maximum usage range (which occurs along the baseline bisector) to 1.866 times the baseline length. (The plane containing two DME ground stations and an aircraft in not strictly horizontal, but usually is nearly so.) Similarly, surveyors select point P in Fig. 1 so that C1-C2-P roughly form an equilateral triangle (where HDOP = 1.633).
Errors in trilateration surveys are discussed in several documents.[24][25] Generally, emphasis is placed on the effects of range measurement errors, rather than on the effects of algorithm numerical errors.
Applications
• Land surveying using the trilateration method
• Aerial surveying
• Maritime archeology surveying[26]
• DME/DME RNAV aircraft navigation[23][27]
• Multiple radar integration (e.g., FAA ERAM)[28]
• Celestial navigation using the altitude intercept method
• Intercept method—Graphical solution to the altitude intercept problem
• Calibrating laser interferometers[12]
• SHORAN, Oboe, Gee-H—Aircraft guidance systems developed for 'blind' bombing
• JTIDS (Joint Tactical Information Distribution System) -- U.S./NATO system that (among other capabilities) locates participants in a network using inter-participant ranges
• USAF SR-71 Blackbird aircraft—Employs astro-inertial navigation
• USAF B-2 Spirit aircraft—Employs astro-inertial navigation
• Experimental Loran-C technique[2]
Advantages and disadvantages for vehicle navigation and surveillance
Navigation and surveillance systems typically involve vehicles and require that a government entity or other organization deploy multiple stations that employ a form of radio technology (i.e., utilize electromagnetic waves). The advantages and disadvantages of employing true-range multilateration for such a system are shown in the following table.
Advantages Disadvantages
• Station locations are flexible; they can be placed centrally or peripherally
• Accuracy degrades slowly with distance from station (generally better than pseudo-range multilateration)
• Requires one fewer station than a pseudo range multilateration system
• Station synchronization is not demanding (based on speed of point of interest, and may be addressed by dead reckoning)
• Often a user is required to have both a transmitter and a receiver
• Cooperative system accuracy is sensitive to equipment turn-around error
• Cannot be used for stealth surveillance
• Non-cooperative surveillance involves path losses to the fourth power of distance
True-range multilateration is often contrasted with (pseudo range) multilateration, as both require a form of user ranges to multiple stations. Complexity and cost of user equipage is likely the most important factor in limiting use of true-range multilateration for vehicle navigation and surveillance. Some uses are not the original purpose for system deployment – e.g., DME/DME aircraft navigation.
See also
• Distance geometry problem, similar technique applied to molecules
• Celestial navigation—ancient technique of navigation based on heavenly bodies
• Distance measuring equipment (DME) -- System used to measure distance between an aircraft and a ground station
• Euclidean distance
• Intercept method—Graphical technique used in celestial navigation
• Laser rangefinder
• Multilateration – Addresses pseudo range multilateration
• Rangefinder—Systems used to measure distance between two points on the ground
• Resection (orientation)
• SHORAN—Developed as a military aircraft navigation system, later used for civil purposes
• Surveying
• Tellurometer—First microwave electronic rangefinder
• Triangulation – Surveying method based on measuring angles
References
1. Accuracy limitations of range-range (spherical) multilateration systems, Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Report Number: DOT/TSC-RA-3-8-(1) (Technical note 1973-43), Oct. 11, 1973
2. "Rho-Rho Loran-C Combined with Satellite Navigation for Offshore Surveys". S.T. Grant, International Hydrographic Review, undated
3. Wirtanen, Theodore H. (1969). "Laser Multilateration". Journal of the Surveying and Mapping Division. American Society of Civil Engineers (ASCE). 95 (1): 81–92. doi:10.1061/jsueax.0000322. ISSN 0569-8073.
4. Escobal, P. R.; Fliegel, H. F.; Jaffe, R. M.; Muller, P. M.; Ong, K. M.; Vonroos, O. H. (2013-08-07). "A 3-D Multilateration: A Precision Geodetic Measurement System". JPL Quart. Tech. Rev. 2 (3). Retrieved 2022-11-06.
5. Impact of Rubidium Clock Aiding on GPS Augmented Vehicular Navigation, Zhaonian Zhang; University of Calgary; December, 1997.
6. Geyer, Michael (June 2016). Earth-Referenced Aircraft Navigation and Surveillance Analysis. U.S. DOT National Transportation Library: U.S. DOT John A. Volpe National Transportation Systems Center.
7. Adastra Aerial Surveys retrieved Jan. 22, 2019.
8. "The Nature of Geographic Information: Trilateration", Pennsylvania State Univ., 2018.
9. "Trilateration and extension to global positioning system navigation", B.T. Fang, Journal of Guidance, Control, and Dynamics, vol. 9 (1986), pp 715–717.
10. "Closed-form Algorithms in Mobile Positioning: Myths and Misconceptions", Niilo Sirola, Proceedings of the 7th Workshop on Positioning, Navigation and Communication 2010 (WPNC'10), March 11, 2010.
11. "An Algebraic Solution of the GPS Equations", Stephen Bancroft, IEEE Transactions on Aerospace and Electronic Systems, Volume: AES-21, Issue: 7 (Jan. 1985), pp 56–59.
12. LaserTracer – A New Type of Self Tracking Laser Interferometer, Carl-Thomas Schneider, IWAA2004, CERN, Geneva, October 2004
13. "How a Chip-Scale Atomic Clock Can Help Mitigate Broadband Interference"; Fang-Cheng Chan, Mathieu Joerger, Samer Khanafseh, Boris Pervan, and Ondrej Jakubov; GPS World -- Innovations; May 2014.
14. "Microwave Landing System"; Thomas E. Evans; IEEE Aerospace and Electronic Systems Magazine; Vol. 1, Issue 5; May 1986.
15. Spherical Trigonometry, Isaac Todhunter, MacMillan; 5th edition, 1886.
16. A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples, John Casey, Dublin, Hodges, Figgis & Co., 1889.
17. "Vector-based geodesy", Chris Veness. 2016.
18. "STELLA (System To Estimate Latitude and Longitude Astronomically)", George Kaplan, John Bangert, Nancy Oliversen; U.S. Naval Observatory, 1999.
19. Tracking and Data Fusion: A Handbook of Algorithms; Y. Bar-Shalom, P.K. Willett, X. Tian; 2011
20. "Alternative Position, Navigation, and Timing: The Need for Robust Radionavigation"; M.J. Narins, L.V. Eldredge, P. Enge, S.C. Lo, M.J. Harrison, and R. Kenagy; Chapter in Global Navigation Satellite SystemsJoint Workshop of the National Academy of Engineering and the Chinese Academy of Engineering (2012).
21. "Dilution of Precision", Richard Langeley, GPS World, May 1999, pp 52–59.
22. Accuracy Limitations of Range-Range (Spherical) Multilateration Systems, Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Technical Note 1973-43, Oct. 11, 1973.
23. "DME/DME for Alternate Position, Navigation, and Timing (APNT)", Robert W. Lilley and Robert Erikson, Federal Aviation Administration, White Paper, July 23, 2012.
24. Statistical Methods in Surveying by Trilateration; William Navidi, William S Murphy, Jr and Willy Hereman; December 20, 1999.
25. Comparison of the Accuracy of Triangulation, Trilateration and Triangulation-Trilateration; K.L. Provoro; Novosibirsk Institnte of Engineers of Geodesy; 1960.
26. "Trilateration in Maritime Archeology", YouTube, U.S. National Oceanic and Atmospheric Administration, 2006.
27. "DME/DME Accuracy", Michael Tran, Proceedings of the 2008 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 2008, pp. 443–451.
28. "Radar Basics", Christian Wolff, undated
External links
• stackexchange.com, PHP / Python Implementation
| Wikipedia |
Truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (true or false).[1][2]
Computing
In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false".
Classical logic
⊤ ·∧·
true conjunction
¬ ↕ ↕
⊥ ·∨·
false disjunction
Negation interchanges
true with false and
conjunction with disjunction.
In classical logic, with its intended semantics, the truth values are true (denoted by 1 or the verum ⊤), and untrue or false (denoted by 0 or the falsum ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws:
¬(p ∧ q) ⇔ ¬p ∨ ¬q
¬(p ∨ q) ⇔ ¬p ∧ ¬q
Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.
Intuitionistic and constructive logic
Main article: Constructivism (mathematics)
In intuitionistic logic, and more generally, constructive mathematics, statements are assigned a truth value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if one can build a proof of the statement from those axioms. A statement is false if one can deduce a contradiction from it. This leaves open the possibility of statements that have not yet been assigned a truth value. Unproven statements in intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted). Indeed, one can prove that they have no third truth value, a result dating back to Glivenko in 1928.[3]
Instead, statements simply remain of unknown truth value, until they are either proven or disproven.
There are various ways of interpreting intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation. See also Intuitionistic logic § Semantics.
Multi-valued logic
Multi-valued logics (such as fuzzy logic and relevance logic) allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval [0,1] such structure is a total order; this may be expressed as the existence of various degrees of truth.
Algebraic semantics
Main article: Algebraic logic
Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the necessary truth of formulae.
But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Boolean algebra semantics of classical propositional calculus.
In other theories
Intuitionistic type theory uses types in the place of truth values.
Topos theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.
See also
• Agnosticism
• Bayesian probability
• Circular reasoning
• Degree of truth
• False dilemma
• History of logic § Algebraic period
• Paradox
• Semantic theory of truth
• Slingshot argument
• Supervaluationism
• Truth-value semantics
• Verisimilitude
References
1. Shramko, Yaroslav; Wansing, Heinrich. "Truth Values". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
2. "Truth value". Lexico UK English Dictionary. Oxford University Press. n.d.
3. Proof that intuitionistic logic has no third truth value, Glivenko 1928
External links
• Shramko, Yaroslav; Wansing, Heinrich. "Truth Values". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
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Logical truth ⊤
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| Wikipedia |
True arithmetic
In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.[1] This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.
Definition
The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the language of first-order arithmetic are built up from these symbols together with the logical symbols in the usual manner of first-order logic.
The structure ${\mathcal {N}}$ is defined to be a model of Peano arithmetic as follows.
• The domain of discourse is the set $\mathbb {N} $ of natural numbers,
• The symbol 0 is interpreted as the number 0,
• The function symbols are interpreted as the usual arithmetical operations on $\mathbb {N} $,
• The equality and less-than relation symbols are interpreted as the usual equality and order relation on $\mathbb {N} $.
This structure is known as the standard model or intended interpretation of first-order arithmetic.
A sentence in the language of first-order arithmetic is said to be true in ${\mathcal {N}}$ if it is true in the structure just defined. The notation ${\mathcal {N}}\models \varphi $ is used to indicate that the sentence $\varphi $ is true in ${\mathcal {N}}.$
True arithmetic is defined to be the set of all sentences in the language of first-order arithmetic that are true in ${\mathcal {N}}$, written Th(${\mathcal {N}}$). This set is, equivalently, the (complete) theory of the structure ${\mathcal {N}}$.[2]
Arithmetic undefinability
The central result on true arithmetic is the undefinability theorem of Alfred Tarski (1936). It states that the set Th(${\mathcal {N}}$) is not arithmetically definable. This means that there is no formula $\varphi (x)$ in the language of first-order arithmetic such that, for every sentence θ in this language,
${\mathcal {N}}\models \theta \quad {\text{if and only if}}\quad {\mathcal {N}}\models \varphi ({\underline {\#(\theta )}}).$
Here ${\underline {\#(\theta )}}$ is the numeral of the canonical Gödel number of the sentence θ.
Post's theorem is a sharper version of the undefinability theorem that shows a relationship between the definability of Th(${\mathcal {N}}$) and the Turing degrees, using the arithmetical hierarchy. For each natural number n, let Thn(${\mathcal {N}}$) be the subset of Th(${\mathcal {N}}$) consisting of only sentences that are $\Sigma _{n}^{0}$ or lower in the arithmetical hierarchy. Post's theorem shows that, for each n, Thn(${\mathcal {N}}$) is arithmetically definable, but only by a formula of complexity higher than $\Sigma _{n}^{0}$. Thus no single formula can define Th(${\mathcal {N}}$), because
${\mbox{Th}}({\mathcal {N}})=\bigcup _{n\in \mathbb {N} }{\mbox{Th}}_{n}({\mathcal {N}})$
but no single formula can define Thn(${\mathcal {N}}$) for arbitrarily large n.
Computability properties
As discussed above, Th(${\mathcal {N}}$) is not arithmetically definable, by Tarski's theorem. A corollary of Post's theorem establishes that the Turing degree of Th(${\mathcal {N}}$) is 0(ω), and so Th(${\mathcal {N}}$) is not decidable nor recursively enumerable.
Th(${\mathcal {N}}$) is closely related to the theory Th(${\mathcal {R}}$) of the recursively enumerable Turing degrees, in the signature of partial orders.[3] In particular, there are computable functions S and T such that:
• For each sentence φ in the signature of first-order arithmetic, φ is in Th(${\mathcal {N}}$) if and only if S(φ) is in Th(${\mathcal {R}}$).
• For each sentence ψ in the signature of partial orders, ψ is in Th(${\mathcal {R}}$) if and only if T(ψ) is in Th(${\mathcal {N}}$).
Model-theoretic properties
True arithmetic is an unstable theory, and so has $2^{\kappa }$ models for each uncountable cardinal $\kappa $. As there are continuum many types over the empty set, true arithmetic also has $2^{\aleph _{0}}$ countable models. Since the theory is complete, all of its models are elementarily equivalent.
True theory of second-order arithmetic
The true theory of second-order arithmetic consists of all the sentences in the language of second-order arithmetic that are satisfied by the standard model of second-order arithmetic, whose first-order part is the structure ${\mathcal {N}}$ and whose second-order part consists of every subset of $\mathbb {N} $.
The true theory of first-order arithmetic, Th(${\mathcal {N}}$), is a subset of the true theory of second-order arithmetic, and Th(${\mathcal {N}}$) is definable in second-order arithmetic. However, the generalization of Post's theorem to the analytical hierarchy shows that the true theory of second-order arithmetic is not definable by any single formula in second-order arithmetic.
Simpson (1977) has shown that the true theory of second-order arithmetic is computably interpretable with the theory of the partial order of all Turing degrees, in the signature of partial orders, and vice versa.
Notes
1. Boolos, Burgess & Jeffrey 2002, p. 295
2. see theories associated with a structure
3. Shore 2011, p. 184
References
• Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002), Computability and logic (4th ed.), Cambridge University Press, ISBN 978-0-521-00758-0.
• Bovykin, Andrey; Kaye, Richard (2001), "On order-types of models of arithmetic", in Zhang, Yi (ed.), Logic and algebra, Contemporary Mathematics, vol. 302, American Mathematical Society, pp. 275–285, ISBN 978-0-8218-2984-4.
• Shore, Richard (2011), "The recursively enumerable degrees", in Griffor, E.R. (ed.), Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland (published 1999), pp. 169–197, ISBN 978-0-444-54701-9.
• Simpson, Stephen G. (1977), "First-order theory of the degrees of recursive unsolvability", Annals of Mathematics, Second Series, Annals of Mathematics, 105 (1): 121–139, doi:10.2307/1971028, ISSN 0003-486X, JSTOR 1971028, MR 0432435
• Tarski, Alfred (1936), "Der Wahrheitsbegriff in den formalisierten Sprachen". An English translation "The Concept of Truth in Formalized Languages" appears in Corcoran, J., ed. (1983), Logic, Semantics and Metamathematics: Papers from 1923 to 1938 (2nd ed.), Hackett Publishing Company, Inc., ISBN 978-0-915144-75-4
External links
• Weisstein, Eric W. "Arithmetic". MathWorld.
• Weisstein, Eric W. "Peano Arithmetic". MathWorld.
• Weisstein, Eric W. "Tarski's Theorem". MathWorld.
| Wikipedia |
Truncated 5-cell
In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell.
5-cell
Truncated 5-cell
Bitruncated 5-cell
Schlegel diagrams centered on [3,3] (cells at opposite at [3,3])
There are two degrees of truncations, including a bitruncation.
Truncated 5-cell
Truncated 5-cell
Schlegel diagram
(tetrahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t0,1{3,3,3}
t{3,3,3}
Coxeter diagram
Cells 10 5 (3.3.3)
5 (3.6.6)
Faces 30 20 {3}
10 {6}
Edges 40
Vertices 20
Vertex figure
Equilateral-triangular pyramid
Symmetry group A4, [3,3,3], order 120
Properties convex, isogonal
Uniform index 2 3 4
The truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron.
Construction
The truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 of its edge length. This transforms the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned near the original vertices.
Structure
The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[1]
A4 k-facefkf0f1f2f3k-figure Notes
A2( ) f0 20133331{3}v( )A4/A2 = 5!/3! = 20
A2A1{ } f1 210*3030{3}A4/A2A1 = 5!/3!/2 = 10
A1A1 2*301221{ }v( )A4/A1A1 = 5!/2/2 = 30
A2A1t{3} f2 63310*20{ }A4/A2A1 = 5!/3!/2 = 10
A2{3} 303*2011A4/A2 = 5!/3! = 20
A3t{3,3} f3 12612445*( )A4/A3 = 5!/4! = 5
{3,3} 40604*5
Projections
The truncated tetrahedron-first Schlegel diagram projection of the truncated 5-cell into 3-dimensional space has the following structure:
• The projection envelope is a truncated tetrahedron.
• One of the truncated tetrahedral cells project onto the entire envelope.
• One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
• Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
• Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining truncated tetrahedral cells.
This layout of cells in projection is analogous to the layout of faces in the face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.
Images
orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
• net
• stereographic projection
(centered on truncated tetrahedron)
Alternate names
• Truncated pentatope
• Truncated 4-simplex
• Truncated pentachoron (Acronym: tip) (Jonathan Bowers)
Coordinates
The Cartesian coordinates for the vertices of an origin-centered truncated 5-cell having edge length 2 are:
$\left({\frac {3}{\sqrt {10}}},\ {\sqrt {3 \over 2}},\ \pm {\sqrt {3}},\ \pm 1\right)$
$\left({\frac {3}{\sqrt {10}}},\ {\sqrt {3 \over 2}},\ 0,\ \pm 2\right)$
$\left({\frac {3}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)$
$\left({\frac {3}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)$
$\left({\frac {3}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$
$\left({\frac {3}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$
$\left(-{\sqrt {2 \over 5}},\ {\sqrt {2 \over 3}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)$
$\left(-{\sqrt {2 \over 5}},\ {\sqrt {2 \over 3}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)$
$\left(-{\sqrt {2 \over 5}},\ -{\sqrt {6}},\ 0,\ 0\right)$
$\left({\frac {-7}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$
$\left({\frac {-7}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$
$\left({\frac {-7}{\sqrt {10}}},\ -{\sqrt {3 \over 2}},\ 0,\ 0\right)$
More simply, the vertices of the truncated 5-cell can be constructed on a hyperplane in 5-space as permutations of (0,0,0,1,2) or of (0,1,2,2,2). These coordinates come from positive orthant facets of the truncated pentacross and bitruncated penteract respectively.
Related polytopes
The convex hull of the truncated 5-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 60 cells: 10 tetrahedra, 20 octahedra (as triangular antiprisms), 30 tetrahedra (as tetragonal disphenoids), and 40 vertices. Its vertex figure is a hexakis triangular cupola.
Vertex figure
Bitruncated 5-cell
Bitruncated 5-cell
Schlegel diagram with alternate cells hidden.
Type Uniform 4-polytope
Schläfli symbol t1,2{3,3,3}
2t{3,3,3}
Coxeter diagram
or or
Cells 10 (3.6.6)
Faces 40 20 {3}
20 {6}
Edges 60
Vertices 30
Vertex figure
({ }v{ })
dual polytope Disphenoidal 30-cell
Symmetry group Aut(A4), [[3,3,3]], order 240
Properties convex, isogonal, isotoxal, isochoric
Uniform index 5 6 7
The bitruncated 5-cell (also called a bitruncated pentachoron, decachoron and 10-cell) is a 4-dimensional polytope, or 4-polytope, composed of 10 cells in the shape of truncated tetrahedra.
Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 uniform truncated tetrahedra. The hexagons are always regular because of the polychoron's inversion symmetry, of which the regular hexagon is the only such case among ditrigons (an isogonal hexagon with 3-fold symmetry).
E. L. Elte identified it in 1912 as a semiregular polytope.
Each hexagonal face of the truncated tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle. Each vertex is surrounded by 4 truncated tetrahedral cells in a tetragonal disphenoid vertex figure.
The bitruncated 5-cell is the intersection of two pentachora in dual configuration. As such, it is also the intersection of a penteract with the hyperplane that bisects the penteract's long diagonal orthogonally. In this sense it is a 4-dimensional analog of the regular octahedron (intersection of regular tetrahedra in dual configuration / tesseract bisection on long diagonal) and the regular hexagon (equilateral triangles / cube). The 5-dimensional analog is the birectified 5-simplex, and the $n$-dimensional analog is the polytope whose Coxeter–Dynkin diagram is linear with rings on the middle one or two nodes.
The bitruncated 5-cell is one of the two non-regular convex uniform 4-polytopes which are cell-transitive. The other is the bitruncated 24-cell, which is composed of 48 truncated cubes.
Symmetry
This 4-polytope has a higher extended pentachoric symmetry (2×A4, [[3,3,3]]), doubled to order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.
Alternative names
• Bitruncated 5-cell (Norman W. Johnson)
• 10-cell as a cell-transitive 4-polytope
• Bitruncated pentachoron
• Bitruncated pentatope
• Bitruncated 4-simplex
• Decachoron (Acronym: deca) (Jonathan Bowers)
Images
orthographic projections
Ak
Coxeter plane
A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]
stereographic projection of spherical 4-polytope
(centred on a hexagon face)
Net (polytope)
Coordinates
The Cartesian coordinates of an origin-centered bitruncated 5-cell having edge length 2 are:
Coordinates
$\pm \left({\sqrt {\frac {5}{2}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)$
$\pm \left({\sqrt {\frac {5}{2}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)$
$\pm \left({\sqrt {\frac {5}{2}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {4}{\sqrt {3}}},\ 0\right)$
$\pm \left({\sqrt {\frac {5}{2}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ \pm 2\right)$
$\pm \left({\sqrt {\frac {5}{2}}},\ -{\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)$
$\pm \left({\sqrt {\frac {5}{2}}},\ -{\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)$
$\pm \left(0,\ 2{\sqrt {\frac {2}{3}}},\ {\frac {4}{\sqrt {3}}},\ 0\right)$
$\pm \left(0,\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ \pm 2\right)$
More simply, the vertices of the bitruncated 5-cell can be constructed on a hyperplane in 5-space as permutations of (0,0,1,2,2). These represent positive orthant facets of the bitruncated pentacross. Another 5-space construction, centered on the origin are all 20 permutations of (-1,-1,0,1,1).
Related polytopes
The bitruncated 5-cell can be seen as the intersection of two regular 5-cells in dual positions. = ∩ .
Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
$\left\{{\begin{array}{l}3\\3\end{array}}\right\}$
Decachoron
2t{33}
Dodecateron
2r{34} = {32,2}
$\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$
Tetradecapeton
3t{35}
Hexadecaexon
3r{36} = {33,3}
$\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$
Octadecazetton
4t{37}
Images
Vertex figure ( )∨( )
{ }×{ }
{ }∨{ }
{3}×{3}
{3}∨{3}
{3,3}×{3,3}
{3,3}∨{3,3}
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}
As
intersecting
dual
simplexes
∩
∩
∩
∩
∩ ∩ ∩
Configuration
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[2]
Elementfk f0 f1 f2 f3
f0 30 2 2 1 4 1 2 2
f1 2 30 * 1 2 0 2 1
2 * 30 0 2 1 1 2
f2 3 3 0 10 * * 2 0
6 3 3 * 20 * 1 1
3 0 3 * * 10 0 2
f3 12 12 6 4 4 0 5 *
12 6 12 0 4 4 * 5
Related regular skew polyhedron
The regular skew polyhedron, {6,4|3}, exists in 4-space with 4 hexagonal around each vertex, in a zig-zagging nonplanar vertex figure. These hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 30 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed. The dual regular skew polyhedron, {4,6|3}, is similarly related to the square faces of the runcinated 5-cell.
Disphenoidal 30-cell
Disphenoidal 30-cell
Type perfect[3] polychoron
Symbol f1,2A4[3]
Coxeter
Cells 30 congruent tetragonal disphenoids
Faces 60 congruent isosceles triangles
(2 short edges)
Edges 40 20 of length $\scriptstyle 1$
20 of length $\scriptstyle {\sqrt {3/5}}$
Vertices 10
Vertex figure
(Triakis tetrahedron)
Dual Bitruncated 5-cell
Coxeter group Aut(A4), [[3,3,3]], order 240
Orbit vector (1, 2, 1, 1)
Properties convex, isochoric
The disphenoidal 30-cell is the dual of the bitruncated 5-cell. It is a 4-dimensional polytope (or polychoron) derived from the 5-cell. It is the convex hull of two 5-cells in opposite orientations.
Being the dual of a uniform polychoron, it is cell-transitive, consisting of 30 congruent tetragonal disphenoids. In addition, it is vertex-transitive under the group Aut(A4).
Related polytopes
These polytope are from a set of 9 uniform 4-polytope constructed from the [3,3,3] Coxeter group.
Name 5-cell truncated 5-cell rectified 5-cell cantellated 5-cell bitruncated 5-cell cantitruncated 5-cell runcinated 5-cell runcitruncated 5-cell omnitruncated 5-cell
Schläfli
symbol
{3,3,3}
3r{3,3,3}
t{3,3,3}
2t{3,3,3}
r{3,3,3}
2r{3,3,3}
rr{3,3,3}
r2r{3,3,3}
2t{3,3,3} tr{3,3,3}
t2r{3,3,3}
t0,3{3,3,3} t0,1,3{3,3,3}
t0,2,3{3,3,3}
t0,1,2,3{3,3,3}
Coxeter
diagram
Schlegel
diagram
A4
Coxeter plane
Graph
A3 Coxeter plane
Graph
A2 Coxeter plane
Graph
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]
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 p. 88 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
• Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• 1. Convex uniform polychora based on the pentachoron - Model 3, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora)". x3x3o3o - tip, o3x3x3o - deca
Specific
1. Klitzing, Richard. "x3x4o3o - tip".
2. Klitzing, Richard. "x3o4x3o - srip".
3. On Perfect 4-Polytopes Gabor Gévay Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 243-259 ] Table 2, page 252
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 |
Cantic 5-cube
In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.
Truncated 5-demicube
Cantic 5-cube
D5 Coxeter plane projection
Typeuniform 5-polytope
Schläfli symbolh2{4,3,3,3}
t{3,32,1}
Coxeter-Dynkin diagram =
4-faces42 total:
16 r{3,3,3}
16 t{3,3,3}
10 t{3,3,4}
Cells280 total:
80 {3,3}
120 t{3,3}
80 {3,4}
Faces640 total:
480 {3}
160 {6}
Edges560
Vertices160
Vertex figure
( )v{ }×{3}
Coxeter groupsD5, [32,1,1]
Propertiesconvex
Cartesian coordinates
The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6√2 are coordinate permutations:
(±1,±1,±3,±3,±3)
with an odd number of plus signs.
Alternate names
• Cantic penteract, truncated demipenteract
• Truncated hemipenteract (thin) (Jonathan Bowers)[1]
Images
orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]
Related polytopes
It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections:
Cantic 5-cube
Cantellated 5-cube
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
Dimensional family of cantic n-cubes
n345678
Symmetry
[1+,4,3n-2]
[1+,4,3]
= [3,3]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Cantic
figure
Coxeter
=
=
=
=
=
=
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}
There are 23 uniform 5-polytope that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
D5 polytopes
h{4,3,3,3}
h2{4,3,3,3}
h3{4,3,3,3}
h4{4,3,3,3}
h2,3{4,3,3,3}
h2,4{4,3,3,3}
h3,4{4,3,3,3}
h2,3,4{4,3,3,3}
Notes
1. Klitzing, (x3x3o *b3o3o - thin)
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. "5D uniform polytopes (polytera) x3x3o *b3o3o - thin".
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• 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 |
Truncated 5-orthoplexes
In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.
5-orthoplex
Truncated 5-orthoplex
Bitruncated 5-orthoplex
5-cube
Truncated 5-cube
Bitruncated 5-cube
Orthogonal projections in B5 Coxeter plane
There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.
Truncated 5-orthoplex
Truncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbolt{3,3,3,4}
t{3,31,1}
Coxeter-Dynkin diagrams
4-faces4210
32
Cells240160
80
Faces400320
80
Edges280240
40
Vertices80
Vertex figure
( )v{3,4}
Coxeter groupsB5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
Propertiesconvex
Alternate names
• Truncated pentacross
• Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)[1]
Coordinates
Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±2,±1,0,0,0)
Images
The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Bitruncated 5-orthoplex
Bitruncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbol2t{3,3,3,4}
2t{3,31,1}
Coxeter-Dynkin diagrams
4-faces4210
32
Cells28040
160
80
Faces720320
320
80
Edges720480
240
Vertices240
Vertex figure
{ }v{4}
Coxeter groupsB5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
Propertiesconvex
The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.
Alternate names
• Bitruncated pentacross
• Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of
(±2,±2,±1,0,0)
Images
The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex.
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Related polytopes
This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
B5 polytopes
β5
t1β5
t2γ5
t1γ5
γ5
t0,1β5
t0,2β5
t1,2β5
t0,3β5
t1,3γ5
t1,2γ5
t0,4γ5
t0,3γ5
t0,2γ5
t0,1γ5
t0,1,2β5
t0,1,3β5
t0,2,3β5
t1,2,3γ5
t0,1,4β5
t0,2,4γ5
t0,2,3γ5
t0,1,4γ5
t0,1,3γ5
t0,1,2γ5
t0,1,2,3β5
t0,1,2,4β5
t0,1,3,4γ5
t0,1,2,4γ5
t0,1,2,3γ5
t0,1,2,3,4γ5
Notes
1. Klitzing, (x3x3o3o4o - tot)
2. Klitzing, (o3x3x3o4o - bittit)
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. "5D uniform polytopes (polytera)". x3x3o3o4o - tot, o3x3x3o4o - bittit
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• 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 |
Truncated 5-simplexes
In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
5-simplex
Truncated 5-simplex
Bitruncated 5-simplex
Orthogonal projections in A5 Coxeter plane
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.
Truncated 5-simplex
Truncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 12 6 {3,3,3}
6 t{3,3,3}
Cells 45 30 {3,3}
15 t{3,3}
Faces 80 60 {3}
20 {6}
Edges 75
Vertices 30
Vertex figure
( )v{3,3}
Coxeter group A5 [3,3,3,3], order 720
Properties convex
The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).
Alternate names
• Truncated hexateron (Acronym: tix) (Jonathan Bowers)[1]
Coordinates
The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.
Images
orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]
Bitruncated 5-simplex
bitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2t{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 12 6 2t{3,3,3}
6 t{3,3,3}
Cells 60 45 {3,3}
15 t{3,3}
Faces 140 80 {3}
60 {6}
Edges 150
Vertices 60
Vertex figure
{ }v{3}
Coxeter group A5 [3,3,3,3], order 720
Properties convex
Alternate names
• Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)[2]
Coordinates
The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.
Images
orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]
Related uniform 5-polytopes
The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes
t0
t1
t2
t0,1
t0,2
t1,2
t0,3
t1,3
t0,4
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,1,2,3,4
Notes
1. Klitizing, (x3x3o3o3o - tix)
2. Klitizing, (o3x3x3o3o - bittix)
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. "5D uniform polytopes (polytera)". x3x3o3o3o - tix, o3x3x3o3o - bittix
External links
• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
• Truncated uniform polytera (tix), Jonathan Bowers
• 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 |
Cyclotruncated 5-simplex honeycomb
In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1.
Cyclotruncated 5-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilyCyclotruncated simplectic honeycomb
Schläfli symbolt0,1{3[6]}
Coxeter diagram or
5-face types{3,3,3,3}
t{3,3,3,3}
2t{3,3,3,3}
4-face types{3,3,3}
t{3,3,3}
Cell types{3,3}
t{3,3}
Face types{3}
t{3}
Vertex figure
Elongated 5-cell antiprism
Coxeter groups${\tilde {A}}_{5}$×22, [[3[6]]]
Propertiesvertex-transitive
Structure
Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.
It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
This honeycomb is one of 12 unique uniform honeycombs[1] constructed by the ${\tilde {A}}_{5}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\tilde {A}}_{5}$ Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:
A5 honeycombs
Hexagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
a1 [3[6]] ${\tilde {A}}_{5}$
d2 <[3[6]]> ${\tilde {A}}_{5}$×21 1, , , ,
p2 [[3[6]]] ${\tilde {A}}_{5}$×22 2,
i4 [<[3[6]]>] ${\tilde {A}}_{5}$×21×22 ,
d6 <3[3[6]]> ${\tilde {A}}_{5}$×61
r12 [6[3[6]]] ${\tilde {A}}_{5}$×12 3
See also
Regular and uniform honeycombs in 5-space:
• 5-cubic honeycomb
• 5-demicubic honeycomb
• 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb
Notes
1. mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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] (1.9 Uniform space-fillings)
• (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 |
Truncated 6-orthoplexes
In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.
6-orthoplex
Truncated 6-orthoplex
Bitruncated 6-orthoplex
Tritruncated 6-cube
6-cube
Truncated 6-cube
Bitruncated 6-cube
Orthogonal projections in B6 Coxeter plane
There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.
Truncated 6-orthoplex
Truncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces76
4-faces576
Cells1200
Faces1120
Edges540
Vertices120
Vertex figure
( )v{3,4}
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex
Alternate names
• Truncated hexacross
• Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)[1]
Construction
There are two Coxeter groups associated with the truncated hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of
(±2,±1,0,0,0,0)
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Bitruncated 6-orthoplex
Bitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol2t{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
{ }v{3,4}
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex
Alternate names
• Bitruncated hexacross
• Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)[2]
Images
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
These polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes
β6
t1β6
t2β6
t2γ6
t1γ6
γ6
t0,1β6
t0,2β6
t1,2β6
t0,3β6
t1,3β6
t2,3γ6
t0,4β6
t1,4γ6
t1,3γ6
t1,2γ6
t0,5γ6
t0,4γ6
t0,3γ6
t0,2γ6
t0,1γ6
t0,1,2β6
t0,1,3β6
t0,2,3β6
t1,2,3β6
t0,1,4β6
t0,2,4β6
t1,2,4β6
t0,3,4β6
t1,2,4γ6
t1,2,3γ6
t0,1,5β6
t0,2,5β6
t0,3,4γ6
t0,2,5γ6
t0,2,4γ6
t0,2,3γ6
t0,1,5γ6
t0,1,4γ6
t0,1,3γ6
t0,1,2γ6
t0,1,2,3β6
t0,1,2,4β6
t0,1,3,4β6
t0,2,3,4β6
t1,2,3,4γ6
t0,1,2,5β6
t0,1,3,5β6
t0,2,3,5γ6
t0,2,3,4γ6
t0,1,4,5γ6
t0,1,3,5γ6
t0,1,3,4γ6
t0,1,2,5γ6
t0,1,2,4γ6
t0,1,2,3γ6
t0,1,2,3,4β6
t0,1,2,3,5β6
t0,1,2,4,5β6
t0,1,2,4,5γ6
t0,1,2,3,5γ6
t0,1,2,3,4γ6
t0,1,2,3,4,5γ6
Notes
1. Klitzing, (x3x3o3o3o4o - tag)
2. Klitzing, (o3x3x3o3o4o - botag)
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. "6D uniform polytopes (polypeta)". x3x3o3o3o4o - tag, o3x3x3o3o4o - botag
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 |
Truncated 6-simplexes
In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.
6-simplex
Truncated 6-simplex
Bitruncated 6-simplex
Tritruncated 6-simplex
Orthogonal projections in A7 Coxeter plane
There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.
Truncated 6-simplex
Truncated 6-simplex
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbolt{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces14:
7 {3,3,3,3}
7 t{3,3,3,3}
4-faces63:
42 {3,3,3}
21 t{3,3,3}
Cells140:
105 {3,3}
35 t{3,3}
Faces175:
140 {3}
35 {6}
Edges126
Vertices42
Vertex figure
( )v{3,3,3}
Coxeter groupA6, [35], order 5040
Dual?
Propertiesconvex
Alternate names
• Truncated heptapeton (Acronym: til) (Jonathan Bowers)[1]
Coordinates
The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.
Images
orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]
Bitruncated 6-simplex
Bitruncated 6-simplex
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbol2t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces14
4-faces84
Cells245
Faces385
Edges315
Vertices105
Vertex figure
{ }v{3,3}
Coxeter groupA6, [35], order 5040
Propertiesconvex
Alternate names
• Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)[2]
Coordinates
The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.
Images
orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]
Tritruncated 6-simplex
Tritruncated 6-simplex
Typeuniform 6-polytope
ClassA6 polytope
Schläfli symbol3t{3,3,3,3,3}
Coxeter-Dynkin diagram
or
5-faces14 2t{3,3,3,3}
4-faces84
Cells280
Faces490
Edges420
Vertices140
Vertex figure
{3}v{3}
Coxeter groupA6, [[35]], order 10080
Propertiesconvex, isotopic
The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.
The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration: and .
Alternate names
• Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)[3]
Coordinates
The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).
Images
orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.
Related polytopes
Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
$\left\{{\begin{array}{l}3\\3\end{array}}\right\}$
Decachoron
2t{33}
Dodecateron
2r{34} = {32,2}
$\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$
Tetradecapeton
3t{35}
Hexadecaexon
3r{36} = {33,3}
$\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$
Octadecazetton
4t{37}
Images
Vertex figure ( )∨( )
{ }×{ }
{ }∨{ }
{3}×{3}
{3}∨{3}
{3,3}×{3,3}
{3,3}∨{3,3}
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}
As
intersecting
dual
simplexes
∩
∩
∩
∩
∩ ∩ ∩
Related uniform 6-polytopes
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
A6 polytopes
t0
t1
t2
t0,1
t0,2
t1,2
t0,3
t1,3
t2,3
t0,4
t1,4
t0,5
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t1,2,4
t0,3,4
t0,1,5
t0,2,5
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,2,3,4
t1,2,3,4
t0,1,2,5
t0,1,3,5
t0,2,3,5
t0,1,4,5
t0,1,2,3,4
t0,1,2,3,5
t0,1,2,4,5
t0,1,2,3,4,5
Notes
1. Klitzing, (o3x3o3o3o3o - til)
2. Klitzing, (o3x3x3o3o3o - batal)
3. Klitzing, (o3o3x3x3o3o - fe)
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. "6D uniform polytopes (polypeta)". o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe
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 |
Cyclotruncated 6-simplex honeycomb
In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
Cyclotruncated 6-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilyCyclotruncated simplectic honeycomb
Schläfli symbolt0,1{3[7]}
Coxeter diagram
6-face types{35}
t{35}
2t{35}
3t{35}
Vertex figureElongated 5-simplex antiprism
Symmetry${\tilde {A}}_{6}$×2, [[3[7]]]
Propertiesvertex-transitive
Structure
It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
This honeycomb is one of 17 unique uniform honeycombs[1] constructed by the ${\tilde {A}}_{6}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
A6 honeycombs
Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[7]] ${\tilde {A}}_{6}$
i2 [[3[7]]] ${\tilde {A}}_{6}$×2
1
2
r14 [7[3[7]]] ${\tilde {A}}_{6}$×14
3
See also
Regular and uniform honeycombs in 6-space:
• 6-cubic honeycomb
• 6-demicubic honeycomb
• 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb
• 222 honeycomb
Notes
• Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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] (1.9 Uniform space-fillings)
• (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 |
Cantic 7-cube
In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.
Truncated 7-demicube
Cantic 7-cube
D7 Coxeter plane projection
Typeuniform 7-polytope
Schläfli symbolt{3,34,1}
h2{4,3,3,3,3,3}
Coxeter diagram
6-faces142
5-faces1428
4-faces5656
Cells11760
Faces13440
Edges7392
Vertices1344
Vertex figure( )v{ }x{3,3,3}
Coxeter groupsD7, [34,1,1]
Propertiesconvex
A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes representing active mirrors. A demihypercube is an alternation of a hypercube.
Its 3-dimensional analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube.
Alternate names
• Truncated demihepteract
• Truncated hemihepteract (thesa) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:
(±1,±1,±3,±3,±3,±3,±3)
with an odd number of plus signs.
Images
It can be visualized as a 2-dimensional orthogonal projections, for example the a D7 Coxeter plane, containing 12-gonal symmetry. Most visualizations in symmetric projections will contain overlapping vertices, so the colors of the vertices are changed based on how many vertices are at each projective position, here shown with red color for no overlaps.
orthographic projections
Coxeter
plane
B7 D7 D6
Graph
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph
Dihedral
symmetry
[6] [4]
Related polytopes
Dimensional family of cantic n-cubes
n345678
Symmetry
[1+,4,3n-2]
[1+,4,3]
= [3,3]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Cantic
figure
Coxeter
=
=
=
=
=
=
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
D7 polytopes
t0(141)
t0,1(141)
t0,2(141)
t0,3(141)
t0,4(141)
t0,5(141)
t0,1,2(141)
t0,1,3(141)
t0,1,4(141)
t0,1,5(141)
t0,2,3(141)
t0,2,4(141)
t0,2,5(141)
t0,3,4(141)
t0,3,5(141)
t0,4,5(141)
t0,1,2,3(141)
t0,1,2,4(141)
t0,1,2,5(141)
t0,1,3,4(141)
t0,1,3,5(141)
t0,1,4,5(141)
t0,2,3,4(141)
t0,2,3,5(141)
t0,2,4,5(141)
t0,3,4,5(141)
t0,1,2,3,4(141)
t0,1,2,3,5(141)
t0,1,2,4,5(141)
t0,1,3,4,5(141)
t0,2,3,4,5(141)
t0,1,2,3,4,5(141)
Notes
1. Klitzing, (x3x3o *b3o3o3o3o - thesa)
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) x3x3o *b3o3o3o3o – thesa".
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• 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 |
Cyclotruncated 7-simplex honeycomb
In seven-dimensional Euclidean geometry, the cyclotruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, truncated 7-simplex, bitruncated 7-simplex, and tritruncated 7-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.
Cyclotruncated 7-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilyCyclotruncated simplectic honeycomb
Schläfli symbolt0,1{3[8]}
Coxeter diagram
7-face types{36}
t0,1{36}
t1,2{36}
t2,3{36}
Vertex figureElongated 6-simplex antiprism
Symmetry${\tilde {A}}_{7}$×22, [[3[8]]]
Propertiesvertex-transitive
Structure
It can be constructed by eight sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 6-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
This honeycomb is one of 29 unique uniform honeycombs[1] constructed by the ${\tilde {A}}_{7}$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
A7 honeycombs
Octagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[8]] ${\tilde {A}}_{7}$
d2 <[3[8]]> ${\tilde {A}}_{7}$×21
1
p2 [[3[8]]] ${\tilde {A}}_{7}$×22
2
d4 <2[3[8]]> ${\tilde {A}}_{7}$×41
p4 [2[3[8]]] ${\tilde {A}}_{7}$×42
d8 [4[3[8]]] ${\tilde {A}}_{7}$×8
r16 [8[3[8]]] ${\tilde {A}}_{7}$×16 3
See also
Regular and uniform honeycombs in 7-space:
• 7-cubic honeycomb
• 7-demicubic honeycomb
• 7-simplex honeycomb
• Omnitruncated 7-simplex honeycomb
• 331 honeycomb
Notes
1. Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 30-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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] (1.9 Uniform space-fillings)
• (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 |
Truncated 7-simplexes
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
7-simplex
Truncated 7-simplex
Bitruncated 7-simplex
Tritruncated 7-simplex
Orthogonal projections in A7 Coxeter plane
There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.
Truncated 7-simplex
Truncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces16
5-faces
4-faces
Cells350
Faces336
Edges196
Vertices56
Vertex figure( )v{3,3,3,3}
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex, Vertex-transitive
In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.
Alternate names
• Truncated octaexon (Acronym: toc) (Jonathan Bowers)[1]
Coordinates
The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.
Images
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Bitruncated 7-simplex
Bitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbol2t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges588
Vertices168
Vertex figure{ }v{3,3,3}
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex, Vertex-transitive
Alternate names
• Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)[2]
Coordinates
The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.
Images
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Tritruncated 7-simplex
Tritruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbol3t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges980
Vertices280
Vertex figure{3}v{3,3}
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex, Vertex-transitive
Alternate names
• Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)[3]
Coordinates
The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.
Images
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Related polytopes
These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.
A7 polytopes
t0
t1
t2
t3
t0,1
t0,2
t1,2
t0,3
t1,3
t2,3
t0,4
t1,4
t2,4
t0,5
t1,5
t0,6
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t1,2,4
t0,3,4
t1,3,4
t2,3,4
t0,1,5
t0,2,5
t1,2,5
t0,3,5
t1,3,5
t0,4,5
t0,1,6
t0,2,6
t0,3,6
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,2,3,4
t1,2,3,4
t0,1,2,5
t0,1,3,5
t0,2,3,5
t1,2,3,5
t0,1,4,5
t0,2,4,5
t1,2,4,5
t0,3,4,5
t0,1,2,6
t0,1,3,6
t0,2,3,6
t0,1,4,6
t0,2,4,6
t0,1,5,6
t0,1,2,3,4
t0,1,2,3,5
t0,1,2,4,5
t0,1,3,4,5
t0,2,3,4,5
t1,2,3,4,5
t0,1,2,3,6
t0,1,2,4,6
t0,1,3,4,6
t0,2,3,4,6
t0,1,2,5,6
t0,1,3,5,6
t0,1,2,3,4,5
t0,1,2,3,4,6
t0,1,2,3,5,6
t0,1,2,4,5,6
t0,1,2,3,4,5,6
See also
• List of A7 polytopes
Notes
1. Klitizing, (x3x3o3o3o3o3o - toc)
2. Klitizing, (o3x3x3o3o3o3o - roc)
3. Klitizing, (o3o3x3x3o3o3o - tattoc)
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)". x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc
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 |
Truncated 8-orthoplexes
In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.
8-orthoplex
Truncated 8-orthoplex
Bitruncated 8-orthoplex
Tritruncated 8-orthoplex
Quadritruncated 8-cube
Tritruncated 8-cube
Bitruncated 8-cube
Truncated 8-cube
8-cube
Orthogonal projections in B8 Coxeter plane
There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.
Truncated 8-orthoplex
Truncated 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt0,1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1456
Vertices224
Vertex figure( )v{3,3,3,4}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex
Alternate names
• Truncated octacross (acronym tek) (Jonthan Bowers)[1]
Construction
There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D8 or [35,1,1] Coxeter group.
Coordinates
Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of
(±2,±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]
Bitruncated 8-orthoplex
Bitruncated 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt1,2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{ }v{3,3,3,4}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex
Alternate names
• Bitruncated octacross (acronym batek) (Jonthan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±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]
Tritruncated 8-orthoplex
Tritruncated 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt2,3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{3}v{3,3,4}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex
Alternate names
• Tritruncated octacross (acronym tatek) (Jonthan Bowers)[3]
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±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. Klitizing, (x3x3o3o3o3o3o4o - tek)
2. Klitizing, (o3x3x3o3o3o3o4o - batek)
3. Klitizing, (o3o3x3x3o3o3o4o - tatek)
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. (1966)
• Klitzing, Richard. "8D uniform polytopes (polyzetta)". x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek
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 |
Truncated 8-simplexes
In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.
8-simplex
Truncated 8-simplex
Rectified 8-simplex
Quadritruncated 8-simplex
Tritruncated 8-simplex
Bitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane
There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex.
Truncated 8-simplex
Truncated 8-simplex
Typeuniform 8-polytope
Schläfli symbolt{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges288
Vertices72
Vertex figure( )v{3,3,3,3,3}
Coxeter groupA8, [37], order 362880
Propertiesconvex
Alternate names
• Truncated enneazetton (Acronym: tene) (Jonathan Bowers)[1]
Coordinates
The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Bitruncated 8-simplex
Bitruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol2t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges1008
Vertices252
Vertex figure{ }v{3,3,3,3}
Coxeter groupA8, [37], order 362880
Propertiesconvex
Alternate names
• Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)[2]
Coordinates
The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Tritruncated 8-simplex
tritruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol3t{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges2016
Vertices504
Vertex figure{3}v{3,3,3}
Coxeter groupA8, [37], order 362880
Propertiesconvex
Alternate names
• Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)[3]
Coordinates
The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Quadritruncated 8-simplex
Quadritruncated 8-simplex
Typeuniform 8-polytope
Schläfli symbol4t{37}
Coxeter-Dynkin diagrams
or
6-faces18 3t{3,3,3,3,3,3}
7-faces
5-faces
4-faces
Cells
Faces
Edges2520
Vertices630
Vertex figure
{3,3}v{3,3}
Coxeter groupA8, [[37]], order 725760
Propertiesconvex, isotopic
The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.
Alternate names
• Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)[4]
Coordinates
The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on facets of the quadritruncated 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]
Related polytopes
Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
$\left\{{\begin{array}{l}3\\3\end{array}}\right\}$
Decachoron
2t{33}
Dodecateron
2r{34} = {32,2}
$\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$
Tetradecapeton
3t{35}
Hexadecaexon
3r{36} = {33,3}
$\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$
Octadecazetton
4t{37}
Images
Vertex figure ( )∨( )
{ }×{ }
{ }∨{ }
{3}×{3}
{3}∨{3}
{3,3}×{3,3}
{3,3}∨{3,3}
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}
As
intersecting
dual
simplexes
∩
∩
∩
∩
∩ ∩ ∩
Related polytopes
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
A8 polytopes
t0
t1
t2
t3
t01
t02
t12
t03
t13
t23
t04
t14
t24
t34
t05
t15
t25
t06
t16
t07
t012
t013
t023
t123
t014
t024
t124
t034
t134
t234
t015
t025
t125
t035
t135
t235
t045
t145
t016
t026
t126
t036
t136
t046
t056
t017
t027
t037
t0123
t0124
t0134
t0234
t1234
t0125
t0135
t0235
t1235
t0145
t0245
t1245
t0345
t1345
t2345
t0126
t0136
t0236
t1236
t0146
t0246
t1246
t0346
t1346
t0156
t0256
t1256
t0356
t0456
t0127
t0137
t0237
t0147
t0247
t0347
t0157
t0257
t0167
t01234
t01235
t01245
t01345
t02345
t12345
t01236
t01246
t01346
t02346
t12346
t01256
t01356
t02356
t12356
t01456
t02456
t03456
t01237
t01247
t01347
t02347
t01257
t01357
t02357
t01457
t01267
t01367
t012345
t012346
t012356
t012456
t013456
t023456
t123456
t012347
t012357
t012457
t013457
t023457
t012367
t012467
t013467
t012567
t0123456
t0123457
t0123467
t0123567
t01234567
Notes
1. Klitizing, (x3x3o3o3o3o3o3o - tene)
2. Klitizing, (o3x3x3o3o3o3o3o - batene)
3. Klitizing, (o3o3x3x3o3o3o3o - tatene)
4. Klitizing, (o3o3o3x3x3o3o3o - be)
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)". x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be
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 |
Cyclotruncated 8-simplex honeycomb
In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.
Cyclotruncated 8-simplex honeycomb
(No image)
TypeUniform honeycomb
FamilyCyclotruncated simplectic honeycomb
Schläfli symbolt0,1{3[9]}
Coxeter diagram
8-face types{37} , t0,1{37}
t1,2{37} , t2,3{37}
t3,4{37}
Vertex figureElongated 7-simplex antiprism
Symmetry${\tilde {A}}_{8}$×2, [[3[9]]]
Propertiesvertex-transitive
Structure
It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
This honeycomb is one of 45 unique uniform honeycombs[1] constructed by the ${\tilde {A}}_{8}$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:
A8 honeycombs
Enneagon
symmetry
Symmetry Extended
diagram
Extended
group
Honeycombs
a1 [3[9]] ${\tilde {A}}_{8}$
i2 [[3[9]]] ${\tilde {A}}_{8}$×2
1 2
i6 [3[3[9]]] ${\tilde {A}}_{8}$×6
r18 [9[3[9]]] ${\tilde {A}}_{8}$×18 3
See also
Regular and uniform honeycombs in 8-space:
• 8-cubic honeycomb
• 8-demicubic honeycomb
• 8-simplex honeycomb
• Omnitruncated 8-simplex honeycomb
• 521 honeycomb
• 251 honeycomb
• 152 honeycomb
Notes
• Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 46-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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] (1.9 Uniform space-fillings)
• (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 |
Truncated Newton method
The truncated Newton method, originated in a paper by Ron Dembo and Trond Steihaug,[1] also known as Hessian-free optimization,[2] are a family of optimization algorithms designed for optimizing non-linear functions with large numbers of independent variables. A truncated Newton method consists of repeated application of an iterative optimization algorithm to approximately solve Newton's equations, to determine an update to the function's parameters. The inner solver is truncated, i.e., run for only a limited number of iterations. It follows that, for truncated Newton methods to work, the inner solver needs to produce a good approximation in a finite number of iterations;[3] conjugate gradient has been suggested and evaluated as a candidate inner loop.[2] Another prerequisite is good preconditioning for the inner algorithm.[4]
References
1. Dembo, Ron S.; Steihaug, Trond (1983). "Truncated-Newton algorithms for large-scale unconstrained optimization". Mathematical Programming. Springer. 26 (2): 190–212. doi:10.1007/BF02592055. S2CID 40537623.. Convergence results for this algorithm can be found in Dembo, Ron S.; Eisenstat, Stanley C.; Steihaug, Trond (1982). "Inexact newton methods". SIAM Journal on Numerical Analysis. 19 (2): 400–408. Bibcode:1982SJNA...19..400D. doi:10.1137/0719025. JSTOR 2156954..
2. Martens, James (2010). Deep learning via Hessian-free optimization (PDF). Proc. International Conference on Machine Learning.
3. Nash, Stephen G. (2000). "A survey of truncated-Newton methods". Journal of Computational and Applied Mathematics. 124 (1–2): 45–59. Bibcode:2000JCoAM.124...45N. doi:10.1016/S0377-0427(00)00426-X.
4. Nash, Stephen G. (1985). "Preconditioning of truncated-Newton methods" (PDF). SIAM J. Sci. Stat. Comput. 6 (3): 599–616. doi:10.1137/0906042.
Further reading
• Grippo, L.; Lampariello, F.; Lucidi, S. (1989). "A Truncated Newton Method with Nonmonotone Line Search for Unconstrained Optimization". J. Optimization Theory and Applications. 60 (3): 401–419. CiteSeerX 10.1.1.455.7495. doi:10.1007/BF00940345. S2CID 18990650.
• Nash, Stephen G.; Nocedal, Jorge (1991). "A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization". SIAM J. Optim. 1 (3): 358–372. CiteSeerX 10.1.1.474.3400. doi:10.1137/0801023.
Optimization: Algorithms, methods, and heuristics
Unconstrained nonlinear
Functions
• Golden-section search
• Interpolation methods
• Line search
• Nelder–Mead method
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| Wikipedia |
Pareto distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[2] is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.[3][4] The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value (α) of log45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena[5] and human activities.[6][7]
Pareto Type I
Probability density function
Pareto Type I probability density functions for various $\alpha $ with $x_{\mathrm {m} }=1.$ As $\alpha \rightarrow \infty ,$ the distribution approaches $\delta (x-x_{\mathrm {m} }),$ where $\delta $ is the Dirac delta function.
Cumulative distribution function
Pareto Type I cumulative distribution functions for various $\alpha $ with $x_{\mathrm {m} }=1.$
Parameters $x_{\mathrm {m} }>0$ scale (real)
$\alpha >0$ shape (real)
Support $x\in [x_{\mathrm {m} },\infty )$
PDF ${\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}$
CDF $1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }$
Quantile $x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}$
Mean ${\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}$
Median $x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}$
Mode $x_{\mathrm {m} }$
Variance ${\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}$
Skewness ${\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3$
Ex. kurtosis ${\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4$
Entropy $\log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)$
MGF does not exist
CF $\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)$
Fisher information
${\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha }{x_{\mathrm {m} }^{2}}}&-{\dfrac {1}{x_{\mathrm {m} }}}\\-{\dfrac {1}{x_{\mathrm {m} }}}&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}$
Right: ${\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}$
CVaR (ES) ${\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}$[1]
bPOE $\left({\frac {x_{m}\alpha }{x(\alpha -1)}}\right)^{\alpha }$[1]
Definitions
If X is a random variable with a Pareto (Type I) distribution,[8] then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by
${\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x<x_{\mathrm {m} },\end{cases}}$
where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
Cumulative distribution function
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is
$F_{X}(x)={\begin{cases}1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}$
Probability density function
It follows (by differentiation) that the probability density function is
$f_{X}(x)={\begin{cases}{\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}&x\geq x_{\mathrm {m} },\\0&x<x_{\mathrm {m} }.\end{cases}}$
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log-log plot, the distribution is represented by a straight line.
Properties
Moments and characteristic function
• The expected value of a random variable following a Pareto distribution is
$\operatorname {E} (X)={\begin{cases}\infty &\alpha \leq 1,\\{\frac {\alpha x_{\mathrm {m} }}{\alpha -1}}&\alpha >1.\end{cases}}$
• The variance of a random variable following a Pareto distribution is
$\operatorname {Var} (X)={\begin{cases}\infty &\alpha \in (1,2],\\\left({\frac {x_{\mathrm {m} }}{\alpha -1}}\right)^{2}{\frac {\alpha }{\alpha -2}}&\alpha >2.\end{cases}}$
(If α ≤ 1, the variance does not exist.)
• The raw moments are
$\mu _{n}'={\begin{cases}\infty &\alpha \leq n,\\{\frac {\alpha x_{\mathrm {m} }^{n}}{\alpha -n}}&\alpha >n.\end{cases}}$
• The moment generating function is only defined for non-positive values t ≤ 0 as
$M\left(t;\alpha ,x_{\mathrm {m} }\right)=\operatorname {E} \left[e^{tX}\right]=\alpha (-x_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-x_{\mathrm {m} }t)$
$M\left(0,\alpha ,x_{\mathrm {m} }\right)=1.$
Thus, since the expectation does not converge on an open interval containing $t=0$ we say that the moment generating function does not exist.
• The characteristic function is given by
$\varphi (t;\alpha ,x_{\mathrm {m} })=\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t),$
where Γ(a, x) is the incomplete gamma function.
The parameters may be solved for using the method of moments.[9]
Conditional distributions
The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number $x_{1}$ exceeding $x_{\text{m}}$, is a Pareto distribution with the same Pareto index $\alpha $ but with minimum $x_{1}$ instead of $x_{\text{m}}$. This implies that the conditional expected value (if it is finite, i.e. $\alpha >1$) is proportional to $x_{1}$. In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called the Lindy effect or Lindy's Law.[10]
A characterization theorem
Suppose $X_{1},X_{2},X_{3},\dotsc $ are independent identically distributed random variables whose probability distribution is supported on the interval $[x_{\text{m}},\infty )$ for some $x_{\text{m}}>0$. Suppose that for all $n$, the two random variables $\min\{X_{1},\dotsc ,X_{n}\}$ and $(X_{1}+\dotsb +X_{n})/\min\{X_{1},\dotsc ,X_{n}\}$ are independent. Then the common distribution is a Pareto distribution.
Geometric mean
The geometric mean (G) is[11]
$G=x_{\text{m}}\exp \left({\frac {1}{\alpha }}\right).$
Harmonic mean
The harmonic mean (H) is[11]
$H=x_{\text{m}}\left(1+{\frac {1}{\alpha }}\right).$
Graphical representation
The characteristic curved 'long tail' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a log-log graph, which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for x ≥ xm,
$\log f_{X}(x)=\log \left(\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}\right)=\log(\alpha x_{\mathrm {m} }^{\alpha })-(\alpha +1)\log x.$
Since α is positive, the gradient −(α + 1) is negative.
Related distributions
Generalized Pareto distributions
There is a hierarchy [8][12] of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.[8][12][13] Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto[12][14] distribution generalizes Pareto Type IV.
Pareto types I–IV
The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).
When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.[15]
In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.
Pareto distributions
${\overline {F}}(x)=1-F(x)$SupportParameters
Type I $\left[{\frac {x}{\sigma }}\right]^{-\alpha }$ $x\geq \sigma $ $\sigma >0,\alpha $
Type II $\left[1+{\frac {x-\mu }{\sigma }}\right]^{-\alpha }$ $x\geq \mu $ $\mu \in \mathbb {R} ,\sigma >0,\alpha $
Lomax $\left[1+{\frac {x}{\sigma }}\right]^{-\alpha }$ $x\geq 0$ $\sigma >0,\alpha $
Type III $\left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-1}$ $x\geq \mu $ $\mu \in \mathbb {R} ,\sigma ,\gamma >0$
Type IV $\left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }$ $x\geq \mu $ $\mu \in \mathbb {R} ,\sigma ,\gamma >0,\alpha $
The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are
$P(IV)(\sigma ,\sigma ,1,\alpha )=P(I)(\sigma ,\alpha ),$
$P(IV)(\mu ,\sigma ,1,\alpha )=P(II)(\mu ,\sigma ,\alpha ),$
$P(IV)(\mu ,\sigma ,\gamma ,1)=P(III)(\mu ,\sigma ,\gamma ).$
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.
Moments of Pareto I–IV distributions (case μ = 0)
$\operatorname {E} [X]$Condition$\operatorname {E} [X^{\delta }]$Condition
Type I ${\frac {\sigma \alpha }{\alpha -1}}$ $\alpha >1$ ${\frac {\sigma ^{\delta }\alpha }{\alpha -\delta }}$ $\delta <\alpha $
Type II ${\frac {\sigma }{\alpha -1}}+\mu $ $\alpha >1$ ${\frac {\sigma ^{\delta }\Gamma (\alpha -\delta )\Gamma (1+\delta )}{\Gamma (\alpha )}}$ $0<\delta <\alpha $
Type III $\sigma \Gamma (1-\gamma )\Gamma (1+\gamma )$ $-1<\gamma <1$ $\sigma ^{\delta }\Gamma (1-\gamma \delta )\Gamma (1+\gamma \delta )$ $-\gamma ^{-1}<\delta <\gamma ^{-1}$
Type IV ${\frac {\sigma \Gamma (\alpha -\gamma )\Gamma (1+\gamma )}{\Gamma (\alpha )}}$ $-1<\gamma <\alpha $ ${\frac {\sigma ^{\delta }\Gamma (\alpha -\gamma \delta )\Gamma (1+\gamma \delta )}{\Gamma (\alpha )}}$ $-\gamma ^{-1}<\delta <\alpha /\gamma $
Feller–Pareto distribution
Feller[12][14] defines a Pareto variable by transformation U = Y−1 − 1 of a beta random variable Y, whose probability density function is
$f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 0<y<1;\gamma _{1},\gamma _{2}>0,$
where B( ) is the beta function. If
$W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,$
then W has a Feller–Pareto distribution FP(μ, σ, γ, γ1, γ2).[8]
If $U_{1}\sim \Gamma (\delta _{1},1)$ and $U_{2}\sim \Gamma (\delta _{2},1)$ are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is[16]
$W=\mu +\sigma \left({\frac {U_{1}}{U_{2}}}\right)^{\gamma }$
and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are
$FP(\sigma ,\sigma ,1,1,\alpha )=P(I)(\sigma ,\alpha )$
$FP(\mu ,\sigma ,1,1,\alpha )=P(II)(\mu ,\sigma ,\alpha )$
$FP(\mu ,\sigma ,\gamma ,1,1)=P(III)(\mu ,\sigma ,\gamma )$
$FP(\mu ,\sigma ,\gamma ,1,\alpha )=P(IV)(\mu ,\sigma ,\gamma ,\alpha ).$
Inverse-Pareto Distribution / Power Distribution
When a random variable $Y$ follows a pareto distribution, then its inverse $X=1/Y$ follows an Inverse Pareto distribution. Inverse Pareto distribution is equivalent to a Power distribution[17]
$Y\sim \mathrm {Pa} (\alpha ,x_{m})={\frac {\alpha x_{m}^{\alpha }}{y^{\alpha +1}}}\quad (y\geq x_{m})\quad \Leftrightarrow \quad X\sim \mathrm {iPa} (\alpha ,x_{m})=\mathrm {Power} (x_{m}^{-1},\alpha )={\frac {\alpha x^{\alpha -1}}{(x_{m}^{-1})^{\alpha }}}\quad (0<x\leq x_{m}^{-1})$
Relation to the exponential distribution
The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then
$Y=\log \left({\frac {X}{x_{\mathrm {m} }}}\right)$
is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then
$x_{\mathrm {m} }e^{Y}$
is Pareto-distributed with minimum xm and index α.
This can be shown using the standard change-of-variable techniques:
${\begin{aligned}\Pr(Y<y)&=\Pr \left(\log \left({\frac {X}{x_{\mathrm {m} }}}\right)<y\right)\\&=\Pr(X<x_{\mathrm {m} }e^{y})=1-\left({\frac {x_{\mathrm {m} }}{x_{\mathrm {m} }e^{y}}}\right)^{\alpha }=1-e^{-\alpha y}.\end{aligned}}$
The last expression is the cumulative distribution function of an exponential distribution with rate α.
Pareto distribution can be constructed by hierarchical exponential distributions.[18] Let $\phi |a\sim {\text{Exp}}(a)$ and $\eta |\phi \sim {\text{Exp}}(\phi )$. Then we have $p(\eta |a)={\frac {a}{(a+\eta )^{2}}}$ and, as a result, $a+\eta \sim {\text{Pareto}}(a,1)$.
More in general, if $\lambda \sim {\text{Gamma}}(\alpha ,\beta )$ (shape-rate parametrization) and $\eta |\lambda \sim {\text{Exp}}(\lambda )$, then $\beta +\eta \sim {\text{Pareto}}(\beta ,\alpha )$.
Equivalently, if $Y\sim {\text{Gamma}}(\alpha ,1)$ and $X\sim {\text{Exp}}(1)$, then $x_{\text{m}}\!\left(1+{\frac {X}{Y}}\right)\sim {\text{Pareto}}(x_{\text{m}},\alpha )$.
Relation to the log-normal distribution
The Pareto distribution and log-normal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution. (See the previous section.)
Relation to the generalized Pareto distribution
The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.
The Pareto distribution with scale $x_{m}$ and shape $\alpha $ is equivalent to the generalized Pareto distribution with location $\mu =x_{m}$, scale $\sigma =x_{m}/\alpha $ and shape $\xi =1/\alpha $. Vice versa one can get the Pareto distribution from the GPD by $x_{m}=\sigma /\xi $ and $\alpha =1/\xi $.
Bounded Pareto distribution
Bounded Pareto
Parameters
$L>0$ location (real)
$H>L$ location (real)
$\alpha >0$ shape (real)
Support $L\leqslant x\leqslant H$
PDF ${\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}$
CDF ${\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}$
Mean
${\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -1}}\right)\cdot \left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),\alpha \neq 1$
${\frac {{H}{L}}{{H}-{L}}}\ln {\frac {H}{L}},\alpha =1$
Median $L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}$
Variance
${\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -2}}\right)\cdot \left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),\alpha \neq 2$ ${\frac {2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln {\frac {H}{L}},\alpha =2$
(this is the second raw moment, not the variance)
Skewness
${\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha (L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j$
(this is the kth raw moment, not the skewness)
The bounded (or truncated) Pareto distribution has three parameters: α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value.
The probability density function is
${\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}$,
where L ≤ x ≤ H, and α > 0.
Generating bounded Pareto random variables
If U is uniformly distributed on (0, 1), then applying inverse-transform method [19]
$U={\frac {1-L^{\alpha }x^{-\alpha }}{1-({\frac {L}{H}})^{\alpha }}}$
$x=\left(-{\frac {UH^{\alpha }-UL^{\alpha }-H^{\alpha }}{H^{\alpha }L^{\alpha }}}\right)^{-{\frac {1}{\alpha }}}$
is a bounded Pareto-distributed.
Symmetric Pareto distribution
The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.[20]
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:[20]
$F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over 2b-X})^{a}&X<b\\1-{\tfrac {1}{2}}({\tfrac {b}{X}})^{a}&X\geq b\end{cases}}$
The corresponding probability density function (PDF) is:[20]
$p(x)={ab^{a} \over 2(b+\left\vert x-b\right\vert )^{a+1}},X\in R$
This distribution has two parameters: a and b. It is symmetric by b. Then the mathematic expectation is b. When, it has variance as following:
$E((x-b)^{2})=\int _{-\infty }^{\infty }(x-b)^{2}p(x)dx={2b^{2} \over (a-2)(a-1)}$
The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:
$F(X)=P(x<X)={\begin{cases}{\tfrac {1}{2}}({b \over b-X})^{a}&X<0\\1-{\tfrac {1}{2}}({\tfrac {b}{b+X}})^{a}&X\geq 0\end{cases}}$
The corresponding PDF is:
$p(x)={ab^{a} \over 2(b+\left\vert x\right\vert )^{a+1}},X\in R$
This distribution is symmetric by zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.[20]
Multivariate Pareto distribution
The univariate Pareto distribution has been extended to a multivariate Pareto distribution.[21]
Statistical inference
Estimation of parameters
The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
$L(\alpha ,x_{\mathrm {m} })=\prod _{i=1}^{n}\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x_{i}^{\alpha +1}}}=\alpha ^{n}x_{\mathrm {m} }^{n\alpha }\prod _{i=1}^{n}{\frac {1}{x_{i}^{\alpha +1}}}.$
Therefore, the logarithmic likelihood function is
$\ell (\alpha ,x_{\mathrm {m} })=n\ln \alpha +n\alpha \ln x_{\mathrm {m} }-(\alpha +1)\sum _{i=1}^{n}\ln x_{i}.$
It can be seen that $\ell (\alpha ,x_{\mathrm {m} })$ is monotonically increasing with xm, that is, the greater the value of xm, the greater the value of the likelihood function. Hence, since x ≥ xm, we conclude that
${\widehat {x}}_{\mathrm {m} }=\min _{i}{x_{i}}.$
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
${\frac {\partial \ell }{\partial \alpha }}={\frac {n}{\alpha }}+n\ln x_{\mathrm {m} }-\sum _{i=1}^{n}\ln x_{i}=0.$
Thus the maximum likelihood estimator for α is:
${\widehat {\alpha }}={\frac {n}{\sum _{i}\ln(x_{i}/{\widehat {x}}_{\mathrm {m} })}}.$
The expected statistical error is:[22]
$\sigma ={\frac {\widehat {\alpha }}{\sqrt {n}}}.$
Malik (1970)[23] gives the exact joint distribution of $({\hat {x}}_{\mathrm {m} },{\hat {\alpha }})$. In particular, ${\hat {x}}_{\mathrm {m} }$ and ${\hat {\alpha }}$ are independent and ${\hat {x}}_{\mathrm {m} }$ is Pareto with scale parameter xm and shape parameter nα, whereas ${\hat {\alpha }}$ has an inverse-gamma distribution with shape and scale parameters n − 1 and nα, respectively.
Occurrence and applications
General
Vilfredo Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.[4] This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.[24] However, the 80-20 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, net worth may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:
• The sizes of human settlements (few cities, many hamlets/villages)[25][26]
• File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)[25]
• Hard disk drive error rates[27]
• Clusters of Bose–Einstein condensate near absolute zero[28]
• The values of oil reserves in oil fields (a few large fields, many small fields)[25]
• The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)[29]
• The standardized price returns on individual stocks [25]
• Sizes of sand particles [25]
• The size of meteorites
• Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.[30][31]
• Amount of time a user on Steam will spend playing different games. (Some games get played a lot, but most get played almost never.)
• In hydrology the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.[32] The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
• In Electric Utility Distribution Reliability (80% of the Customer Minutes Interrupted occur on approximately 20% of the days in a given year).
Relation to Zipf's law
The Pareto distribution is a continuous probability distribution. Zipf's law, also sometimes called the zeta distribution, is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the $x$ values (incomes) are binned into $N$ ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining $x_{m}$ so that $\alpha x_{\mathrm {m} }^{\alpha }={\frac {1}{H(N,\alpha -1)}}$ where $H(N,\alpha -1)$ is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's.
$f(x)={\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}={\frac {1}{x^{s}H(N,s)}}$
where $s=\alpha -1$ and $x$ is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has $f(x)$ probability of ranking $x$.
Relation to the "Pareto principle"
The "80–20 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is $\alpha =\log _{4}5={\cfrac {\log _{10}5}{\log _{10}4}}\approx 1.161$. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown[33] to be mathematically equivalent:
• Income is distributed according to a Pareto distribution with index α > 1.
• There is some number 0 ≤ p ≤ 1/2 such that 100p % of all people receive 100(1 − p)% of all income, and similarly for every real (not necessarily integer) n > 0, 100pn % of all people receive 100(1 − p)n percentage of all income. α and p are related by
$1-{\frac {1}{\alpha }}={\frac {\ln(1-p)}{\ln(p)}}={\frac {\ln((1-p)^{n})}{\ln(p^{n})}}$
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have an infinite expected value, and so cannot reasonably model income distribution.
Relation to Price's law
Price's square root law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that $\alpha =1$. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.
Lorenz curve and Gini coefficient
The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF f or the CDF F as
$L(F)={\frac {\int _{x_{\mathrm {m} }}^{x(F)}xf(x)\,dx}{\int _{x_{\mathrm {m} }}^{\infty }xf(x)\,dx}}={\frac {\int _{0}^{F}x(F')\,dF'}{\int _{0}^{1}x(F')\,dF'}}$
where x(F) is the inverse of the CDF. For the Pareto distribution,
$x(F)={\frac {x_{\mathrm {m} }}{(1-F)^{\frac {1}{\alpha }}}}$
and the Lorenz curve is calculated to be
$L(F)=1-(1-F)^{1-{\frac {1}{\alpha }}},$
For $0<\alpha \leq 1$ the denominator is infinite, yielding L=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
According to Oxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population.[34] We can estimate the Pareto index that would apply to this situation. Letting ε equal $62/(7\times 10^{9})$ we have:
$L(1/2)=1-L(1-\varepsilon )$
or
$1-(1/2)^{1-{\frac {1}{\alpha }}}=\varepsilon ^{1-{\frac {1}{\alpha }}}$
The solution is that α equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.[35]
The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated (for $\alpha \geq 1$) to be
$G=1-2\left(\int _{0}^{1}L(F)\,dF\right)={\frac {1}{2\alpha -1}}$
(see Aaberge 2005).
Random variate generation
Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0, 1], the variate T given by
$T={\frac {x_{\mathrm {m} }}{U^{1/\alpha }}}$
is Pareto-distributed.[36] If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).
See also
• Bradford's law – Pattern of references in science journals
• Gutenberg–Richter law
• Matthew effect – The rich get richer and the poor get poorer
• Pareto analysis – Statistical concept
• Pareto efficiency – Concept in studies of efficiency
• Pareto interpolation
• Power law probability distributions – Functional relationship between two quantities
• Sturgeon's law – "Ninety percent of everything is crap"
• Traffic generation model – simulated flow of data in a communications networkPages displaying wikidata descriptions as a fallback
• Zipf's law – Probability distribution
• Heavy-tailed distribution – Probability distribution
References
1. Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. Springer. 299 (1–2): 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Retrieved 2023-02-27.
2. Amoroso, Luigi (1938). "VILFREDO PARETO". Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest. 6.
3. Pareto, Vilfredo (1898). "Cours d'economie politique". Journal of Political Economy. 6. doi:10.1086/250536.
4. Pareto, Vilfredo, Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964, pp. 299–345. Original book archived
5. VAN MONTFORT, M.A.J. (1986). "The Generalized Pareto distribution applied to rainfall depths". Hydrological Sciences Journal. 31 (2): 151–162. doi:10.1080/02626668609491037.
6. Oancea, Bogdan (2017). "Income inequality in Romania: The exponential-Pareto distribution". Physica A: Statistical Mechanics and Its Applications. 469: 486–498. Bibcode:2017PhyA..469..486O. doi:10.1016/j.physa.2016.11.094.
7. Morella, Matteo. "Pareto Distribution". academia.edu.
8. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 978-0-89974-012-6.
9. S. Hussain, S.H. Bhatti (2018). Parameter estimation of Pareto distribution: Some modified moment estimators. Maejo International Journal of Science and Technology 12(1):11-27
10. Eliazar, Iddo (November 2017). "Lindy's Law". Physica A: Statistical Mechanics and Its Applications. 486: 797–805. Bibcode:2017PhyA..486..797E. doi:10.1016/j.physa.2017.05.077. S2CID 125349686.
11. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.
12. Johnson, Kotz, and Balakrishnan (1994), (20.4).
13. Christian Kleiber & Samuel Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 978-0-471-15064-0.
14. Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. II (2nd ed.). New York: Wiley. p. 50. "The densities (4.3) are sometimes called after the economist Pareto. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ Ax−α as x → ∞."
15. Lomax, K. S. (1954). "Business failures. Another example of the analysis of failure data". Journal of the American Statistical Association. 49 (268): 847–52. doi:10.1080/01621459.1954.10501239.
16. Chotikapanich, Duangkamon (16 September 2008). "Chapter 7: Pareto and Generalized Pareto Distributions". Modeling Income Distributions and Lorenz Curves. pp. 121–22. ISBN 9780387727967.
17. Dallas, A. C. "Characterizing the Pareto and power distributions." Annals of the Institute of Statistical Mathematics 28.1 (1976): 491-497.
18. White, Gentry (2006). Bayesian semiparametric spatial and joint spatio-temporal modeling (Thesis thesis). University of Missouri--Columbia. section 5.3.1
19. "Inverse Transform Method". Archived from the original on 2012-01-17. Retrieved 2012-08-27.
20. Huang, Xiao-dong (2004). "A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic". IEEE Transactions on Broadcasting. 50 (3): 323–334. doi:10.1109/TBC.2004.834013.
21. Rootzén, Holger; Tajvidi, Nader (2006). "Multivariate generalized Pareto distributions". Bernoulli. 12 (5): 917–30. CiteSeerX 10.1.1.145.2991. doi:10.3150/bj/1161614952. S2CID 16504396.
22. M. E. J. Newman (2005). "Power laws, Pareto distributions and Zipf's law". Contemporary Physics. 46 (5): 323–51. arXiv:cond-mat/0412004. Bibcode:2005ConPh..46..323N. doi:10.1080/00107510500052444. S2CID 202719165.
23. H. J. Malik (1970). "Estimation of the Parameters of the Pareto Distribution". Metrika. 15: 126–132. doi:10.1007/BF02613565. S2CID 124007966.
24. For a two-quantile population, where approximately 18% of the population owns 82% of the wealth, the Theil index takes the value 1.
25. Reed, William J.; et al. (2004). "The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions". Communications in Statistics – Theory and Methods. 33 (8): 1733–53. CiteSeerX 10.1.1.70.4555. doi:10.1081/sta-120037438. S2CID 13906086.
26. Reed, William J. (2002). "On the rank‐size distribution for human settlements". Journal of Regional Science. 42 (1): 1–17. doi:10.1111/1467-9787.00247. S2CID 154285730.
27. Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24). "Understanding latent sector error and how to protect against them" (PDF). 8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved 2010-09-10. We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ2 statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.
28. Yuji Ijiri; Simon, Herbert A. (May 1975). "Some Distributions Associated with Bose–Einstein Statistics". Proc. Natl. Acad. Sci. USA. 72 (5): 1654–57. Bibcode:1975PNAS...72.1654I. doi:10.1073/pnas.72.5.1654. PMC 432601. PMID 16578724.
29. Harchol-Balter, Mor; Downey, Allen (August 1997). "Exploiting Process Lifetime Distributions for Dynamic Load Balancing" (PDF). ACM Transactions on Computer Systems. 15 (3): 253–258. doi:10.1145/263326.263344. S2CID 52861447.
30. Kleiber and Kotz (2003): p. 94.
31. Seal, H. (1980). "Survival probabilities based on Pareto claim distributions". ASTIN Bulletin. 11: 61–71. doi:10.1017/S0515036100006620.
32. CumFreq, software for cumulative frequency analysis and probability distribution fitting
33. Hardy, Michael (2010). "Pareto's Law". Mathematical Intelligencer. 32 (3): 38–43. doi:10.1007/s00283-010-9159-2. S2CID 121797873.
34. "62 people own the same as half the world, reveals Oxfam Davos report". Oxfam. Jan 2016.
35. "Global Wealth Report 2013". Credit Suisse. Oct 2013. p. 22. Archived from the original on 2015-02-14. Retrieved 2016-01-24.
36. Tanizaki, Hisashi (2004). Computational Methods in Statistics and Econometrics. CRC Press. p. 133. ISBN 9780824750886.
Notes
• M. O. Lorenz (1905). "Methods of measuring the concentration of wealth". Publications of the American Statistical Association. 9 (70): 209–19. Bibcode:1905PAmSA...9..209L. doi:10.2307/2276207. JSTOR 2276207. S2CID 154048722.
• Pareto, Vilfredo (1965). Librairie Droz (ed.). Ecrits sur la courbe de la répartition de la richesse. Œuvres complètes : T. III. p. 48. ISBN 9782600040211.
• Pareto, Vilfredo (1895). "La legge della domanda". Giornale Degli Economisti. 10: 59–68.
• Pareto, Vilfredo (1896). "Cours d'économie politique". doi:10.1177/000271629700900314. S2CID 143528002. {{cite journal}}: Cite journal requires |journal= (help)
External links
• "Pareto distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Pareto distribution". MathWorld.
• Aabergé, Rolf (May 2005). "Gini's Nuclear Family". International Conference to Honor Two Eminent Social Scientists (PDF).
• Crovella, Mark E.; Bestavros, Azer (December 1997). Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes (PDF). IEEE/ACM Transactions on Networking. Vol. 5. pp. 835–846. Archived from the original (PDF) on 2016-03-04. Retrieved 2019-02-25.
• syntraf1.c is a C program to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.
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
Authority control: National
• Germany
| Wikipedia |
Bifrustum
An n-agonal bifrustum is a polyhedron composed of three parallel planes of n-agons, with the middle plane largest and usually the top and bottom congruent.
Set of bifrusta
Example hexagonal bifrustum
Faces2 n-gons, 2n trapezoids
Edges5n
Vertices3n
Symmetry groupDnh, [n,2], (*n22)
Dual polyhedronElongated bipyramids
Propertiesconvex
It can be constructed as two congruent frusta combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated.[1]
They are duals to the family of elongated bipyramids.
Formulae
For a regular n-gonal bifrustum with the equatorial polygon sides a, bases sides b and semi-height (half the distance between the planes of bases) h, the lateral surface area Al, total area A and volume V are:[2]
$A_{l}=n(a+b){\sqrt {\left({\frac {a-b}{2}}\cot {\frac {\pi }{n}}\right)^{2}+h^{2}}}\,,$
$A=A_{l}+n{\frac {b^{2}}{2\tan {\frac {\pi }{n}}}}\,,$
$V=n{\frac {a^{2}-b^{2}}{6\tan {\frac {\pi }{n}}}}h\,.$
Forms
Three bifrusta are duals to three Johnson solids, J14-16. In general, a n-agonal bifrustum has 2n trapezoids, 2 n-agons, and is dual to the elongated dipyramids.
Triangular bifrustum Square bifrustum Pentagonal bifrustum
6 trapezoids, 2 triangles. Dual to elongated triangular bipyramid, J14 8 trapezoids, 2 squares. Dual to elongated square bipyramid, J15 10 trapezoids, 2 pentagons. Dual to elongated pentagonal bipyramid, J16
References
1. "Octagonal Bifrustum". etc.usf.edu. Retrieved 2022-06-16.
2. "Regelmäßiges Bifrustum - Rechner". RECHNERonline (in German). Retrieved 2022-06-30.
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Compound of five truncated tetrahedra
The compound of five truncated tetrahedra is a uniform polyhedron compound. It's composed of 5 truncated tetrahedra rotated around a common axis. It may be formed by truncating each of the tetrahedra in the compound of five tetrahedra. A far-enough truncation creates the compound of five octahedra. Its convex hull is a nonuniform snub dodecahedron.
Compound of five truncated tetrahedra
TypeUniform compound
IndexUC55
Polyhedra5 truncated tetrahedra
Faces20 triangles, 20 hexagons
Edges90
Vertices60
DualCompound of five triakis tetrahedra
Symmetry groupchiral icosahedral (I)
Subgroup restricting to one constituentchiral tetrahedral (T)
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
(±1, ±1, ±3)
(±τ−1, ±(−τ−2), ±2τ)
(±τ, ±(−2τ−1), ±τ2)
(±τ2, ±(−τ−2), ±2)
(±(2τ−1), ±1, ±(2τ − 1))
with an even number of minuses in the choices for '±', where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.
| Wikipedia |
Truncated cubic prism
In geometry, a truncated cubic prism is a convex uniform polychoron (four-dimensional polytope).
Truncated cubic prism
Schlegel diagram
TypePrismatic uniform polychoron
Uniform index52
Schläfli symbolt0,1,3{4,3,2} or t{4,3}×{}
Coxeter-Dynkin
Cells16 total:
2 3.8.8
8 3.4.4
6 4.4.8
Faces65 total:
16 {3}
36 {4}
12 {8}
Edges96
Vertices48
Vertex figure
Square pyramid
Symmetry group[4,3,2], order 96
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
Net
Alternative names
• Truncated-cubic hyperprism
• Truncated-cubic dyadic prism (Norman W. Johnson)
• Ticcup (Jonathan Bowers: for truncated-cube prism)
See also
• Truncated tesseract,
External links
• 6. Convex uniform prismatic polychora - Model 52, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) o3x4x x - ticcup".
| Wikipedia |
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
Truncated cube
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 14, E = 36, V = 24 (χ = 2)
Faces by sides8{3}+6{8}
Conway notationtC
Schläfli symbolst{4,3}
t0,1{4,3}
Wythoff symbol2 3 | 4
Coxeter diagram
Symmetry groupOh, B3, [4,3], (*432), order 48
Rotation groupO, [4,3]+, (432), order 24
Dihedral angle3-8: 125°15′51″
8-8: 90°
ReferencesU09, C21, W8
PropertiesSemiregular convex
Colored faces
3.8.8
(Vertex figure)
Triakis octahedron
(dual polyhedron)
Net
If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + √2.
Area and volume
The area A and the volume V of a truncated cube of edge length a are:
${\begin{aligned}A&=2\left(6+6{\sqrt {2}}+{\sqrt {3}}\right)a^{2}&&\approx 32.434\,6644a^{2}\\V&={\frac {21+14{\sqrt {2}}}{3}}a^{3}&&\approx 13.599\,6633a^{3}.\end{aligned}}$
Orthogonal projections
The truncated cube has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge
3-8
Edge
8-8
Face
Octagon
Face
Triangle
Solid
Wireframe
Dual
Projective
symmetry
[2] [2] [2] [4] [6]
Spherical tiling
The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
octagon-centered
triangle-centered
Orthographic projection Stereographic projections
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated hexahedron centered at the origin with edge length 2ξ are all the permutations of
(±ξ, ±1, ±1),
where ξ = √2 − 1.
The parameter ξ can be varied between ±1. A value of 1 produces a cube, 0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces.
If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, truncated octahedra are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.
Dissection
The truncated cube can be dissected into a central cube, with six square cupolae around each of the cube's faces, and 8 regular tetrahedra in the corners. This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells.
This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupolae and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.[1][2]
Vertex arrangement
It shares the vertex arrangement with three nonconvex uniform polyhedra:
Truncated cube
Nonconvex great rhombicuboctahedron
Great cubicuboctahedron
Great rhombihexahedron
Related polyhedra
The truncated cube is related to other polyhedra and tilings in symmetry.
The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
=
=
=
=
or
=
or
=
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
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, and a series of polyhedra and tilings n.8.8.
*n32 symmetry mutation of truncated spherical tilings: t{n,3}
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]
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}
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.∞.∞
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
Alternated truncation
Tetrahedron, its edge truncation, and the truncated cube
Truncating alternating vertices of the cube gives the chamfered tetrahedron, i.e. the edge truncation of the tetrahedron.
The truncated triangular trapezohedron is another polyhedron which can be formed from cube edge truncation.
Related polytopes
The truncated cube, is second in a sequence of truncated hypercubes:
Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}
Truncated cubical graph
Truncated cubical graph
4-fold symmetry Schlegel diagram
Vertices24
Edges36
Automorphisms48
Chromatic number3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters
In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the truncated cube, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[3]
Orthographic
See also
• Spinning truncated cube
• Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube
References
1. B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4
2. "Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1".
3. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p. 79-86 Archimedean solids
External links
• Eric W. Weisstein, Truncated cube (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Truncated cubical graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra o3x4x - tic".
• Editable printable net of a truncated cube with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
• VRML model
• Conway Notation for Polyhedra Try: "tC"
Archimedean solids
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Catalan duals
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Cantic 6-cube
In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope.
Cantic 6-cube
Truncated 6-demicube
D6 Coxeter plane projection
Typeuniform polypeton
Schläfli symbolt0,1{3,33,1}
h2{4,34}
Coxeter-Dynkin diagram =
5-faces76
4-faces636
Cells2080
Faces3200
Edges2160
Vertices480
Vertex figure( )v[{ }x{3,3}]
Coxeter groupsD6, [33,1,1]
Propertiesconvex
Alternate names
• Truncated 6-demicube/demihexeract (Acronym thax) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a cantic 6-cube centered at the origin and edge length 6√2 are coordinate permutations:
(±1,±1,±3,±3,±3,±3)
with an odd number of plus signs.
Images
orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
Dimensional family of cantic n-cubes
n345678
Symmetry
[1+,4,3n-2]
[1+,4,3]
= [3,3]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Cantic
figure
Coxeter
=
=
=
=
=
=
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
D6 polytopes
h{4,34}
h2{4,34}
h3{4,34}
h4{4,34}
h5{4,34}
h2,3{4,34}
h2,4{4,34}
h2,5{4,34}
h3,4{4,34}
h3,5{4,34}
h4,5{4,34}
h2,3,4{4,34}
h2,3,5{4,34}
h2,4,5{4,34}
h3,4,5{4,34}
h2,3,4,5{4,34}
Notes
1. Klitizing, (x3x3o *b3o3o3o – thax)
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. "6D uniform polytopes (polypeta)". x3x3o *b3o3o3o – thax
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 |
Steric 5-cubes
In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.
• 5-cube
• Steric 5-cube
• Stericantic 5-cube
• Half 5-cube
• Steriruncic 5-cube
• Steriruncicantic 5-cube
Orthogonal projections in B5 Coxeter plane
Steric 5-cube
Steric 5-cube
Typeuniform polyteron
Schläfli symbol
• t0,3{3,32,1}
• h4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells480
Faces720
Edges400
Vertices80
Vertex figure{3,3}-t1{3,3} antiprism
Coxeter groupsD5, [32,1,1]
Propertiesconvex
Alternate names
• Steric penteract, runcinated demipenteract
• Small prismated hemipenteract (siphin) (Jonathan Bowers)[1]: (x3o3o *b3o3x - siphin)
Cartesian coordinates
The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of
(±1,±1,±1,±1,±3)
with an odd number of plus signs.
Images
orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]
Related polytopes
Dimensional family of steric n-cubes
n5678
[1+,4,3n-2]
= [3,3n-3,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Steric
figure
Coxeter
=
=
=
=
Schläfli h4{4,33} h4{4,34} h4{4,35} h4{4,36}
Stericantic 5-cube
Stericantic 5-cube
Typeuniform polyteron
Schläfli symbol
• t0,1,3{3,32,1}
• h2,4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells720
Faces1840
Edges1680
Vertices480
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex
Alternate names
• Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)[1]: (x3x3o *b3o3x - pithin)
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:
(±1,±1,±3,±3,±5)
with an odd number of plus signs.
Images
orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]
Steriruncic 5-cube
Steriruncic 5-cube
Typeuniform polyteron
Schläfli symbol
• t0,2,3{3,32,1}
• h3,4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells560
Faces1280
Edges1120
Vertices320
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex
Alternate names
• Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)[1]: (x3o3o *b3x3x - pirhin)
Cartesian coordinates
The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±5)
with an odd number of plus signs.
Images
orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]
Steriruncicantic 5-cube
Steriruncicantic 5-cube
Typeuniform polyteron
Schläfli symbol
• t0,1,2,3{3,32,1}
• h2,3,4{4,3,3,3
}
Coxeter-Dynkin diagram
4-faces82
Cells720
Faces2080
Edges2400
Vertices960
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex
Alternate names
• Great prismated hemipenteract (giphin) (Jonathan Bowers)[1]: (x3x3o *b3x3x - giphin)
Cartesian coordinates
The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:
(±1,±1,±3,±5,±7)
with an odd number of plus signs.
Images
orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]
Related polytopes
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
D5 polytopes
h{4,3,3,3}
h2{4,3,3,3}
h3{4,3,3,3}
h4{4,3,3,3}
h2,3{4,3,3,3}
h2,4{4,3,3,3}
h3,4{4,3,3,3}
h2,3,4{4,3,3,3}
References
1. Klitzing, Richard. "5D uniform polytopes (polytera)".
Further reading
• Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York City: Dover. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1995-05-17). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivić (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons. ISBN 978-0-471-01003-6. LCCN 94047368. OCLC 632987525. OL 7598569M. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1940-12-01). "Regular and Semi Regular Polytopes I". Mathematische Zeitschrift. Springer Nature. 46: 380–407. doi:10.1007/BF01181449. ISSN 1432-1823. S2CID 186237114. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1985-12-01). "Regular and Semi-Regular Polytopes II". Mathematische Zeitschrift. Springer Nature. 188 (4): 559–591. doi:10.1007/BF01161657. ISSN 1432-1823. S2CID 120429557. Retrieved 2022-05-19.
• Coxeter, H. S. M. (1988-03-01). "Regular and Semi-Regular Polytopes III". Mathematische Zeitschrift. Springer Nature. 200 (1): 3–45. doi:10.1007/BF01161745. ISSN 1432-1823. S2CID 186237142. Retrieved 2022-05-19.
• Johnson, Norman W. (1991). Uniform Polytopes (Unfinished manuscript thesis).
• Johnson, Norman W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD thesis). University of Toronto. Retrieved 2022-05-19.
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• 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 |
Truncated dodecahedral prism
In geometry, a truncated dodecahedral prism is a convex uniform polychoron (four-dimensional polytope).
Truncated dodecahedral prism
Schlegel diagram
Decagonal prisms hidden
TypePrismatic uniform polychoron
Uniform index60
Schläfli symbolt0,1,3{3,5,2} or t{3,5}×{}
Coxeter-Dynkin
Cells34 total:
2 t0,1{5,3}
12 {}x{10}
20 {}x{3}
Faces154 total:
40 {3}
90 {4}
24 {10}
Edges240
Vertices120
Vertex figure
Isosceles-triangular pyramid
Symmetry group[5,3,2], order 240
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
Alternative names
• Truncated-dodecahedral dyadic prism (Norman W. Johnson)
• Tiddip (Jonathan Bowers: for truncated-dodecahedral prism)
• Truncated-dodecahedral hyperprism
See also
• Truncated 120-cell,
External links
• 6. Convex uniform prismatic polychora - Model 60, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x o3x5x - tiddip".
| Wikipedia |
Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
Truncated dodecahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 90, V = 60 (χ = 2)
Faces by sides20{3}+12{10}
Conway notationtD
Schläfli symbolst{5,3}
t0,1{5,3}
Wythoff symbol2 3 | 5
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle10-10: 116.57°
3-10: 142.62°
ReferencesU26, C29, W10
PropertiesSemiregular convex
Colored faces
3.10.10
(Vertex figure)
Triakis icosahedron
(dual polyhedron)
Net
Geometric relations
This polyhedron can be formed from a regular dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles.
It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.
Area and volume
The area A and the volume V of a truncated dodecahedron of edge length a are:
${\begin{aligned}A&=5\left({\sqrt {3}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 100.990\,76a^{2}\\V&={\tfrac {5}{12}}\left(99+47{\sqrt {5}}\right)a^{3}&&\approx 85.039\,6646a^{3}\end{aligned}}$
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2φ − 2, centered at the origin,[1] are all even permutations of:
(0, ±1/φ, ±(2 + φ))
(±1/φ, ±φ, ±2φ)
(±φ, ±2, ±(φ + 1))
where φ = 1 + √5/2 is the golden ratio.
Orthogonal projections
The truncated dodecahedron has five special orthogonal projections, centered: on a vertex, on two types of edges, and two types of faces. The last two correspond to the A2 and H2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge
3-3
Edge
10-10
Face
Triangle
Face
Decagon
Solid
Wireframe
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
Spherical tilings and Schlegel diagrams
The truncated dodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Schlegel diagrams are similar, with a perspective projection and straight edges.
Orthographic projection Stereographic projections
Decagon-centered
Triangle-centered
Vertex arrangement
It shares its vertex arrangement with three nonconvex uniform polyhedra:
Truncated dodecahedron
Great icosicosidodecahedron
Great ditrigonal dodecicosidodecahedron
Great dodecicosahedron
Related polyhedra and tilings
It is part of a truncation process between a dodecahedron and icosahedron:
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
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 spherical tilings: t{n,3}
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]
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}
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.∞.∞
Truncated dodecahedral graph
Truncated dodecahedral graph
5-fold symmetry Schlegel diagram
Vertices60
Edges90
Automorphisms120
Chromatic number3
Chromatic index3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters
In the mathematical field of graph theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.[2]
Circular
Notes
1. Weisstein, Eric W. "Icosahedral group". MathWorld.
2. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
References
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
• Eric W. Weisstein, Truncated dodecahedron (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Truncated dodecahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra o3x5x - tid".
• Editable printable net of a truncated dodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Archimedean solids
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Catalan duals
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Truncated great dodecahedron
In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices.[1] It is given a Schläfli symbol t{5,5/2}.
Truncated great dodecahedron
TypeUniform star polyhedron
ElementsF = 24, E = 90
V = 60 (χ = −6)
Faces by sides12{5/2}+12{10}
Coxeter diagram
Wythoff symbol2 5/2 | 5
2 5/3 | 5
Symmetry groupIh, [5,3], *532
Index referencesU37, C47, W75
Dual polyhedronSmall stellapentakis dodecahedron
Vertex figure
10.10.5/2
Bowers acronymTigid
Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron; and with the uniform compounds of 6 or 12 pentagonal prisms.
Nonconvex great rhombicosidodecahedron
Great dodecicosidodecahedron
Great rhombidodecahedron
Truncated great dodecahedron
Compound of six pentagonal prisms
Compound of twelve pentagonal prisms
This polyhedron is the truncation of the great dodecahedron:
The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces, 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams).
Name Small stellated dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated
great
dodecahedron
Great
dodecahedron
Coxeter-Dynkin
diagram
Picture
Small stellapentakis dodecahedron
Small stellapentakis dodecahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 90
V = 24 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU37
dual polyhedronTruncated great dodecahedron
The small stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "37: truncated great dodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Truncated great dodecahedron". MathWorld.
• Weisstein, Eric W. "Small stellapentakis dodecahedron". MathWorld.
• Uniform polyhedra and duals
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
| Wikipedia |
Truncated order-4 heptagonal tiling
In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}.
Truncated heptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.14.14
Schläfli symbolt{7,4}
Wythoff symbol2 4 | 7
2 7 7 |
Coxeter diagram
or
Symmetry group[7,4], (*742)
[7,7], (*772)
DualOrder-7 tetrakis square tiling
PropertiesVertex-transitive
Constructions
There are two uniform constructions of this tiling, first by the [7,4] kaleidoscope, and second by removing the last mirror, [7,4,1+], gives [7,7], (*772).
Two uniform constructions of 4.7.4.7
Name Tetraheptagonal Truncated heptaheptagonal
Image
Symmetry [7,4]
(*742)
[7,7] = [7,4,1+]
(*772)
=
Symbol t{7,4} tr{7,7}
Coxeter diagram
Symmetry
There is only one simple subgroup [7,7]+, index 2, removing all the mirrors. This symmetry can be doubled to 742 symmetry by adding a bisecting mirror.
Small index subgroups of [7,7]
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)
[7,7] =
(*772)
[7,7]+ =
(772)
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{7,7} t{7,7}
r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7
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.
See also
Wikimedia Commons has media related to Uniform tiling 4-14-14.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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 |
Truncated 16-cell honeycomb
In four-dimensional Euclidean geometry, the truncated 16-cell honeycomb (or cantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by 24-cell and truncated 16-cell facets.
Truncated 16-cell honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolst{3,3,4,3}
h2{4,3,3,4}
t{3,31,1,1}
Coxeter diagrams
=
4-face type{3,4,3}
t{3,3,4}
Cell type{3,3}
t{3,3}
Face type{3}
{6}
Vertex figurecubic pyramid
Coxeter group${\tilde {F}}_{4}$ = [3,3,4,3]
${\tilde {B}}_{4}$ = [4,3,31,1]
${\tilde {D}}_{4}$ = [31,1,1,1]
Dual?
Propertiesvertex-transitive
Alternate names
• Truncated hexadecachoric tetracomb / Truncated hexadecachoric honeycomb
Related honeycombs
The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.
F4 honeycombs
Extended
symmetry
Extended
diagram
OrderHoneycombs
[3,3,4,3]×1
1, 3, 5, 6, 8,
9, 10, 11, 12
[3,4,3,3]×1
2, 4, 7, 13,
14, 15, 16, 17,
18, 19, 20, 21,
22 23, 24, 25,
26, 27, 28, 29
[(3,3)[3,3,4,3*]]
=[(3,3)[31,1,1,1]]
=[3,4,3,3]
=
=
×4
(2), (4), (7), (13)
The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.
C4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1
1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13
[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×6
14, 15, 16, 17
The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.
B4 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]: ×1
5, 6, 7, 8
<[4,3,31,1]>:
↔[4,3,3,4]
↔
×2
9, 10, 11, 12, 13, 14,
(10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
↔ [3,3,4,3]
↔
↔
×3
1, 2, 3, 4
[(3,3)[1+,4,3,31,1]]
↔ [(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
↔
×12
20, 21, 22, 23
There are ten uniform honeycombs constructed by the ${\tilde {D}}_{4}$ Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1].
The ten permutations are listed with its highest extended symmetry relation:
D4 honeycombs
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
[31,1,1,1] ${\tilde {D}}_{4}$ (none)
<[31,1,1,1]>
↔ [31,1,3,4]
↔
${\tilde {D}}_{4}$×2 = ${\tilde {B}}_{4}$ (none)
<2[1,131,1]>
↔ [4,3,3,4]
↔
${\tilde {D}}_{4}$×4 = ${\tilde {C}}_{4}$ 1, 2
[3[3,31,1,1]]
↔ [3,3,4,3]
↔
${\tilde {D}}_{4}$×6 = ${\tilde {F}}_{4}$ 3, 4, 5, 6
[4[1,131,1]]
↔ [[4,3,3,4]]
↔
${\tilde {D}}_{4}$×8 = ${\tilde {C}}_{4}$×2 7, 8, 9
[(3,3)[31,1,1,1]]
↔ [3,4,3,3]
↔
${\tilde {D}}_{4}$×24 = ${\tilde {F}}_{4}$
[(3,3)[31,1,1,1]]+
↔ [3+,4,3,3]
↔
½${\tilde {D}}_{4}$×24 = ½${\tilde {F}}_{4}$ 10
See also
Regular and uniform honeycombs in 4-space:
• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Rectified 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
Notes
References
• 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]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Klitzing, Richard. "4D Euclidean tesselations". (x3x3o *b3o4o), (x3x3o *b3o *b3o), x3x3o4o3o - thext - O105
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 |
Truncated hexagonal trapezohedron
In geometry, the truncated hexagonal trapezohedron is the fourth in an infinite series of truncated trapezohedra. It has 12 pentagon and 2 hexagon faces. It can be constructed by taking a hexagonal trapezohedron and truncating the polar axis vertices.
Truncated hexagonal trapezohedron
TypeTruncated trapezohedron
Faces12 pentagons
2 hexagons
Edges36
Vertices24
Conway notationt6dA6
Symmetry groupD6d, [12,2+], 2*6, order 24
Rotation groupD6, [6,2]+, 226, order 12
Dual polyhedronGyroelongated hexagonal dipyramid
Propertiesconvex
Weaire–Phelan structure
Another form of this polyhedron has D2d symmetry and is a part of a space-filling honeycomb along with an irregular dodecahedron, called Weaire–Phelan structure.
irregular tetradecahedron
(Truncated hexagonal trapezohedron)
Weaire–Phelan honeycomb
See also
• Goldberg polyhedron
External links
• Conway Notation for Polyhedra Try: "t6dA6".
| Wikipedia |
Truncated order-4 hexagonal tiling
In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.
Truncated order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.12.12
Schläfli symbolt{6,4}
tr{6,6} or $t{\begin{Bmatrix}6\\6\end{Bmatrix}}$
Wythoff symbol2 4 | 6
2 6 6 |
Coxeter diagram
or
Symmetry group[6,4], (*642)
[6,6], (*662)
DualOrder-6 tetrakis square tiling
PropertiesVertex-transitive
Constructions
There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662).
Two uniform constructions of 4.6.4.6
Name Tetrahexagonal Truncated hexahexagonal
Image
Symmetry [6,4]
(*642)
[6,6] = [6,4,1+]
(*662)
=
Symbol t{6,4} tr{6,6}
Coxeter diagram
Dual tiling
The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group.
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663)
[6+,6]
(6*3)
[6,1+,6]
(*3232)
[6,6+]
(6*3)
[6,6,1+]
(*663)
[(6,6,2+)]
(2*33)
[6,6]+
(662)
= = =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
Symmetry
The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,6,1+,6,1+] (3333) is the commutator subgroup of [6,6].
Larger subgroup constructed as [6,6*], removing the gyration points of (6*3), index 12 becomes (*333333).
The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.
Small index subgroups of [6,6] (*662)
Index 1 2 4
Diagram
Coxeter [6,6]
[1+,6,6]
=
[6,6,1+]
=
[6,1+,6]
=
[1+,6,6,1+]
=
[6+,6+]
Orbifold *662 *663 *3232 *3333 33×
Direct subgroups
Diagram
Coxeter [6,6+]
[6+,6]
[(6,6,2+)]
[6,1+,6,1+]
= =
= =
[1+,6,1+,6]
= =
= =
Orbifold 6*3 2*33 3*33
Direct subgroups
Index 2 4 8
Diagram
Coxeter [6,6]+
[6,6+]+
=
[6+,6]+
=
[6,1+,6]+
=
[6+,6+]+ = [1+,6,1+,6]+
= = =
Orbifold 662 663 3232 3333
Radical subgroups
Index 12 24
Diagram
Coxeter [6,6*]
[6*,6]
[6,6*]+
[6*,6]+
Orbifold *333333 333333
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.
See also
Wikimedia Commons has media related to Uniform tiling 4-12-12.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Truncated hexaoctagonal tiling
In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.
Truncated hexaoctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.12.16
Schläfli symboltr{8,6} or $t{\begin{Bmatrix}8\\6\end{Bmatrix}}$
Wythoff symbol2 8 6 |
Coxeter diagram or
Symmetry group[8,6], (*862)
DualOrder-6-8 kisrhombille tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling is called an order-6-8 kisrhombille tiling, made as a complete bisection of the order-6 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,6] (*862) symmetry.
Symmetry
There are six reflective subgroup kaleidoscopic constructed from [8,6] by removing one or two of three mirrors. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,8,1+,6,1+] (4343) is the commutator subgroup of [8,6].
A radical subgroup is constructed as [8,6*], index 12, as [8,6+], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8+,6], (8*3) with gyration points removed as (*33333333).
Small index subgroups of [8,6] (*862)
Index 1 2 4
Diagram
Coxeter [8,6]
=
[1+,8,6]
=
[8,6,1+]
= =
[8,1+,6]
=
[1+,8,6,1+]
=
[8+,6+]
Orbifold *862 *664 *883 *4232 *4343 43×
Semidirect subgroups
Diagram
Coxeter [8,6+]
[8+,6]
[(8,6,2+)]
[8,1+,6,1+]
= =
= =
[1+,8,1+,6]
= =
= =
Orbifold 6*4 8*3 2*43 3*44 4*33
Direct subgroups
Index 2 4 8
Diagram
Coxeter [8,6]+
=
[8,6+]+
=
[8+,6]+
=
[8,1+,6]+
=
[8+,6+]+ = [1+,8,1+,6,1+]
= = =
Orbifold 862 664 883 4232 4343
Radical subgroups
Index 12 24 16 32
Diagram
Coxeter [8,6*]
[8*,6]
[8,6*]+
[8*,6]+
Orbifold *444444 *33333333 444444 33333333
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 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 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8
See also
Wikimedia Commons has media related to Uniform tiling 4-12-16.
• Tilings of regular polygons
• List of uniform planar 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 |
Truncated hexagonal tiling
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
Truncated hexagonal tiling
TypeSemiregular tiling
Vertex configuration
3.12.12
Schläfli symbolt{6,3}
Wythoff symbol2 3 | 6
Coxeter diagram
Symmetryp6m, [6,3], (*632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymToxat
DualTriakis triangular tiling
PropertiesVertex-transitive
As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}.
Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille).
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)
Topologically identical tilings
The dodecagonal faces can be distorted into different geometries, such as:
Related polyhedra and tilings
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
Symmetry mutations
This tiling 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.∞.∞
Related 2-uniform tilings
Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares.[1][2]
1-uniform Dissection 2-uniform dissections
(3.122)
(3.4.6.4) & (33.42)
(3.4.6.4) & (32.4.3.4)
Dual Tilings
O
to DB
to DC
Circle packing
The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point.[3] Every circle is in contact with 3 other circles in the packing (kissing number). This is the lowest density packing that can be created from a uniform tiling.
Triakis triangular tiling
Triakis triangular tiling
TypeDual semiregular tiling
Facestriangle
Coxeter diagram
Symmetry groupp6m, [6,3], (*632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronTruncated hexagonal tiling
Face configurationV3.12.12
Propertiesface-transitive
The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles.
Conway calls it a kisdeltille,[4] constructed as a kis operation applied to a triangular tiling (deltille).
In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron.[5]
It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.[6]
It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile.[7]
Related duals to uniform tilings
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.
Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6
See also
Wikimedia Commons has media related to Uniform tiling 3-12-12 (truncated hexagonal tiling).
• Tilings of regular polygons
• List of uniform tilings
References
1. Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
2. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
3. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G
4. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 "A K Peters, LTD. - the Symmetries of Things". Archived from the original on 2010-09-19. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
5. Inose, Mikio. "mikworks.com : Original Work : Asanoha". www.mikworks.com. Retrieved 20 April 2018.
6. Weisstein, Eric W. "Dual tessellation". MathWorld.
7. Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", Mathematics Magazine, 84 (4): 283–289, arXiv:0908.3257, doi:10.4169/math.mag.84.4.283, MR 2843659.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• 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)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 39. ISBN 0-486-23729-X.
• Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern E, Dual p. 77-76, pattern 1
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 117
External links
• Weisstein, Eric W. "Semiregular tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings o3x6x - toxat - O7".
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 |
Truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. In general usage, the degree of truncation is assumed to be uniform unless specified.
Truncated icosahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 90, V = 60 (χ = 2)
Faces by sides12{5}+20{6}
Conway notationtI
Schläfli symbolst{3,5}
t0,1{3,5}
Wythoff symbol2 5 | 3
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle6-6: 138.189685°
6-5: 142.62°
ReferencesU25, C27, W9
PropertiesSemiregular convex
Colored faces
5.6.6
(Vertex figure)
Pentakis dodecahedron
(dual polyhedron)
Net
It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.
It is the Goldberg polyhedron GPV(1,1) or {5+,3}1,1, containing pentagonal and hexagonal faces.
This geometry is associated with footballs (soccer balls) typically patterned with white hexagons and black pentagons. Geodesic domes such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It also corresponds to the geometry of the fullerene C60 ("buckyball") molecule.
It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb.
Construction
This polyhedron can be constructed from an icosahedron by truncating, or cutting off, each of the 12 vertices at the one-third mark of each edge, creating 12 pentagonal faces and transforming the original 20 triangle faces into regular hexagons.[1][2]
Characteristics
In geometry and graph theory, there are some standard polyhedron characteristics.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of:
(0, ±1, ±3φ)
(±1, ±(2 + φ), ±2φ)
(±φ, ±2, ±(2φ + 1))
where φ = 1 + √5/2 is the golden mean. The circumradius is √9φ + 10 ≈ 4.956 and the edges have length 2.[3]
Orthogonal projections
The truncated icosahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge
5-6
Edge
6-6
Face
Hexagon
Face
Pentagon
Solid
Wireframe
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
Spherical tiling
The truncated icosahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
pentagon-centered
hexagon-centered
Orthographic projection Stereographic projections
Dimensions
If the edge length of a truncated icosahedron is a, the radius of a circumscribed sphere (one that touches the truncated icosahedron at all vertices) is:
$r_{\mathrm {u} }={\frac {a}{2}}{\sqrt {1+9\varphi ^{2}}}={\frac {a}{4}}{\sqrt {58+18{\sqrt {5}}}}\approx 2.478\,018\,66a$
where φ is the golden ratio.
This result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron (before cut off) as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge (calculated on the basis of this construction) is approximately 23.281446°.
Area and volume
The area A and the volume V of the truncated icosahedron of edge length a are:
${\begin{aligned}A&=\left(20\cdot {\frac {3}{2}}{\sqrt {3}}+12\cdot {\frac {5}{4}}{\sqrt {1+{\frac {2}{\sqrt {5}}}}}\right)a^{2}&&\approx 72.607\,253a^{2}\\V&={\frac {125+43{\sqrt {5}}}{4}}a^{3}&&\approx 55.287\,7308a^{3}.\end{aligned}}$
With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons).
The truncated icosahedron easily demonstrates the Euler characteristic:
32 + 60 − 90 = 2.
Applications
The balls used in association football and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life.[4] The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball. This ball type was introduced to the World Cup in 1970 (starting in 2006, this iconic design has been superseded by alternative patterns).
British traffic signs indicating football grounds use a uniformly-colored hexagonal tiling section to represent a football, rather than a truncated icosahedron. This angered mathematician and comedian Matt Parker, who started a petition to the UK government to have these signs changed to be geometrically accurate. The petition was ultimately declined.
Geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller.[5]
A variation of the icosahedron was used as the basis of the honeycomb wheels (made from a polycast material) used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix.
This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs.[6]
The truncated icosahedron can also be described as a model of the Buckminsterfullerene (fullerene) (C60), or "buckyball", molecule – an allotrope of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and about 0.71 nm, respectively, hence the size ratio is ≈31,000,000:1.
In popular craft culture, large sparkleballs can be made using a icosahedron pattern and plastic, styrofoam or paper cups.
In the arts
• Gallery
• The truncated icosahedron (left) compared with an association football.
• Fullerene C60 molecule
• Truncated icosahedral radome on a weather station
• Truncated icosahedron machined out of 6061-T6 aluminum
• A wooden truncated icosahedron artwork by George W. Hart.
Related polyhedra
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. 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
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6
These uniform star-polyhedra, and one icosahedral stellation have nonuniform truncated icosahedra convex hulls:
Uniform star polyhedra with truncated icosahedra convex hulls
Nonuniform
truncated icosahedron
2 5 | 3
U37
2 5/2 | 5
U61
5/2 3 | 5/3
U67
5/3 3 | 2
U73
2 5/3 (3/2 5/4)
Complete stellation
Nonuniform
truncated icosahedron
2 5 | 3
U38
5/2 5 | 2
U44
5/3 5 | 3
U56
2 3 (5/4 5/2) |
Nonuniform
truncated icosahedron
2 5 | 3
U32
| 5/2 3 3
This polyhedron looks similar to the uniform chamfered dodecahedron which has 12 pentagons, but 30 hexagons.
Truncated icosahedral graph
Truncated icosahedral graph
6-fold symmetry schlegel diagram
Vertices60
Edges90
Automorphisms120
Chromatic number3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters
In the mathematical field of graph theory, a truncated icosahedral graph is the graph of vertices and edges of the truncated icosahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.[7][8][9]
Orthographic projection
5-fold symmetry
5-fold Schlegel diagram
History
The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus,[10] which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509. Although Albrecht Dürer omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, Underweysung der Messung, a description of it was found in his posthumous papers, published in 1538. Johannes Kepler later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, Harmonices Mundi.[11]
See also
Look up truncated icosahedron in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Truncated icosahedron.
• Fullerene
• Buckminsterfullerene (C60)
• Hyperbolic soccerball
• Snyder equal-area projection
• Soccer ball
• Adidas Telstar
Notes
1. Mednikov, Evgueni G.; Jewell, Matthew C.; Dahl, Lawrence F. (2007-09-01). "Nanosized (μ 12 -Pt)Pd 164- x Pt x (CO) 72 (PPh 3 ) 20 ( x ≈ 7) Containing Pt-Centered Four-Shell 165-Atom Pd−Pt Core with Unprecedented Intershell Bridging Carbonyl Ligands: Comparative Analysis of Icosahedral Shell-Growth Patterns with Geometrically Related Pd 145 (CO) x (PEt 3 ) 30 ( x ≈ 60) Containing Capped Three-Shell Pd 145 Core". Journal of the American Chemical Society. 129 (37): 11624. doi:10.1021/ja073945q. ISSN 0002-7863.
2. Kotschick, Dieter (July–August 2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350. doi:10.1511/2006.60.350.{{cite journal}}: CS1 maint: date format (link)
3. Weisstein, Eric W. "Icosahedral group". MathWorld.
4. Kotschick, Dieter (2006). "The Topology and Combinatorics of Soccer Balls". American Scientist. 94 (4): 350–357. doi:10.1511/2006.60.350.
5. Krebs, Albin (July 2, 1983). "R. Buckminster Fuller Dead; Futurist Built Geodesic Dome". The New York Times. New York, N.Y. p. 1. Retrieved 7 November 2021.
6. Rhodes, Richard (1996). Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books. pp. 195. ISBN 0-684-82414-0.
7. Read, R. C.; Wilson, R. J. (1998). An Atlas of Graphs. Oxford University Press. p. 268.
8. Godsil, C. and Royle, G. Algebraic Graph Theory New York: Springer-Verlag, p. 211, 2001
9. Kostant, B. The Graph of the Truncated Icosahedron and the Last Letter of Galois. Notices Amer. Math. Soc. 42, 1995, pp. 959-968 PDF
10. Katz, Eugene A. (2011). "Bridges between mathematics, natural sciences, architecture and art: case of fullerenes". Art, Science, and Technology: Interaction Between Three Cultures, Proceedings of the First International Conference. pp. 60–71.
11. Field, J. V. (1997). "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler". Archive for History of Exact Sciences. 50 (3–4): 241–289. doi:10.1007/BF00374595. JSTOR 41134110. MR 1457069. S2CID 118516740.
References
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Cromwell, P. (1997). "Archimedean solids". Polyhedra: "One of the Most Charming Chapters of Geometry". Cambridge: Cambridge University Press. pp. 79–86. ISBN 0-521-55432-2. OCLC 180091468.
External links
• Eric W. Weisstein, Truncated icosahedron (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Truncated icosahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x5o - ti".
• Editable printable net of a truncated icosahedron with interactive 3D view
• The Uniform Polyhedra
• "Virtual Reality Polyhedra"—The Encyclopedia of Polyhedra
• 3D paper data visualization World Cup ball
Archimedean solids
Tetrahedron
(Seed)
Tetrahedron
(Dual)
Cube
(Seed)
Octahedron
(Dual)
Dodecahedron
(Seed)
Icosahedron
(Dual)
Truncated tetrahedron
(Truncate)
Truncated tetrahedron
(Zip)
Truncated cube
(Truncate)
Truncated octahedron
(Zip)
Truncated dodecahedron
(Truncate)
Truncated icosahedron
(Zip)
Tetratetrahedron
(Ambo)
Cuboctahedron
(Ambo)
Icosidodecahedron
(Ambo)
Rhombitetratetrahedron
(Expand)
Truncated tetratetrahedron
(Bevel)
Rhombicuboctahedron
(Expand)
Truncated cuboctahedron
(Bevel)
Rhombicosidodecahedron
(Expand)
Truncated icosidodecahedron
(Bevel)
Snub tetrahedron
(Snub)
Snub cube
(Snub)
Snub dodecahedron
(Snub)
Catalan duals
Tetrahedron
(Dual)
Tetrahedron
(Seed)
Octahedron
(Dual)
Cube
(Seed)
Icosahedron
(Dual)
Dodecahedron
(Seed)
Triakis tetrahedron
(Needle)
Triakis tetrahedron
(Kis)
Triakis octahedron
(Needle)
Tetrakis hexahedron
(Kis)
Triakis icosahedron
(Needle)
Pentakis dodecahedron
(Kis)
Rhombic hexahedron
(Join)
Rhombic dodecahedron
(Join)
Rhombic triacontahedron
(Join)
Deltoidal dodecahedron
(Ortho)
Disdyakis hexahedron
(Meta)
Deltoidal icositetrahedron
(Ortho)
Disdyakis dodecahedron
(Meta)
Deltoidal hexecontahedron
(Ortho)
Disdyakis triacontahedron
(Meta)
Pentagonal dodecahedron
(Gyro)
Pentagonal icositetrahedron
(Gyro)
Pentagonal hexecontahedron
(Gyro)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Icosahedral honeycomb
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.
Icosahedral honeycomb
Poincaré disk model
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol{3,5,3}
Coxeter diagram
Cells{5,3} (regular icosahedron)
Faces{3} (triangle)
Edge figure{3} (triangle)
Vertex figure
dodecahedron
DualSelf-dual
Coxeter groupJ3, [3,5,3]
PropertiesRegular
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.
Description
The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.
Related regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
Four regular compact honeycombs in H3
{5,3,4}
{4,3,5}
{3,5,3}
{5,3,5}
Related regular polytopes and honeycombs
It is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:
{3,p,3} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
{3,p,3} {3,3,3} {3,4,3} {3,5,3} {3,6,3} {3,7,3} {3,8,3} ... {3,∞,3}
Image
Cells
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
Vertex
figure
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:
{p,5,p} regular honeycombs
Space H3
Form Compact Noncompact
Name {3,5,3} {4,5,4} {5,5,5} {6,5,6} {7,5,7} {8,5,8} ...{∞,5,∞}
Image
Cells
{p,5}
{3,5}
{4,5}
{5,5}
{6,5}
{7,5}
{8,5}
{∞,5}
Vertex
figure
{5,p}
{5,3}
{5,4}
{5,5}
{5,6}
{5,7}
{5,8}
{5,∞}
Uniform honeycombs
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.
[3,5,3] family honeycombs
{3,5,3}
t1{3,5,3}
t0,1{3,5,3}
t0,2{3,5,3}
t0,3{3,5,3}
t1,2{3,5,3}
t0,1,2{3,5,3}
t0,1,3{3,5,3}
t0,1,2,3{3,5,3}
Rectified icosahedral honeycomb
Rectified icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{3,5,3} or t1{3,5,3}
Coxeter diagram
Cellsr{3,5}
{5,3}
Facestriangle {3}
pentagon {5}
Vertex figure
triangular prism
Coxeter group${\overline {J}}_{3}$, [3,5,3]
PropertiesVertex-transitive, edge-transitive
The rectified icosahedral honeycomb, t1{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:
Perspective projections from center of Poincaré disk model
Related honeycomb
There are four rectified compact regular honeycombs:
Four rectified regular compact honeycombs in H3
Image
Symbols r{5,3,4}
r{4,3,5}
r{3,5,3}
r{5,3,5}
Vertex
figure
Truncated icosahedral honeycomb
Truncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{3,5,3} or t0,1{3,5,3}
Coxeter diagram
Cellst{3,5}
{5,3}
Facespentagon {5}
hexagon {6}
Vertex figure
triangular pyramid
Coxeter group${\overline {J}}_{3}$, [3,5,3]
PropertiesVertex-transitive
The truncated icosahedral honeycomb, t0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.
Related honeycombs
Four truncated regular compact honeycombs in H3
Image
Symbols t{5,3,4}
t{4,3,5}
t{3,5,3}
t{5,3,5}
Vertex
figure
Bitruncated icosahedral honeycomb
Bitruncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbol2t{3,5,3} or t1,2{3,5,3}
Coxeter diagram
Cellst{5,3}
Facestriangle {3}
decagon {10}
Vertex figure
tetragonal disphenoid
Coxeter group$2\times {\overline {J}}_{3}$, [[3,5,3]]
PropertiesVertex-transitive, edge-transitive, cell-transitive
The bitruncated icosahedral honeycomb, t1,2{3,5,3}, , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.
Related honeycombs
Three bitruncated compact honeycombs in H3
Image
Symbols 2t{4,3,5}
2t{3,5,3}
2t{5,3,5}
Vertex
figure
Cantellated icosahedral honeycomb
Cantellated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{3,5,3} or t0,2{3,5,3}
Coxeter diagram
Cellsrr{3,5}
r{5,3}
{}x{3}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter group${\overline {J}}_{3}$, [3,5,3]
PropertiesVertex-transitive
The cantellated icosahedral honeycomb, t0,2{3,5,3}, , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.
Related honeycombs
Four cantellated regular compact honeycombs in H3
Image
Symbols rr{5,3,4}
rr{4,3,5}
rr{3,5,3}
rr{5,3,5}
Vertex
figure
Cantitruncated icosahedral honeycomb
Cantitruncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{3,5,3} or t0,1,2{3,5,3}
Coxeter diagram
Cellstr{3,5}
t{5,3}
{}x{3}
Facestriangle {3}
square {4}
hexagon {6}
decagon {10}
Vertex figure
mirrored sphenoid
Coxeter group${\overline {J}}_{3}$, [3,5,3]
PropertiesVertex-transitive
The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.
Related honeycombs
Four cantitruncated regular compact honeycombs in H3
Image
Symbols tr{5,3,4}
tr{4,3,5}
tr{3,5,3}
tr{5,3,5}
Vertex
figure
Runcinated icosahedral honeycomb
Runcinated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,3{3,5,3}
Coxeter diagram
Cells{3,5}
{}×{3}
Facestriangle {3}
square {4}
Vertex figure
pentagonal antiprism
Coxeter group$2\times {\overline {J}}_{3}$, [[3,5,3]]
PropertiesVertex-transitive, edge-transitive
The runcinated icosahedral honeycomb, t0,3{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.
Viewed from center of triangular prism
Related honeycombs
Three runcinated regular compact honeycombs in H3
Image
Symbols t0,3{4,3,5}
t0,3{3,5,3}
t0,3{5,3,5}
Vertex
figure
Runcitruncated icosahedral honeycomb
Runcitruncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,3{3,5,3}
Coxeter diagram
Cellst{3,5}
rr{3,5}
{}×{3}
{}×{6}
Facestriangle {3}
square {4}
pentagon {5}
hexagon {6}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter group${\overline {J}}_{3}$, [3,5,3]
PropertiesVertex-transitive
The runcitruncated icosahedral honeycomb, t0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism, and triangular prism cells, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb.
Viewed from center of triangular prism
Related honeycombs
Four runcitruncated regular compact honeycombs in H3
Image
Symbols t0,1,3{5,3,4}
t0,1,3{4,3,5}
t0,1,3{3,5,3}
t0,1,3{5,3,5}
Vertex
figure
Omnitruncated icosahedral honeycomb
Omnitruncated icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,2,3{3,5,3}
Coxeter diagram
Cellstr{3,5}
{}×{6}
Facessquare {4}
hexagon {6}
dodecagon {10}
Vertex figure
phyllic disphenoid
Coxeter group$2\times {\overline {J}}_{3}$, [[3,5,3]]
PropertiesVertex-transitive
The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3}, , has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.
Centered on hexagonal prism
Related honeycombs
Three omnitruncated regular compact honeycombs in H3
Image
Symbols t0,1,2,3{4,3,5}
t0,1,2,3{3,5,3}
t0,1,2,3{5,3,5}
Vertex
figure
Omnisnub icosahedral honeycomb
Omnisnub icosahedral honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh(t0,1,2,3{3,5,3})
Coxeter diagram
Cellssr{3,5}
s{2,3}
irr. {3,3}
Facestriangle {3}
pentagon {5}
Vertex figure
Coxeter group[[3,5,3]]+
PropertiesVertex-transitive
The omnisnub icosahedral honeycomb, h(t0,1,2,3{3,5,3}), , has snub dodecahedron, octahedron, and tetrahedron cells, with an irregular vertex figure. It is vertex-transitive, but cannot be made with uniform cells.
Partially diminished icosahedral honeycomb
Partially diminished icosahedral honeycomb
Parabidiminished icosahedral honeycomb
TypeUniform honeycombs
Schläfli symbolpd{3,5,3}
Coxeter diagram-
Cells{5,3}
s{2,5}
Facestriangle {3}
pentagon {5}
Vertex figure
tetrahedrally diminished
dodecahedron
Coxeter group1/5[3,5,3]+
PropertiesVertex-transitive
The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a non-Wythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the {3,5,3} are diminished at opposite vertices (parabidiminished), leaving a pentagonal antiprism (parabidiminished icosahedron) core, and creating new dodecahedron cells above and below.[1][2]
See also
• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Seifert–Weber space
• 11-cell - An abstract regular polychoron which shares the {3,5,3} Schläfli symbol.
References
1. Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) Archived 2013-10-07 at the Wayback Machine
2. "Pd{3,5,3".}
• 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)
• 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
• Klitzing, Richard. "Hyperbolic H3 honeycombs hyperbolic order 3 icosahedral tesselation".
| Wikipedia |
Truncated icosahedral prism
In geometry, a truncated icosahedral prism is a convex uniform polychoron (four-dimensional polytope).
Truncated icosahedral prism
Schlegel diagram
TypePrismatic uniform polychoron
Uniform index62
Schläfli symbolt0,1,3{3,5,2} or t{3,5}×{}
Coxeter-Dynkin
Cells34 total:
2 5.6.6
12 4.4.5
20 4.4.6
Faces154 total:
90 {4}
24 {5}
40 {6}
Edges240
Vertices120
Vertex figure
Isosceles-triangular pyramid
Symmetry group[5,3,2], order 240
Propertiesconvex
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.
Alternative names
• Truncated-icosahedral dyadic prism (Norman W. Johnson)
• Tipe (Jonathan Bowers: for truncated-icosahedral prism)
• Truncated-icosahedral hyperprism
See also
• Truncated 600-cell,
External links
• 6. Convex uniform prismatic polychora - Model 62, George Olshevsky.
• Klitzing, Richard. "4D uniform polytopes (polychora) x x3o5x - tipe".
| Wikipedia |
Compound of ten truncated tetrahedra
This uniform polyhedron compound is a composition of 10 truncated tetrahedra, formed by truncating each of the tetrahedra in the compound of 10 tetrahedra. It also results from composing the two enantiomers of the compound of 5 truncated tetrahedra.
Compound of ten truncated tetrahedra
TypeUniform compound
IndexUC56
Polyhedra10 truncated tetrahedra
Faces40 triangles, 40 hexagons
Edges180
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentchiral tetrahedral (T)
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the even permutations of
(±1, ±1, ±3)
(±τ−1, ±(−τ−2), ±2τ)
(±τ, ±(−2τ−1), ±τ2)
(±τ2, ±(−τ−2), ±2)
(±(2τ−1), ±1, ±(2τ − 1))
where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR 0397554, S2CID 123279687.
| Wikipedia |
Truncated 24-cell honeycomb
In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.
Truncated 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt{3,4,3,3}
tr{3,3,4,3}
t2r{4,3,3,4}
t2r{4,3,31,1}
t{31,1,1,1}
Coxeter-Dynkin diagrams
4-face typeTesseract
Truncated 24-cell
Cell typeCube
Truncated octahedron
Face typeSquare
Triangle
Vertex figure
Tetrahedral pyramid
Coxeter groups${\tilde {F}}_{4}$, [3,4,3,3]
${\tilde {B}}_{4}$, [4,3,31,1]
${\tilde {C}}_{4}$, [4,3,3,4]
${\tilde {D}}_{4}$, [31,1,1,1]
PropertiesVertex transitive
It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the ${\tilde {D}}_{4}$ construction. This truncated 24-cell has Schläfli symbol t{31,1,1,1}, and its snub is represented as s{31,1,1,1}.
Alternate names
• Truncated icositetrachoric tetracomb
• Truncated icositetrachoric honeycomb
• Cantitruncated 16-cell honeycomb
• Bicantitruncated tesseractic honeycomb
Symmetry constructions
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.
Coxeter group Coxeter
diagram
Facets Vertex figure Vertex
figure
symmetry
(order)
${\tilde {F}}_{4}$
= [3,4,3,3]
4:
1:
, [3,3]
(24)
${\tilde {F}}_{4}$
= [3,3,4,3]
3:
1:
1:
, [3]
(6)
${\tilde {C}}_{4}$
= [4,3,3,4]
2,2:
1:
, [2]
(4)
${\tilde {B}}_{4}$
= [31,1,3,4]
1,1:
2:
1:
, [ ]
(2)
${\tilde {D}}_{4}$
= [31,1,1,1]
1,1,1,1:
1:
[ ]+
(1)
See also
Regular and uniform honeycombs in 4-space:
• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Rectified 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb
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]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 99
• Klitzing, Richard. "4D Euclidean tesselations". o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99
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 |
Truncated infinite-order square tiling
In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.
Infinite-order truncated square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration∞.8.8
Schläfli symbolt{4,∞}
Wythoff symbol2 ∞ | 4
Coxeter diagram
Symmetry group[∞,4], (*∞42)
Dualapeirokis apeirogonal tiling
PropertiesVertex-transitive
Uniform color
In (*∞44) symmetry this tiling has 3 colors. Bisecting the isosceles triangle domains can double the symmetry to *∞42 symmetry.
Symmetry
The dual of the tiling represents the fundamental domains of (*∞44) orbifold symmetry. From [(∞,4,4)] (*∞44) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to *∞42 by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,∞,1+,4,1+,4)] (∞22∞22) is the commutator subgroup of [(∞,4,4)].
Small index subgroups of [(∞,4,4)] (*∞44)
Fundamental
domains
Subgroup index 1 2 4
Coxeter
(orbifold)
[(4,4,∞)]
(*∞44)
[(1+,4,4,∞)]
(*∞424)
[(4,4,1+,∞)]
(*∞424)
[(4,1+,4,∞)]
(*∞2∞2)
[(4,1+,4,1+,∞)]
2*∞2∞2
[(1+,4,4,1+,∞)]
(∞*2222)
[(4,4+,∞)]
(4*∞2)
[(4+,4,∞)]
(4*∞2)
[(4,4,∞+)]
(∞*22)
[(1+,4,1+,4,∞)]
2*∞2∞2
[(4+,4+,∞)]
(∞22×)
Rotational subgroups
Subgroup index 2 4 8
Coxeter
(orbifold)
[(4,4,∞)]+
(∞44)
[(1+,4,4+,∞)]
(∞323)
[(4+,4,1+,∞)]
(∞424)
[(4,1+,4,∞+)]
(∞434)
[(1+,4,1+,4,1+,∞)] = [(4+,4+,∞+)]
(∞22∞22)
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
=
=
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 8-8-i.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
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 |
Truncated infinite-order triangular tiling
In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.
Infinite-order truncated triangular tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration∞.6.6
Schläfli symbolt{3,∞}
Wythoff symbol2 ∞ | 3
Coxeter diagram
Symmetry group[∞,3], (*∞32)
Dualapeirokis apeirogonal tiling
PropertiesVertex-transitive
Symmetry
The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.
Small index subgroups of [(∞,3,3)], (*∞33)
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)
[(∞,3,3)]
(*∞33)
[(∞,3,3)]+
(∞33)
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. 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
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6
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 6-6-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.
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 |
Truncated normal distribution
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics.
Probability density function
Probability density function for the truncated normal distribution for different sets of parameters. In all cases, a = −10 and b = 10. For the black: μ = −8, σ = 2; blue: μ = 0, σ = 2; red: μ = 9, σ = 10; orange: μ = 0, σ = 10.
Cumulative distribution function
Cumulative distribution function for the truncated normal distribution for different sets of parameters. In all cases, a = −10 and b = 10. For the black: μ = −8, σ = 2; blue: μ = 0, σ = 2; red: μ = 9, σ = 10; orange: μ = 0, σ = 10.
Notation $\xi ={\frac {x-\mu }{\sigma }},\ \alpha ={\frac {a-\mu }{\sigma }},\ \beta ={\frac {b-\mu }{\sigma }}$
$Z=\Phi (\beta )-\Phi (\alpha )$
Parameters $\mu \in \mathbb {R} $
$\sigma ^{2}\geq 0$ (but see definition)
$a\in \mathbb {R} $ — minimum value of $x$
$b\in \mathbb {R} $ — maximum value of $x$ ($b>a$)
Support $x\in [a,b]$
PDF $f(x;\mu ,\sigma ,a,b)={\frac {\varphi (\xi )}{\sigma Z}}\,$[1]
CDF $F(x;\mu ,\sigma ,a,b)={\frac {\Phi (\xi )-\Phi (\alpha )}{Z}}$
Mean $\mu +{\frac {\varphi (\alpha )-\varphi (\beta )}{Z}}\sigma $
Median $\mu +\Phi ^{-1}\left({\frac {\Phi (\alpha )+\Phi (\beta )}{2}}\right)\sigma $
Mode $\left\{{\begin{array}{ll}a,&\mathrm {if} \ \mu <a\\\mu ,&\mathrm {if} \ a\leq \mu \leq b\\b,&\mathrm {if} \ \mu >b\end{array}}\right.$
Variance $\sigma ^{2}\left[1-{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{Z}}-\left({\frac {\varphi (\alpha )-\varphi (\beta )}{Z}}\right)^{2}\right]$
Entropy $\ln({\sqrt {2\pi e}}\sigma Z)+{\frac {\alpha \varphi (\alpha )-\beta \varphi (\beta )}{2Z}}$
MGF $e^{\mu t+\sigma ^{2}t^{2}/2}\left[{\frac {\Phi (\beta -\sigma t)-\Phi (\alpha -\sigma t)}{\Phi (\beta )-\Phi (\alpha )}}\right]$
Definitions
Suppose $X$ has a normal distribution with mean $\mu $ and variance $\sigma ^{2}$ and lies within the interval $(a,b),{\text{with}}\;-\infty \leq a<b\leq \infty $. Then $X$ conditional on $a<X<b$ has a truncated normal distribution.
Its probability density function, $f$, for $a\leq x\leq b$, is given by
$f(x;\mu ,\sigma ,a,b)={\frac {1}{\sigma }}\,{\frac {\varphi ({\frac {x-\mu }{\sigma }})}{\Phi ({\frac {b-\mu }{\sigma }})-\Phi ({\frac {a-\mu }{\sigma }})}}$
and by $f=0$ otherwise.
Here,
$\varphi (\xi )={\frac {1}{\sqrt {2\pi }}}\exp \left(-{\frac {1}{2}}\xi ^{2}\right)$
is the probability density function of the standard normal distribution and $\Phi (\cdot )$ is its cumulative distribution function
$\Phi (x)={\frac {1}{2}}\left(1+\operatorname {erf} (x/{\sqrt {2}})\right).$
By definition, if $b=\infty $, then $\Phi \left({\tfrac {b-\mu }{\sigma }}\right)=1$, and similarly, if $a=-\infty $, then $\Phi \left({\tfrac {a-\mu }{\sigma }}\right)=0$.
The above formulae show that when $-\infty <a<b<+\infty $ the scale parameter $\sigma ^{2}$ of the truncated normal distribution is allowed to assume negative values. The parameter $\sigma $ is in this case imaginary, but the function $f$ is nevertheless real, positive, and normalizable. The scale parameter $\sigma ^{2}$ of the untruncated normal distribution must be positive because the distribution would not be normalizable otherwise. The doubly truncated normal distribution, on the other hand, can in principle have a negative scale parameter (which is different from the variance, see summary formulae), because no such integrability problems arise on a bounded domain. In this case the distribution cannot be interpreted as an untruncated normal conditional on $a<X<b$, of course, but can still be interpreted as a maximum-entropy distribution with first and second moments as constraints, and has an additional peculiar feature: it presents two local maxima instead of one, located at $x=a$ and $x=b$.
Properties
The truncated normal is the maximum entropy probability distribution for a fixed mean and variance with the random variate X constrained to be in the interval [a,b].[2] Truncated normals with fixed support form an exponential family. Nielsen[3] reported closed-form formula for calculating the Kullback-Leibler divergence and the Bhattacharyya distance between two truncated normal distributions with the support of the first distribution nested into the support of the second distribution.
Moments
If the random variable has been truncated only from below, some probability mass has been shifted to higher values, giving a first-order stochastically dominating distribution and hence increasing the mean to a value higher than the mean $\mu $ of the original normal distribution. Likewise, if the random variable has been truncated only from above, the truncated distribution has a mean less than $\mu .$
Regardless of whether the random variable is bounded above, below, or both, the truncation is a mean-preserving contraction combined with a mean-changing rigid shift, and hence the variance of the truncated distribution is less than the variance $\sigma ^{2}$ of the original normal distribution.
Two sided truncation[4]
Let $\alpha =(a-\mu )/\sigma $ and $\beta =(b-\mu )/\sigma $. Then:
$\operatorname {E} (X\mid a<X<b)=\mu -\sigma {\frac {\varphi (\beta )-\varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}$
and
$\operatorname {Var} (X\mid a<X<b)=\sigma ^{2}\left[1-{\frac {\beta \varphi (\beta )-\alpha \varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}-\left({\frac {\varphi (\beta )-\varphi (\alpha )}{\Phi (\beta )-\Phi (\alpha )}}\right)^{2}\right]$
Care must be taken in the numerical evaluation of these formulas, which can result in catastrophic cancellation when the interval $[a,b]$ does not include $\mu $. There are better ways to rewrite them that avoid this issue.[5]
One sided truncation (of lower tail)[6]
In this case $\;b=\infty ,\;\varphi (\beta )=0,\;\Phi (\beta )=1,$ then
$\operatorname {E} (X\mid X>a)=\mu +\sigma \varphi (\alpha )/Z,\!$
and
$\operatorname {Var} (X\mid X>a)=\sigma ^{2}[1+\alpha \varphi (\alpha )/Z-(\varphi (\alpha )/Z)^{2}],$
where $Z=1-\Phi (\alpha ).$
One sided truncation (of upper tail)
In this case $\;a=\alpha =-\infty ,\;\varphi (\alpha )=0,\;\Phi (\alpha )=0,$ then
$\operatorname {E} (X\mid X<b)=\mu -\sigma {\frac {\varphi (\beta )}{\Phi (\beta )}},$
$\operatorname {Var} (X\mid X<b)=\sigma ^{2}\left[1-\beta {\frac {\varphi (\beta )}{\Phi (\beta )}}-\left({\frac {\varphi (\beta )}{\Phi (\beta )}}\right)^{2}\right].$
Barr & Sherrill (1999) give a simpler expression for the variance of one sided truncations. Their formula is in terms of the chi-square CDF, which is implemented in standard software libraries. Bebu & Mathew (2009) provide formulas for (generalized) confidence intervals around the truncated moments.
A recursive formula
As for the non-truncated case, there is a recursive formula for the truncated moments.[7]
Multivariate
Computing the moments of a multivariate truncated normal is harder.
Generating values from the truncated normal distribution
A random variate $x$ defined as $x=\Phi ^{-1}(\Phi (\alpha )+U\cdot (\Phi (\beta )-\Phi (\alpha )))\sigma +\mu $ with $\Phi $ the cumulative distribution function and $\Phi ^{-1}$ its inverse, $U$ a uniform random number on $(0,1)$, follows the distribution truncated to the range $(a,b)$. This is simply the inverse transform method for simulating random variables. Although one of the simplest, this method can either fail when sampling in the tail of the normal distribution,[8] or be much too slow.[9] Thus, in practice, one has to find alternative methods of simulation.
One such truncated normal generator (implemented in Matlab and in R (programming language) as trandn.R ) is based on an acceptance rejection idea due to Marsaglia.[10] Despite the slightly suboptimal acceptance rate of Marsaglia (1964) in comparison with Robert (1995), Marsaglia's method is typically faster,[9] because it does not require the costly numerical evaluation of the exponential function.
For more on simulating a draw from the truncated normal distribution, see Robert (1995), Lynch (2007), Devroye (1986). The MSM package in R has a function, rtnorm, that calculates draws from a truncated normal. The truncnorm package in R also has functions to draw from a truncated normal.
Chopin (2011) proposed (arXiv) an algorithm inspired from the Ziggurat algorithm of Marsaglia and Tsang (1984, 2000), which is usually considered as the fastest Gaussian sampler, and is also very close to Ahrens's algorithm (1995). Implementations can be found in C, C++, Matlab and Python.
Sampling from the multivariate truncated normal distribution is considerably more difficult.[11] Exact or perfect simulation is only feasible in the case of truncation of the normal distribution to a polytope region.[11][12] In more general cases, Damien & Walker (2001) introduce a general methodology for sampling truncated densities within a Gibbs sampling framework. Their algorithm introduces one latent variable and, within a Gibbs sampling framework, it is more computationally efficient than the algorithm of Robert (1995).
See also
• Folded normal distribution
• Half-normal distribution
• Modified half-normal distribution[13] with the pdf on $(0,\infty )$ is given as $f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}$, where $\Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)$ denotes the Fox–Wright Psi function.
• Normal distribution
• Rectified Gaussian distribution
• Truncated distribution
• PERT distribution
Notes
1. "Lecture 4: Selection" (PDF). web.ist.utl.pt. Instituto Superior Técnico. November 11, 2002. p. 1. Retrieved 14 July 2015.
2. Dowson, D.; Wragg, A. (September 1973). "Maximum-entropy distributions having prescribed first and second moments (Corresp.)". IEEE Transactions on Information Theory. 19 (5): 689–693. doi:10.1109/TIT.1973.1055060. ISSN 1557-9654.
3. Frank Nielsen (2022). "Statistical Divergences between Densities of Truncated Exponential Families with Nested Supports: Duo Bregman and Duo Jensen Divergences". Entropy. MDPI. 24 (3): 421. Bibcode:2022Entrp..24..421N. doi:10.3390/e24030421. PMC 8947456. PMID 35327931.
4. Johnson, Norman Lloyd; Kotz, Samuel; Balakrishnan, N. (1994). Continuous Univariate Distributions. Vol. 1 (2nd ed.). New York: Wiley. Section 10.1. ISBN 0-471-58495-9. OCLC 29428092.
5. Fernandez-de-Cossio-Diaz, Jorge (2017-12-06), TruncatedNormal.jl: Compute mean and variance of the univariate truncated normal distribution (works far from the peak), retrieved 2017-12-06
6. Greene, William H. (2003). Econometric Analysis (5th ed.). Prentice Hall. ISBN 978-0-13-066189-0.
7. Document by Eric Orjebin, "https://people.smp.uq.edu.au/YoniNazarathy/teaching_projects/studentWork/EricOrjebin_TruncatedNormalMoments.pdf"
8. Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). Handbook of Monte Carlo methods. John Wiley & Sons.
9. Botev, Z. I.; L'Ecuyer, P. (2017). "Simulation from the Normal Distribution Truncated to an Interval in the Tail". 10th EAI International Conference on Performance Evaluation Methodologies and Tools. 25th–28th Oct 2016 Taormina, Italy: ACM. pp. 23–29. doi:10.4108/eai.25-10-2016.2266879. ISBN 978-1-63190-141-6.{{cite conference}}: CS1 maint: location (link)
10. Marsaglia, George (1964). "Generating a variable from the tail of the normal distribution". Technometrics. 6 (1): 101–102. doi:10.2307/1266749. JSTOR 1266749.
11. Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. doi:10.1111/rssb.12162. S2CID 88515228.
12. Botev, Zdravko & L'Ecuyer, Pierre (2018). "Chapter 8: Simulation from the Tail of the Univariate and Multivariate Normal Distribution". In Puliafito, Antonio (ed.). Systems Modeling: Methodologies and Tools. EAI/Springer Innovations in Communication and Computing. Springer, Cham. pp. 115–132. doi:10.1007/978-3-319-92378-9_8. ISBN 978-3-319-92377-2. S2CID 125554530.
13. Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics - Theory and Methods: 1–23. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.
References
• Botev, Zdravko & L'Ecuyer, Pierre (2018). "Chapter 8: Simulation from the Tail of the Univariate and Multivariate Normal Distribution". In Puliafito, Antonio (ed.). Systems Modeling: Methodologies and Tools. EAI/Springer Innovations in Communication and Computing. Springer, Cham. pp. 115–132. doi:10.1007/978-3-319-92378-9_8. ISBN 978-3-319-92377-2. S2CID 125554530.
• Devroye, Luc (1986). Non-Uniform Random Variate Generation (PDF). New York: Springer-Verlag. Archived from the original (PDF) on 2014-08-18. Retrieved 2012-04-12.
• Greene, William H. (2003). Econometric Analysis (5th ed.). Prentice Hall. ISBN 978-0-13-066189-0.
• Norman L. Johnson and Samuel Kotz (1970). Continuous univariate distributions-1, chapter 13. John Wiley & Sons.
• Lynch, Scott (2007). Introduction to Applied Bayesian Statistics and Estimation for Social Scientists. New York: Springer. ISBN 978-1-4419-2434-6.
• Robert, Christian P. (1995). "Simulation of truncated normal variables". Statistics and Computing. 5 (2): 121–125. arXiv:0907.4010. doi:10.1007/BF00143942. S2CID 15943491.
• Barr, Donald R.; Sherrill, E.Todd (1999). "Mean and variance of truncated normal distributions". The American Statistician. 53 (4): 357–361. doi:10.1080/00031305.1999.10474490.
• Bebu, Ionut; Mathew, Thomas (2009). "Confidence intervals for limited moments and truncated moments in normal and lognormal models". Statistics and Probability Letters. 79 (3): 375–380. doi:10.1016/j.spl.2008.09.006.
• Damien, Paul; Walker, Stephen G. (2001). "Sampling truncated normal, beta, and gamma densities". Journal of Computational and Graphical Statistics. 10 (2): 206–215. doi:10.1198/10618600152627906. S2CID 123156320.
• Chopin, Nicolas (2011-04-01). "Fast simulation of truncated Gaussian distributions". Statistics and Computing. 21 (2): 275–288. arXiv:1201.6140. doi:10.1007/s11222-009-9168-1. ISSN 1573-1375.
• Burkardt, John. "The Truncated Normal Distribution" (PDF). Department of Scientific Computing website. Florida State University. Retrieved 15 February 2018.
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
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• inverse
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Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
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wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
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and singular
Degenerate
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• Commons
| Wikipedia |
Truncated normal hurdle model
In econometrics, the truncated normal hurdle model is a variant of the Tobit model and was first proposed by Cragg in 1971.[1]
In a standard Tobit model, represented as $y=(x\beta +u)1[x\beta +u>0]$, where $u|x\sim N(0,\sigma ^{2})$This model construction implicitly imposes two first order assumptions:[2]
1. Since: $\partial P[y>0]/\partial x_{j}=\varphi (x\beta /\sigma )\beta _{j}/\sigma $ and $\partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}=\beta _{j}\{1-\theta (x\beta /\sigma \}$, the partial effect of $x_{j}$ on the probability $P[y>0]$ and the conditional expectation: $\operatorname {E} [y\mid x,y>0]$ has the same sign:[3]
2. The relative effects of $x_{h}$ and $x_{j}$ on $P[y>0]$ and $\operatorname {E} [y\mid x,y>0]$ are identical, i.e.:
${\frac {\partial P[y>0]/\partial x_{h}}{\partial P[y>0]/\partial x_{j}}}={\frac {\partial \operatorname {E} [y\mid x,y>0]/\partial x_{h}}{\partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}}}={\frac {\beta _{h}}{\beta _{j}}}|$
However, these two implicit assumptions are too strong and inconsistent with many contexts in economics. For instance, when we need to decide whether to invest and build a factory, the construction cost might be more influential than the product price; but once we have already built the factory, the product price is definitely more influential to the revenue. Hence, the implicit assumption (2) doesn't match this context.[4] The essence of this issue is that the standard Tobit implicitly models a very strong link between the participation decision $(y=0$ or $y>0)$ and the amount decision (the magnitude of $y$ when $y>0$). If a corner solution model is represented in a general form: $y=s\centerdot w,$ , where $s$ is the participate decision and $w$ is the amount decision, standard Tobit model assumes:
$s=1[x\beta +u>0];$
$w=x\beta +u.$
To make the model compatible with more contexts, a natural improvement is to assume:
$s=1[x\gamma +u>0],{\text{ where }}u\sim N(0,1);$
$w=x\beta +e,$ where the error term ($e$) is distributed as a truncated normal distribution with a density as $\varphi (\cdot )/\Phi \left({\frac {x\beta }{\sigma }}\right)/\sigma ;$ ;}
$s$ and $w$ are independent conditional on $x$.
This is called Truncated Normal Hurdle Model, which is proposed in Cragg (1971).[1] By adding one more parameter and detach the amount decision with the participation decision, the model can fit more contexts. Under this model setup, the density of the $y$ given $x$ can be written as:
$f(y\mid x)=[1-\Phi (\chi \gamma )]^{1[y=0]}\cdot \left[{\frac {\Phi \ (\chi \gamma )}{\Phi (\chi \beta /\sigma )}}\left.\varphi \left({\frac {y-\chi \beta }{\sigma }}\right)\right/\sigma \right]^{1[y>0]}$
From this density representation, it is obvious that it will degenerate to the standard Tobit model when $\gamma =\beta /\sigma .$ This also shows that Truncated Normal Hurdle Model is more general than the standard Tobit model.
The Truncated Normal Hurdle Model is usually estimated through MLE. The log-likelihood function can be written as:
${\begin{aligned}\ell (\beta ,\gamma ,\sigma )={}&\sum _{i=1}^{N}1[y_{i}=0]\log[1-\Phi (x_{i}\gamma )]+1[y_{i}>0]\log[\Phi (x_{i}\gamma )]\\[5pt]&{}+1[y_{i}>0]\left[-\log \left[\Phi \left({\frac {x_{i}\beta }{\sigma }}\right)\right]+\log \left(\varphi \left({\frac {y_{i}-x_{i}\beta }{\sigma }}\right)\right)-\log(\sigma )\right]\end{aligned}}$
From the log-likelihood function, $\gamma $ can be estimated by a probit model and $(\beta ,\sigma )$ can be estimated by a truncated normal regression model.[5] Based on the estimates, consistent estimates for the Average Partial Effect can be estimated correspondingly.
See also
• Hurdle model
• Tobit model
References
1. Cragg, John G. (September 1971). "Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods". Econometrica. 39 (5): 829–844. doi:10.2307/1909582. JSTOR 1909582.
2. Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 690.
3. Here, the notation follows Wooldrige (2002). Function $\theta (x)=\lambda '$ where $\lambda (x)=\varphi (\chi )/\Phi (\chi ),$ can be proved to be between 0 and 1.
4. For more application example of corner solution model, refer to: Daniel J. Phaneuf, (1999): “A Dual Approach to Modeling Corner Solutions in Recreation Demand”,Journal of Environmental Economics and Management, Volume 37, Issue 1, Pages 85-105, ISSN 0095-0696.
5. Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 692-694.
| Wikipedia |
Truncated octagonal tiling
In geometry, the truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of t{8,3}.
Truncated octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.16.16
Schläfli symbolt{8,3}
Wythoff symbol2 3 | 8
Coxeter diagram
Symmetry group[8,3], (*832)
DualOrder-8 triakis triangular tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling has face configuration V3.16.16.
Related polyhedra and tilings
This hyperbolic tiling 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.∞.∞
From a Wythoff construction there are ten 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
See also
Wikimedia Commons has media related to Uniform tiling 3-16-16.
• Truncated hexagonal tiling
• Octagonal 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 |
Truncated order-4 octagonal tiling
In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.
Truncated order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.16.16
Schläfli symbolt{8,4}
tr{8,8} or $t{\begin{Bmatrix}8\\8\end{Bmatrix}}$
Wythoff symbol2 8 | 8
2 8 8 |
Coxeter diagram
or
Symmetry group[8,4], (*842)
[8,8], (*882)
DualOrder-8 tetrakis square tiling
PropertiesVertex-transitive
Constructions
There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882).
Two uniform constructions of 4.8.4.8
Name Tetraoctagonal Truncated octaoctagonal
Image
Symmetry [8,4]
(*842)
[8,8] = [8,4,1+]
(*882)
=
Symbol t{8,4} tr{8,8}
Coxeter diagram
Dual tiling
The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group.
Symmetry
The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images unique mirrors are colored red, green, and blue, and alternatively colored triangles show the location of gyration points. The [8+,8+], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,8,1+,8,1+] (4444) is the commutator subgroup of [8,8].
One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]+, index 32, (44444444).
The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.
Small index subgroups of [8,8] (*882)
Index 1 2 4
Diagram
Coxeter [8,8]
[1+,8,8]
=
[8,8,1+]
=
[8,1+,8]
=
[1+,8,8,1+]
=
[8+,8+]
Orbifold *882 *884 *4242 *4444 44×
Semidirect subgroups
Diagram
Coxeter [8,8+]
[8+,8]
[(8,8,2+)]
[8,1+,8,1+]
= =
= =
[1+,8,1+,8]
= =
= =
Orbifold 8*4 2*44 4*44
Direct subgroups
Index 2 4 8
Diagram
Coxeter [8,8]+
[8,8+]+
=
[8+,8]+
=
[8,1+,8]+
=
[8+,8+]+ = [1+,8,1+,8,1+]
= = =
Orbifold 882 884 4242 4444
Radical subgroups
Index 16 32
Diagram
Coxeter [8,8*]
[8*,8]
[8,8*]+
[8*,8]+
Orbifold *44444444 44444444
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
= = = =
=
=
=
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8
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.
See also
Wikimedia Commons has media related to Uniform tiling 4-16-16.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Truncated order-3 apeirogonal tiling
In geometry, the truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}.
Truncated order-3 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.∞.∞
Schläfli symbolt{∞,3}
Wythoff symbol2 3 | ∞
Coxeter diagram
Symmetry group[∞,3], (*∞32)
DualInfinite-order triakis triangular tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling, the infinite-order triakis triangular tiling, has face configuration V3.∞.∞.
Related polyhedra and tiling
This hyperbolic tiling 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.∞.∞
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.∞
See also
Wikimedia Commons has media related to Uniform tiling 3-i-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.
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 |
Truncated heptagonal tiling
In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex. It has Schläfli symbol of t{7,3}. The tiling has a vertex configuration of 3.14.14.
Truncated heptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.14.14
Schläfli symbolt{7,3}
Wythoff symbol2 3 | 7
Coxeter diagram
Symmetry group[7,3], (*732)
DualOrder-7 triakis triangular tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling is called an order-7 triakis triangular tiling, seen as an order-7 triangular tiling with each triangle divided into three by a center point.
Related polyhedra and tilings
This hyperbolic tiling 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.∞.∞
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms.
Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
See also
Wikimedia Commons has media related to Uniform tiling 3-14-14.
• Truncated hexagonal tiling
• Heptagonal 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 |
Truncated order-4 apeirogonal tiling
In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.
Truncated order-4 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.∞.∞
Schläfli symbolt{∞,4}
tr{∞,∞} or $t{\begin{Bmatrix}\infty \\\infty \end{Bmatrix}}$
Wythoff symbol2 4 | ∞
2 ∞ ∞ |
Coxeter diagram
or
Symmetry group[∞,4], (*∞42)
[∞,∞], (*∞∞2)
DualInfinite-order tetrakis square tiling
PropertiesVertex-transitive
Uniform colorings
A half symmetry coloring is tr{∞,∞}, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below. Coxeter diagram are shown with dotted lines for these divergent, ultraparallel mirrors.
(Vertex centered)
(Square centered)
Symmetry
From [∞,∞] symmetry, there are 15 small index subgroup by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞42 symmetry by adding a mirror bisecting the fundamental domain. The subgroup index-8 group, [1+,∞,1+,∞,1+] (∞∞∞∞) is the commutator subgroup of [∞,∞].
Small index subgroups of [∞,∞] (*∞∞2)
Index 1 2 4
Diagram
Coxeter [∞,∞]
=
[1+,∞,∞]
=
[∞,∞,1+]
=
[∞,1+,∞]
=
[1+,∞,∞,1+]
=
[∞+,∞+]
Orbifold *∞∞2 *∞∞∞ *∞2∞2 *∞∞∞∞ ∞∞×
Semidirect subgroups
Diagram
Coxeter [∞,∞+]
[∞+,∞]
[(∞,∞,2+)]
[∞,1+,∞,1+]
= =
= =
[1+,∞,1+,∞]
= =
= =
Orbifold ∞*∞ 2*∞∞ ∞*∞∞
Direct subgroups
Index 2 4 8
Diagram
Coxeter [∞,∞]+
=
[∞,∞+]+
=
[∞+,∞]+
=
[∞,1+,∞]+
=
[∞+,∞+]+ = [1+,∞,1+,∞,1+]
= = =
Orbifold ∞∞2 ∞∞∞ ∞2∞2 ∞∞∞∞
Radical subgroups
Index ∞ ∞
Diagram
Coxeter [∞,∞*]
[∞*,∞]
[∞,∞*]+
[∞*,∞]+
Orbifold *∞∞ ∞∞
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
=
=
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
Paracompact uniform tilings in [∞,∞] family
=
=
=
=
=
=
=
=
=
=
=
=
{∞,∞} t{∞,∞} r{∞,∞} 2t{∞,∞}=t{∞,∞} 2r{∞,∞}={∞,∞} rr{∞,∞} tr{∞,∞}
Dual tilings
V∞∞ V∞.∞.∞ V(∞.∞)2 V∞.∞.∞ V∞∞ V4.∞.4.∞ V4.4.∞
Alternations
[1+,∞,∞]
(*∞∞2)
[∞+,∞]
(∞*∞)
[∞,1+,∞]
(*∞∞∞∞)
[∞,∞+]
(∞*∞)
[∞,∞,1+]
(*∞∞2)
[(∞,∞,2+)]
(2*∞∞)
[∞,∞]+
(2∞∞)
h{∞,∞} s{∞,∞} hr{∞,∞} s{∞,∞} h2{∞,∞} hrr{∞,∞} sr{∞,∞}
Alternation duals
V(∞.∞)∞ V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞∞ V(4.∞.4)2 V3.3.∞.3.∞
See also
Wikimedia Commons has media related to Uniform tiling 4-i-i.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
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 |
Truncated order-5 pentagonal tiling
In geometry, the truncated order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,5}, constructed from one pentagons and two decagons around every vertex.
Truncated order-5 pentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration5.10.10
Schläfli symbolt{5,5}
Wythoff symbol2 5 | 5
Coxeter diagram
Symmetry group[5,5], (*552)
DualOrder-5 pentakis pentagonal tiling
PropertiesVertex-transitive
Related tilings
Uniform pentapentagonal tilings
Symmetry: [5,5], (*552) [5,5]+, (552)
=
=
=
=
=
=
=
=
Order-5 pentagonal tiling
{5,5}
Truncated order-5 pentagonal tiling
t{5,5}
Order-4 pentagonal tiling
r{5,5}
Truncated order-5 pentagonal tiling
2t{5,5} = t{5,5}
Order-5 pentagonal tiling
2r{5,5} = {5,5}
Tetrapentagonal tiling
rr{5,5}
Truncated order-4 pentagonal tiling
tr{5,5}
Snub pentapentagonal tiling
sr{5,5}
Uniform duals
Order-5 pentagonal tiling
V5.5.5.5.5
V5.10.10 Order-5 square tiling
V5.5.5.5
V5.10.10 Order-5 pentagonal tiling
V5.5.5.5.5
V4.5.4.5 V4.10.10 V3.3.5.3.5
See also
Wikimedia Commons has media related to Uniform tiling 5-10-10.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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 |
Truncated order-5 square tiling
In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,5}.
Truncated order-5 square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.8.5
Schläfli symbolt{4,5}
Wythoff symbol2 5 | 4
Coxeter diagram
Symmetry group[5,4], (*542)
DualOrder-4 pentakis pentagonal tiling
PropertiesVertex-transitive
Related polyhedra and tiling
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
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.
See also
• Uniform tilings in hyperbolic plane
• List of regular polytopes
External links
Wikimedia Commons has media related to Uniform tiling 5-8-8.
• 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 |
Truncated order-6 hexagonal tiling
In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling, h2{4,6}
Truncated order-6 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration6.12.12
Schläfli symbolt{6,6} or h2{4,6}
t(6,6,3)
Wythoff symbol2 6 | 6
3 6 6 |
Coxeter diagram =
=
Symmetry group[6,6], (*662)
[(6,6,3)], (*663)
DualOrder-6 hexakis hexagonal tiling
PropertiesVertex-transitive
Uniform colorings
By *663 symmetry, this tiling can be constructed as an omnitruncation, t{(6,6,3)}:
Symmetry
The dual to this tiling represent the fundamental domains of [(6,6,3)] (*663) symmetry. There are 3 small index subgroup symmetries constructed from [(6,6,3)] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
The symmetry can be doubled as 662 symmetry by adding a mirror bisecting the fundamental domain.
Small index subgroups of [(6,6,3)] (*663)
Index 1 2 6
Diagram
Coxeter
(orbifold)
[(6,6,3)] =
(*663)
[(6,1+,6,3)] = =
(*3333)
[(6,6,3+)] =
(3*33)
[(6,6,3*)] =
(*333333)
Direct subgroups
Index 2 4 12
Diagram
Coxeter
(orbifold)
[(6,6,3)]+ =
(663)
[(6,6,3+)]+ = =
(3333)
[(6,6,3*)]+ =
(333333)
Related polyhedra and tiling
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663)
[6+,6]
(6*3)
[6,1+,6]
(*3232)
[6,6+]
(6*3)
[6,6,1+]
(*663)
[(6,6,2+)]
(2*33)
[6,6]+
(662)
= = =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
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.
See also
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Truncated order-6 octagonal tiling
In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.
Truncated order-6 octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration6.16.16
Schläfli symbolt{8,6}
Wythoff symbol2 6 | 8
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-8 hexakis hexagonal tiling
PropertiesVertex-transitive
Uniform colorings
A secondary construction t{(8,8,3)} is called a truncated trioctaoctagonal tiling:
Symmetry
The dual to this tiling represent the fundamental domains of [(8,8,3)] (*883) symmetry. There are 3 small index subgroup symmetries constructed from [(8,8,3)] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
The symmetry can be doubled as 862 symmetry by adding a mirror bisecting the fundamental domain.
Small index subgroups of [(8,8,3)] (*883)
Index 1 2 6
Diagram
Coxeter
(orbifold)
[(8,8,3)] =
(*883)
[(8,1+,8,3)] = =
(*4343)
[(8,8,3+)] =
(3*44)
[(8,8,3*)] =
(*444444)
Direct subgroups
Index 2 4 12
Diagram
Coxeter
(orbifold)
[(8,8,3)]+ =
(883)
[(8,8,3+)]+ = =
(4343)
[(8,8,3*)]+ =
(444444)
Related polyhedra and tiling
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8
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.
See also
Wikimedia Commons has media related to Uniform tiling 6-16-16.
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Truncated order-6 pentagonal tiling
In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.
Truncated order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration6.10.10
Schläfli symbolt{5,6}
t(5,5,3)
Wythoff symbol2 6 | 5
3 5 5 |
Coxeter diagram
Symmetry group[6,5], (*652)
[(5,5,3)], (*553)
DualOrder-5 hexakis hexagonal tiling
PropertiesVertex-transitive
Uniform colorings
t012(5,5,3)
With mirrors
An alternate construction exists from the [(5,5,3)] family, as the omnitruncation t012(5,5,3). It is shown with two (colors) of decagons.
Symmetry
The dual of this tiling represents the fundamental domains of the *553 symmetry. There are no mirror removal subgroups of [(5,5,3)], but this symmetry group can be doubled to 652 symmetry by adding a bisecting mirror to the fundamental domains.
Small index subgroups of [(5,5,3)]
Type Reflective domains Rotational symmetry
Index 1 2
Diagram
Coxeter
(orbifold)
[(5,5,3)] =
(*553)
[(5,5,3)]+ =
(553)
Related polyhedra and tiling
Uniform hexagonal/pentagonal tilings
Symmetry: [6,5], (*652) [6,5]+, (652) [6,5+], (5*3) [1+,6,5], (*553)
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6} h{6,5}
Uniform duals
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5
[(5,5,3)] reflective symmetry 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.
See also
Wikimedia Commons has media related to Uniform tiling 6-10-10.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Truncated order-6 square tiling
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.
Truncated order-6 square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.8.6
Schläfli symbolt{4,6}
Wythoff symbol2 6 | 4
Coxeter diagram
Symmetry group[6,4], (*642)
[(3,3,4)], (*334)
DualOrder-4 hexakis hexagonal tiling
PropertiesVertex-transitive
Uniform colorings
The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram .
Symmetry
The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.
A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).
The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.
Small index subgroups of [(4,4,3)] (*443)
Index 1 2 6
Diagram
Coxeter
(orbifold)
[(4,4,3)] =
(*443)
[(4,1+,4,3)] = =
(*3232)
[(4,4,3+)] =
(3*22)
[(4,4,3*)] =
(*222222)
Direct subgroups
Index 2 4 12
Diagram
Coxeter
(orbifold)
[(4,4,3)]+ =
(443)
[(4,4,3+)]+ = =
(3232)
[(4,4,3*)]+ =
(222222)
Related polyhedra and tilings
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal 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 tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
=
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
=
=
=
=
=
=
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
It can also be generated from the (4 4 3) hyperbolic tilings:
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n
Sym.
*n43
[(n,4,3)]
Spherical Compact hyperbolic Paraco.
*243
[4,3]
*343
[(3,4,3)]
*443
[(4,4,3)]
*543
[(5,4,3)]
*643
[(6,4,3)]
*743
[(7,4,3)]
*843
[(8,4,3)]
*∞43
[(∞,4,3)]
Figures
Config. 4.8.6 6.8.6 8.8.6 10.8.6 12.8.6 14.8.6 16.8.6 ∞.8.6
Duals
Config. V4.8.6 V6.8.6 V8.8.6 V10.8.6 V12.8.6 V14.8.6 V16.8.6 V6.8.∞
See also
Wikimedia Commons has media related to Uniform tiling 6-8-8.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Truncated order-7 heptagonal tiling
In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from one heptagons and two tetrakaidecagons around every vertex.
Truncated order-7 heptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration7.14.14
Schläfli symbolt{7,7}
Wythoff symbol2 7 | 7
Coxeter diagram
Symmetry group[7,7], (*772)
DualOrder-7 heptakis heptagonal tiling
PropertiesVertex-transitive
Related tilings
Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{7,7} t{7,7}
r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7
See also
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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
Wikimedia Commons has media related to Uniform tiling 7-14-14.
• 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 |
Truncated order-7 square tiling
In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}.
Truncated order-7 square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.8.7
Schläfli symbolt{4,7}
Wythoff symbol2 7 | 4
Coxeter diagram
Symmetry group[7,4], (*742)
DualOrder-4 heptakis heptagonal tiling
PropertiesVertex-transitive
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
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.
See also
Wikimedia Commons has media related to Uniform tiling 7-8-8.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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 |
Truncated order-8 hexagonal tiling
In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.
Truncated order-8 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.12.12
Schläfli symbolt{6,8}
Wythoff symbol2 8 | 6
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-6 octakis octagonal tiling
PropertiesVertex-transitive
Uniform colorings
This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 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 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8
Symmetry
The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,6,1+,6,1+,4)] (332332) is the commutator subgroup of [(6,6,4)].
A large subgroup is constructed [(6,6,4*)], index 8, as (4*33) with gyration points removed, becomes (*38), and another large subgroup is constructed [(6,6*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)6).
Small index subgroups of [(6,6,4)] (*664)
Fundamental
domains
Subgroup index 1 2 4
Coxeter [(6,6,4)]
[(1+,6,6,4)]
[(6,6,1+,4)]
[(6,1+,6,4)]
[(1+,6,6,1+,4)]
[(6+,6+,4)]
Orbifold *664 *6362 *4343 2*3333 332×
Coxeter [(6,6+,4)]
[(6+,6,4)]
[(6,6,4+)]
[(6,1+,6,1+,4)]
[(1+,6,1+,6,4)]
Orbifold 6*32 4*33 3*3232
Direct subgroups
Subgroup index 2 4 8
Coxeter [(6,6,4)]+
[(1+,6,6+,4)]
[(6+,6,1+,4)]
[(6,1+,6,4+)]
[(6+,6+,4+)] = [(1+,6,1+,6,1+,4)]
=
Orbifold 664 6362 4343 332332
See also
Wikimedia Commons has media related to Uniform tiling 8-12-12.
• Tilings of regular polygons
• List of uniform planar 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 |
Truncated order-8 octagonal tiling
In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.
Truncated order-8 octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.16.16
Schläfli symbolt{8,8}
t(8,8,4)
Wythoff symbol2 8 | 4
Coxeter diagram
Symmetry group[8,8], (*882)
[(8,8,4)], (*884)
DualOrder-8 octakis octagonal tiling
PropertiesVertex-transitive
Uniform colorings
This tiling can also be constructed in *884 symmetry with 3 colors of faces:
Related polyhedra and tiling
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
= = = =
=
=
=
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8
Symmetry
The dual of the tiling represents the fundamental domains of (*884) orbifold symmetry. From [(8,8,4)] (*884) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 882 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,8,1+,8,1+,4)] (442442) is the commutator subgroup of [(8,8,4)].
Small index subgroups of [(8,8,4)] (*884)
Fundamental
domains
Subgroup index 1 2 4
Coxeter [(8,8,4)]
[(1+,8,8,4)]
[(8,8,1+,4)]
[(8,1+,8,4)]
[(1+,8,8,1+,4)]
[(8+,8+,4)]
orbifold *884 *8482 *4444 2*4444 442×
Coxeter [(8,8+,4)]
[(8+,8,4)]
[(8,8,4+)]
[(8,1+,8,1+,4)]
[(1+,8,1+,8,4)]
Orbifold 8*42 4*44 4*4242
Direct subgroups
Subgroup index 2 4 8
Coxeter [(8,8,4)]+
[(1+,8,8+,4)]
[(8+,8,1+,4)]
[(8,1+,8,4+)]
[(1+,8,1+,8,1+,4)] = [(8+,8+,4+)]
Orbifold 844 8482 4444 442442
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.
See also
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
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 |
Octagonal tiling
In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.
For other uses, see truncated square tiling.
Octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration83
Schläfli symbol{8,3}
t{4,8}
Wythoff symbol3 | 8 2
2 8 | 4
4 4 4 |
Coxeter diagram
Symmetry group[8,3], (*832)
[8,4], (*842)
[(4,4,4)], (*444)
DualOrder-8 triangular tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Uniform colorings
Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.
Regular Truncations
{8,3}
t{4,8}
t{4[3]}
= =
Dual tiling
{3,8}
=
=
= =
Regular maps
The regular map {8,3}2,0 can be seen as a 6-coloring of the {8,3} hyperbolic tiling. Within the regular map, octagons of the same color are considered the same face shown in multiple locations. The 2,0 subscripts show the same color will repeat by moving 2 steps in a straight direction following opposite edges. This regular map also has a representation as a double covering of a cube, represented by Schläfli symbol {8/2,3}, with 6 octagonal faces, double wrapped {8/2}, with 24 edges, and 16 vertices. It was described by Branko Grünbaum in his 2003 paper Are Your Polyhedra the Same as My Polyhedra?[1]
Related polyhedra and tilings
This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.
*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}
And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.
n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling
Config. 8.8 83 84 85 86 87 88 ...8∞
From a Wythoff construction there are ten 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 10 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
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3
See also
Wikimedia Commons has media related to Order-3 octagonal tiling.
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
References
1. Grünbaum, Branko (2003). "Are Your Polyhedra the Same as My Polyhedra?" (PDF). Discrete and Computational Geometry. 25: 461–488. doi:10.1007/978-3-642-55566-4_21. Retrieved 27 April 2023.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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 |
Truncated order-8 triangular tiling
In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.
Truncated order-8 triangular tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.6.6
Schläfli symbolt{3,8}
Wythoff symbol2 8 | 3
4 3 3 |
Coxeter diagram
Symmetry group[8,3], (*832)
[(4,3,3)], (*433)
DualOctakis octagonal tiling
PropertiesVertex-transitive
Uniform colors
The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of hexagons
Dual tiling
Symmetry
The dual of this tiling represents the fundamental domains of *443 symmetry. It only has one subgroup 443, replacing mirrors with gyration points.
This symmetry can be doubled to 832 symmetry by adding a bisecting mirror to the fundamental domain.
Small index subgroups of [(4,3,3)], (*433)
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)
[(4,3,3)] =
(*433)
[(4,3,3)]+ =
(433)
Related tilings
From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.
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
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. 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
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n
Sym.
*n43
[(n,4,3)]
Spherical Compact hyperbolic Paraco.
*243
[4,3]
*343
[(3,4,3)]
*443
[(4,4,3)]
*543
[(5,4,3)]
*643
[(6,4,3)]
*743
[(7,4,3)]
*843
[(8,4,3)]
*∞43
[(∞,4,3)]
Figures
Config. 4.8.6 6.8.6 8.8.6 10.8.6 12.8.6 14.8.6 16.8.6 ∞.8.6
Duals
Config. V4.8.6 V6.8.6 V8.8.6 V10.8.6 V12.8.6 V14.8.6 V16.8.6 V6.8.∞
See also
Wikimedia Commons has media related to Uniform tiling 6-6-8.
• Triangular tiling
• Order-3 octagonal tiling
• Order-8 triangular 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 |
Truncated pentahexagonal tiling
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.
Truncated pentahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.10.12
Schläfli symboltr{6,5} or $t{\begin{Bmatrix}6\\5\end{Bmatrix}}$
Wythoff symbol2 6 5 |
Coxeter diagram
Symmetry group[6,5], (*652)
DualOrder 5-6 kisrhombille
PropertiesVertex-transitive
Dual tiling
The dual tiling is called an order-5-6 kisrhombille tiling, made as a complete bisection of the order-5 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,5] (*652) symmetry.
Symmetry
There are four small index subgroup from [6,5] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
Small index subgroups of [6,5], (*652)
Index 1 2 6
Diagram
Coxeter
(orbifold)
[6,5] =
(*652)
[1+,6,5] = =
(*553)
[6,5+] =
(5*3)
[6,5*] =
(*33333)
Direct subgroups
Index 2 4 12
Diagram
Coxeter
(orbifold)
[6,5]+ =
(652)
[6,5+]+ = =
(553)
[6,5*]+ =
(33333)
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,5] symmetry, and 3 with subsymmetry.
Uniform hexagonal/pentagonal tilings
Symmetry: [6,5], (*652) [6,5]+, (652) [6,5+], (5*3) [1+,6,5], (*553)
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6} h{6,5}
Uniform duals
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5
See also
• Tilings of regular polygons
• List of uniform planar 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 |
Truncated pentakis dodecahedron
The truncated pentakis dodecahedron is a convex polyhedron constructed as a truncation of the pentakis dodecahedron. It is Goldberg polyhedron GV(3,0), with pentagonal faces separated by an edge-direct distance of 3 steps.
Truncated pentakis dodecahedron
Conway notationtkD
Goldberg polyhedronGPV(3,0) or {5+,3}3,0
FullereneC180[1]
Faces92:
12 pentagons
20+60 hexagons
Edges270 (2 types)
Vertices180 (2 types)
Vertex configuration(60) 5.6.6
(120) 6.6.6
Symmetry groupIcosahedral (Ih)
Dual polyhedronPentahexakis truncated icosahedron
Propertiesconvex
Related polyhedra
It is in an infinite sequence of Goldberg polyhedra:
Index GP(1,0) GP(2,0) GP(3,0) GP(4,0) GP(5,0) GP(6,0) GP(7,0) GP(8,0)...
Image
D
kD
tkD
Duals
I
cD
ktI
See also
• Near-miss Johnson solid
• Truncated tetrakis cube
References
1. C180 Isomers
• Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus half-cube embeddings", Discrete Mathematics, 192 (1): 41–80, doi:10.1016/S0012-365X(98)00065-X, archived from the original on 2007-02-06.
• Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF
External links
• VTML polyhedral generator Try "tkD" (Conway polyhedron notation)
| Wikipedia |
Truncated order-4 pentagonal tiling
In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}.
Truncated pentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.10.10
Schläfli symbolt{5,4}
Wythoff symbol2 4 | 5
2 5 5 |
Coxeter diagram
or
Symmetry group[5,4], (*542)
[5,5], (*552)
DualOrder-5 tetrakis square tiling
PropertiesVertex-transitive
Uniform colorings
A half symmetry [1+,4,5] = [5,5] coloring can be constructed with two colors of decagons. This coloring is called a truncated pentapentagonal tiling.
Symmetry
There is only one subgroup of [5,5], [5,5]+, removing all the mirrors. This symmetry can be doubled to 542 symmetry by adding a bisecting mirror.
Small index subgroups of [5,5]
Type Reflective domains Rotational symmetry
Index 1 2
Diagram
Coxeter
(orbifold)
[5,5] = =
(*552)
[5,5]+ = =
(552)
Related polyhedra and tiling
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
Uniform pentapentagonal tilings
Symmetry: [5,5], (*552) [5,5]+, (552)
=
=
=
=
=
=
=
=
Order-5 pentagonal tiling
{5,5}
Truncated order-5 pentagonal tiling
t{5,5}
Order-4 pentagonal tiling
r{5,5}
Truncated order-5 pentagonal tiling
2t{5,5} = t{5,5}
Order-5 pentagonal tiling
2r{5,5} = {5,5}
Tetrapentagonal tiling
rr{5,5}
Truncated order-4 pentagonal tiling
tr{5,5}
Snub pentapentagonal tiling
sr{5,5}
Uniform duals
Order-5 pentagonal tiling
V5.5.5.5.5
V5.10.10 Order-5 square tiling
V5.5.5.5
V5.10.10 Order-5 pentagonal tiling
V5.5.5.5.5
V4.5.4.5 V4.10.10 V3.3.5.3.5
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.
See also
Wikimedia Commons has media related to Uniform tiling 4-10-10.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
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 |
Truncated 5-cubes
In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.
5-cube
Truncated 5-cube
Bitruncated 5-cube
5-orthoplex
Truncated 5-orthoplex
Bitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane
There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.
Truncated 5-cube
Truncated 5-cube
Typeuniform 5-polytope
Schläfli symbolt{4,3,3,3}
Coxeter-Dynkin diagram
4-faces4210
32
Cells20040
160
Faces40080
320
Edges40080
320
Vertices160
Vertex figure
( )v{3,3}
Coxeter groupB5, [3,3,3,4], order 3840
Propertiesconvex
Alternate names
• Truncated penteract (Acronym: tan) (Jonathan Bowers)
Construction and coordinates
The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at $1/({\sqrt {2}}+2)$ of the edge length. A regular 5-cell is formed at each truncated vertex.
The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
$\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)$
Images
The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Related polytopes
The truncated 5-cube, is fourth in a sequence of truncated hypercubes:
Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}
Bitruncated 5-cube
Bitruncated 5-cube
Typeuniform 5-polytope
Schläfli symbol2t{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces4210
32
Cells28040
160
80
Faces72080
320
320
Edges800320
480
Vertices320
Vertex figure
{ }v{3}
Coxeter groupsB5, [3,3,3,4], order 3840
Propertiesconvex
Alternate names
• Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)
Construction and coordinates
The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at ${\sqrt {2}}$ of the edge length.
The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
$\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)$
Images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
Related polytopes
The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:
Bitruncated hypercubes
Image ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v{ }
{ }v{ }
{ }v{3}
{ }v{3,3}
{ }v{3,3,3} { }v{3,3,3,3}
Related polytopes
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.
B5 polytopes
β5
t1β5
t2γ5
t1γ5
γ5
t0,1β5
t0,2β5
t1,2β5
t0,3β5
t1,3γ5
t1,2γ5
t0,4γ5
t0,3γ5
t0,2γ5
t0,1γ5
t0,1,2β5
t0,1,3β5
t0,2,3β5
t1,2,3γ5
t0,1,4β5
t0,2,4γ5
t0,2,3γ5
t0,1,4γ5
t0,1,3γ5
t0,1,2γ5
t0,1,2,3β5
t0,1,2,4β5
t0,1,3,4γ5
t0,1,2,4γ5
t0,1,2,3γ5
t0,1,2,3,4γ5
Notes
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. "5D uniform polytopes (polytera)". o3o3o3x4x - tan, o3o3x3x4o - bittin
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 |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.
Truncated square is a regular octagon:
t{4} = {8}
=
Truncated cube
t{4,3} or
Truncated cubic honeycomb
t{4,3,4} or
Uniform truncation
In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation.
A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra.
In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r{5,3} or ${\begin{Bmatrix}5\\3\end{Bmatrix}}$, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented as tr{5,3} or $t{\begin{Bmatrix}5\\3\end{Bmatrix}}$, . In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the nodes adjacent to the ringed node.
A uniform truncation performed on the regular triangular tiling {3,6} results in the regular hexagonal tiling {6,3}.
Truncation of polygons
A truncated n-sided polygon will have 2n sides (edges). A regular polygon uniformly truncated will become another regular polygon: t{n} is {2n}. A complete truncation (or rectification), r{3}, is another regular polygon in its dual position.
A regular polygon can also be represented by its Coxeter-Dynkin diagram, , and its uniform truncation , and its complete truncation . The graph represents Coxeter group I2(n), with each node representing a mirror, and the edge representing the angle π/n between the mirrors, and a circle is given around one or both mirrors to show which ones are active.
Parametric truncations of a triangle
{3}
t{3} = {6}
r{3} = {3}
Star polygons can also be truncated. A truncated pentagram {5/2} will look like a pentagon, but is actually a double-covered (degenerate) decagon ({10/2}) with two sets of overlapping vertices and edges. A truncated great heptagram {7/3} gives a tetradecagram {14/3}.
Uniform truncation in regular polyhedra and tilings and higher
When "truncation" applies to platonic solids or regular tilings, usually "uniform truncation" is implied, which means truncating until the original faces become regular polygons with twice as many sides as the original form.
This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron. The middle image is the uniform truncated cube; it is represented by a Schläfli symbol t{p,q,...}.
A bitruncation is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. Example: a truncated octahedron is a bitruncated cube: t{3,4} = 2t{4,3}.
A complete bitruncation, called a birectification, reduces original faces to points. For polyhedra, this becomes the dual polyhedron. Example: an octahedron is a birectification of a cube: {3,4} = 2r{4,3}.
Another type of truncation, cantellation, cuts edges and vertices, removing the original edges, replacing them with rectangles, removing the original vertices, and replacing them with the faces of the dual of the original regular polyhedra or tiling.
Higher dimensional polytopes have higher truncations. Runcination cuts faces, edges, and vertices. In 5 dimensions, sterication cuts cells, faces, and edges.
Edge-truncation
Truncating the edges of a cube, creating a chamfered cube
Edge-truncation is a beveling, or chamfer for polyhedra, similar to cantellation, but retaining the original vertices, and replacing edges by hexagons. In 4-polytopes, edge-truncation replaces edges with elongated bipyramid cells.
Alternation or partial truncation
A uniform alternation of a truncated cuboctahedron gives a nonuniform snub cube.
Alternation or partial truncation removes only some of the original vertices.
In partial truncation, or alternation, half of the vertices and connecting edges are completely removed. The operation applies only to polytopes with even-sided faces. Faces are reduced to half as many sides, and square faces degenerate into edges. For example, the tetrahedron is an alternated cube, h{4,3}.
Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices. For example, the tridiminished icosahedron starts with a regular icosahedron with 3 vertices removed.
Other partial truncations are symmetry-based; for example, the tetrahedrally diminished dodecahedron.
Generalized truncations
The linear truncation process can be generalized by allowing parametric truncations that are negative, or that go beyond the midpoint of the edges, causing self-intersecting star polyhedra, and can parametrically relate to some of the regular star polygons and uniform star polyhedra.
• Shallow truncation - Edges are reduced in length, faces are truncated to have twice as many sides, while new facets are formed, centered at the old vertices.
• Uniform truncation are a special case of this with equal edge lengths. The truncated cube, t{4,3}, with square faces becoming octagons, with new triangular faces are the vertices.
• Antitruncation A reverse shallow truncation, truncated outwards off the original edges, rather than inward. This results in a polytope which looks like the original, but has parts of the dual dangling off its corners, instead of the dual cutting into its own corners.
• Complete truncation or rectification - The limit of a shallow truncation, where edges are reduced to points. The cuboctahedron, r{4,3}, is an example.
• Hypertruncation A form of truncation that goes past the rectification, inverting the original edges, and causing self-intersections to appear.
• Quasitruncation A form of truncation that goes even farther than hypertruncation where the inverted edge becomes longer than the original edge. It can be generated from the original polytope by treating all the faces as retrograde, i.e. going backwards round the vertex. For example, quasitruncating the square gives a regular octagram (t{4,3}={8/3}), and quasitruncating the cube gives the uniform stellated truncated hexahedron, t{4/3,3}.
Truncations on a square
Types of truncation on a square, {4}, showing red original edges, and new truncated edges in cyan. A uniform truncated square is a regular octagon, t{4}={8}. A complete truncated square becomes a new square, with a diagonal orientation. Vertices are sequenced around counterclockwise, 1-4, with truncated pairs of vertices as a and b.
Truncations of the cube
⇨ taC
Cube
{4,3} C
⇨ tC
Truncation
t{4,3} tC
⇨ tC
Complete truncation
r{4,3} aC
⇩ thC
Antitruncation taC
Hypertruncation thC
⇧ taC
Complete quasitruncation
aqC
⇦
Quasitruncation
t{4/3,3} tqC
⇦ tqC
Complete hypertruncation ahC
⇦ thC
See also
• Uniform polyhedron
• Uniform 4-polytope
• Bitruncation (geometry)
• Rectification (geometry)
• Alternation (geometry)
• Conway polyhedron notation
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
• Weisstein, Eric W. "Truncation". MathWorld.
• Olshevsky, George. "Truncation". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polyhedra Names, truncation
Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}
| Wikipedia |
Truncated power function
In mathematics, the truncated power function[1] with exponent $n$ is defined as
$x_{+}^{n}={\begin{cases}x^{n}&:\ x>0\\0&:\ x\leq 0.\end{cases}}$
In particular,
$x_{+}={\begin{cases}x&:\ x>0\\0&:\ x\leq 0.\end{cases}}$
and interpret the exponent as conventional power.
Relations
• Truncated power functions can be used for construction of B-splines.
• $x\mapsto x_{+}^{0}$ is the Heaviside function.
• $\chi _{[a,b)}(x)=(b-x)_{+}^{0}-(a-x)_{+}^{0}$ where $\chi $ is the indicator function.
• Truncated power functions are refinable.
See also
• Macaulay brackets
External links
• Truncated Power Function on MathWorld
References
1. Massopust, Peter (2010). Interpolation and Approximation with Splines and Fractals. Oxford University Press, USA. p. 46. ISBN 0-19-533654-2.
| Wikipedia |
Frustum
In geometry, a frustum (Latin for 'morsel');[lower-alpha 1] (PL: frusta or frustums) is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal. A right frustum is a right pyramid or a right cone truncated perpendicularly to its axis;[3] otherwise, it is an oblique frustum.
Set of pyramidal right n-gonal frustums
Examples: right pentagonal and square frustums
(n = 5 and n = 4)
Facesn isosceles trapezoids, 2 regular n-gons
Edges3n
Vertices2n
Symmetry groupCnv, [1,n], (*nn)
Dual polyhedronconvex asymmetric right n-gonal bipyramid
Propertiesconvex
Net
Example: net of right trigonal frustum (n = 3)
If all its edges are forced to become of the same length, then a frustum becomes a prism (possibly oblique or/and with irregular bases).
In computer graphics, the viewing frustum is the three-dimensional region which is visible on the screen. It is formed by a clipped pyramid; in particular, frustum culling is a method of hidden-surface determination.
In the aerospace industry, a frustum is the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.
Elements, special cases, and related concepts
A frustum's axis is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.
The height of a frustum is the perpendicular distance between the planes of the two bases.
Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of prismatoids.
Two frusta with two congruent bases joined at these congruent bases make a bifrustum.
Formulas
Volume
The formula for the volume of a pyramidal square frustum was introduced by the ancient Egyptian mathematics in what is called the Moscow Mathematical Papyrus, written in the 13th dynasty (c. 1850 BC):
$V={\frac {h}{3}}\left(a^{2}+ab+b^{2}\right),$
where a and b are the base and top side lengths, and h is the height.
The Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.
The volume of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":
$V={\frac {h_{1}B_{1}-h_{2}B_{2}}{3}},$
where B1 and B2 are the base and top areas, and h1 and h2 are the perpendicular heights from the apex to the base and top planes.
Considering that
${\frac {B_{1}}{h_{1}^{2}}}={\frac {B_{2}}{h_{2}^{2}}}={\frac {\sqrt {B_{1}B_{2}}}{h_{1}h_{2}}}=\alpha ,$
the formula for the volume can be expressed as the third of the product of this proportionality, $\alpha $, and of the difference of the cubes of the heights h1 and h2 only:
$V={\frac {h_{1}\alpha h_{1}^{2}-h_{2}\alpha h_{2}^{2}}{3}}=\alpha {\frac {h_{1}^{3}-h_{2}^{3}}{3}}.$
By using the identity a3 − b3 = (a − b)(a2 + ab + b2), one gets:
$V=(h_{1}-h_{2})\alpha {\frac {h_{1}^{2}+h_{1}h_{2}+h_{2}^{2}}{3}},$
where h1 − h2 = h is the height of the frustum.
Distributing $\alpha $ and substituting from its definition, the Heronian mean of areas B1 and B2 is obtained:
${\frac {B_{1}+{\sqrt {B_{1}B_{2}}}+B_{2}}{3}};$
the alternative formula is therefore:
$V={\frac {h}{3}}\left(B_{1}+{\sqrt {B_{1}B_{2}}}+B_{2}\right).$
Heron of Alexandria is noted for deriving this formula, and with it, encountering the imaginary unit: the square root of negative one.[4]
In particular:
• The volume of a circular cone frustum is:
$V={\frac {\pi h}{3}}\left(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}\right),$
where r1 and r2 are the base and top radii.
• The volume of a pyramidal frustum whose bases are regular n-gons is:
$V={\frac {nh}{12}}\left(a_{1}^{2}+a_{1}a_{2}+a_{2}^{2}\right)\cot {\frac {\pi }{n}},$
where a1 and a2 are the base and top side lengths.
Surface area
For a right circular conical frustum[5][6]
${\begin{aligned}{\text{Lateral surface area}}&=\pi \left(r_{1}+r_{2}\right)s\\&=\pi \left(r_{1}+r_{2}\right){\sqrt {\left(r_{1}-r_{2}\right)^{2}+h^{2}}}\end{aligned}}$
and
${\begin{aligned}{\text{Total surface area}}&=\pi \left(\left(r_{1}+r_{2}\right)s+r_{1}^{2}+r_{2}^{2}\right)\\&=\pi \left(\left(r_{1}+r_{2}\right){\sqrt {\left(r_{1}-r_{2}\right)^{2}+h^{2}}}+r_{1}^{2}+r_{2}^{2}\right)\end{aligned}}$
where r1 and r2 are the base and top radii respectively, and s is the slant height of the frustum.
The surface area of a right frustum whose bases are similar regular n-sided polygons is
$A={\frac {n}{4}}\left[\left(a_{1}^{2}+a_{2}^{2}\right)\cot {\frac {\pi }{n}}+{\sqrt {\left(a_{1}^{2}-a_{2}^{2}\right)^{2}\sec ^{2}{\frac {\pi }{n}}+4h^{2}\left(a_{1}+a_{2}\right)^{2}}}\right]$
where a1 and a2 are the sides of the two bases.
Examples
• On the back (the reverse) of a United States one-dollar bill, a pyramidal frustum appears on the reverse of the Great Seal of the United States, surmounted by the Eye of Providence.
• Ziggurats, step pyramids, and certain ancient Native American mounds also form the frustum of one or more pyramids, with additional features such as stairs added.
• Chinese pyramids.
• The John Hancock Center in Chicago, Illinois is a frustum whose bases are rectangles.
• The Washington Monument is a narrow square-based pyramidal frustum topped by a small pyramid.
• The viewing frustum in 3D computer graphics is a virtual photographic or video camera's usable field of view modeled as a pyramidal frustum.
• In the English translation of Stanislaw Lem's short-story collection The Cyberiad, the poem Love and tensor algebra claims that "every frustum longs to be a cone".
• Buckets and typical lampshades are everyday examples of conical frustums.
• Drinking glasses and some space capsules are also some examples.
• Garsų Gaudyklė wooden structure or statue in Lithuania.
• Valençay cheese
• Rolo candies
See also
• Spherical frustum
Notes
1. The term frustum comes from Latin frustum, meaning 'piece' or 'morsel". The English word is often misspelled as frustrum, a different Latin word cognate to the English word "frustrate".[1] The confusion between these two words is very old: a warning about them can be found in the Appendix Probi, and the works of Plautus include a pun on them.[2]
References
1. Clark, John Spencer (1895). Teachers' Manual: Books I–VIII. For Prang's complete course in form-study and drawing, Books 7–8. Prang Educational Company. p. 49.
2. Fontaine, Michael (2010). Funny Words in Plautine Comedy. Oxford University Press. pp. 117, 154. ISBN 9780195341447.
3. Kern, William F.; Bland, James R. (1938). Solid Mensuration with Proofs. p. 67.
4. Nahin, Paul. An Imaginary Tale: The story of √−1. Princeton University Press. 1998
5. "Mathwords.com: Frustum". Retrieved 17 July 2011.
6. Al-Sammarraie, Ahmed T.; Vafai, Kambiz (2017). "Heat transfer augmentation through convergence angles in a pipe". Numerical Heat Transfer, Part A: Applications. 72 (3): 197−214. Bibcode:2017NHTA...72..197A. doi:10.1080/10407782.2017.1372670. S2CID 125509773.
External links
Look up frustum in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Frustums.
• Derivation of formula for the volume of frustums of pyramid and cone (Mathalino.com)
• Weisstein, Eric W. "Pyramidal frustum". MathWorld.
• Weisstein, Eric W. "Conical frustum". MathWorld.
• Paper models of frustums (truncated pyramids)
• Paper model of frustum (truncated cone)
• Design paper models of conical frustum (truncated cones)
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Truncated square tiling
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.
Truncated square tiling
TypeSemiregular tiling
Vertex configuration
4.8.8
Schläfli symbolt{4,4}
tr{4,4} or $t{\begin{Bmatrix}4\\4\end{Bmatrix}}$
Wythoff symbol2 | 4 4
4 4 2 |
Coxeter diagram
or
Symmetryp4m, [4,4], (*442)
Rotation symmetryp4, [4,4]+, (442)
Bowers acronymTosquat
DualTetrakis square tiling
PropertiesVertex-transitive
Conway calls it a truncated quadrille, constructed as a truncation operation applied to a square tiling (quadrille).
Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges.
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There are two distinct uniform colorings of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)
2 colors: 122
3 colors: 123
Circle packing
The truncated square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number).[1]
Variations
One variations on this pattern, often called a Mediterranean pattern, is shown in stone tiles with smaller squares and diagonally aligned with the borders. Other variations stretch the squares or octagons.
The Pythagorean tiling alternates large and small squares, and may be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices.
A weaving pattern also has the same topology, with octagons flattened rectangles.
p4m, (*442) p4, (442) p4g, (4*2) pmm (*2222)
p4m, (*442) p4, (442) cmm, (2*22) pmm (*2222)
Mediterranean Pythagorean Flemish bond Weaving Twisted Rectangular/rhombic
Related polyhedra and tilings
The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
The 3-dimensional bitruncated cubic honeycomb projected into the plane shows two copies of a truncated tiling. In the plane it can be represented by a compound tiling, or combined can be seen as a chamfered square tiling.
+
Wythoff constructions from square tiling
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling, snub square tiling.
Uniform tilings based on square tiling symmetry
Symmetry: [4,4], (*442) [4,4]+, (442) [4,4+], (4*2)
{4,4} t{4,4} r{4,4} t{4,4} {4,4} rr{4,4} tr{4,4} sr{4,4} s{4,4}
Uniform duals
V4.4.4.4 V4.8.8 V4.4.4.4 V4.8.8 V4.4.4.4 V4.4.4.4 V4.8.8 V3.3.4.3.4
Related tilings in other symmetries
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Tetrakis square tiling
Main article: Tetrakis square tiling
The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2.
Conway calls it a kisquadrille,[2] represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.[3]
See also
Wikimedia Commons has media related to Uniform tiling 4-8-8 (truncated square tiling).
• Tilings of regular polygons
• List of uniform tilings
• Percolation threshold
References
1. Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern H
2. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 "A K Peters, LTD. - the Symmetries of Things". Archived from the original on 2010-09-19. Retrieved 2012-01-20. (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
3. Stephenson, John (1970), "Ising Model with Antiferromagnetic Next-Nearest-Neighbor Coupling: Spin Correlations and Disorder Points", Phys. Rev. B, 1 (11): 4405–4409, doi:10.1103/PhysRevB.1.4405.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
• 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)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 40. ISBN 0-486-23729-X.
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56
External links
• Weisstein, Eric W. "Semiregular tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings o4x4x - tosquat - O6".
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 |
Chamfer (geometry)
In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge.
Unchamfered, slightly chamfered and chamfered cube
Historical crystal models of slightly chamfered Platonic solids
In Conway polyhedron notation it is represented by the letter c. A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and e new hexagonal faces.
Chamfered Platonic solids
In the chapters below the chamfers of the five Platonic solids are described in detail. Each is shown in a version with edges of equal length and in a canonical version where all edges touch the same midsphere. (They only look noticeably different for solids containing triangles.) The shown duals are dual to the canonical versions.
Seed
{3,3}
{4,3}
{3,4}
{5,3}
{3,5}
Chamfered
Chamfered tetrahedron
Chamfered tetrahedron
(with equal edge length)
Conway notationcT
Goldberg polyhedronGPIII(2,0) = {3+,3}2,0
Faces4 triangles
6 hexagons
Edges24 (2 types)
Vertices16 (2 types)
Vertex configuration(12) 3.6.6
(4) 6.6.6
Symmetry groupTetrahedral (Td)
Dual polyhedronAlternate-triakis tetratetrahedron
Propertiesconvex, equilateral-faced
net
The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons.
It is the Goldberg polyhedron GIII(2,0), containing triangular and hexagonal faces.
Tetrahedral chamfers and related solids
chamfered tetrahedron (canonical)
dual of the tetratetrahedron
chamfered tetrahedron (canonical)
alternate-triakis tetratetrahedron
tetratetrahedron
alternate-triakis tetratetrahedron
Chamfered cube
Chamfered cube
(with equal edge length)
Conway notationcC = t4daC
Goldberg polyhedronGPIV(2,0) = {4+,3}2,0
Faces6 squares
12 hexagons
Edges48 (2 types)
Vertices32 (2 types)
Vertex configuration(24) 4.6.6
(8) 6.6.6
SymmetryOh, [4,3], (*432)
Th, [4,3+], (3*2)
Dual polyhedronTetrakis cuboctahedron
Propertiesconvex, equilateral-faced
net
The chamfered cube is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 12 hexagons and 6 squares. It is constructed as a chamfer of a cube. The squares are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the tetrakis cuboctahedron.
It is also inaccurately called a truncated rhombic dodecahedron, although that name rather suggests a rhombicuboctahedron. It can more accurately be called a tetratruncated rhombic dodecahedron because only the order-4 vertices are truncated.
The hexagonal faces are equilateral but not regular. They are formed by a truncated rhombus, have 2 internal angles of about 109.47°, or $\cos ^{-1}(-{\frac {1}{3}})$, and 4 internal angles of about 125.26°, while a regular hexagon would have all 120° angles.
Because all its faces have an even number of sides with 180° rotation symmetry, it is a zonohedron. It is also the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0, containing square and hexagonal faces.
The chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at $(\pm 1,\pm 1,\pm 1)$ and its six vertices are at the permutations of $(\pm {\sqrt {3}},0,0)$.
A topological equivalent with pyritohedral symmetry and rectangular faces can be constructed by chamfering the axial edges of a pyritohedron. This occurs in pyrite crystals.
Pyritohedron and its axis truncation
Historical crystallographic models
Octahedral chamfers and related solids
chamfered cube (canonical)
rhombic dodecahedron
chamfered octahedron (canonical)
tetrakis cuboctahedron
cuboctahedron
triakis cuboctahedron
Chamfered octahedron
Chamfered octahedron
(with equal edge length)
Conway notationcO = t3daO
Faces8 triangles
12 hexagons
Edges48 (2 types)
Vertices30 (2 types)
Vertex configuration(24) 3.6.6
(6) 6.6.6
SymmetryOh, [4,3], (*432)
Dual polyhedronTriakis cuboctahedron
Propertiesconvex
In geometry, the chamfered octahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 8 (order 3) vertices.
It can also be called a tritruncated rhombic dodecahedron, a truncation of the order-3 vertices of the rhombic dodecahedron.
The 8 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles.
The hexagonal faces are equilateral but not regular.
Historical models of triakis cuboctahedron and chamfered octahedron
Chamfered dodecahedron
Chamfered dodecahedron
(with equal edge length)
Conway notationcD] = t5daD = dk5aD
Goldberg polyhedronGV(2,0) = {5+,3}2,0
FullereneC80[1]
Faces12 pentagons
30 hexagons
Edges120 (2 types)
Vertices80 (2 types)
Vertex configuration(60) 5.6.6
(20) 6.6.6
Symmetry groupIcosahedral (Ih)
Dual polyhedronPentakis icosidodecahedron
Propertiesconvex, equilateral-faced
Main article: Chamfered dodecahedron
The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.
It is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a rhombicosidodecahedron. It can more accurately be called a pentatruncated rhombic triacontahedron because only the order-5 vertices are truncated.
Icosahedral chamfers and related solids
chamfered dodecahedron (canonical)
rhombic triacontahedron
chamfered icosahedron (canonical)
pentakis icosidodecahedron
icosidodecahedron
triakis icosidodecahedron
Chamfered icosahedron
Chamfered icosahedron
(with equal edge length)
Conway notationcI = t3daI
Faces20 triangles
30 hexagons
Edges120 (2 types)
Vertices72 (2 types)
Vertex configuration(24) 3.6.6
(12) 6.6.6
SymmetryIh, [5,3], (*532)
Dual polyhedrontriakis icosidodecahedron
Propertiesconvex
In geometry, the chamfered icosahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the 20 order-3 vertices. The hexagonal faces can be made equilateral but not regular.
It can also be called a tritruncated rhombic triacontahedron, a truncation of the order-3 vertices of the rhombic triacontahedron.
Chamfered regular tilings
Chamfered regular and quasiregular tilings
Square tiling, Q
{4,4}
Triangular tiling, Δ
{3,6}
Hexagonal tiling, H
{6,3}
Rhombille, daH
dr{6,3}
cQ cΔ cH cdaH
Relation to Goldberg polyhedra
The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. The chamfer operator transforms GP(m,n) to GP(2m,2n).
A regular polyhedron, GP(1,0), create a Goldberg polyhedra sequence: GP(1,0), GP(2,0), GP(4,0), GP(8,0), GP(16,0)...
GP(1,0) GP(2,0) GP(4,0) GP(8,0) GP(16,0)...
GPIV
{4+,3}
C
cC
ccC
cccC
GPV
{5+,3}
D
cD
ccD
cccD
ccccD
GPVI
{6+,3}
H
cH
ccH
cccH
ccccH
The truncated octahedron or truncated icosahedron, GP(1,1) creates a Goldberg sequence: GP(1,1), GP(2,2), GP(4,4), GP(8,8)....
GP(1,1) GP(2,2) GP(4,4)...
GPIV
{4+,3}
tO
ctO
cctO
GPV
{5+,3}
tI
ctI
cctI
GPVI
{6+,3}
tH
ctH
cctH
A truncated tetrakis hexahedron or pentakis dodecahedron, GP(3,0), creates a Goldberg sequence: GP(3,0), GP(6,0), GP(12,0)...
GP(3,0) GP(6,0) GP(12,0)...
GPIV
{4+,3}
tkC
ctkC
cctkC
GPV
{5+,3}
tkD
ctkD
cctkD
GPVI
{6+,3}
tkH
ctkH
cctkH
Chamfered polytopes and honeycombs
Like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. For polychora, new cells are created around the original edges. The cells are prisms, containing two copies of the original face, with pyramids augmented onto the prism sides.
See also
• Conway polyhedron notation
• Near-miss Johnson solid
• Cantellation (geometry)
References
1. "C80 Isomers". Archived from the original on 2014-08-12. Retrieved 2014-08-09.
• Goldberg, Michael (1937). "A class of multi-symmetric polyhedra". Tohoku Mathematical Journal. 43: 104–108.
• Joseph D. Clinton, Clinton’s Equal Central Angle Conjecture
• Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed.). Shaping Space (2nd ed.). Springer. pp. 125–138. doi:10.1007/978-0-387-92714-5_9. ISBN 978-0-387-92713-8.
• Hart, George (June 18, 2013). "Mathematical Impressions: Goldberg Polyhedra". Simons Science News.
• Antoine Deza, Michel Deza, Viatcheslav Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings, 1998 PDF (p. 72 Fig. 26. Chamfered tetrahedron)
• Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus half-cube embeddings", Discrete Mathematics, 192 (1): 41–80, doi:10.1016/S0012-365X(98)00065-X.
External links
• Chamfered Tetrahedron
• Chamfered Solids
• Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro
• VRML polyhedral generator (Conway polyhedron notation)
• VRML model Chamfered cube
• 3.2.7. Systematic numbering for (C80-Ih) [5,6] fullerene
• Fullerene C80
1. (Number 7 -Ih)
• How to make a chamfered cube
| Wikipedia |
Truncated rhombicosidodecahedron
In geometry, the truncated rhombicosidodecahedron is a polyhedron, constructed as a truncated rhombicosidodecahedron. It has 122 faces: 12 decagons, 30 octagons, 20 hexagons, and 60 squares.
Truncated rhombicosidodecahedron
Schläfli symboltrr{5,3} = $tr{\begin{Bmatrix}5\\3\end{Bmatrix}}$
Conway notationtaD = baD
Faces122:
60 {4}
20 {6}
30 {8}
12 {10}
Edges360
Vertices240
Symmetry groupIh, [5,3], (*532) order 120
Rotation groupI, [5,3]+, (532), order 60
Dual polyhedronDisdyakis hexecontahedron
Propertiesconvex
Other names
• Truncated small rhombicosidodecahedron
• Beveled icosidodecahedron
Zonohedron
As a zonohedron, it can be constructed with all but 30 octagons as regular polygons. It is 2-uniform, with 2 sets of 120 vertices existing on two distances from its center.
This polyhedron represents the Minkowski sum of a truncated icosidodecahedron, and a rhombic triacontahedron.[1]
Related polyhedra
The truncated icosidodecahedron is similar, with all regular faces, and 4.6.10 vertex figure. Also see the truncated rhombirhombicosidodecahedron.
truncated icosidodecahedron Truncated rhombicosidodecahedron
4.6.10
4.8.10 and 4.6.8
The truncated rhombicosidodecahedron can be seen in sequence of rectification and truncation operations from the icosidodecahedron. A further alternation step leads to the snub rhombicosidodecahedron.
Name Icosidodeca-
hedron
Rhomb-
icosidodeca-
hedron
Truncated rhomb-
icosidodeca-
hedron
Snub rhomb-
icosidodeca-
hedron
Coxeter ID (rD) rID (rrD) trID (trrD) srID (htrrD)
Conway aD aaD = eD taaD = baD saD
Image
Conway jD oD maD gaD
Dual
See also
• Expanded icosidodecahedron
• Truncated rhombicuboctahedron
References
1. Eppstein (1996)
• Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.
• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
External links
• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input
| Wikipedia |
Cyclotruncated simplectic honeycomb
In geometry, the cyclotruncated simplectic honeycomb (or cyclotruncated n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the symmetry of the ${\tilde {A}}_{n}$ affine Coxeter group. It is given a Schläfli symbol t0,1{3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with two adjacent nodes ringed. It is composed of n-simplex facets, along with all truncated n-simplices.
It is also called a Kagome lattice in two and three dimensions, although it is not a lattice.
In n-dimensions, each can be seen as a set of n+1 sets of parallel hyperplanes that divide space. Each hyperplane contains the same honeycomb of one dimension lower.
In 1-dimension, the honeycomb represents an apeirogon, with alternately colored line segments. In 2-dimensions, the honeycomb represents the trihexagonal tiling, with Coxeter graph . In 3-dimensions it represents the quarter cubic honeycomb, with Coxeter graph filling space with alternately tetrahedral and truncated tetrahedral cells. In 4-dimensions it is called a cyclotruncated 5-cell honeycomb, with Coxeter graph , with 5-cell, truncated 5-cell, and bitruncated 5-cell facets. In 5-dimensions it is called a cyclotruncated 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets. In 6-dimensions it is called a cyclotruncated 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets.
n ${\tilde {A}}_{n}$ Name
Coxeter diagram
Vertex figure Image and facets
1 ${\tilde {A}}_{1}$ Apeirogon
Yellow and cyan line segments
2 ${\tilde {A}}_{2}$ Trihexagonal tiling
Rectangle
With yellow and blue equilateral triangles,
and red hexagons
3 ${\tilde {A}}_{3}$ quarter cubic honeycomb
Elongated
triangular antiprism
With yellow and blue tetrahedra,
and red and purple truncated tetrahedra
4 ${\tilde {A}}_{4}$ Cyclotruncated 5-cell honeycomb
Elongated
tetrahedral antiprism
5-cell, truncated 5-cell,
bitruncated 5-cell
5 ${\tilde {A}}_{5}$ Cyclotruncated 5-simplex honeycomb
5-simplex, truncated 5-simplex,
bitruncated 5-simplex
6 ${\tilde {A}}_{6}$ Cyclotruncated 6-simplex honeycomb
6-simplex, truncated 6-simplex,
bitruncated 6-simplex, tritruncated 6-simplex
7 ${\tilde {A}}_{7}$ Cyclotruncated 7-simplex honeycomb
7-simplex, truncated 7-simplex,
bitruncated 7-simplex
8 ${\tilde {A}}_{8}$ Cyclotruncated 8-simplex honeycomb
8-simplex, truncated 8-simplex,
bitruncated 8-simplex, tritruncated 8-simplex,
quadritruncated 8-simplex
Projection by folding
The cyclotruncated (2n+1)- and 2n-simplex honeycombs and (2n-1)-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
${\tilde {A}}_{3}$ ${\tilde {A}}_{5}$ ${\tilde {A}}_{7}$ ${\tilde {A}}_{9}$ ${\tilde {A}}_{11}$ ...
${\tilde {A}}_{2}$ ${\tilde {A}}_{4}$ ${\tilde {A}}_{6}$ ${\tilde {A}}_{8}$ ${\tilde {A}}_{10}$ ...
${\tilde {A}}_{3}$ ${\tilde {A}}_{5}$ ${\tilde {A}}_{7}$ ${\tilde {A}}_{9}$ ...
${\tilde {C}}_{1}$ ${\tilde {C}}_{2}$ ${\tilde {C}}_{3}$ ${\tilde {C}}_{4}$ ${\tilde {C}}_{5}$ ...
See also
• Hypercubic honeycomb
• Alternated hypercubic honeycomb
• Quarter hypercubic honeycomb
• Simplectic honeycomb
• Omnitruncated simplectic honeycomb
References
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• 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] (1.9 Uniform space-fillings)
• (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 |
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