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Resource-dependent branching process
A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal populations, bacteria and others which reproduce in a biological sense, cascade process, or particles which split in a physical sense, and others. Members of a BP-population are called individuals, or particles.
If the times of reproductions are discrete (usually denoted by 1,2, ...) then the totality of individuals present at time n and living to time n+1 excluded are thought of as forming the nth generation. Simple BPs are defined by an initial state (number of individuals at time 0) and a law of reproduction, usually denoted by pk, k = 1,2,....
A resource-dependent branching process (RDBP) is a discrete-time BP which models the development of a population in which individuals are supposed to have to work in order to be able to live and to reproduce. The population decides on a society form which determines the rules how available resources are distributed among the individuals. For this purpose a RDBP should incorporate at least four additional model components, namely the individual demands for resources, the creation of new resources for the next generation, the notion of a policy to distribute resources, and a control option for individuals for interactions with the society.
Definition
A (discrete-time) resource-dependent branching process is a stochastic process Γ defined on the non-negative integers which is a BP defined by
• an initial state Γ0;
• a law of reproduction of individuals;
• a law of individual creation of resources;
• a law of individual resource demands (claims);
• a policy to distribute available resources to individuals which are present in the population
• a tool of interaction between individuals and the society.
History and objectives of RDBPs
RDBPs may be seen (in a wider sense) as so-called controlled branching processes. They were introduced by F. Thomas Bruss (1983)) with the objective to model different society structures and to compare the advantages and disadvantages of different forms of human societies. In these processes individuals have a means of interaction with the society which determines the rules how the current available resources should be distributed among them. This interaction (as e.g. in form of emigration) changes the effective rate of reproduction of the individuals remaining in the society. In that respect RDBPs have some parts in common with so-called population-size dependent BPs (see Klebaner (1984) and Klebaner & Jagers (2000)) in which the law of individual independent reproduction (see Galton-Watson process) is a function of the current population size.
Tractable RDBPs
Realistic models for human societies ask for a bisexual mode of reproduction whereas in the definition of an RDBP one simply speaks of a law of reproduction. However the notion of an average reproduction rate per individual (Bruss 1984) for bisexual processes shows that for all relevant questions for the long-term behavior of human societies it is justified for simplicity to assume asexual reproduction. This is why certain limiting results of Klebaner (1984) and Jagers & Klebaner (2000) bear over to RDBPs. Models for the development of a human society in time must allow for interdependence between the different components. Such models are in general very complicated and risk to become intractable. This led to the idea not to try to model the development of a society with a (single) realistic RDBP but rather by a sequence of control actions defining a sequence of relevant short-horizon RDBPs.
Two special policies stand out as guidelines for the development of any society. The two policies are the so-called weakest-first policy (wf-policy) and the so-called strongest-first policy (sf-policy).
Definition
The wf-policy is the rule to serve in each generation, as long as the accumulated resource space allows for it, with priority always the individuals with the smallest individual claims. The sf-policy is the rule to serve in each generation always with priority the largest individual resource claims, again as long as the accumulated resource space suffices. The societies adapting these policies strictly are called the wf-society, respectively the sf-society.
Survival criteria
In the theory of BPs it is of interest to know whether survival of a process is possible in the long run. For RDBPs this question depends also strongly on a feature on which individuals have a great influence, namely the policy to distribute resources.
Let:
m = mean reproduction (descendants) per individual
r = mean production (resource creation) per individual
F = the individual probability distribution of claims (resources)
Further suppose that all individuals which will not obtain their resource claim will either die or emigrate before reproduction. Then using results on expected stopping times for sums of order statistics (1991) the survival criteria can be explicitly computed for both the wf-society and the sf-society as a function of m, r and F.
The arguably strongest result known for RDBPs is the theorem of the envelopment of societies (Bruss and Duerinckx 2015). It says that, in the long run, any society which would like to survive and in which individuals prefer in general a higher standard of living to a lower one is bound to live in the long run between the wf-society and the sf-society. Intuition why this should be true, is wrong. The mathematical proof depends on the mentioned results on expected stopping times for sums of order statistics (1991) and fine-tuned balancing acts between model assumptions and different notions of Convergence of random variables.
See also
• Controlled branching process
• Bisexual Galton–Watson process
• Bruss–Duerinckx theorem
References
• Jagers, Peter (1975). Branching processes with biological applications. London: Wiley-Interscience [John Wiley & Sons].
• Bruss, F. Thomas (1983). "Resource-dependent branching processes". Stochastic Processes and Their Applications. 16: 36.
• Bruss, F. Thomas (1984). "A note on extinction criteria for bisexual Galton–Watson processes". Journal of Applied Probability. 21: 915–919. doi:10.2307/3213707.
• Klebaner, Fima C. (1984). "On population-size dependent branching processes". Advances in Applied Probability. 16: 30–55. doi:10.2307/1427223.
• Bruss, F. Thomas; Robertson, James (1991). "Wald's Lemma for the sum of order statistics of i.i.d. random variables". Advances in Applied Probability. 23: 612–623. doi:10.2307/1427625.
• Jagers, Peter; Klebaner, Fima C. (2000). "Population-size-dependent and age-dependent branching processes". Stochastic Processes and Their Applications. 87: 235–254. doi:10.1016/s0304-4149(99)00111-8.
• Bruss, F. Thomas; Duerinckx, Mitia (2015). "Resource–dependent branching processes and the envelope of societies". Annals of Applied Probability. 25: 324–372. arXiv:1212.0693. doi:10.1214/13-aap998.
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Resource-bounded measure
Lutz's resource-bounded measure is a generalisation of Lebesgue measure to complexity classes. It was originally developed by Jack Lutz. Just as Lebesgue measure gives a method to quantify the size of subsets of the Euclidean space $\mathbb {R} ^{n}$, resource bounded measure gives a method to classify the size of subsets of complexity classes.
For instance, computer scientists generally believe that the complexity class P (the set of all decision problems solvable in polynomial time) is not equal to the complexity class NP (the set of all decision problems checkable, but not necessarily solvable, in polynomial time). Since P is a subset of NP, this would mean that NP contains more problems than P. A stronger hypothesis than "P is not NP" is the statement "NP does not have p-measure 0". Here, p-measure is a generalization of Lebesgue measure to subsets of the complexity class E, in which P is contained. P is known to have p-measure 0, and so the hypothesis "NP does not have p-measure 0" would imply not only that NP and P are unequal, but that NP is, in a measure-theoretic sense, "much bigger than P".
Definition
$\{0,1\}^{\infty }$ is the set of all infinite, binary sequences. We can view a real number in the unit interval as an infinite binary sequence, by considering its binary expansion. We may also view a language (a set of binary strings) as an infinite binary sequence, by setting the nth bit of the sequence to 1 if and only if the nth binary string (in lexicographical order) is contained in the language. Thus, sets of real numbers in the unit interval and complexity classes (which are sets of languages) may both be viewed as sets of infinite binary sequences, and thus the techniques of measure theory used to measure the size of sets of real numbers may be applied to measure complexity classes. However, since each computable complexity class contains only a countable number of elements(because the number of computable languages is countable), each complexity class has Lebesgue measure 0. Thus, to do measure theory inside of complexity classes, we must define an alternative measure that works meaningfully on countable sets of infinite sequences. For this measure to be meaningful, it should reflect something about the underlying definition of each complexity class; namely, that they are defined by computational problems that can be solved within a given resource bound.
The foundation of resource-bounded measure is Ville's formulation of martingales. A martingale is a function $d:\{0,1\}^{*}\to [0,\infty )$ such that, for all finite strings w,
$d(w)={\frac {d(w0)+d(w1)}{2}}$.
(This is Ville's original definition of a martingale, later extended by Joseph Leo Doob.) A martingale d is said to succeed on a sequence $S\in \{0,1\}^{\infty }$ if $\limsup _{n\to \infty }d(S\upharpoonright n)=\infty ,$ where $S\upharpoonright n$ is the first n bits of S. A martingale succeeds on a set of sequences $X\subseteq \{0,1\}^{\infty }$ if it succeeds on every sequence in X.
Intuitively, a martingale is a gambler that starts with some finite amount of money (say, one dollar). It reads a sequence of bits indefinitely. After reading the finite prefix $w\in \{0,1\}^{*}$, it bets some of its current money that the next bit will be a 0, and the remainder of its money that the next bit will be a 1. It doubles whatever money was placed on the bit that appears next, and it loses the money placed on the bit that did not appear. It must bet all of its money, but it may "bet nothing" by placing half of its money on each bit. For a martingale d, d(w) represents the amount of money d has after reading the string w. Although the definition of a martingale has the martingale calculating how much money it will have, rather than calculating what bets to place, because of the constrained nature of the game, knowledge the values d(w), d(w0), and d(w1) suffices to calculate the bets that d placed on 0 and 1 after seeing the string w. The fact that the martingale is a function that takes as input the string seen so far means that the bets placed are solely a function of the bits already read; no other information may affect the bets (other information being the so-called filtration in the generalized theory of martingales).
The key result relating measure to martingales is Ville's observation that a set $X\subseteq \{0,1\}^{\infty }$ has Lebesgue measure 0 if and only if there is a martingale that succeeds on X. Thus, we can define a measure 0 set to be one for which there exists a martingale that succeeds on all elements of the set.
To extend this type of measure to complexity classes, Lutz considered restricting the computational power of the martingale. For instance, if instead of allowing any martingale, we require the martingale to be polynomial-time computable, then we obtain a definition of p-measure: a set of sequences has p-measure 0 if there is a polynomial-time computable martingale that succeeds on the set. We define a set to have p-measure 1 if its complement has p-measure 0. For example, proving the above-mentioned conjecture, that NP does not have p-measure 0, amounts to proving that no polynomial-time martingale succeeds on all of NP.
Almost complete
A problem is almost complete for a complexity class C if it is in C and "many" other problems in C reduce to it. More specifically, the subset of problems of C which reduce to the problem is a measure one set, in terms of the resource bounded measure. This is a weaker requirement than the problem being complete for the class.
References
• van Melkebeek, Dieter (2001), Randomness and Completeness in Computational Complexity, Springer, ISBN 3-540-41492-4, archived from the original on 2011-07-19
External links
• Resource-Bounded Measure Bibliography
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Resource monotonicity
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM principle has been studied in various division problems.[1]: 46–51 [2]
Allocating divisible resources
Single homogeneous resource, general utilities
Suppose society has $m$ units of some homogeneous divisible resource, such as water or flour. The resource should be divided among $n$ agents with different utilities. The utility of agent $i$ is represented by a function $u_{i}$; when agent $i$ receives $y_{i}$ units of resource, he derives from it a utility of $u_{i}(y_{i})$. Society has to decide how to divide the resource among the agents, i.e, to find a vector $y_{1},\dots ,y_{n}$ such that: $y_{1}+\cdots +y_{n}=m$.
Two classic allocation rules are the egalitarian rule - aiming to equalize the utilities of all agents (equivalently: maximize the minimum utility), and the utilitarian rule - aiming to maximize the sum of utilities.
The egalitarian rule is always RM:[1]: 47 when there is more resource to share, the minimum utility that can be guaranteed to all agents increases, and all agents equally share the increase. In contrast, the utilitarian rule might be not RM.
For example, suppose there are two agents, Alice and Bob, with the following utilities:
• $u_{A}(y_{A})=y_{A}^{2}$
• $u_{B}(y_{B})=y_{B}$
The egalitarian allocation is found by solving the equation: $y_{A}^{2}=(m-y_{A})$, which is equivalent to $m=y_{A}^{2}+y_{A}$, so $y_{A}$ is monotonically increasing with $m$. An equivalent equation is: $y_{B}=(m-y_{B})^{2}$, which is equivalent to $m={\sqrt {y_{B}}}+y_{B}$, so $y_{B}$ too is monotonically increasing with $m$. So in this example (as always) the egalitarian rule is RM.
In contrast, the utilitarian rule is not RM. This is because Alice has increasing returns: her marginal utility is small when she has few resources, but it increases fast when she has many resources. Hence, when the total amount of resource is small (specifically, $m<1$), the utilitarian sum is maximized when all resources are given to Bob; but when there are many resources ($m>1$), the maximum is attained when all resources are given to Alice. Mathematically, if $y$ is the amount given to Alice, then the utilitarian sum is $y^{2}+(m-y)$. This function has only an internal minimum point but not an internal maximum point; its maximum point in the range $[0,m]$ is attained in one of the endpoints. It is the left endpoint when $m<1$ and the right endpoint when $m>1$. In general, the utilitarian allocation rule is RM when all agents have diminishing returns, but it may be not RM when some agents have increasing returns (as in the example).[1]: 46–47
Thus, if society uses the utilitarian rule to allocate resources, then Bob loses value when the amount of resources increases. This is bad because it gives Bob an incentive against economic growth: Bob will try to keep the total amount small in order to keep his own share large.
Two complementary resources, Leontief utilities
Consider a cloud server with some units of RAM and CPU. There are two users with different types of tasks:
• The tasks of Alice need 1 unit of RAM and 2 units of CPU;
• The tasks of Bob need 2 units of RAM and 1 unit of CPU.
Thus, the utility functions (=number of tasks), denoting RAM by r and CPU by c, are Leontief utilities:
• $u_{A}(r,c)=\min(r,c/2)$
• $u_{B}(r,c)=\min(r/2,c)$
If the server has 12 RAM and 12 CPU, then both the utilitarian and the egalitarian allocations (and also the Nash-optimal, max-product allocation) are:
• $r_{A}=4,c_{A}=8\implies u_{A}=4$
• $r_{B}=8,c_{B}=4\implies u_{B}=4$
Now, suppose 12 more units of CPU become available. The egalitarian allocation does not change, but the utilitarian allocation now gives all resources to Alice:
• $r_{A}=12,c_{A}=24\implies u_{A}=12$
• $r_{B}=0,c_{B}=0\implies u_{B}=0$
so Bob loses value from the increase in resources.
The Nash-optimal (max-product) allocation becomes:
• $r_{A}=6,c_{A}=12\implies u_{A}=6$
• $r_{B}=6,c_{B}=3\implies u_{B}=3$
so Bob loses value here too, but the loss is less severe.[1]: 83–84
Cake cutting, additive utilities
In the fair cake-cutting problem, classic allocation rules such as divide and choose are not RM. Several rules are known to be RM:
• When the pieces may be disconnected, any allocation rule maximizing a concave welfare function of the absolute (not normalized) utilities is RM. In particular, the Nash-optimal rule, absolute-leximin rule and absolute-utilitarian rule are all RM. However, if the maximization uses the relative utilities (utilities divided by total cake value) then most of these rules are not RM; the only one that remains RM is the Nash-optimal rule.[3]
• When the pieces must be connected, no Pareto-optimal proportional division rule is RM. The absolute-equitable rule is weakly Pareto-optimal and RM, but not proportional. The relative-equitable rule is weakly Pareto-optimal and proportional, but not RM. The so-called rightmost mark rule, which is an variant of divide-and-choose, is proportional, weakly Pareto-optimal and RM - but it works only for two agents. It is an open question whether there exist division procedures that are both proportional and RM for three or more agents.[4]
Single-peaked preferences
Resource-monotonicity was studied in problems of fair division with single-peaked preferences.[5][6]
Allocating discrete items
Identical items, general utilities
The egalitarian rule (maximizing the leximin vector of utilities) might be not RM when the resource to divide consists of several indivisible (discrete) units.
For example,[1]: 82 suppose there are $m$ tennis rackets. Alice gets a utility of 1 whenever she has a racket, since she enjoys playing against the wall. But Bob and Carl get a utility of 1 only when they have two rackets, since they only enjoy playing against each other or against Alice. Hence, if there is only one racket, the egalitarian rule gives it entirely to Alice. But if there are two rackets, they are divided equally between the agents (each agent gets a racket for 2/3 of the time). Hence, Alice loses utility when the total amount of rackets increases. Alice has an incentive to oppose growth.
Different items, additive utilities
In the fair item allocation problem, classic allocation procedures such as adjusted winner and envy-graph are not RM. Moreover, even the Nash-optimal rule, which is RM in cake-cutting, is not RM in item allocation. In contrast, round-robin item allocation is RM. Moreover, round-robin can be adapted to yield picking sequences appropriate for agents with different entitlements; all these picking sequences are RM too.[7]
Identical items, additive utilities
The special case in which all items are identical and each agent's utility is simply the number of items he receives is known as apportionment. It originated from the task of allocating seats in a parliament among states or among parties. Therefore, it is often called house monotonicity.
Facility location
Facility location is the social choice question is where a certain facility should be located. Consider the following network of roads, where the letters denote junctions and the numbers denote distances:
A---6---B--5--C--5--D---6---E
The population is distributed uniformly along the roads. People want to be as close as possible to the facility, so they have "dis-utility" (negative utility) measured by their distance to the facility.
In the initial situation, the egalitarian rule locates the facility at C, since it minimizes the maximum distance to the facility, which is 11 (the utilitarian and Nash rules also locate the facility at C).
Now, there is a new junction X and some new roads (the previous roads do not change):
B--3--X--3--D
..........|.........
..........4.........
..........|.........
..........C.........
The egalitarian rule now locates the facility at X, since it allows to decrease the maximum distance from 11 to 9 (the utilitarian and Nash rules also locate the facility at X).
The increase in resources helped most people, but decreased the utility of those living in or near C.[1]: 84–85
Bargaining
A monotonicity axiom closely related to resource-monotonicity appeared first in the context of the bargaining problem. A bargaining problem is defined by a set of alternatives; a bargaining solution should select a single alternative from the set, subject to some axioms. The resource-monotonicity axiom was presented in two variants:
1. "If, for every utility level that player 1 may demand, the maximum feasible utility level that player 2 can simultaneously reach is increased, then the utility level assigned to player 2 according to the solution should also be increased". This axiom leads to a characterization of the Kalai–Smorodinsky bargaining solution.
2. "Let T and S be bargaining games; if T contains S then for all agents, the utility in T is weakly larger than the utility in S". In other words, if the set of alternatives grows, the selected solution should be at least as good for all agents as the previous solution. This axiom, in addition to Pareto optimality and symmetry and Independence of irrelevant alternatives, leads to a characterization of the egalitarian bargaining solution.[8]
See also
• Throw away paradox
[9] [10] [11] [12] [13] [14] [15] [16]
References
1. Herve Moulin (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press. ISBN 9780262134231.
2. Thomson, William (2011). Fair Allocation Rules. Handbook of Social Choice and Welfare. Vol. 2. pp. 393–506. doi:10.1016/s0169-7218(10)00021-3. ISBN 9780444508942.
3. Segal-Halevi, Erel; Sziklai, Balázs R. (2019-09-01). "Monotonicity and competitive equilibrium in cake-cutting". Economic Theory. 68 (2): 363–401. arXiv:1510.05229. doi:10.1007/s00199-018-1128-6. ISSN 1432-0479. S2CID 179618.
4. Segal-Halevi, Erel; Sziklai, Balázs R. (2018-09-01). "Resource-monotonicity and population-monotonicity in connected cake-cutting". Mathematical Social Sciences. 95: 19–30. arXiv:1703.08928. doi:10.1016/j.mathsocsci.2018.07.001. ISSN 0165-4896. S2CID 16282641.
5. Thomson, William (1994). "Resource-monotonic solutions to the problem of fair division when preferences are single-peaked". Social Choice and Welfare. 11 (3). doi:10.1007/bf00193807. S2CID 122306487.
6. Thomson, William (1997). "The Replacement Principle in Economies with Single-Peaked Preferences". Journal of Economic Theory. 76: 145–168. doi:10.1006/jeth.1997.2294.
7. Chakraborty, Mithun; Schmidt-Kraepelin, Ulrike; Suksompong, Warut (2021-04-29). "Picking sequences and monotonicity in weighted fair division". Artificial Intelligence. 301: 103578. arXiv:2104.14347. doi:10.1016/j.artint.2021.103578. S2CID 233443832.
8. Kalai, Ehud (1977). "Proportional solutions to bargaining situations: Intertemporal utility comparisons" (PDF). Econometrica. 45 (7): 1623–1630. doi:10.2307/1913954. JSTOR 1913954.
9. Mantel, Rolf R. (1984). "Substitutability and the welfare effects of endowment increases". Journal of International Economics. 17 (3–4): 325–334. doi:10.1016/0022-1996(84)90027-8.
10. Moulin, Hervé (1992). "Welfare bounds in the cooperative production problem". Games and Economic Behavior. 4 (3): 373–401. doi:10.1016/0899-8256(92)90045-t.
11. Polterovich, V.M.; Spivak, V.A. (1983). "Gross substitutability of point-to-set correspondences". Journal of Mathematical Economics. 11 (2): 117. doi:10.1016/0304-4068(83)90032-0.
12. Sobel, Joel (1979). "Fair allocations of a renewable resource". Journal of Economic Theory. 21 (2): 235–248. CiteSeerX 10.1.1.394.9698. doi:10.1016/0022-0531(79)90029-2.
13. Moulin, Hervé; Thomson, William (1988). "Can everyone benefit from growth?". Journal of Mathematical Economics. 17 (4): 339. doi:10.1016/0304-4068(88)90016-x.
14. Moulin, Herve (1992). "An Application of the Shapley Value to Fair Division with Money". Econometrica. 60 (6): 1331–1349. doi:10.2307/2951524. JSTOR 2951524.
15. Moulin, H. (1990). "Fair division under joint ownership: Recent results and open problems". Social Choice and Welfare. 7 (2): 149–170. doi:10.1007/bf01560582. S2CID 154300207.
16. Moulin, Hervé (1991). "Welfare bounds in the fair division problem". Journal of Economic Theory. 54 (2): 321–337. doi:10.1016/0022-0531(91)90125-n.
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Lorenzo Respighi
Lorenzo Respighi (7 October 1824 – 10 December 1889) was an Italian mathematician and natural philosopher. Born at Cortemaggiore, Piacenza, to Luigi Respighi and Giuseppina Rossetti. He studied mathematics and natural philosophy, first at Parma and then at the University of Bologna, where he obtained his degree ad honorem in 1845. From 1855 to 1864 he was director of the Astronomic Observatory of Bologna, and during these years he discovered three comets, #1862 IV, #1863 III and #1863 V. In 1865 he was nominated director of the Astronomic Observatory of the Campidoglio, in Rome.
Lorenzo Respighi
Born(1824-10-07)7 October 1824
Cortemaggiore, Italy
Died10 December 1889(1889-12-10) (aged 65)
Rome, Italy
NationalityItalian
Scientific career
FieldsMathematics
The crater Respighi on the Moon is named after him.
Sources
• Herbermann, Charles, ed. (1913). "Lorenzo Respighi" . Catholic Encyclopedia. New York: Robert Appleton Company.
This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "Lorenzo Respighi". Catholic Encyclopedia. New York: Robert Appleton Company.
Authority control
International
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National
• Germany
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Responsive set extension
In utility theory, the responsive set (RS) extension is an extension of a preference-relation on individual items, to a partial preference-relation of item-bundles.
Example
Suppose there are four items: $w,x,y,z$. A person states that he ranks the items according to the following total order:
$w\prec x\prec y\prec z$
(i.e., z is his best item, then y, then x, then w). Assuming the items are independent goods, one can deduce that:
$\{w,x\}\prec \{y,z\}$ – the person prefers his two best items to his two worst items;
$\{w,y\}\prec \{x,z\}$ – the person prefers his best and third-best items to his second-best and fourth-best items.
But, one cannot deduce anything about the bundles $\{w,z\},\{x,y\}$; we do not know which of them the person prefers.
The RS extension of the ranking $w\prec x\prec y\prec z$ is a partial order on the bundles of items, that includes all relations that can be deduced from the item-ranking and the independence assumption.
Definitions
Let $O$ be a set of objects and $\preceq $ a total order on $O$.
The RS extension of $\preceq $ is a partial order on $2^{O}$. It can be defined in several equivalent ways.[1]
Responsive set (RS)
The original RS extension[2]: 44–48 is constructed as follows. For every bundle $X\subseteq O$, every item $x\in X$ and every item $y\notin X$, take the following relations:
• $X\setminus \{x\}\prec ^{RS}X$ (- adding an item improves the bundle)
• If $x\preceq y$ then $X\preceq ^{RS}(X\setminus \{x\})\cup \{y\}$ (- replacing an item with a better item improves the bundle).
The RS extension is the transitive closure of these relations.
Pairwise dominance (PD)
The PD extension is based on a pairing of the items in one bundle with the items in the other bundle.
Formally, $X\preceq ^{PD}Y$ if-and-only-if there exists an Injective function $f$ from $X$ to $Y$ such that, for each $x\in X$, $x\preceq f(x)$.
Stochastic dominance (SD)
The SD extension (named after stochastic dominance) is defined not only on discrete bundles but also on fractional bundles (bundles that contains fractions of items). Informally, a bundle Y is SD-preferred to a bundle X if, for each item z, the bundle Y contains at least as many objects, that are at least as good as z, as the bundle X.
Formally, $X\preceq ^{SD}Y$ iff, for every item $z$:
$\sum _{x\succeq z}X[x]\leq \sum _{y\succeq z}Y[y]$
where $X[x]$ is the fraction of item $x$ in the bundle $X$.
If the bundles are discrete, the definition has a simpler form. $X\preceq ^{SD}Y$ iff, for every item $z$:
$|\{x\in X|x\succeq z\}|\leq |\{y\in Y|y\succeq z\}|$
Additive utility (AU)
The AU extension is based on the notion of an additive utility function.
Many different utility functions are compatible with a given ordering. For example, the order $w\prec x\prec y\prec z$ is compatible with the following utility functions:
$u_{1}(w)=0,u_{1}(x)=2,u_{1}(y)=4,u_{1}(z)=7$
$u_{2}(w)=0,u_{2}(x)=2,u_{2}(y)=4,u_{2}(z)=5$
Assuming the items are independent, the utility function on bundles is additive, so the utility of a bundle is the sum of the utilities of its items, for example:
$u_{1}(\{w,x\})=2,u_{1}(\{w,z\})=7,u_{1}(\{x,y\})=6$
$u_{2}(\{w,x\})=2,u_{2}(\{w,z\})=5,u_{2}(\{x,y\})=6$
The bundle $\{w,x\}$ has less utility than $\{w.z\}$ according to both utility functions. Moreover, for every utility function $u$ compatible with the above ranking:
$u(\{w,x\})<u(\{w,z\})$.
In contrast, the utility of the bundle $\{w,z\}$ can be either less or more than the utility of $\{x,y\}$.
This motivates the following definition:
$X\preceq ^{AU}Y$ iff, for every additive utility function $u$ compatible with $\preceq $:
$u(X)\leq u(Y)$
Equivalence
• $X\preceq ^{SD}Y$ implies $X\preceq ^{RS}Y$.[1]
• $X\preceq ^{RS}Y$ and $X\preceq ^{PD}Y$ are equivalent.[1]
• $X\preceq ^{PD}Y$ implies $X\preceq ^{AU}Y$. Proof: If $X\preceq ^{PD}Y$, then there is an injection $f:X\to Y$ such that, for all $x\in X$, $x\preceq f(x)$. Therefore, for every utility function $u$ compatible with $\preceq $, $u(x)\leq u(f(x))$. Therefore, if $u$ is additive, then $u(X)\leq u(Y)$.[1]
• It is known that $\preceq ^{AU}$ and $\preceq ^{SD}$ are equivalent, see e.g.[3]
Therefore, the four extensions $\preceq ^{RS}$ and $\preceq ^{PD}$ and $\preceq ^{SD}$ and $\preceq ^{AU}$ are all equivalent.
Responsive orders and valuations
A total order on bundles is called responsive[4]: 287–288 if it is contains the responsive-set-extension of some total order on items. I.e., it contains all the relations that are implied by the underlying ordering of the items, and adds some more relations that are not implied nor contradicted.
Similarly, a utility function on bundles is called responsive if it induces a responsive order. To be more explicit,[5] a utility function u is responsive if for every bundle X and every two items y,z that are not in X: $u(y)\geq u(z)\implies u(X\cup \{y\})\geq u(X\cup \{z\})$.
Responsiveness is implied by additivity, but not vice versa:
• If a total order is additive (represented by an additive function) then by definition it contains the AU extension $\preceq ^{AU}$, which is equivalent to $\preceq ^{RS}$, so it is responsive. Similarly, if a utility function is additive, then $u(X\cup \{y\})-u(X\cup \{z\})=u(y)-u(z)$, so responsiveness is satisfied.
• On the other hand, a total order may responsive but not additive: it may contain the AU extension which is consistent with all additive functions, but may also contain other relations that are inconsistent with a single additive function.
For example,[6] suppose there are four items with $w\prec x\prec y\prec z$. Responsiveness constrains only the relation between bundles of the same size with one item replaced, or bundles of different sizes where the small is contained in the large. It says nothing about bundles of different sizes that are not subsets of each other. So, for example, a responsive order can have both $\{z\}\prec \{x,y\}$ and $\{w,z\}\succ \{w,x,y\}$. But this is incompatible with additivity: there is no additive function for which $u(\{z\})<u(\{x,y\})$ while $u(\{w,z\})>u(\{w,x,y\})$.
See also
• Weakly additive
• Picking sequence
References
1. Aziz, Haris; Gaspers, Serge; MacKenzie, Simon; Walsh, Toby (2015). "Fair assignment of indivisible objects under ordinal preferences". Artificial Intelligence. 227: 71–92. arXiv:1312.6546. doi:10.1016/j.artint.2015.06.002. S2CID 1408197.
2. Barberà, S., Bossert, W., Pattanaik, P. K. (2004). "Ranking sets of objects." (PDF). Handbook of utility theory. Springer US.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Katta, Akshay-Kumar; Sethuraman, Jay (2006). "A solution to the random assignment problem on the full preference domain". Journal of Economic Theory. 131 (1): 231. doi:10.1016/j.jet.2005.05.001.
4. Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432. (free online version)
5. Kyropoulou, Maria; Suksompong, Warut; Voudouris, Alexandros A. (2020-11-12). "Almost envy-freeness in group resource allocation". Theoretical Computer Science. 841: 110–123. doi:10.1016/j.tcs.2020.07.008. ISSN 0304-3975. S2CID 59222796.
6. Moshe, Babaioff; Noam, Nisan; Inbal, Talgam-Cohen (2017-03-23). "Competitive Equilibrium with Indivisible Goods and Generic Budgets". arXiv:1703.08150. Bibcode:2017arXiv170308150B. {{cite journal}}: Cite journal requires |journal= (help)
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Restricted root system
In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group.
Lie groups and Lie algebras
Classical groups
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
Simple Lie groups
Classical
• An
• Bn
• Cn
• Dn
Exceptional
• G2
• F4
• E6
• E7
• E8
Other Lie groups
• Circle
• Lorentz
• Poincaré
• Conformal group
• Diffeomorphism
• Loop
• Euclidean
Lie algebras
• Lie group–Lie algebra correspondence
• Exponential map
• Adjoint representation
• Killing form
• Index
• Simple Lie algebra
• Loop algebra
• Affine Lie algebra
Semisimple Lie algebra
• Dynkin diagrams
• Cartan subalgebra
• Root system
• Weyl group
• Real form
• Complexification
• Split Lie algebra
• Compact Lie algebra
Representation theory
• Lie group representation
• Lie algebra representation
• Representation theory of semisimple Lie algebras
• Representations of classical Lie groups
• Theorem of the highest weight
• Borel–Weil–Bott theorem
Lie groups in physics
• Particle physics and representation theory
• Lorentz group representations
• Poincaré group representations
• Galilean group representations
Scientists
• Sophus Lie
• Henri Poincaré
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Claude Chevalley
• Harish-Chandra
• Armand Borel
• Glossary
• Table of Lie groups
See also
• Satake diagram
References
• Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics, vol. 225, Springer, ISBN 0387211543
• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0821828487
• Onishchik, A. L.; Vinberg, E. B. (1994), Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, vol. 41, Springer, ISBN 9783540546832
• Wolf, Joseph A. (2010), Spaces of constant curvature, AMS Chelsea Publishing (6th ed.), American Mathematical Society, ISBN 0821852825
• Wolf, Joseph A. (1972), "Fine structure of Hermitian symmetric spaces", in Boothby, William; Weiss, Guido (eds.), Symmetric spaces (Short Courses, Washington University), Pure and Applied Mathematics, vol. 8, Dekker, pp. 271–357, ISBN 978-0-608-30568-4
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Restricted isometry property
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence Tao[1] and is used to prove many theorems in the field of compressed sensing.[2] There are no known large matrices with bounded restricted isometry constants (computing these constants is strongly NP-hard,[3] and is hard to approximate as well[4]), but many random matrices have been shown to remain bounded. In particular, it has been shown that with exponentially high probability, random Gaussian, Bernoulli, and partial Fourier matrices satisfy the RIP with number of measurements nearly linear in the sparsity level.[5] The current smallest upper bounds for any large rectangular matrices are for those of Gaussian matrices.[6] Web forms to evaluate bounds for the Gaussian ensemble are available at the Edinburgh Compressed Sensing RIC page.[7]
Definition
Let A be an m × p matrix and let 1 ≤ s ≤ p be an integer. Suppose that there exists a constant $\delta _{s}\in (0,1)$ such that, for every m × s submatrix As of A and for every s-dimensional vector y,
$(1-\delta _{s})\|y\|_{2}^{2}\leq \|A_{s}y\|_{2}^{2}\leq (1+\delta _{s})\|y\|_{2}^{2}.\,$
Then, the matrix A is said to satisfy the s-restricted isometry property with restricted isometry constant $\delta _{s}$.
This condition is equivalent to the statement that for every m × s submatrix As of A we have
$\|A_{s}^{*}A_{s}-I_{s\times s}\|_{2\to 2}\leq \delta _{s},$
where $I_{s\times s}$ is the $s\times s$ identity matrix and $\|X\|_{2\to 2}$ is the operator norm. See for example [8] for a proof.
Finally this is equivalent to stating that all eigenvalues of $A_{s}^{*}A_{s}$ are in the interval $[1-\delta _{s},1+\delta _{s}]$.
Restricted Isometric Constant (RIC)
The RIC Constant is defined as the infimum of all possible $\delta $ for a given $A\in \mathbb {R} ^{n\times m}$.
$\delta _{K}=\inf \left[\delta :(1-\delta )\|y\|_{2}^{2}\leq \|A_{s}y\|_{2}^{2}\leq (1+\delta )\|y\|_{2}^{2}\right],\ \forall |s|\leq K,\forall y\in R^{|s|}$ :(1-\delta )\|y\|_{2}^{2}\leq \|A_{s}y\|_{2}^{2}\leq (1+\delta )\|y\|_{2}^{2}\right],\ \forall |s|\leq K,\forall y\in R^{|s|}}
It is denoted as $\delta _{K}$.
Eigenvalues
For any matrix that satisfies the RIP property with a RIC of $\delta _{K}$, the following condition holds:[1]
$1-\delta _{K}\leq \lambda _{min}(A_{\tau }^{*}A_{\tau })\leq \lambda _{max}(A_{\tau }^{*}A_{\tau })\leq 1+\delta _{K}$.
The tightest upper bound on the RIC can be computed for Gaussian matrices. This can be achieved by computing the exact probability that all the eigenvalues of Wishart matrices lie within an interval.
See also
• Compressed sensing
• Mutual coherence (linear algebra)
• Terence Tao's website on compressed sensing lists several related conditions, such as the 'Exact reconstruction principle' (ERP) and 'Uniform uncertainty principle' (UUP)[9]
• Nullspace property, another sufficient condition for sparse recovery
• Generalized restricted isometry property,[10] a generalized sufficient condition for sparse recovery, where mutual coherence and restricted isometry property are both its special forms.
• Johnson-Lindenstrauss lemma
References
1. E. J. Candes and T. Tao, "Decoding by Linear Programming," IEEE Trans. Inf. Th., 51(12): 4203–4215 (2005).
2. E. J. Candes, J. K. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Communications on Pure and Applied Mathematics, Vol. LIX, 1207–1223 (2006).
3. A. M. Tillmann and M. E. Pfetsch, "The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing," IEEE Trans. Inf. Th., 60(2): 1248–1259 (2014)
4. Abhiram Natarajan and Yi Wu, "Computational Complexity of Certifying Restricted Isometry Property," Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014) (2014)
5. F. Yang, S. Wang, and C. Deng, "Compressive sensing of image reconstruction using multi-wavelet transform", IEEE 2010
6. B. Bah and J. Tanner "Improved Bounds on Restricted Isometry Constants for Gaussian Matrices"
7. "Edinburgh University - School of Mathematics - Compressed Sensing Group - Restricted Isometry Constants". Archived from the original on 2010-04-27. Retrieved 2010-03-31.
8. "A Mathematical Introduction to Compressive Sensing" (PDF). Cis.pku.edu.cn. Retrieved 15 May 2018.
9. "Compressed sensing". Math.ucla.edu. Retrieved 15 May 2018.
10. Yu Wang, Jinshan Zeng, Zhimin Peng, Xiangyu Chang and Zongben Xu (2015). "On Linear Convergence of Adaptively Iterative Thresholding Algorithms for Compressed Sensing". IEEE Transactions on Signal Processing. 63 (11): 2957–2971. arXiv:1408.6890. Bibcode:2015ITSP...63.2957W. doi:10.1109/TSP.2015.2412915. S2CID 10734058.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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Restricted Lie algebra
In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."
Definition
Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map $X\mapsto X^{[p]}$ satisfying
• $\mathrm {ad} (X^{[p]})=\mathrm {ad} (X)^{p}$ for all $X\in L$,
• $(tX)^{[p]}=t^{p}X^{[p]}$ for all $t\in k,X\in L$,
• $(X+Y)^{[p]}=X^{[p]}+Y^{[p]}+\sum _{i=1}^{p-1}{\frac {s_{i}(X,Y)}{i}}$, for all $X,Y\in L$, where $s_{i}(X,Y)$ is the coefficient of $t^{i-1}$ in the formal expression $\mathrm {ad} (tX+Y)^{p-1}(X)$.
If the characteristic of k is 0, then L is a restricted Lie algebra where the p operation is the identity map.
Examples
For any associative algebra A defined over a field of characteristic p, the bracket operation $[X,Y]:=XY-YX$ and p operation $X^{[p]}:=X^{p}$ make A into a restricted Lie algebra $\mathrm {Lie} (A)$.
Let G be an algebraic group over a field k of characteristic p, and $\mathrm {Lie} (G)$ be the Zariski tangent space at the identity element of G. Each element of $\mathrm {Lie} (G)$ uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on $\mathrm {Lie} (G)$ just as in the Lie group case. If p>0, the Frobenius map $x\mapsto x^{p}$ defines a p operation on $\mathrm {Lie} (G)$.
Restricted universal enveloping algebra
The functor $A\mapsto \mathrm {Lie} (A)$ has a left adjoint $L\mapsto U^{[p]}(L)$ called the restricted universal enveloping algebra. To construct this, let $U(L)$ be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form $x^{p}-x^{[p]}$, we set $U^{[p]}(L)=U(L)/I$. It satisfies a form of the PBW theorem.
See also
Restricted Lie algebras are used in Jacobson's Galois correspondence for purely inseparable extensions of fields of exponent 1.
References
• Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Springer-Verlag, Zbl 0726.20030.
• Block, Richard E.; Wilson, Robert Lee (1988), "Classification of the restricted simple Lie algebras", Journal of Algebra, 114 (1): 115–259, doi:10.1016/0021-8693(88)90216-5, ISSN 0021-8693, MR 0931904.
• Montgomery, Susan (1993), Hopf algebras and their actions on rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992, Regional Conference Series in Mathematics, vol. 82, Providence, RI: American Mathematical Society, p. 23, ISBN 978-0-8218-0738-5, Zbl 0793.16029.
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Restricted product
In mathematics, the restricted product is a construction in the theory of topological groups.
Let $I$ be an index set; $S$ a finite subset of $I$. If $G_{i}$ is a locally compact group for each $i\in I$, and $K_{i}\subset G_{i}$ is an open compact subgroup for each $i\in I\setminus S$, then the restricted product
$\prod _{i}\nolimits 'G_{i}\,$
is the subset of the product of the $G_{i}$'s consisting of all elements $(g_{i})_{i\in I}$ such that $g_{i}\in K_{i}$ for all but finitely many $i\in I\setminus S$.
This group is given the topology whose basis of open sets are those of the form
$\prod _{i}A_{i}\,,$
where $A_{i}$ is open in $G_{i}$ and $A_{i}=K_{i}$ for all but finitely many $i$.
One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field.
See also
• Direct sum
References
• Fröhlich, A.; Cassels, J. W. (1967), Algebraic number theory, Boston, MA: Academic Press, ISBN 978-0-12-163251-9
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
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Restriction (mathematics)
In mathematics, the restriction of a function $f$ is a new function, denoted $f\vert _{A}$ or $f{\upharpoonright _{A}},$ obtained by choosing a smaller domain $A$ for the original function $f.$ The function $f$ is then said to extend $f\vert _{A}.$
Function
x ↦ f (x)
Examples of domains and codomains
• $X$ → $\mathbb {B} $, $\mathbb {B} $ → $X$, $\mathbb {B} ^{n}$ → $X$
• $X$ → $\mathbb {Z} $, $\mathbb {Z} $ → $X$
• $X$ → $\mathbb {R} $, $\mathbb {R} $ → $X$, $\mathbb {R} ^{n}$ → $X$
• $X$ → $\mathbb {C} $, $\mathbb {C} $ → $X$, $\mathbb {C} ^{n}$ → $X$
Classes/properties
• Constant
• Identity
• Linear
• Polynomial
• Rational
• Algebraic
• Analytic
• Smooth
• Continuous
• Measurable
• Injective
• Surjective
• Bijective
Constructions
• Restriction
• Composition
• λ
• Inverse
Generalizations
• Partial
• Multivalued
• Implicit
• space
Formal definition
Let $f:E\to F$ be a function from a set $E$ to a set $F.$ If a set $A$ is a subset of $E,$ then the restriction of $f$ to $A$ is the function[1]
${f|}_{A}:A\to F$
given by ${f|}_{A}(x)=f(x)$ for $x\in A.$ Informally, the restriction of $f$ to $A$ is the same function as $f,$ but is only defined on $A$.
If the function $f$ is thought of as a relation $(x,f(x))$ on the Cartesian product $E\times F,$ then the restriction of $f$ to $A$ can be represented by its graph $G({f|}_{A})=\{(x,f(x))\in G(f):x\in A\}=G(f)\cap (A\times F),$ where the pairs $(x,f(x))$ represent ordered pairs in the graph $G.$
Extensions
A function $F$ is said to be an extension of another function $f$ if whenever $x$ is in the domain of $f$ then $x$ is also in the domain of $F$ and $f(x)=F(x).$ That is, if $\operatorname {domain} f\subseteq \operatorname {domain} F$ and $F{\big \vert }_{\operatorname {domain} f}=f.$
A linear extension (respectively, continuous extension, etc.) of a function $f$ is an extension of $f$ that is also a linear map (respectively, a continuous map, etc.).
Examples
1. The restriction of the non-injective function$f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto x^{2}$ to the domain $\mathbb {R} _{+}=[0,\infty )$ is the injection$f:\mathbb {R} _{+}\to \mathbb {R} ,\ x\mapsto x^{2}.$
2. The factorial function is the restriction of the gamma function to the positive integers, with the argument shifted by one: ${\Gamma |}_{\mathbb {Z} ^{+}}\!(n)=(n-1)!$
Properties of restrictions
• Restricting a function $f:X\rightarrow Y$ to its entire domain $X$ gives back the original function, that is, $f|_{X}=f.$
• Restricting a function twice is the same as restricting it once, that is, if $A\subseteq B\subseteq \operatorname {dom} f,$ then $\left(f|_{B}\right)|_{A}=f|_{A}.$
• The restriction of the identity function on a set $X$ to a subset $A$ of $X$ is just the inclusion map from $A$ into $X.$[2]
• The restriction of a continuous function is continuous.[3][4]
Applications
Inverse functions
Main article: Inverse function
For a function to have an inverse, it must be one-to-one. If a function $f$ is not one-to-one, it may be possible to define a partial inverse of $f$ by restricting the domain. For example, the function
$f(x)=x^{2}$
defined on the whole of $\mathbb {R} $ is not one-to-one since $x^{2}=(-x)^{2}$ for any $x\in \mathbb {R} .$ However, the function becomes one-to-one if we restrict to the domain $\mathbb {R} _{\geq 0}=[0,\infty ),$ in which case
$f^{-1}(y)={\sqrt {y}}.$
(If we instead restrict to the domain $(-\infty ,0],$ then the inverse is the negative of the square root of $y.$) Alternatively, there is no need to restrict the domain if we allow the inverse to be a multivalued function.
Selection operators
In relational algebra, a selection (sometimes called a restriction to avoid confusion with SQL's use of SELECT) is a unary operation written as $\sigma _{a\theta b}(R)$ or $\sigma _{a\theta v}(R)$ where:
• $a$ and $b$ are attribute names,
• $\theta $ is a binary operation in the set $\{<,\leq ,=,\neq ,\geq ,>\},$
• $v$ is a value constant,
• $R$ is a relation.
The selection $\sigma _{a\theta b}(R)$ selects all those tuples in $R$ for which $\theta $ holds between the $a$ and the $b$ attribute.
The selection $\sigma _{a\theta v}(R)$ selects all those tuples in $R$ for which $\theta $ holds between the $a$ attribute and the value $v.$
Thus, the selection operator restricts to a subset of the entire database.
The pasting lemma
Main article: Pasting lemma
The pasting lemma is a result in topology that relates the continuity of a function with the continuity of its restrictions to subsets.
Let $X,Y$ be two closed subsets (or two open subsets) of a topological space $A$ such that $A=X\cup Y,$ and let $B$ also be a topological space. If $f:A\to B$ is continuous when restricted to both $X$ and $Y,$ then $f$ is continuous.
This result allows one to take two continuous functions defined on closed (or open) subsets of a topological space and create a new one.
Sheaves
Main article: Sheaf theory
Sheaves provide a way of generalizing restrictions to objects besides functions.
In sheaf theory, one assigns an object $F(U)$ in a category to each open set $U$ of a topological space, and requires that the objects satisfy certain conditions. The most important condition is that there are restriction morphisms between every pair of objects associated to nested open sets; that is, if $V\subseteq U,$ then there is a morphism $\operatorname {res} _{V,U}:F(U)\to F(V)$ satisfying the following properties, which are designed to mimic the restriction of a function:
• For every open set $U$ of $X,$ the restriction morphism $\operatorname {res} _{U,U}:F(U)\to F(U)$ is the identity morphism on $F(U).$
• If we have three open sets $W\subseteq V\subseteq U,$ then the composite $\operatorname {res} _{W,V}\circ \operatorname {res} _{V,U}=\operatorname {res} _{W,U}.$
• (Locality) If $\left(U_{i}\right)$ is an open covering of an open set $U,$ and if $s,t\in F(U)$ are such that $s{\big \vert }_{U_{i}}=t{\big \vert }_{U_{i}}$s|Ui = t|Ui for each set $U_{i}$ of the covering, then $s=t$; and
• (Gluing) If $\left(U_{i}\right)$ is an open covering of an open set $U,$ and if for each $i$ a section $x_{i}\in F\left(U_{i}\right)$ is given such that for each pair $U_{i},U_{j}$ of the covering sets the restrictions of $s_{i}$ and $s_{j}$ agree on the overlaps: $s_{i}{\big \vert }_{U_{i}\cap U_{j}}=s_{j}{\big \vert }_{U_{i}\cap U_{j}},$ then there is a section $s\in F(U)$ such that $s{\big \vert }_{U_{i}}=s_{i}$ for each $i.$
The collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf.
Left- and right-restriction
More generally, the restriction (or domain restriction or left-restriction) $A\triangleleft R$ of a binary relation $R$ between $E$ and $F$ may be defined as a relation having domain $A,$ codomain $F$ and graph $G(A\triangleleft R)=\{(x,y)\in F(R):x\in A\}.$ Similarly, one can define a right-restriction or range restriction $R\triangleright B.$ Indeed, one could define a restriction to $n$-ary relations, as well as to subsets understood as relations, such as ones of the Cartesian product $E\times F$ for binary relations. These cases do not fit into the scheme of sheaves.
Anti-restriction
The domain anti-restriction (or domain subtraction) of a function or binary relation $R$ (with domain $E$ and codomain $F$) by a set $A$ may be defined as $(E\setminus A)\triangleleft R$; it removes all elements of $A$ from the domain $E.$ It is sometimes denoted $A$ ⩤ $R.$[5] Similarly, the range anti-restriction (or range subtraction) of a function or binary relation $R$ by a set $B$ is defined as $R\triangleright (F\setminus B)$; it removes all elements of $B$ from the codomain $F.$ It is sometimes denoted $R$ ⩥ $B.$
See also
• Constraint – Condition of an optimization problem which the solution must satisfy
• Deformation retract – Continuous, position-preserving mapping from a topological space into a subspacePages displaying short descriptions of redirect targets
• Local property – property which occurs on sufficiently small or arbitrarily small neighborhoods of pointsPages displaying wikidata descriptions as a fallback
• Function (mathematics) § Restriction and extension
• Binary relation § Restriction
• Relational algebra § Selection (σ)
References
1. Stoll, Robert (1974). Sets, Logic and Axiomatic Theories (2nd ed.). San Francisco: W. H. Freeman and Company. pp. [36]. ISBN 0-7167-0457-9.
2. Halmos, Paul (1960). Naive Set Theory. Princeton, NJ: D. Van Nostrand. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
3. Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River: Prentice Hall. ISBN 0-13-181629-2.
4. Adams, Colin Conrad; Franzosa, Robert David (2008). Introduction to Topology: Pure and Applied. Pearson Prentice Hall. ISBN 978-0-13-184869-6.
5. Dunne, S. and Stoddart, Bill Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 5–7, 2006, Revised Selected ... Computer Science and General Issues). Springer (2006)
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Results in Mathematics
Results in Mathematics/Resultate der Mathematik is a peer-reviewed scientific journal that covers all aspects of pure and applied mathematics and is published by Birkhäuser. It was established in 1978 and the editor-in-chief is Catalin Badea (University of Lille).
Results in Mathematics
DisciplineMathematics
LanguageEnglish
Edited byCatalin Badea
Publication details
History1978-present
Publisher
Birkhäuser
Frequency8/year
Impact factor
2.214 (2021)
Standard abbreviations
ISO 4 (alt) · Bluebook (alt1 · alt2)
NLM (alt) · MathSciNet (alt )
ISO 4Results Math.
Indexing
CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt)
MIAR · NLM (alt) · Scopus
ISSN1422-6383 (print)
1420-9012 (web)
LCCN2007204710
OCLC no.609909072
Links
• Journal homepage
• Online access
Abstracting and indexing
This journal is abstracted and indexed by:
• Science Citation Index Expanded
• Mathematical Reviews
• Scopus
• Zentralblatt Math
• Academic OneFile
• Current Contents/Physical, Chemical and Earth Sciences
• Mathematical Reviews
According to the Journal Citation Reports, the journal has a 2013 impact factor of 0.642.[1]
References
1. "Results in Mathematics". 2013 Journal Citation Reports. Web of Science (Science ed.). Thomson Reuters. 2014.
External links
• Official website
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Retract (group theory)
In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup. In symbols, $H$ is a retract of $G$ if and only if there is an endomorphism $\sigma :G\to G$ such that $\sigma (h)=h$ for all $h\in H$ and $\sigma (g)\in H$ for all $g\in G$.[1][2]
The endomorphism itself (having this property) is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism[1][3] or a retraction.[2]
The following is known about retracts:
• A subgroup is a retract if and only if it has a normal complement.[4] The normal complement, specifically, is the kernel of the retraction.
• Every direct factor is a retract.[1] Conversely, any retract which is a normal subgroup is a direct factor.[5]
• Every retract has the congruence extension property.
• Every regular factor, and in particular, every free factor, is a retract.
See also
• Retraction (category theory)
References
1. Baer, Reinhold (1946), "Absolute retracts in group theory", Bulletin of the American Mathematical Society, 52 (6): 501–506, doi:10.1090/S0002-9904-1946-08601-2, MR 0016419.
2. Lyndon, Roger C.; Schupp, Paul E. (2001), Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, p. 2, ISBN 3-540-41158-5, MR 1812024
3. Krylov, Piotr A.; Mikhalev, Alexander V.; Tuganbaev, Askar A. (2003), Endomorphism rings of abelian groups, Algebras and Applications, vol. 2, Kluwer Academic Publishers, Dordrecht, p. 24, doi:10.1007/978-94-017-0345-1, ISBN 1-4020-1438-4, MR 2013936.
4. Myasnikov, Alexei G.; Roman'kov, Vitaly (2014), "Verbally closed subgroups of free groups", Journal of Group Theory, 17 (1): 29–40, arXiv:1201.0497, doi:10.1515/jgt-2013-0034, MR 3176650, S2CID 119323021.
5. For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see García, O. C.; Larrión, F. (1982), "Injectivity in varieties of groups", Algebra Universalis, 14 (3): 280–286, doi:10.1007/BF02483931, MR 0654396, S2CID 122193204.
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Wikipedia
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Constructible set (topology)
In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure. They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms) of algebraic varieties (or more generally schemes). In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible. Constructible sets also feature in the definition of various types of constructible sheaves in algebraic geometry and intersection cohomology.
For a Gödel constructive set, see Constructible universe.
Definitions
A simple definition, adequate in many situations, is that a constructible set is a finite union of locally closed sets. (A set is locally closed if it is the intersection of an open set and closed set.) However, a modification and another slightly weaker definition are needed to have definitions that behave better with "large" spaces:
Definitions: A subset $Z$ of a topological space $X$ is called retrocompact if $Z\cap U$ is compact for every compact open subset $U\subset X$. A subset of $X$ is constructible if it is a finite union of subsets of the form $U\cap (X-V)$ where both $U$ and $V$ are open and retrocompact subsets of $X$. A subset $Z\subset X$ is locally constructible if there is a cover $(U_{i})_{i\in I}$ of $X$ consisting of open subsets with the property that each $Z\cap U_{i}$ is a constructible subset of $U_{i}$. [1][2]
Equivalently the constructible subsets of a topological space $X$ are the smallest collection ${\mathfrak {C}}$ of subsets of $X$ that (i) contains all open retrocompact subsets and (ii) contains all complements and finite unions (and hence also finite intersections) of sets in it. In other words, constructible sets are precisely the Boolean algebra generated by retrocompact open subsets.
In a locally noetherian topological space, all subsets are retrocompact,[3] and so for such spaces the simplified definition given first above is equivalent to the more elaborate one. Most of the commonly met schemes in algebraic geometry (including all algebraic varieties) are locally Noetherian, but there are important constructions that lead to more general schemes.
In any (not necessarily noetherian) topological space, every constructible set contains a dense open subset of its closure.[4]
Terminology: The definition given here is the one used by the first edition of EGA and the Stacks Project. In the second edition of EGA constructible sets (according to the definition above) are called "globally constructible" while the word "constructible" is reserved for what are called locally constructible above. [5]
Chevalley's theorem
A major reason for the importance of constructible sets in algebraic geometry is that the image of a (locally) constructible set is also (locally) constructible for a large class of maps (or "morphisms"). The key result is:
Chevalley's theorem. If $f:X\to Y$ is a finitely presented morphism of schemes and $Z\subset X$ is a locally constructible subset, then $f(Z)$ is also locally constructible in $Y$.[6][7][8]
In particular, the image of an algebraic variety need not be a variety, but is (under the assumptions) always a constructible set. For example, the map $\mathbf {A} ^{2}\rightarrow \mathbf {A} ^{2}$ that sends $(x,y)$ to $(x,xy)$ has image the set $\{x\neq 0\}\cup \{x=y=0\}$, which is not a variety, but is constructible.
Chevalley's theorem in the generality stated above would fail if the simplified definition of constructible sets (without restricting to retrocompact open sets in the definition) were used.[9]
Constructible properties
A large number of "local" properties of morphisms of schemes and quasicoherent sheaves on schemes hold true over a locally constructible subset. EGA IV § 9[10] covers a large number of such properties. Below are some examples (where all references point to EGA IV):
• If $f\colon X\rightarrow S$ is an finitely presented morphism of schemes and ${\mathcal {F}}'\rightarrow {\mathcal {F}}\rightarrow {\mathcal {F}}''$ is a sequence of finitely presented quasi-coherent ${\mathcal {O}}_{X}$-modules, then the set of $s\in S$ for which ${\mathcal {F}}'_{s}\rightarrow {\mathcal {F}}_{s}\rightarrow {\mathcal {F}}''_{s}$ is exact is locally constructible. (Proposition (9.4.4))
• If $f\colon X\rightarrow S$ is an finitely presented morphism of schemes and ${\mathcal {F}}$ is a finitely presented quasi-coherent ${\mathcal {O}}_{X}$-module, then the set of $s\in S$ for which ${\mathcal {F}}_{s}$ is locally free is locally constructible. (Proposition (9.4.7))
• If $f\colon X\rightarrow S$ is an finitely presented morphism of schemes and $Z\subset X$ is a locally constructible subset, then the set of $s\in S$ for which $f^{-1}(s)\cap Z$ is closed (or open) in $f^{-1}(s)$ is locally constructible. (Corollary (9.5.4))
• Let $S$ be a scheme and $f\colon X\rightarrow Y$ a morphism of $S$-schemes. Consider the set $P\subset S$ of $s\in S$ for which the induced morphism $f_{s}\colon X_{s}\rightarrow Y_{s}$ of fibres over $s$ has some property $\mathbf {P} $. Then $P$ is locally constructible if $\mathbf {P} $ is any of the following properties: surjective, proper, finite, immersion, closed immersion, open immersion, isomorphism. (Proposition (9.6.1))
• Let $f\colon X\rightarrow S$ be an finitely presented morphism of schemes and consider the set $P\subset S$ of $s\in S$ for which the fibre $f^{-1}(s)$ has a property $\mathbf {P} $. Then $P$ is locally constructible if $\mathbf {P} $ is any of the following properties: geometrically irreducible, geometrically connected, geometrically reduced. (Theorem (9.7.7))
• Let $f\colon X\rightarrow S$ be an locally finitely presented morphism of schemes and consider the set $P\subset X$ of $x\in X$ for which the fibre $f^{-1}(f(x))$ has a property $\mathbf {P} $. Then $P$ is locally constructible if $\mathbf {P} $ is any of the following properties: geometrically regular, geometrically normal, geometrically reduced. (Proposition (9.9.4))
One important role that these constructibility results have is that in most cases assuming the morphisms in questions are also flat it follows that the properties in question in fact hold in an open subset. A substantial number of such results is included in EGA IV § 12.[11]
See also
• Constructible topology
• Constructible sheaf
Notes
1. Grothendieck & Dieudonné 1961, Ch. 0III, Définitions (9.1.1), (9.1.2) and (9.1.11), pp. 12-14
2. "Definition 5.15.1 (tag 005G)". stacks.math.columbia.edu. Retrieved 2022-10-04.
3. Grothendieck & Dieudonné 1961, Ch. 0III, Sect. (9.1), p. 12
4. Jinpeng An (2012). "Rigid geometric structures, isometric actions, and algebraic quotients". Geom. Dedicata 157: 153–185.
5. Grothendieck & Dieudonné 1971, Ch. 0I, Définitions (2.3.1), (2.3.2) and (2.3.10), pp. 55-57
6. Grothendieck & Dieudonné 1964, Ch. I, Théorème (1.8.4), p. 239.
7. "Theorem 29.22.3 (Chevalley's Theorem) (tag 054K)". stacks.math.columbia.edu. Retrieved 2022-10-04.
8. Grothendieck & Dieudonné 1971, Ch. I, Théorème (7.1.4), p. 329.
9. "Section 109.24 Images of locally closed subsets (tag 0GZL)". stacks.math.columbia.edu. Retrieved 2022-10-04.
10. Grothendieck & Dieudonné 1966, Ch. IV, § 9 Propriétés constructibles, pp. 54-94.
11. Grothendieck & Dieudonné 1966, Ch. IV, § 12 Étude des fibres des morphismes plats de présentation finie, pp. 173-187.
References
• Allouche, Jean Paul. Note on the constructible sets of a topological space.
• Andradas, Carlos; Bröcker, Ludwig; Ruiz, Jesús M. (1996). Constructible sets in real geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) --- Results in Mathematics and Related Areas (3). Vol. 33. Berlin: Springer-Verlag. pp. x+270. ISBN 3-540-60451-0. MR 1393194.
• Borel, Armand. Linear algebraic groups.
• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20. doi:10.1007/bf02684747. MR 0173675.
• Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28. doi:10.1007/bf02684343. MR 0217086.
• Grothendieck, Alexandre; Dieudonné, Jean (1971). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften (in French). Vol. 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8.
• Mostowski, A. (1969). Constructible sets with applications. Studies in Logic and the Foundations of Mathematics. Amsterdam --- Warsaw: North-Holland Publishing Co. ---- PWN-Polish Scientific Publishers. pp. ix+269. MR 0255390.
External links
• https://stacks.math.columbia.edu/tag/04ZC Topological definition of (local) constructibility
• https://stacks.math.columbia.edu/tag/054H Constructibility properties of morphisms of schemes (incl. Chevalley's theorem)
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Wikipedia
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Great icosahedron
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
Great icosahedron
TypeKepler–Poinsot polyhedron
Stellation coreicosahedron
ElementsF = 20, E = 30
V = 12 (χ = 2)
Faces by sides20{3}
Schläfli symbol{3,5⁄2}
Face configurationV(53)/2
Wythoff symbol5⁄2 | 2 3
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532)
ReferencesU53, C69, W41
PropertiesRegular nonconvex deltahedron
(35)/2
(Vertex figure)
Great stellated dodecahedron
(dual polyhedron)
The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the (n–1)-dimensional simplex faces of the core n-polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process.
Images
Transparent model Density Stellation diagram Net
A transparent model of the great icosahedron (See also Animation)
It has a density of 7, as shown in this cross-section.
It is a stellation of the icosahedron, counted by Wenninger as model [W41] and the 16th of 17 stellations of the icosahedron and 7th of 59 stellations by Coxeter.
× 12
Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines.
Spherical tiling
This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)
As a snub
The great icosahedron can be constructed as a uniform snub, with different colored faces and only tetrahedral symmetry: . This construction can be called a retrosnub tetrahedron or retrosnub tetratetrahedron,[1] similar to the snub tetrahedron symmetry of the icosahedron, as a partial faceting of the truncated octahedron (or omnitruncated tetrahedron): . It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, or , and is called a retrosnub octahedron.
Tetrahedral Pyritohedral
Related polyhedra
It shares the same vertex arrangement as the regular convex icosahedron. It also shares the same edge arrangement as the small stellated dodecahedron.
A truncation operation, repeatedly applied to the great icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a rectified great icosahedron. The process completes as a birectification, reducing the original faces down to points, and producing the great stellated dodecahedron.
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two great dodecahedra inscribed within and sharing the edges of the icosahedron.
Name Great
stellated
dodecahedron
Truncated great stellated dodecahedron Great
icosidodecahedron
Truncated
great
icosahedron
Great
icosahedron
Coxeter-Dynkin
diagram
Picture
References
1. Klitzing, Richard. "uniform polyhedra Great icosahedron".
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
• Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR 0676126. (1st Edn University of Toronto (1938))
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp. 96–104
External links
• Eric W. Weisstein, Great icosahedron (Uniform polyhedron) at MathWorld.
• Weisstein, Eric W. "Fifteen stellations of the icosahedron". 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
Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
(Convex) icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.
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Wikipedia
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Reuleaux triangle
A Reuleaux triangle [ʁœlo] is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle.[1] It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"[2]
Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. They are named after Franz Reuleaux,[3] a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs.[4] However, these shapes were known before his time, for instance by the designers of Gothic church windows, by Leonardo da Vinci, who used it for a map projection, and by Leonhard Euler in his study of constant-width shapes. Other applications of the Reuleaux triangle include giving the shape to guitar picks, fire hydrant nuts, pencils, and drill bits for drilling filleted square holes, as well as in graphic design in the shapes of some signs and corporate logos.
Among constant-width shapes with a given width, the Reuleaux triangle has the minimum area and the sharpest (smallest) possible angle (120°) at its corners. By several numerical measures it is the farthest from being centrally symmetric. It provides the largest constant-width shape avoiding the points of an integer lattice, and is closely related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. It can perform a complete rotation within a square while at all times touching all four sides of the square, and has the smallest possible area of shapes with this property. However, although it covers most of the square in this rotation process, it fails to cover a small fraction of the square's area, near its corners. Because of this property of rotating within a square, the Reuleaux triangle is also sometimes known as the Reuleaux rotor.[5]
The Reuleaux triangle is the first of a sequence of Reuleaux polygons whose boundaries are curves of constant width formed from regular polygons with an odd number of sides. Some of these curves have been used as the shapes of coins. The Reuleaux triangle can also be generalized into three dimensions in multiple ways: the Reuleaux tetrahedron (the intersection of four balls whose centers lie on a regular tetrahedron) does not have constant width, but can be modified by rounding its edges to form the Meissner tetrahedron, which does. Alternatively, the surface of revolution of the Reuleaux triangle also has constant width.
Construction
The Reuleaux triangle may be constructed either directly from three circles, or by rounding the sides of an equilateral triangle.[6]
The three-circle construction may be performed with a compass alone, not even needing a straightedge. By the Mohr–Mascheroni theorem the same is true more generally of any compass-and-straightedge construction,[7] but the construction for the Reuleaux triangle is particularly simple. The first step is to mark two arbitrary points of the plane (which will eventually become vertices of the triangle), and use the compass to draw a circle centered at one of the marked points, through the other marked point. Next, one draws a second circle, of the same radius, centered at the other marked point and passing through the first marked point. Finally, one draws a third circle, again of the same radius, with its center at one of the two crossing points of the two previous circles, passing through both marked points.[8] The central region in the resulting arrangement of three circles will be a Reuleaux triangle.[6]
Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle T by drawing three arcs of circles, each centered at one vertex of T and connecting the other two vertices.[9] Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with radius equal to the side length of T.[10]
Mathematical properties
The most basic property of the Reuleaux triangle is that it has constant width, meaning that for every pair of parallel supporting lines (two lines of the same slope that both touch the shape without crossing through it) the two lines have the same Euclidean distance from each other, regardless of the orientation of these lines.[9] In any pair of parallel supporting lines, one of the two lines will necessarily touch the triangle at one of its vertices. The other supporting line may touch the triangle at any point on the opposite arc, and their distance (the width of the Reuleaux triangle) equals the radius of this arc.[11]
The first mathematician to discover the existence of curves of constant width, and to observe that the Reuleaux triangle has constant width, may have been Leonhard Euler.[5] In a paper that he presented in 1771 and published in 1781 entitled De curvis triangularibus, Euler studied curvilinear triangles as well as the curves of constant width, which he called orbiforms.[12][13]
Extremal measures
By many different measures, the Reuleaux triangle is one of the most extreme curves of constant width.
By the Blaschke–Lebesgue theorem, the Reuleaux triangle has the smallest possible area of any curve of given constant width. This area is
${\frac {1}{2}}(\pi -{\sqrt {3}})s^{2}\approx 0.705s^{2},$
where s is the constant width. One method for deriving this area formula is to partition the Reuleaux triangle into an inner equilateral triangle and three curvilinear regions between this inner triangle and the arcs forming the Reuleaux triangle, and then add the areas of these four sets. At the other extreme, the curve of constant width that has the maximum possible area is a circular disk, which has area $\pi s^{2}/4\approx 0.785s^{2}$.[14]
The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°. This is the sharpest possible angle at any vertex of any curve of constant width.[9] Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles.[15] The largest equilateral triangle inscribed in a Reuleaux triangle is the one connecting its three corners, and the smallest one is the one connecting the three midpoints of its sides. The subset of the Reuleaux triangle consisting of points belonging to three or more diameters is the interior of the larger of these two triangles; it has a larger area than the set of three-diameter points of any other curve of constant width.[16]
Although the Reuleaux triangle has sixfold dihedral symmetry, the same as an equilateral triangle, it does not have central symmetry. The Reuleaux triangle is the least symmetric curve of constant width according to two different measures of central asymmetry, the Kovner–Besicovitch measure (ratio of area to the largest centrally symmetric shape enclosed by the curve) and the Estermann measure (ratio of area to the smallest centrally symmetric shape enclosing the curve). For the Reuleaux triangle, the two centrally symmetric shapes that determine the measures of asymmetry are both hexagonal, although the inner one has curved sides.[17] The Reuleaux triangle has diameters that split its area more unevenly than any other curve of constant width. That is, the maximum ratio of areas on either side of a diameter, another measure of asymmetry, is bigger for the Reuleaux triangle than for other curves of constant width.[18]
Among all shapes of constant width that avoid all points of an integer lattice, the one with the largest width is a Reuleaux triangle. It has one of its axes of symmetry parallel to the coordinate axes on a half-integer line. Its width, approximately 1.54, is the root of a degree-6 polynomial with integer coefficients.[17][19][20]
Just as it is possible for a circle to be surrounded by six congruent circles that touch it, it is also possible to arrange seven congruent Reuleaux triangles so that they all make contact with a central Reuleaux triangle of the same size. This is the maximum number possible for any curve of constant width.[21]
Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite that can be inscribed into a Reuleaux triangle.[22]
Other measures
By Barbier's theorem all curves of the same constant width including the Reuleaux triangle have equal perimeters. In particular this perimeter equals the perimeter of the circle with the same width, which is $\pi s$.[23][24][9]
The radii of the largest inscribed circle of a Reuleaux triangle with width s, and of the circumscribed circle of the same triangle, are
$\displaystyle \left(1-{\frac {1}{\sqrt {3}}}\right)s\approx 0.423s\quad {\text{and}}\quad \displaystyle {\frac {s}{\sqrt {3}}}\approx 0.577s$
respectively; the sum of these radii equals the width of the Reuleaux triangle. More generally, for every curve of constant width, the largest inscribed circle and the smallest circumscribed circle are concentric, and their radii sum to the constant width of the curve.[25]
Unsolved problem in mathematics:
How densely can Reuleaux triangles be packed in the plane?
(more unsolved problems in mathematics)
The optimal packing density of the Reuleaux triangle in the plane remains unproven, but is conjectured to be
${\frac {2(\pi -{\sqrt {3}})}{{\sqrt {15}}+{\sqrt {7}}-{\sqrt {12}}}}\approx 0.923,$
which is the density of one possible double lattice packing for these shapes. The best proven upper bound on the packing density is approximately 0.947.[26] It has also been conjectured, but not proven, that the Reuleaux triangles have the highest packing density of any curve of constant width.[27]
Rotation within a square
Any curve of constant width can form a rotor within a square, a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square. However, the Reuleaux triangle is the rotor with the minimum possible area.[9] As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four ellipses.[28] Because of its 120° angles, the rotating Reuleaux triangle cannot reach some points near the sharper angles at the square's vertices, but rather covers a shape with slightly rounded corners, also formed by elliptical arcs.[9]
One of the four ellipses followed by the center of a rotating Reuleaux triangle in a square
Ellipse separating one of the corners (lower left) of a square from the region swept by a rotating Reuleaux triangle
At any point during this rotation, two of the corners of the Reuleaux triangle touch two adjacent sides of the square, while the third corner of the triangle traces out a curve near the opposite vertex of the square. The shape traced out by the rotating Reuleaux triangle covers approximately 98.8% of the area of the square.[29]
As a counterexample
Reuleaux's original motivation for studying the Reuleaux triangle was as a counterexample, showing that three single-point contacts may not be enough to fix a planar object into a single position.[30] The existence of Reuleaux triangles and other curves of constant width shows that diameter measurements alone cannot verify that an object has a circular cross-section.[31]
In connection with the inscribed square problem, Eggleston (1958) observed that the Reuleaux triangle provides an example of a constant-width shape in which no regular polygon with more than four sides can be inscribed, except the regular hexagon, and he described a small modification to this shape that preserves its constant width but also prevents regular hexagons from being inscribed in it. He generalized this result to three dimensions using a cylinder with the same shape as its cross section.[32]
Applications
Reaching into corners
Several types of machinery take the shape of the Reuleaux triangle, based on its property of being able to rotate within a square.
The Watts Brothers Tool Works square drill bit has the shape of a Reuleaux triangle, modified with concavities to form cutting surfaces. When mounted in a special chuck which allows for the bit not having a fixed centre of rotation, it can drill a hole that is nearly square.[33] Although patented by Henry Watts in 1914, similar drills invented by others were used earlier.[9] Other Reuleaux polygons are used to drill pentagonal, hexagonal, and octagonal holes.[9][33]
Panasonic's RULO robotic vacuum cleaner has its shape based on the Reuleaux triangle in order to ease cleaning up dust in the corners of rooms.[34][35]
Rolling cylinders
Another class of applications of the Reuleaux triangle involves cylindrical objects with a Reuleaux triangle cross section. Several pencils are manufactured in this shape, rather than the more traditional round or hexagonal barrels.[36] They are usually promoted as being more comfortable or encouraging proper grip, as well as being less likely to roll off tables (since the center of gravity moves up and down more than a rolling hexagon).
A Reuleaux triangle (along with all other curves of constant width) can roll but makes a poor wheel because it does not roll about a fixed center of rotation. An object on top of rollers that have Reuleaux triangle cross-sections would roll smoothly and flatly, but an axle attached to Reuleaux triangle wheels would bounce up and down three times per revolution.[9][37] This concept was used in a science fiction short story by Poul Anderson titled "The Three-Cornered Wheel".[11][38] A bicycle with floating axles and a frame supported by the rim of its Reuleaux triangle shaped wheel was built and demonstrated in 2009 by Chinese inventor Guan Baihua, who was inspired by pencils with the same shape.[39]
Mechanism design
Another class of applications of the Reuleaux triangle involves using it as a part of a mechanical linkage that can convert rotation around a fixed axis into reciprocating motion.[10] These mechanisms were studied by Franz Reuleaux. With the assistance of the Gustav Voigt company, Reuleaux built approximately 800 models of mechanisms, several of which involved the Reuleaux triangle.[40] Reuleaux used these models in his pioneering scientific investigations of their motion.[41] Although most of the Reuleaux–Voigt models have been lost, 219 of them have been collected at Cornell University, including nine based on the Reuleaux triangle.[40][42] However, the use of Reuleaux triangles in mechanism design predates the work of Reuleaux; for instance, some steam engines from as early as 1830 had a cam in the shape of a Reuleaux triangle.[43][44]
One application of this principle arises in a film projector. In this application, it is necessary to advance the film in a jerky, stepwise motion, in which each frame of film stops for a fraction of a second in front of the projector lens, and then much more quickly the film is moved to the next frame. This can be done using a mechanism in which the rotation of a Reuleaux triangle within a square is used to create a motion pattern for an actuator that pulls the film quickly to each new frame and then pauses the film's motion while the frame is projected.[45]
The rotor of the Wankel engine is shaped as a curvilinear triangle that is often cited as an example of a Reuleaux triangle.[3][5][9][44] However, its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width.[46]
Architecture
In Gothic architecture, beginning in the late 13th century or early 14th century,[47] the Reuleaux triangle became one of several curvilinear forms frequently used for windows, window tracery, and other architectural decorations.[3] For instance, in English Gothic architecture, this shape was associated with the decorated period, both in its geometric style of 1250–1290 and continuing into its curvilinear style of 1290–1350.[47] It also appears in some of the windows of the Milan Cathedral.[48] In this context, the shape is more frequently called a spherical triangle,[47][49][50] but the more usual mathematical meaning of a spherical triangle is a triangle on the surface of a sphere (a shape also commonly used in architecture as a pendentive). In its use in Gothic church architecture, the three-cornered shape of the Reuleaux triangle may be seen both as a symbol of the Trinity,[51] and as "an act of opposition to the form of the circle".[52]
The Reuleaux triangle has also been used in other styles of architecture. For instance, Leonardo da Vinci sketched this shape as the plan for a fortification.[42] Modern buildings that have been claimed to use a Reuleaux triangle shaped floorplan include the MIT Kresge Auditorium, the Kölntriangle, the Donauturm, the Torre de Collserola, and the Mercedes-Benz Museum.[53] However in many cases these are merely rounded triangles, with different geometry than the Reuleaux triangle.
Mapmaking
Main article: octant projection
Another early application of the Reuleaux triangle, da Vinci's world map from circa 1514, was a world map in which the spherical surface of the earth was divided into eight octants, each flattened into the shape of a Reuleaux triangle.[54][55][56]
Similar maps also based on the Reuleaux triangle were published by Oronce Finé in 1551 and by John Dee in 1580.[56]
Other objects
Many guitar picks employ the Reuleaux triangle, as its shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre. Because all three points of the shape are usable, it is easier to orient and wears less quickly compared to a pick with a single tip.[57]
Illicit use of a fire hydrant, Philadelphia, 1996, and a newer Philadelphia hydrant with a Reuleaux triangle shaped nut to prevent such use.
The Reuleaux triangle has been used as the shape for the cross section of a fire hydrant valve nut. The constant width of this shape makes it difficult to open the fire hydrant using standard parallel-jawed wrenches; instead, a wrench with a special shape is needed. This property allows the fire hydrants to be opened only by firefighters (who have the special wrench) and not by other people trying to use the hydrant as a source of water for other activities.[58]
Following a suggestion of Keto (1997),[59] the antennae of the Submillimeter Array, a radio-wave astronomical observatory on Mauna Kea in Hawaii, are arranged on four nested Reuleaux triangles.[60][61] Placing the antennae on a curve of constant width causes the observatory to have the same spatial resolution in all directions, and provides a circular observation beam. As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the Fourier transform of the signal from the array.[59][61] The antennae may be moved from one Reuleaux triangle to another for different observations, according to the desired angular resolution of each observation.[60][61] The precise placement of the antennae on these Reuleaux triangles was optimized using a neural network. In some places the constructed observatory departs from the preferred Reuleaux triangle shape because that shape was not possible within the given site.[61]
Signs and logos
The shield shapes used for many signs and corporate logos feature rounded triangles. However, only some of these are Reuleaux triangles.
The corporate logo of Petrofina (Fina), a Belgian oil company with major operations in Europe, North America and Africa, used a Reuleaux triangle with the Fina name from 1950 until Petrofina's merger with Total S.A. (today TotalEnergies) in 2000.[62][63] Another corporate logo framed in the Reuleaux triangle, the south-pointing compass of Bavaria Brewery, was part of a makeover by design company Total Identity that won the SAN 2010 Advertiser of the Year award.[64] The Reuleaux triangle is also used in the logo of Colorado School of Mines.[65]
In the United States, the National Trails System and United States Bicycle Route System both mark routes with Reuleaux triangles on signage.[66]
In nature
According to Plateau's laws, the circular arcs in two-dimensional soap bubble clusters meet at 120° angles, the same angle found at the corners of a Reuleaux triangle. Based on this fact, it is possible to construct clusters in which some of the bubbles take the form of a Reuleaux triangle.[67]
The shape was first isolated in crystal form in 2014 as Reuleaux triangle disks.[68] Basic bismuth nitrate disks with the Reuleaux triangle shape were formed from the hydrolysis and precipitation of bismuth nitrate in an ethanol–water system in the presence of 2,3-bis(2-pyridyl)pyrazine.
Generalizations
Triangular curves of constant width with smooth rather than sharp corners may be obtained as the locus of points at a fixed distance from the Reuleaux triangle.[69] Other generalizations of the Reuleaux triangle include surfaces in three dimensions, curves of constant width with more than three sides, and the Yanmouti sets which provide extreme examples of an inequality between width, diameter, and inradius.
Three-dimensional version
The intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s is called the Reuleaux tetrahedron, but its surface is not a surface of constant width.[70] It can, however, be made into a surface of constant width, called Meissner's tetrahedron, by replacing three of its edge arcs by curved surfaces, the surfaces of rotation of a circular arc. Alternatively, the surface of revolution of a Reuleaux triangle through one of its symmetry axes forms a surface of constant width, with minimum volume among all known surfaces of revolution of given constant width.[71]
Reuleaux polygons
Main article: Reuleaux polygon
Reuleaux polygons
Botswana 2 pula Reuleaux heptagon coin
The Reuleaux triangle can be generalized to regular or irregular polygons with an odd number of sides, yielding a Reuleaux polygon, a curve of constant width formed from circular arcs of constant radius. The constant width of these shapes allows their use as coins that can be used in coin-operated machines.[9] Although coins of this type in general circulation usually have more than three sides, a Reuleaux triangle has been used for a commemorative coin from Bermuda.[53]
Similar methods can be used to enclose an arbitrary simple polygon within a curve of constant width, whose width equals the diameter of the given polygon. The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in linear time, and can be drawn with compass and straightedge.[72] Although the Reuleaux polygons all have an odd number of circular-arc sides, it is possible to construct constant-width shapes with an even number of circular-arc sides of varying radii.[73]
Yanmouti sets
The Yanmouti sets are defined as the convex hulls of an equilateral triangle together with three circular arcs, centered at the triangle vertices and spanning the same angle as the triangle, with equal radii that are at most equal to the side length of the triangle. Thus, when the radius is small enough, these sets degenerate to the equilateral triangle itself, but when the radius is as large as possible they equal the corresponding Reuleaux triangle. Every shape with width w, diameter d, and inradius r (the radius of the largest possible circle contained in the shape) obeys the inequality
$w-r\leq {\frac {d}{\sqrt {3}}},$
and this inequality becomes an equality for the Yanmouti sets, showing that it cannot be improved.[74]
Related figures
In the classical presentation of a three-set Venn diagram as three overlapping circles, the central region (representing elements belonging to all three sets) takes the shape of a Reuleaux triangle.[3] The same three circles form one of the standard drawings of the Borromean rings, three mutually linked rings that cannot, however, be realized as geometric circles.[75] Parts of these same circles are used to form the triquetra, a figure of three overlapping semicircles (each two of which form a vesica piscis symbol) that again has a Reuleaux triangle at its center;[76] just as the three circles of the Venn diagram may be interlaced to form the Borromean rings, the three circular arcs of the triquetra may be interlaced to form a trefoil knot.[77]
Relatives of the Reuleaux triangle arise in the problem of finding the minimum perimeter shape that encloses a fixed amount of area and includes three specified points in the plane. For a wide range of choices of the area parameter, the optimal solution to this problem will be a curved triangle whose three sides are circular arcs with equal radii. In particular, when the three points are equidistant from each other and the area is that of the Reuleaux triangle, the Reuleaux triangle is the optimal enclosure.[78]
Circular triangles are triangles with circular-arc edges, including the Reuleaux triangle as well as other shapes. The deltoid curve is another type of curvilinear triangle, but one in which the curves replacing each side of an equilateral triangle are concave rather than convex. It is not composed of circular arcs, but may be formed by rolling one circle within another of three times the radius.[79] Other planar shapes with three curved sides include the arbelos, which is formed from three semicircles with collinear endpoints,[80] and the Bézier triangle.[81]
The Reuleaux triangle may also be interpreted as the conformal image of a spherical triangle with 120° angles.[67] This spherical triangle is one of the Schwarz triangles (with parameters 3/2, 3/2, 3/2), triangles bounded by great-circle arcs on the surface of a sphere that can tile the sphere by reflection.[82]
References
1. Gardner (2014) calls it the simplest, while Gruber (1983, p. 59) calls it "the most notorious".
2. Klee, Victor (1971), "Shapes of the future", The Two-Year College Mathematics Journal, 2 (2): 14–27, doi:10.2307/3026963, JSTOR 3026963.
3. Alsina, Claudi; Nelsen, Roger B. (2011), Icons of Mathematics: An Exploration of Twenty Key Images, Dolciani Mathematical Expositions, vol. 45, Mathematical Association of America, p. 155, ISBN 978-0-88385-352-8.
4. Moon, F. C. (2007), The Machines of Leonardo Da Vinci and Franz Reuleaux: Kinematics of Machines from the Renaissance to the 20th Century, History of Mechanism and Machine Science, vol. 2, Springer, ISBN 978-1-4020-5598-0.
5. Bryant, John; Sangwin, Chris (2011), How Round Is Your Circle?: Where Engineering and Mathematics Meet, Princeton University Press, p. 190, ISBN 978-0-691-14992-9.
6. Hann, Michael (2014), Structure and Form in Design: Critical Ideas for Creative Practice, A&C Black, p. 34, ISBN 978-1-4725-8431-1.
7. Hungerbühler, Norbert (1994), "A short elementary proof of the Mohr-Mascheroni theorem", American Mathematical Monthly, 101 (8): 784–787, CiteSeerX 10.1.1.45.9902, doi:10.2307/2974536, JSTOR 2974536, MR 1299166.
8. This construction is briefly described by Maor & Jost (2014) and may be seen, for instance, in the video Fun with Reuleaux triangles by Alex Franke, August 21, 2011.
9. Gardner, Martin (2014), "Chapter 18: Curves of Constant Width", Knots and Borromean Rings, Rep-Tiles, and Eight Queens, The New Martin Gardner Mathematical Library, vol. 4, Cambridge University Press, pp. 223–245, ISBN 978-0-521-75613-6.
10. Klee, Victor; Wagon, S. (1991), Old and New Unsolved Problems in Plane Geometry and Number Theory, Dolciani mathematical expositions, vol. 11, Cambridge University Press, p. 21, ISBN 978-0-88385-315-3.
11. Maor, Eli; Jost, Eugen (2014), "46 The Reuleaux Triangle", Beautiful Geometry, Princeton University Press, pp. 154–156, ISBN 978-1-4008-4833-1.
12. Reich, Karin (2007), "Euler's contribution to differential geometry and its reception", in Bradley, Robert E.; Sandifer, Ed (eds.), Leonhard Euler: Life, Work and Legacy, Studies in the History and Philosophy of Mathematics, vol. 5, Elsevier, pp. 479–502, doi:10.1016/S0928-2017(07)80026-0, ISBN 9780444527288. See in particular section 1.4, "Orbiforms, 1781", pp. 484–485.
13. Euler, Leonhard (1781), "De curvis triangularibus", Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin), 1778: 3–30. See in particular p. 7 for the definition of orbiforms.
14. Gruber, Peter M. (1983), Convexity and its Applications, Birkhäuser, p. 67, ISBN 978-3-7643-1384-5
15. Gruber (1983, p. 76)
16. Makeev, V. V. (2000), "An extremal property of the Reuleaux triangle", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 267 (Geom. i Topol. 5): 152–155, 329, doi:10.1023/A:1021287302603, MR 1809823, S2CID 116027099.
17. Finch, Steven R. (2003), "8.10 Reuleaux Triangle Constants" (PDF), Mathematical Constants, Encyclopedia of Mathematics and its Applications, Cambridge University Press, pp. 513–514, ISBN 978-0-521-81805-6.
18. Groemer, H.; Wallen, L. J. (2001), "A measure of asymmetry for domains of constant width", Beiträge zur Algebra und Geometrie, 42 (2): 517–521, MR 1865537.
19. Gruber (1983, p. 78)
20. Sallee, G. T. (1969), "The maximal set of constant width in a lattice", Pacific Journal of Mathematics, 28 (3): 669–674, doi:10.2140/pjm.1969.28.669, MR 0240724.
21. Fejes Tóth, L. (1967), "On the number of equal discs that can touch another of the same kind", Studia Scientiarum Mathematicarum Hungarica, 2: 363–367, MR 0221388; Schopp, J. (1970), "Über die Newtonsche Zahl einer Scheibe konstanter Breite", Studia Scientiarum Mathematicarum Hungarica (in German), 5: 475–478, MR 0285983.
22. Ball, D.G. (1973), "A generalisation of π", The Mathematical Gazette, 57 (402): 298–303, doi:10.2307/3616052, JSTOR 3616052, S2CID 125396664; Griffiths, David; Culpin, David (1975), "Pi-optimal polygons", The Mathematical Gazette, 59 (409): 165–175, doi:10.2307/3617699, JSTOR 3617699, S2CID 126325288.
23. Lay, Steven R. (2007), Convex Sets and Their Applications, Dover, Theorem 11.11, pp. 81–82, ISBN 978-0-486-45803-8.
24. Barbier, E. (1860), "Note sur le problème de l'aiguille et le jeu du joint couvert" (PDF), Journal de Mathématiques Pures et Appliquées, 2e série (in French), 5: 273–286. See in particular pp. 283–285.
25. Lay (2007), Theorem 11.8, pp. 80–81.
26. Blind, G.; Blind, R. (1983), "Eine Abschätzung für die Dichte der dichtesten Packung mit Reuleaux-Dreiecken", Studia Scientiarum Mathematicarum Hungarica (in German), 18 (2–4): 465–469, MR 0787951. See also Blind, G.; Blind, R. (1987), "Reguläre Packungen mit Reuleaux-Dreiecken", Results in Mathematics (in German), 11 (1–2): 1–7, doi:10.1007/BF03323256, MR 0880190, S2CID 121633860.
27. Resnikoff, Howard L. (2015), On Curves and Surfaces of Constant Width, arXiv:1504.06733, Bibcode:2015arXiv150406733R.
28. Gleiftner, Winfried; Zeitler, Herbert (May 2000), "The Reuleaux triangle and its center of mass", Results in Mathematics, 37 (3–4): 335–344, doi:10.1007/bf03322004, S2CID 119600507.
29. Pickover, Clifford A. (2009), "Reuleaux Triangle", The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, p. 266, ISBN 978-1-4027-5796-9.
30. Moon (2007), p. 239.
31. Granovsky, V. A.; Siraya, T. N., "Metrological traceability and quality of industrial tests measurements", in Pavese, F.; Bär, M.; Filtz, J.-R.; Forbes, A. B.; Pendrill, L.; Shirono, K. (eds.), Advanced Mathematical and Computational Tools in Metrology and Testing IX, World Scientific, pp. 194–201. See in particular p. 200.
32. Eggleston, H. G. (1958), "Figures inscribed in convex sets", American Mathematical Monthly, 65 (2): 76–80, doi:10.2307/2308878, JSTOR 2308878, MR 0097768.
33. How to drill square hexagon octagon pentagon holes, Wilmerding, Pennsylvania: Watts Brothers Tool Works, 1950–1951 (27 page brochure).
34. Mochizuki, Takashi (January 22, 2015), "Panasonic Rolls Out Triangular Robot Vacuum", Japan Real Time, Wall Street Journal.
35. Coxworth, Ben (March 3, 2015), "Panasonic enters the robo-vac game, with the triangular Rulo", Gizmag.
36. Gamber, Johnny (April 26, 2006), "Review of Staedtler Noris Ergosoft HB", Pencil Revolution, retrieved 2015-05-22.
37. Masferrer León, Claudia; von Wuthenau Mayer, Sebastián (December 2005), "Reinventing the wheel: Non-circular wheels", The Mathematical Intelligencer, 27 (4): 7–13, doi:10.1007/bf02985852.
38. Anderson, Poul (October 1963), "The Three-Cornered Wheel", Analog, pp. 50–69
39. Dempster, Tyra (June 17, 2009), Chinese man reinvents the wheel, Reuters
40. Moon, Francis C. (July 1999), The Reuleaux Collection of Kinematic Mechanisms at Cornell University (PDF), Cornell University Library, archived from the original (PDF) on June 14, 2020.
41. Henderson, David W.; Taimina, Daina (2007), "Experiencing meanings in geometry", in Sinclair, Nathalie; Pimm, David; Higginson, William (eds.), Mathematics and the Aesthetic: New Approaches to an Ancient Affinity, CMS Books in Mathematics, Springer, pp. 58–83, doi:10.1007/978-0-387-38145-9_4, hdl:1813/2714, ISBN 978-0-387-38145-9. See in particular p. 81.
42. Moon (2007, p. 241).
43. Moon (2007, p. 240)
44. Peterson, Ivars (October 19, 1996), "Rolling with Reuleaux", MathTrek, ScienceNews. Reprinted in Peterson, Ivars (2002), Mathematical Treks: From Surreal Numbers to Magic Circles, MAA spectrum, Mathematical Association of America, pp. 141–144, ISBN 978-0-88385-537-9.
45. Lay (2007), p. 83.
46. Gruber (1983, p. 80); Nash, David H. (March 1977), "Rotary engine geometry", Mathematics Magazine, 50 (2): 87–89, doi:10.1080/0025570x.1977.11976621; Badr, O.; Naik, S.; O'Callaghan, P. W.; Probert, S. D. (1991), "Rotary Wankel engines as expansion devices in steam Rankine-cycle engines", Applied Energy, 39 (1): 59–76, doi:10.1016/0306-2619(91)90063-4.
47. Hart, Stephen (2010), Medieval Church Window Tracery in England, Boydell & Brewer Ltd, pp. 63–64, ISBN 978-1-84383-533-2.
48. Marchetti, Elena; Costa, Luisa Rossi (2014), "What geometries in Milan Cathedral?", in Williams, Kim; Ostwald, Michael J. (eds.), Architecture and Mathematics from Antiquity to the Future, Volume I: Antiquity to the 1500s, Birkhäuser, pp. 509–534, doi:10.1007/978-3-319-00137-1_35
49. Parker, John Henry (1850), A glossary of terms used in Grecian, Roman, Italian, and Gothic architecture, vol. 1 (5th ed.), London: David Rogue, p. 202.
50. Burchett, E. S. (1876), Practical plane geometry, London and Glasgow: William Collins, Sons, and Co., Caption to Plate LV, Fig. 6.
51. Durand, Guillaume (1906), The Symbolism of Churches and Church Ornaments: A Translation of the First Book of the Rationale Divinorum Officiorum (3rd ed.), Gibbings, p. lxxxviii.
52. Frankl, Paul; Crossley, Paul (2000), Gothic Architecture, Pelican history of art, vol. 19, Yale University Press, p. 146, ISBN 978-0-300-08799-4.
53. Conti, Giuseppe; Paoletti, Raffaella (October 2019), "Reuleaux triangle in architecture and applications", in Magnaghi-Delfino, Paola; Mele, Giampiero; Norando, Tullia (eds.), Faces of Geometry: From Agnesi to Mirzakhani, Lecture Notes in Networks and Systems, Springer, pp. 79–89, doi:10.1007/978-3-030-29796-1_7, S2CID 209976466
54. Snyder, John P. (1997), Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, p. 40, ISBN 978-0-226-76747-5.
55. Keuning, Johannes (January 1955), "The history of geographical map projections until 1600", Imago Mundi, 12 (1): 1–24, doi:10.1080/03085695508592085, JSTOR 1150090.
56. Bower, David I. (February 2012), "The unusual projection for one of John Dee's maps of 1580" (PDF), The Cartographic Journal, 49 (1): 55–61, doi:10.1179/1743277411y.0000000015, S2CID 129873912.
57. Hoover, Will (November 1995), Picks!: The Colorful Saga of Vintage Celluloid Guitar Plectrums, Backbeat Books, pp. 32–33, ISBN 978-0-87930-377-8.
58. Martini, Horst; Montejano, Luis; Oliveros, Déborah (2019), Bodies of Constant Width: An Introduction to Convex Geometry with Applications, Birkhäuser, p. 3, doi:10.1007/978-3-030-03868-7, ISBN 978-3-030-03866-3, MR 3930585, S2CID 127264210
59. Keto, Eric (1997), "The shapes of cross-correlation interferometers", The Astrophysical Journal, 475 (2): 843–852, Bibcode:1997ApJ...475..843K, doi:10.1086/303545.
60. Blundell, Raymond (2007), "The submillimeter array" (PDF), Proc. 2007 IEEE/MTT-S International Microwave Symposium, pp. 1857–1860, doi:10.1109/mwsym.2007.380132, ISBN 978-1-4244-0687-6, S2CID 41312640.
61. Ho, Paul T. P.; Moran, James M.; Lo, Kwok Yung (2004), "The submillimeter array", The Astrophysical Journal, 616 (1): L1–L6, arXiv:astro-ph/0406352, Bibcode:2004ApJ...616L...1H, doi:10.1086/423245, S2CID 115133614.
62. Gwillian, Sam (May 16, 2015), Interesting Stuff: Curves of Constant Width, Newport City Radio, archived from the original on June 16, 2016
63. "Fina Logo History: from Petrofina to Fina", Total: Group Presentation, Total S.A., archived from the original on December 26, 2012, retrieved 31 October 2015.
64. "Global: Bavaria, Fundamental Rebranding Operation at Bavaria", Total Identity, archived from the original on 2015-06-30, retrieved 2015-06-27{{citation}}: CS1 maint: unfit URL (link)
65. Fisher, Roland B. (Spring 2002), "M-blems: Explaining the logo" (PDF), Mines: The Magazine of Colorado School of Mines, vol. 92, no. 2, p. 29, archived from the original on 2010-07-10{{citation}}: CS1 maint: unfit URL (link)
66. Lindley, Jeffrey A. (June 1, 2012), "Information: MUTCD — Interim Approval for the Optional Use of an Alternative Design for the U.S. Bicycle Route (M1-9) Sign (IA-15)", Manual on Uniform Traffic Control Devices for Streets and Highways: Resources, US Department of Transportation, Federal Highway Administration, retrieved August 20, 2018
67. Modes, Carl D.; Kamien, Randall D. (2013), "Spherical foams in flat space", Soft Matter, 9 (46): 11078–11084, arXiv:0810.5724, Bibcode:2013SMat....911078M, doi:10.1039/c3sm51585k, S2CID 96591302.
68. Ng, C. H. B.; Fan, W. Y. (2014), "Reuleaux triangle disks: New shape on the block", Journal of the American Chemical Society, 136 (37): 12840–12843, doi:10.1021/ja506625y, PMID 25072943.
69. Banchoff, Thomas; Giblin, Peter (1994), "On the geometry of piecewise circular curves", American Mathematical Monthly, 101 (5): 403–416, doi:10.2307/2974900, JSTOR 2974900, MR 1272938.
70. Weber, Christof (2009), What does this solid have to do with a ball? (PDF) Weber also has films of both types of Meissner body rotating as well as interactive images.
71. Campi, Stefano; Colesanti, Andrea; Gronchi, Paolo (1996), "Minimum problems for volumes of convex bodies", Partial Differential Equations and Applications: Collected Papers in Honor of Carlo Pucci, Lecture Notes in Pure and Applied Mathematics, no. 177, Marcel Dekker, pp. 43–55.
72. Chandru, V.; Venkataraman, R. (1991), "Circular hulls and orbiforms of simple polygons", Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '91), Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, pp. 433–440, ISBN 978-0-89791-376-8.
73. Peterson, Bruce B. (1973), "Intersection properties of curves of constant width", Illinois Journal of Mathematics, 17 (3): 411–420, doi:10.1215/ijm/1256051608, MR 0320885.
74. Hernández Cifre, M. A. (2000), "Is there a planar convex set with given width, diameter, and inradius?", American Mathematical Monthly, 107 (10): 893–900, doi:10.2307/2695582, JSTOR 2695582, MR 1806918.
75. Lindström, Bernt; Zetterström, Hans-Olov (1991), "Borromean circles are impossible", American Mathematical Monthly, 98 (4): 340–341, doi:10.2307/2323803, JSTOR 2323803.
76. Weisstein, Eric W., "Triquetra", MathWorld
77. Hoy, Jessica; Millett, Kenneth C. (2014), "A mathematical analysis of knotting and linking in Leonardo da Vinci's cartelle of the Accademia Vinciana" (PDF), Journal of Mathematics and the Arts.
78. Courant, Richard; Robbins, Herbert (1996), What Is Mathematics? An Elementary Approach to Ideas and Methods (2nd ed.), Oxford University Press, pp. 378–379, ISBN 978-0-19-975487-8.
79. Lockwood, E. H. (1961), "Chapter 8: The Deltoid", A Book of Curves, Cambridge University Press
80. Mackay, J. S. (February 1884), "The shoemaker's knife", Proceedings of the Edinburgh Mathematical Society, 3: 2, doi:10.1017/s0013091500037196.
81. Bruijns, J. (1998), "Quadratic Bezier triangles as drawing primitives", Proceedings of the ACM SIGGRAPH/EUROGRAPHICS Workshop on Graphics Hardware (HWWS '98), New York, NY, USA: ACM, pp. 15–24, doi:10.1145/285305.285307, ISBN 978-1-58113-097-3, S2CID 28967106.
82. Wenninger, Magnus J. (2014), Spherical Models, Dover, p. 134, ISBN 978-0-486-14365-1.
External links
Wikimedia Commons has media related to Reuleaux triangles.
• Weisstein, Eric W., "Reuleaux Triangle", MathWorld
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Wikipedia
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Reuleaux polygon
In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius.[1] These shapes are named after their prototypical example, the Reuleaux triangle, which in turn, is named after 19th-century German engineer Franz Reuleaux.[2] The Reuleaux triangle can be constructed from an equilateral triangle by connecting each two vertices by a circular arc centered on the third vertex, and Reuleaux polygons can be formed by a similar construction from any regular polygon with an odd number of sides, or from certain irregular polygons. Every curve of constant width can be accurately approximated by Reuleaux polygons. They have been applied in coinage shapes.
Regular Reuleaux polygons
Irregular Reuleaux heptagon
Construction
If $P$ is a convex polygon with an odd number of sides, in which each vertex is equidistant to the two opposite vertices and closer to all other vertices, then replacing each side of $P$ by an arc centered at its opposite vertex produces a Reuleaux polygon. As a special case, this construction is possible for every regular polygon with an odd number of sides.[1]
Every Reuleaux polygon must have an odd number of circular-arc sides, and can be constructed in this way from a polygon, the convex hull of its arc endpoints. However, it is possible for other curves of constant width to be made of an even number of arcs with varying radii.[1]
Properties
The Reuleaux polygons based on regular polygons are the only curves of constant width whose boundaries are formed by finitely many circular arcs of equal length.[3]
Every curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width.[1]
A regular Reuleaux polygon has sides of equal length. More generally, when a Reuleaux polygon has sides that can be split into arcs of equal length, the convex hull of the arc endpoints is a Reinhardt polygon. These polygons are optimal in multiple ways: they have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.[4]
Applications
The constant width of these shapes allows their use as coins that can be used in coin-operated machines. For instance, the United Kingdom has made 20-pence and 50-pence coins in the shape of a regular Reuleaux heptagon.[5] The Canadian loonie dollar coin uses another regular Reuleaux polygon with 11 sides.[6] However, some coins with rounded-polygon sides, such as the 12-sided 2017 British pound coin, do not have constant width and are not Reuleaux polygons.[7]
Although Chinese inventor Guan Baihua has made a bicycle with Reuleaux polygon wheels, the invention has not caught on.[8]
References
1. Martini, Horst; Montejano, Luis; Oliveros, Déborah (2019), "Section 8.1: Reuleaux Polygons", Bodies of Constant Width: An Introduction to Convex Geometry with Applications, Birkhäuser, pp. 167–169, doi:10.1007/978-3-030-03868-7, ISBN 978-3-030-03866-3, MR 3930585, S2CID 127264210
2. Alsina, Claudi; Nelsen, Roger B. (2011), Icons of Mathematics: An Exploration of Twenty Key Images, Dolciani Mathematical Expositions, vol. 45, Mathematical Association of America, p. 155, ISBN 978-0-88385-352-8
3. Firey, W. J. (1960), "Isoperimetric ratios of Reuleaux polygons", Pacific Journal of Mathematics, 10 (3): 823–829, doi:10.2140/pjm.1960.10.823, MR 0113176
4. Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic", Geometriae Dedicata, 198: 1–18, arXiv:1405.5233, doi:10.1007/s10711-018-0326-5, MR 3933447, S2CID 119629098
5. Gardner, Martin (1991), "Chapter 18: Curves of Constant Width", The Unexpected Hanging and Other Mathematical Diversions, University of Chicago Press, pp. 212–221, ISBN 0-226-28256-2
6. Chamberland, Marc (2015), Single Digits: In Praise of Small Numbers, Princeton University Press, pp. 104–105, ISBN 9781400865697
7. Freiberger, Marianne (December 13, 2016), "New £1 coin gets even", Plus Magazine
8. du Sautoy, Marcus (May 27, 2009), "A new bicycle reinvents the wheel, with a pentagon and triangle", The Times. See also Newitz, Annalee (September 30, 2014), "Inventor creates seriously cool wheels", Gizmodo
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Wikipedia
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Exchange matrix
In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the identity matrix.[1]
$J_{2}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};\quad J_{3}={\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}};\quad J_{n}={\begin{pmatrix}0&0&\cdots &0&0&1\\0&0&\cdots &0&1&0\\0&0&\cdots &1&0&0\\\vdots &\vdots &&\vdots &\vdots &\vdots \\0&1&\cdots &0&0&0\\1&0&\cdots &0&0&0\end{pmatrix}}.$
Definition
If J is an n × n exchange matrix, then the elements of J are
$J_{i,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}$
Properties
• Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,
${\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}={\begin{pmatrix}7&8&9\\4&5&6\\1&2&3\end{pmatrix}}.$
• Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,
${\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}{\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}}={\begin{pmatrix}3&2&1\\6&5&4\\9&8&7\end{pmatrix}}.$
• Exchange matrices are symmetric; that is, JnT = Jn.
• For any integer k, Jnk = I if k is even and Jnk = Jn if k is odd. In particular, Jn is an involutory matrix; that is, Jn−1 = Jn.
• The trace of Jn is 1 if n is odd and 0 if n is even. In other words, the trace of Jn equals $n{\bmod {2}}$.
• The determinant of Jn equals $(-1)^{n(n-1)/2}$. As a function of n, it has period 4, giving 1, 1, −1, −1 when n is congruent modulo 4 to 0, 1, 2, and 3 respectively.
• The characteristic polynomial of Jn is $\det(\lambda I-J_{n})={\big (}(\lambda +1)(\lambda -1){\big )}^{n/2}$ when n is even, and $(\lambda -1)^{(n+1)/2}(\lambda +1)^{(n-1)/2}$ when n is odd.
• The adjugate matrix of Jn is $\operatorname {adj} (J_{n})=\operatorname {sgn}(\pi _{n})J_{n}$.
Relationships
• An exchange matrix is the simplest anti-diagonal matrix.
• Any matrix A satisfying the condition AJ = JA is said to be centrosymmetric.
• Any matrix A satisfying the condition AJ = JAT is said to be persymmetric.
• Symmetric matrices A that satisfy the condition AJ = JA are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.
See also
• Pauli matrices (the first Pauli matrix is a 2 × 2 exchange matrix)
References
1. Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis (2nd ed.), Cambridge University Press, p. 33, ISBN 9781139788885.
Matrix classes
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Reverse-delete algorithm
The reverse-delete algorithm is an algorithm in graph theory used to obtain a minimum spanning tree from a given connected, edge-weighted graph. It first appeared in Kruskal (1956), but it should not be confused with Kruskal's algorithm which appears in the same paper. If the graph is disconnected, this algorithm will find a minimum spanning tree for each disconnected part of the graph. The set of these minimum spanning trees is called a minimum spanning forest, which contains every vertex in the graph.
This algorithm is a greedy algorithm, choosing the best choice given any situation. It is the reverse of Kruskal's algorithm, which is another greedy algorithm to find a minimum spanning tree. Kruskal’s algorithm starts with an empty graph and adds edges while the Reverse-Delete algorithm starts with the original graph and deletes edges from it. The algorithm works as follows:
• Start with graph G, which contains a list of edges E.
• Go through E in decreasing order of edge weights.
• For each edge, check if deleting the edge will further disconnect the graph.
• Perform any deletion that does not lead to additional disconnection.
Pseudocode
function ReverseDelete(edges[] E) is
sort E in decreasing order
Define an index i ← 0
while i < size(E) do
Define edge ← E[i]
delete E[i]
if graph is not connected then
E[i] ← edge
i ← i + 1
return edges[] E
In the above the graph is the set of edges E with each edge containing a weight and connected vertices v1 and v2.
Example
In the following example green edges are being evaluated by the algorithm and red edges have been deleted.
This is our original graph. The numbers near the edges indicate their edge weight.
The algorithm will start with the maximum weighted edge, which in this case is DE with an edge weight of 15. Since deleting edge DE does not further disconnect the graph, it is deleted.
The next largest edge is FG so the algorithm will check if deleting this edge will further disconnect the graph. Since deleting the edge will not further disconnect the graph, the edge is then deleted.
The next largest edge is edge BD so the algorithm will check this edge and delete the edge.
The next edge to check is edge EG, which will not be deleted since it would disconnect node G from the graph. Therefore, the next edge to delete is edge BC.
The next largest edge is edge EF so the algorithm will check this edge and delete the edge.
The algorithm will then search the remaining edges and will not find another edge to delete; therefore this is the final graph returned by the algorithm.
Running time
The algorithm can be shown to run in O(E log V (log log V)3) time (using big-O notation), where E is the number of edges and V is the number of vertices. This bound is achieved as follows:
• Sorting the edges by weight using a comparison sort takes O(E log E) time, which can be simplified to O(E log V) using the fact that the largest E can be is V2.
• There are E iterations of the loop.
• Deleting an edge, checking the connectivity of the resulting graph, and (if it is disconnected) re-inserting the edge can be done in O(logV (log log V)3) time per operation (Thorup 2000).
Proof of correctness
It is recommended to read the proof of the Kruskal's algorithm first.
The proof consists of two parts. First, it is proved that the edges that remain after the algorithm is applied form a spanning tree. Second, it is proved that the spanning tree is of minimal weight.
Spanning tree
The remaining sub-graph (g) produced by the algorithm is not disconnected since the algorithm checks for that in line 7. The result sub-graph cannot contain a cycle since if it does then when moving along the edges we would encounter the max edge in the cycle and we would delete that edge. Thus g must be a spanning tree of the main graph G.
Minimality
We show that the following proposition P is true by induction: If F is the set of edges remained at the end of the while loop, then there is some minimum spanning tree that (its edges) are a subset of F.
1. Clearly P holds before the start of the while loop . since a weighted connected graph always has a minimum spanning tree and since F contains all the edges of the graph then this minimum spanning tree must be a subset of F.
2. Now assume P is true for some non-final edge set F and let T be a minimum spanning tree that is contained in F. we must show that after deleting edge e in the algorithm there exists some (possibly other) spanning tree T' that is a subset of F.
1. if the next deleted edge e doesn't belong to T then T=T' is a subset of F and P holds. .
2. otherwise, if e belongs to T: first note that the algorithm only removes the edges that do not cause a disconnectedness in the F. so e does not cause a disconnectedness. But deleting e causes a disconnectedness in tree T (since it is a member of T). assume e separates T into sub-graphs t1 and t2. Since the whole graph is connected after deleting e then there must exists a path between t1 and t2 (other than e) so there must exist a cycle C in the F (before removing e). now we must have another edge in this cycle (call it f) that is not in T but it is in F (since if all the cycle edges were in tree T then it would not be a tree anymore). we now claim that T' = T - e + f is the minimum spanning tree that is a subset of F.
3. firstly we prove that T' is a spanning tree . we know by deleting an edge in a tree and adding another edge that does not cause a cycle we get another tree with the same vertices. since T was a spanning tree so T' must be a spanning tree too. since adding " f " does not cause any cycles since "e" is removed.(note that tree T contains all the vertices of the graph).
4. secondly we prove T' is a minimum spanning tree . we have three cases for the edges "e" and " f ". wt is the weight function.
1. wt( f ) < wt( e ) this is impossible since this causes the weight of tree T' to be strictly less than T . since T is the minimum spanning tree, this is simply impossible.
2. wt( f ) > wt( e ) this is also impossible. since then when we are going through edges in decreasing order of edge weights we must see " f " first . since we have a cycle C so removing " f " would not cause any disconnectedness in the F. so the algorithm would have removed it from F earlier . so " f " does not exist in F which is impossible( we have proved f exists in step 4 .
3. so wt(f) = wt(e) so T' is also a minimum spanning tree. so again P holds.
3. so P holds when the while loop is done ( which is when we have seen all the edges ) and we proved at the end F becomes a spanning tree and we know F has a minimum spanning tree as its subset . so F must be the minimum spanning tree itself .
See also
• Kruskal's algorithm
• Prim's algorithm
• Borůvka's algorithm
• Dijkstra's algorithm
References
• Kleinberg, Jon; Tardos, Éva (2006), Algorithm Design, New York: Pearson Education, Inc..
• Kruskal, Joseph B. (1956), "On the shortest spanning subtree of a graph and the traveling salesman problem", Proceedings of the American Mathematical Society, 7 (1): 48–50, doi:10.2307/2033241, JSTOR 2033241.
• Thorup, Mikkel (2000), "Near-optimal fully-dynamic graph connectivity", Proc. 32nd ACM Symposium on Theory of Computing, pp. 343–350, doi:10.1145/335305.335345.
Graph and tree traversal algorithms
• α–β pruning
• A*
• IDA*
• LPA*
• SMA*
• Best-first search
• Beam search
• Bidirectional search
• Breadth-first search
• Lexicographic
• Parallel
• B*
• Depth-first search
• Iterative Deepening
• D*
• Fringe search
• Jump point search
• Monte Carlo tree search
• SSS*
Shortest path
• Bellman–Ford
• Dijkstra's
• Floyd–Warshall
• Johnson's
• Shortest path faster
• Yen's
Minimum spanning tree
• Borůvka's
• Kruskal's
• Prim's
• Reverse-delete
List of graph search algorithms
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Wikipedia
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Reverse-search algorithm
Reverse-search algorithms are a class of algorithms for generating all objects of a given size, from certain classes of combinatorial objects. In many cases, these methods allow the objects to be generated in polynomial time per object, using only enough memory to store a constant number of objects (polynomial space). (Generally, however, they are not classed as polynomial-time algorithms, because the number of objects they generate is exponential.) They work by organizing the objects to be generated into a spanning tree of their state space, and then performing a depth-first search of this tree.
Reverse-search algorithms were introduced by David Avis and Komei Fukuda in 1991, for problems of generating the vertices of convex polytopes and the cells of arrangements of hyperplanes.[1] They were formalized more broadly by Avis and Fukuda in 1996.[2]
Principles
A reverse-search algorithm generates the combinatorial objects in a state space, an implicit graph whose vertices are the objects to be listed and whose edges represent certain "local moves" connecting pairs of objects, typically by making small changes to their structure. It finds each objects using a depth-first search in a rooted spanning tree of this state space, described by the following information:[2]
• The root of the spanning tree, one of the objects
• A subroutine for generating the parent of each object in the tree, with the property that if repeated enough times it will eventually reach the root
• A subroutine for listing all of the neighbors in the state space (not all of which may be neighbors in the tree)
From this information it is possible to find the children of any given node in the tree, reversing the links given by the parent subroutine: they are simply the neighbors whose parent is the given node. It is these reversed links to child nodes that the algorithm searches.[2]
A classical depth-first search of this spanning tree would traverse the tree recursively, starting from the root, at each node listing all of the children and making a recursive call for each one. Unlike a depth-first search of a graph with cycles, it is not necessary to maintain the set of already-visited nodes to avoid repeated visits; such repetition is not possible in a tree. However, this recursive algorithm may still require a large amount of memory for its call stack, in cases when the tree is very deep. Instead, reverse search traverses the spanning tree in the same order while only storing two objects: the current object of the traversal, and the previously traversed object. Initially, the current object is set to the root of the tree, and there is no previous object. From this information, it is possible to determine the next step of the traversal by the following case analysis:[2]
• If there is no previous object, or the previous object is the parent of the current object, then this is the first time the traversal has reached the current object, so it is output from the search. The next object is its first child or, if it has no children, its parent.
• In all other cases, the previous object must be a child of the current object. The algorithm lists the children (that is, state-space neighbors of the current object that have the current object as their parent) one at a time until reaching this previous child, and then takes one more step in this list of children. If another child is found in this way, it is the next object. If there is no next child and the current object is not the root, the next object is the parent of the current object. In the remaining case, when there is no next child and the current object is the root, the reverse search terminates.
This algorithm involves listing the neighbors of an object once for each step in the search. However, if there are $N$ objects to be listed, then the search performs $2N-1$ steps, so the number of times it generates neighbors of objects is within a factor of two of the number of times the recursive depth-first search would do the same thing.[2]
Applications
Examples of the problems to which reverse search has been applied include the following combinatorial generation problems:
Vertices of simple convex polytopes
If a $d$-dimensional convex polytope is defined as an intersection of half-spaces, then its vertices can be described as the points of intersection of $d$ or more hyperplanes bounding the halfspaces; it is a simple polytope if no vertex is the intersection of more than $d$ of these hyperplanes. The vertex enumeration problem is the problem of listing all of these vertices. The edges of the polytope connect pairs of vertices that have $d-1$ hyperplanes in common, so the vertices and edges form a state space in which each vertex has $d$ neighbors. The simplex algorithm from the theory of linear programming finds a vertex maximizing a given linear function of the coordinates, by walking from vertex to vertex, choosing at each step a vertex with a greater value of the function; there are several standard choices of "pivot rule" that specify more precisely which vertex to choose. Any such pivot rule can be interpreted as defining the parent function of a spanning tree of the polytope, whose root is the optimal vertex. Applying reverse search to this data generates all vertices of the polytope. A similar algorithm can also enumerate all bases of a linear program, without requiring that it defines a polytope that is simple.[2][3]
Cells of hyperplane arrangements
A hyperplane arrangement decomposes Euclidean space into cells, each described by a "sign vector" that describes whether its points belong to one of the hyperplanes (sign 0), are on one side of the hyperplane (sign +1), or are on the other side (sign −1). The cells form a connected state space under local moves that change a single sign by one unit, and it is possible to check that this operation produces a valid cell by solving a linear programming feasibility problem. A spanning tree can be constructed for any choice of root cell by defining a parent operator that makes the first possible change that would bring the sign vector closer to that of the root. Using reverse search for this state space and parent operator produces an algorithm for listing all cells in polynomial time per cell.[2][4]
Point-set triangulations
The triangulations of a planar point set are connected by "flip" moves that remove one diagonal from a triangulation and replace it by another. If the Delaunay triangulation is chosen as the root, then every triangulation can be flipped to the Delaunay triangulation by steps in which the triangulation of some subset of four points is replaced by its Delaunay triangulation.[5][6] Choosing the first Delaunay flip as the parent of each triangulation, and applying local search, produces an algorithm for listing all triangulations in polynomial time per triangulation.[2]
Connected subgraphs
The connected subgraphs, and connected induced subgraphs, of a given connected graph form a state space whose local moves are the addition or removal of a single edge or vertex of the graph, respectively. A spanning tree of this state space can be obtained by adding the first edge or vertex (in some ordering of the edges or vertices) whose addition produces another connected subgraph; its root is the whole graph. Applying local search to this state space and parent operator produces an algorithm for listing all connected subgraphs in polynomial time per subgraph.[2]
Other applications include algorithms for generating the following structures:
• Polyominos,[7] polyiamond prototiles,[8] and polyhex (mathematics) hydrocarbon molecules.[9]
• Topological orderings of directed acyclic graphs, using a state space whose local moves reverse the ordering of two elements.[2]
• Spanning trees of graphs, non-crossing spanning trees of planar point sets, and more generally bases of matroids, using a state space that swaps one edge for another.[2]
• Euler tours in graphs.[10]
• The maximal independent sets of sparse graphs.[11]
• Maximal planar graphs[12] and polyhedral graphs.[13]
• Non-crossing minimally rigid graphs on a given point set.[14]
• Surrounding polygons, polygons that have some of a given set of points as vertices and surround the rest, using a state space that adds or removes one vertex of the polygon.[15]
• Vertices or facets of the Minkowski sum of convex polytopes.[16][17]
• The corners (multidegrees) of monomial ideals.[18]
References
1. Avis, David; Fukuda, Komei (1992), "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra", Discrete & Computational Geometry, 8 (3): 295–313, doi:10.1007/BF02293050, MR 1174359; preliminary version in Seventh Annual Symposium on Computational Geometry, 1991, doi:10.1145/109648.109659
2. Avis, David; Fukuda, Komei (1996), "Reverse search for enumeration", Discrete Applied Mathematics, 65 (1–3): 21–46, doi:10.1016/0166-218X(95)00026-N, MR 1380066
3. Avis, David (2000), "A revised implementation of the reverse search vertex enumeration algorithm", in Kalai, Gil; Ziegler, Günter M. (eds.), Polytopes—combinatorics and computation: Including papers from the DMV-Seminar "Polytopes and Optimization" held in Oberwolfach, November 1997, DMV Seminar, vol. 29, Basel: Birkhäuser, pp. 177–198, MR 1785299
4. Sleumer, Nora H. (1999), "Output-sensitive cell enumeration in hyperplane arrangements", Nordic Journal of Computing, 6 (2): 137–147, MR 1709978
5. Lawson, C. L. (1972), Generation of a triangular grid with applications to contour plotting, Memo 299, Jet Propulsion Laboratory
6. Sibson, R. (1973), "Locally equiangular triangulations", The Computer Journal, 21 (3): 243–245, doi:10.1093/comjnl/21.3.243, MR 0507358
7. Liang, Xiaodong; Wang, Rui; Meng, Ji xiang (2017), "Code for polyomino and computer search of isospectral polyominoes", Journal of Combinatorial Optimization, 33 (1): 254–264, doi:10.1007/s10878-015-9953-z, MR 3595411
8. Horiyama, Takashi; Yamane, Shogo (2010), "Generation of polyiamonds for p6 tiling by the reverse search", in Akiyama, Jin; Jiang, Bo; Kano, Mikio; Tan, Xuehou (eds.), Computational Geometry, Graphs and Applications - 9th International Conference, CGGA 2010, Dalian, China, November 3-6, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 7033, Springer, pp. 96–107, doi:10.1007/978-3-642-24983-9_10, MR 2927314
9. Caporossi, Gilles; Hansen, Pierre (May 1998), "Enumeration of polyhex hydrocarbons to $h=21$", Journal of Chemical Information and Computer Sciences, 38 (4): 610–619, doi:10.1021/ci970116n
10. Kurita, Kazuhiro; Wasa, Kunihiro (2022), "Constant amortized time enumeration of Eulerian trails", Theoretical Computer Science, 923: 1–12, doi:10.1016/j.tcs.2022.04.048, MR 4436557
11. Eppstein, David (2009), "All maximal independent sets and dynamic dominance for sparse graphs", ACM Transactions on Algorithms, 5 (4): A38:1–A38:14, arXiv:cs/0407036, doi:10.1145/1597036.1597042, MR 2571901
12. Avis, David (1996), "Generating rooted triangulations without repetitions", Algorithmica, 16 (6): 618–632, doi:10.1007/s004539900067, MR 1412663
13. Deza, Antoine; Fukuda, Komei; Rosta, Vera (1994), "Wagner's theorem and combinatorial enumeration of 3-polytopes", Proceedings of a symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, May 17–19, 1993, RIMS Kôkyûroku Bessatsu, vol. 872, pp. 30–34, MR 1330480
14. Avis, David; Katoh, Naoki; Ohsaki, Makoto; Streinu, Ileana; Tanigawa, Shin-ichi (June 2007), "Enumerating non-crossing minimally rigid frameworks" (PDF), Graphs and Combinatorics, 23 (S1): 117–134, doi:10.1007/s00373-007-0709-0
15. Yamanaka, Katsuhisa; Avis, David; Horiyama, Takashi; Okamoto, Yoshio; Uehara, Ryuhei; Yamauchi, Tanami (2021), "Algorithmic enumeration of surrounding polygons" (PDF), Discrete Applied Mathematics, 303: 305–313, doi:10.1016/j.dam.2020.03.034, MR 4310502
16. Fukuda, Komei (2004), "From the zonotope construction to the Minkowski addition of convex polytopes", Journal of Symbolic Computation, 38 (4): 1261–1272, doi:10.1016/j.jsc.2003.08.007, MR 2094220
17. Weibel, Christophe (2010), "Implementation and parallelization of a reverse-search algorithm for Minkowski sums", in Blelloch, Guy E.; Halperin, Dan (eds.), Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments, ALENEX 2010, Austin, Texas, USA, January 16, 2010, Society for Industrial and Applied Mathematics, pp. 34–42, doi:10.1137/1.9781611972900.4
18. Bayer, Dave; Taylor, Amelia (2009), "Reverse search for monomial ideals", Journal of Symbolic Computation, 44 (10): 1477–1486, doi:10.1016/j.jsc.2009.05.002, MR 2543431
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Wikipedia
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Bessel polynomials
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series[1]: 101
$y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,\left({\frac {x}{2}}\right)^{k}.$
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials[2]: 8 [3]: 15
$\theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,{\frac {x^{n-k}}{2^{k}}}.$
The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
$y_{3}(x)=15x^{3}+15x^{2}+6x+1$
while the third-degree reverse Bessel polynomial is
$\theta _{3}(x)=x^{3}+6x^{2}+15x+15.$
The reverse Bessel polynomial is used in the design of Bessel electronic filters.
Properties
Definition in terms of Bessel functions
The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.
$y_{n}(x)=\,x^{n}\theta _{n}(1/x)\,$
$y_{n}(x)={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{n+{\frac {1}{2}}}(1/x)$
$\theta _{n}(x)={\sqrt {\frac {2}{\pi }}}\,x^{n+1/2}e^{x}K_{n+{\frac {1}{2}}}(x)$
where Kn(x) is a modified Bessel function of the second kind, yn(x) is the ordinary polynomial, and θn(x) is the reverse polynomial .[2]: 7, 34 For example:[4]
$y_{3}(x)=15x^{3}+15x^{2}+6x+1={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{3+{\frac {1}{2}}}(1/x)$
Definition as a hypergeometric function
The Bessel polynomial may also be defined as a confluent hypergeometric function[5]: 8
$y_{n}(x)=\,_{2}F_{0}(-n,n+1;;-x/2)=\left({\frac {2}{x}}\right)^{-n}U\left(-n,-2n,{\frac {2}{x}}\right)=\left({\frac {2}{x}}\right)^{n+1}U\left(n+1,2n+2,{\frac {2}{x}}\right).$
A similar expression holds true for the generalized Bessel polynomials (see below):[2]: 35
$y_{n}(x;a,b)=\,_{2}F_{0}(-n,n+a-1;;-x/b)=\left({\frac {b}{x}}\right)^{n+a-1}U\left(n+a-1,2n+a,{\frac {b}{x}}\right).$
The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:
$\theta _{n}(x)={\frac {n!}{(-2)^{n}}}\,L_{n}^{-2n-1}(2x)$
from which it follows that it may also be defined as a hypergeometric function:
$\theta _{n}(x)={\frac {(-2n)_{n}}{(-2)^{n}}}\,\,_{1}F_{1}(-n;-2n;2x)$
where (−2n)n is the Pochhammer symbol (rising factorial).
Generating function
The Bessel polynomials, with index shifted, have the generating function
$\sum _{n=0}^{\infty }{\sqrt {\frac {2}{\pi }}}x^{n+{\frac {1}{2}}}e^{x}K_{n-{\frac {1}{2}}}(x){\frac {t^{n}}{n!}}=1+x\sum _{n=1}^{\infty }\theta _{n-1}(x){\frac {t^{n}}{n!}}=e^{x(1-{\sqrt {1-2t}})}.$
Differentiating with respect to $t$, cancelling $x$, yields the generating function for the polynomials $\{\theta _{n}\}_{n\geq 0}$
$\sum _{n=0}^{\infty }\theta _{n}(x){\frac {t^{n}}{n!}}={\frac {1}{\sqrt {1-2t}}}e^{x(1-{\sqrt {1-2t}})}.$
Similar generating function exists for the $y_{n}$ polynomials as well:[1]: 106
$\sum _{n=0}^{\infty }y_{n-1}(x){\frac {t^{n}}{n!}}=\exp \left({\frac {1-{\sqrt {1-2xt}}}{x}}\right).$
Upon setting $t=z-xz^{2}/2$, one has the following representation for the exponential function:[1]: 107
$e^{z}=\sum _{n=0}^{\infty }y_{n-1}(x){\frac {(z-xz^{2}/2)^{n}}{n!}}.$
Recursion
The Bessel polynomial may also be defined by a recursion formula:
$y_{0}(x)=1\,$
$y_{1}(x)=x+1\,$
$y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,$
and
$\theta _{0}(x)=1\,$
$\theta _{1}(x)=x+1\,$
$\theta _{n}(x)=(2n\!-\!1)\theta _{n-1}(x)+x^{2}\theta _{n-2}(x)\,$
Differential equation
The Bessel polynomial obeys the following differential equation:
$x^{2}{\frac {d^{2}y_{n}(x)}{dx^{2}}}+2(x\!+\!1){\frac {dy_{n}(x)}{dx}}-n(n+1)y_{n}(x)=0$
and
$x{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=0$
Orthogonality
The Bessel polynomials are orthogonal with respect to the weight $e^{-2/x}$ integrated over the unit circle of the complex plane.[1]: 104 In other words, if $n\neq m$,
$\int _{0}^{2\pi }y_{n}\left(e^{i\theta }\right)y_{m}\left(e^{i\theta }\right)ie^{i\theta }\mathrm {d} \theta =0$
Generalization
Explicit Form
A generalization of the Bessel polynomials have been suggested in literature, as following:
$y_{n}(x;\alpha ,\beta ):=(-1)^{n}n!\left({\frac {x}{\beta }}\right)^{n}L_{n}^{(-1-2n-\alpha )}\left({\frac {\beta }{x}}\right),$
the corresponding reverse polynomials are
$\theta _{n}(x;\alpha ,\beta ):={\frac {n!}{(-\beta )^{n}}}L_{n}^{(-1-2n-\alpha )}(\beta x)=x^{n}y_{n}\left({\frac {1}{x}};\alpha ,\beta \right).$
The explicit coefficients of the $y_{n}(x;\alpha ,\beta )$ polynomials are:[1]: 108
$y_{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(n+k+\alpha -2)^{\underline {k}}\left({\frac {x}{\beta }}\right)^{k}.$
Consequently, the $\theta _{n}(x;\alpha ,\beta )$ polynomials can explicitly be written as follows:
$\theta _{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(2n-k+\alpha -2)^{\underline {n-k}}{\frac {x^{k}}{\beta ^{n-k}}}.$
For the weighting function
$\rho (x;\alpha ,\beta ):=\,_{1}F_{1}\left(1,\alpha -1,-{\frac {\beta }{x}}\right)$
they are orthogonal, for the relation
$0=\oint _{c}\rho (x;\alpha ,\beta )y_{n}(x;\alpha ,\beta )y_{m}(x;\alpha ,\beta )\mathrm {d} x$
holds for m ≠ n and c a curve surrounding the 0 point.
They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(x) = exp(−2 / x).
Rodrigues formula for Bessel polynomials
The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :
$B_{n}^{(\alpha ,\beta )}(x)={\frac {a_{n}^{(\alpha ,\beta )}}{x^{\alpha }e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})$
where a(α, β)
n
are normalization coefficients.
Associated Bessel polynomials
According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
$x^{2}{\frac {d^{2}B_{n,m}^{(\alpha ,\beta )}(x)}{dx^{2}}}+[(\alpha +2)x+\beta ]{\frac {dB_{n,m}^{(\alpha ,\beta )}(x)}{dx}}-\left[n(\alpha +n+1)+{\frac {m\beta }{x}}\right]B_{n,m}^{(\alpha ,\beta )}(x)=0$
where $0\leq m\leq n$. The solutions are,
$B_{n,m}^{(\alpha ,\beta )}(x)={\frac {a_{n,m}^{(\alpha ,\beta )}}{x^{\alpha +m}e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n-m}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})$
Zeros
If one denotes the zeros of $y_{n}(x;\alpha ,\beta )$ as $\alpha _{k}^{(n)}(\alpha ,\beta )$, and that of the $\theta _{n}(x;\alpha ,\beta )$ by $\beta _{k}^{(n)}(\alpha ,\beta )$, then the following estimates exist:[2]: 82
${\frac {2}{n(n+\alpha -1)}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}},$
and
${\frac {n+\alpha -1}{2}}\leq \beta _{k}^{(n)}(\alpha ,2)\leq {\frac {n(n+\alpha -1)}{2}},$
for all $\alpha \geq 2$. Moreover, all these zeros have negative real part.
Sharper results can be said if one resorts to more powerful theorems regarding the estimates of zeros of polynomials (more concretely, the Parabola Theorem of Saff and Varga, or differential equations techniques).[2]: 88 [6] One result is the following:[7]
${\frac {2}{2n+\alpha -{\frac {2}{3}}}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}}.$
Particular values
The Bessel polynomials $y_{n}(x)$ up to $n=5$ are[8]
${\begin{aligned}y_{0}(x)&=1\\y_{1}(x)&=x+1\\y_{2}(x)&=3x^{2}+3x+1\\y_{3}(x)&=15x^{3}+15x^{2}+6x+1\\y_{4}(x)&=105x^{4}+105x^{3}+45x^{2}+10x+1\\y_{5}(x)&=945x^{5}+945x^{4}+420x^{3}+105x^{2}+15x+1\end{aligned}}$
No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.[9] The reverse Bessel polynomials are obtained by reversing the coefficients. Equivalently, $ \theta _{k}(x)=x^{k}y_{k}(1/x)$. This results in the following:
${\begin{aligned}\theta _{0}(x)&=1\\\theta _{1}(x)&=x+1\\\theta _{2}(x)&=x^{2}+3x+3\\\theta _{3}(x)&=x^{3}+6x^{2}+15x+15\\\theta _{4}(x)&=x^{4}+10x^{3}+45x^{2}+105x+105\\\theta _{5}(x)&=x^{5}+15x^{4}+105x^{3}+420x^{2}+945x+945\\\end{aligned}}$
See also
• Bessel function
• Neumann polynomial
• Lommel polynomial
• Hankel transform
• Fourier–Bessel series
References
1. Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516.
2. Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4.
3. Berg, Christian; Vignat, Christophe (2008). "Linearization coefficients of Bessel polynomials and properties of Student-t distributions" (PDF). Constructive Approximation. 27: 15–32. doi:10.1007/s00365-006-0643-6. Retrieved 2006-08-16.
4. Wolfram Alpha example
5. Dita, Petre; Grama, Nicolae (May 14, 1997). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008.
6. Saff, E. B.; Varga, R. S. (1976). "Zero-free parabolic regions for sequences of polynomials". SIAM J. Math. Anal. 7 (3): 344–357. doi:10.1137/0507028.
7. de Bruin, M. G.; Saff, E. B.; Varga, R. S. (1981). "On the zeros of generalized Bessel polynomials. I". Indag. Math. 84 (1): 1–13.
• Sloane, N. J. A. (ed.). "Sequence A001498 (Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik. 2002 (550): 125–140. CiteSeerX 10.1.1.6.9538. doi:10.1515/crll.2002.069.
• Carlitz, Leonard (1957). "A Note on the Bessel Polynomials". Duke Math. J. 24 (2): 151–162. doi:10.1215/S0012-7094-57-02421-3. MR 0085360.
• Fakhri, H.; Chenaghlou, A. (2006). "Ladder operators and recursion relations for the associated Bessel polynomials". Physics Letters A. 358 (5–6): 345–353. Bibcode:2006PhLA..358..345F. doi:10.1016/j.physleta.2006.05.070.
• Roman, S. (1984). The Umbral Calculus (The Bessel Polynomials §4.1.7). New York: Academic Press. ISBN 978-0-486-44139-9.
External links
• "Bessel polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Bessel Polynomial". MathWorld.
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Wikipedia
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Cuthill–McKee algorithm
In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee,[1] is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse Cuthill–McKee algorithm (RCM) due to Alan George and Joseph Liu is the same algorithm but with the resulting index numbers reversed.[2] In practice this generally results in less fill-in than the CM ordering when Gaussian elimination is applied.[3]
The Cuthill McKee algorithm is a variant of the standard breadth-first search algorithm used in graph algorithms. It starts with a peripheral node and then generates levels $R_{i}$ for $i=1,2,..$ until all nodes are exhausted. The set $R_{i+1}$ is created from set $R_{i}$ by listing all vertices adjacent to all nodes in $R_{i}$. These nodes are ordered according to predecessors and degree.
Algorithm
Given a symmetric $n\times n$ matrix we visualize the matrix as the adjacency matrix of a graph. The Cuthill–McKee algorithm is then a relabeling of the vertices of the graph to reduce the bandwidth of the adjacency matrix.
The algorithm produces an ordered n-tuple $R$ of vertices which is the new order of the vertices.
First we choose a peripheral vertex (the vertex with the lowest degree) $x$ and set $R:=(\{x\})$.
Then for $i=1,2,\dots $ we iterate the following steps while $|R|<n$
• Construct the adjacency set $A_{i}$ of $R_{i}$ (with $R_{i}$ the i-th component of $R$) and exclude the vertices we already have in $R$
$A_{i}:=\operatorname {Adj} (R_{i})\setminus R$
• Sort $A_{i}$ ascending by minimum predecessor (the already-visited neighbor with the earliest position in R), and as a tiebreak ascending by vertex degree.[4]
• Append $A_{i}$ to the Result set $R$.
In other words, number the vertices according to a particular level structure (computed by breadth-first search) where the vertices in each level are visited in order of their predecessor's numbering from lowest to highest. Where the predecessors are the same, vertices are distinguished by degree (again ordered from lowest to highest).
See also
• Graph bandwidth
• Sparse matrix
References
1. E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices In Proc. 24th Nat. Conf. ACM, pages 157–172, 1969.
2. "Ciprian Zavoianu - weblog: Tutorial: Bandwidth reduction - The CutHill-McKee Algorithm".
3. J. A. George and J. W-H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981
4. The Reverse Cuthill-McKee Algorithm in Distributed-Memory , slide 8, 2016
• Cuthill–McKee documentation for the Boost C++ Libraries.
• A detailed description of the Cuthill–McKee algorithm.
• symrcm MATLAB's implementation of RCM.
• reverse_cuthill_mckee RCM routine from SciPy written in Cython.
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Wikipedia
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Fatou's lemma
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.
Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.
Standard statement
In what follows, $\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}$ denotes the $\sigma $-algebra of Borel sets on $[0,+\infty ]$.
Theorem — Fatou's lemma. Given a measure space $(\Omega ,{\mathcal {F}},\mu )$ and a set $X\in {\mathcal {F}},$ let $\{f_{n}\}$ be a sequence of $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$-measurable non-negative functions $f_{n}:X\to [0,+\infty ]$. Define the function $f:X\to [0,+\infty ]$ by setting $f(x)=\liminf _{n\to \infty }f_{n}(x),$ for every $x\in X$.
Then $f$ is $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$-measurable, and also $\int _{X}f\,d\mu \leq \liminf _{n\to \infty }\int _{X}f_{n}\,d\mu $, where the integrals may be infinite.
Fatou's lemma remains true if its assumptions hold $\mu $-almost everywhere. In other words, it is enough that there is a null set $N$ such that the values $\{f_{n}(x)\}$ are non-negative for every ${x\in X\setminus N}.$ To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on $N$.
Proof
Fatou's lemma does not require the monotone convergence theorem, but the latter can be used to provide a quick proof. A proof directly from the definitions of integrals is given further below.
In each case, the proof begins by analyzing the properties of $\textstyle g_{n}(x)=\inf _{k\geq n}f_{k}(x)$. These satisfy:
1. the sequence $\{g_{n}(x)\}_{n}$ is pointwise non-decreasing at any x and
2. $g_{n}\leq f_{n}$, $\forall n\in \mathbb {N} $.
Since $f(x)=\liminf _{n\to \infty }f_{n}(x)=\lim _{n\to \infty }\inf _{k\geq n}f_{k}(x)=\lim _{n\to \infty }{g_{n}(x)}$, we immediately see that f is measurable.
Via the Monotone Convergence Theorem
Moreover,
$\int _{X}f\,d\mu =\int _{X}\lim _{n\to \infty }g_{n}\,d\mu $
By the Monotone Convergence Theorem and property (1), the limit and integral may be interchanged:
${\begin{aligned}\int _{X}f\,d\mu &=\lim _{n\to \infty }\int _{X}g_{n}\,d\mu \\&=\liminf _{n\to \infty }\int _{X}g_{n}\,d\mu \\&\leq \liminf _{n\to \infty }\int _{X}f_{n}\,d\mu ,\end{aligned}}$
where the last step used property (2).
From "first principles"
To demonstrate that the monotone convergence theorem is not "hidden", the proof below does not use any properties of Lebesgue integral except those established here.
Denote by $\operatorname {SF} (f)$ the set of simple $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$-measurable functions $s:X\to [0,\infty )$ such that $0\leq s\leq f$ on $X$.
Monotonicity —
• If $f\leq g$ everywhere on $X,$ then
$\int _{X}f\,d\mu \leq \int _{X}g\,d\mu .$
• If $X_{1},X_{2}\in {\mathcal {F}}$ and $X_{1}\subseteq X_{2},$ then
$\int _{X_{1}}f\,d\mu \leq \int _{X_{2}}f\,d\mu .$
• If f is nonnegative and $S=\cup _{i=1}^{\infty }S_{i}$, where $S_{1}\subseteq \ldots \subseteq S_{i}\subseteq \ldots \subseteq S$ is a non-decreasing chain of $\mu $-measurable sets, then
$\int _{S}{f\,d\mu }=\lim _{n\to \infty }{\int _{S_{n}}{f\,d\mu }}$
Proof
1. Since $f\leq g,$ we have
$\operatorname {SF} (f)\subseteq \operatorname {SF} (g).$
By definition of Lebesgue integral and the properties of supremum,
$\int _{X}f\,d\mu =\sup _{s\in {\rm {SF}}(f)}\int _{X}s\,d\mu \leq \sup _{s\in {\rm {SF}}(g)}\int _{X}s\,d\mu =\int _{X}g\,d\mu .$
2. Let ${\mathbf {1} }_{X_{1}}$ be the indicator function of the set $X_{1}.$ It can be deduced from the definition of Lebesgue integral that
$\int _{X_{2}}f\cdot {\mathbf {1} }_{X_{1}}\,d\mu =\int _{X_{1}}f\,d\mu $
if we notice that, for every $s\in {\rm {SF}}(f\cdot {\mathbf {1} }_{X_{1}}),$ $s=0$ outside of $X_{1}.$ Combined with the previous property, the inequality $f\cdot {\mathbf {1} }_{X_{1}}\leq f$ implies
$\int _{X_{1}}f\,d\mu =\int _{X_{2}}f\cdot {\mathbf {1} }_{X_{1}}\,d\mu \leq \int _{X_{2}}f\,d\mu .$
3. First note that the claim holds if f is the indicator function of a set, by monotonicity of measures. By linearity, this also immediately implies the claim for simple functions.
Since any simple function supported on Sn is simple and supported on X, we must have
$\int _{X}{f\,d\mu }\geq \lim _{n\to \infty }{\int _{S_{n}}{f\,d\mu }}$.
For the reverse, suppose g ∈ SF(f) with $\textstyle \int _{X}{f\,d\mu }-\epsilon \leq \int _{X}{g\,d\mu }$ By the above,
$\int _{X}{f\,d\mu }-\epsilon \leq \int _{X}{g\,d\mu }=\lim _{n\to \infty }{\int _{S_{n}}{g\,d\mu }}\leq \lim _{n\to \infty }{\int _{S_{n}}{f\,d\mu }}$
Now we turn to the main theorem
Step 1 — $g_{n}=g_{n}(x)$ is $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$-measurable, for every $n\geq 1$, as is $f$.
Proof
Recall the closed intervals generate the Borel σ-algebra. Thus it suffices to show, for every $t\in [-\infty ,+\infty ]$, that $g_{n}^{-1}([t,+\infty ])\in {\mathcal {F}}$. Now observe that
${\begin{aligned}g_{n}^{-1}([t,+\infty ])&=\left\{x\in X\mid g_{n}(x)\geq t\right\}\\[3pt]&=\left\{x\in X\;\left|\;\inf _{k\,\geq \,n}f_{k}(x)\geq t\right.\right\}\\[3pt]&=\bigcap _{k\,\geq \,n}\left\{x\in X\mid f_{k}(x)\geq t\right\}\\[3pt]&=\bigcap _{k\,\geq \,n}f_{k}^{-1}([t,+\infty ])\end{aligned}}$
Every set on the right-hand side is from ${\mathcal {F}}$, which is closed under countable intersections. Thus the left-hand side is also a member of ${\mathcal {F}}$.
Similarly, it is enough to verify that $f^{-1}([0,t])\in {\mathcal {F}}$, for every $t\in [-\infty ,+\infty ]$. Since the sequence $\{g_{n}(x)\}$ pointwise non-decreases,
$f^{-1}([0,t])=\bigcap _{n}g_{n}^{-1}([0,t])\in {\mathcal {F}}$.
Step 2 — Given a simple function $s\in \operatorname {SF} (f)$ and a real number $t\in (0,1)$, define
$B_{k}^{s,t}=\{x\in X\mid t\cdot s(x)\leq g_{k}(x)\}\subseteq X.$
Then $B_{k}^{s,t}\in {\mathcal {F}}$, $B_{k}^{s,t}\subseteq B_{k+1}^{s,t}$, and $\textstyle X=\bigcup _{k}B_{k}^{s,t}$.
Proof
Step 2a. To prove the first claim, write s as a weighted sum of indicator functions of disjoint sets:
$s=\sum _{i=1}^{m}c_{i}\cdot \mathbf {1} _{A_{i}}$.
Then
$B_{k}^{s,t}=\bigcup _{i=1}^{m}{\Bigl (}g_{k}^{-1}{\Bigl (}[t\cdot c_{i},+\infty ]{\Bigr )}\cap A_{i}{\Bigr )}$.
Since the pre-image $g_{k}^{-1}{\Bigl (}[t\cdot c_{i},+\infty ]{\Bigr )}$ of the Borel set $[t\cdot c_{i},+\infty ]$ under the measurable function $g_{k}$ is measurable, and $\sigma $-algebras are closed under finite intersection and unions, the first claim follows.
Step 2b. To prove the second claim, note that, for each $k$ and every $x\in X$, $g_{k}(x)\leq g_{k+1}(x).$
Step 2c. To prove the third claim, suppose for contradiction there exists
$x_{0}\in X\setminus \bigcup _{k}B_{k}^{s,t}=\bigcap _{k}(X\setminus B_{k}^{s,t})$
Then $g_{k}(x_{0})<t\cdot s(x_{0})$, for every $k$. Taking the limit as $k\to \infty $,
$f(x_{0})\leq t\cdot s(x_{0})<s(x_{0}).$
This contradicts our initial assumption that $s\leq f$.
Step 3 — From step 2 and monotonicity,
$\lim _{n}\int _{B_{n}^{s,t}}s\,d\mu =\int _{X}s\,d\mu .$
Step 4 — For every $s\in \operatorname {SF} (f)$,
$\int _{X}s\,d\mu \leq \lim _{k}\int _{X}g_{k}\,d\mu $.
Proof
Indeed, using the definition of $B_{k}^{s,t}$, the non-negativity of $g_{k}$, and the monotonicity of Lebesgue integral, we have
$\forall k\geq 1\qquad \int _{B_{k}^{s,t}}t\cdot s\,d\mu \leq \int _{B_{k}^{s,t}}g_{k}\,d\mu \leq \int _{X}g_{k}\,d\mu $.
In accordance with Step 4, as $k\to \infty $ the inequality becomes
$t\int _{X}s\,d\mu \leq \lim _{k}\int _{X}g_{k}\,d\mu $.
Taking the limit as $t\uparrow 1$ yields
$\int _{X}s\,d\mu \leq \lim _{k}\int _{X}g_{k}\,d\mu $,
as required.
Step 5 — To complete the proof, we apply the definition of Lebesgue integral to the inequality established in Step 4 and take into account that $g_{n}\leq f_{n}$:
${\begin{aligned}\int _{X}f\,d\mu &=\sup _{s\in \operatorname {SF} (f)}\int _{X}s\,d\mu \\&\leq \lim _{k}\int _{X}g_{k}\,d\mu \\&=\liminf _{k}\int _{X}g_{k}\,d\mu \\&\leq \liminf _{k}\int _{X}f_{k}\,d\mu \end{aligned}}$
The proof is complete.
Examples for strict inequality
Equip the space $S$ with the Borel σ-algebra and the Lebesgue measure.
• Example for a probability space: Let $S=[0,1]$ denote the unit interval. For every natural number $n$ define
$f_{n}(x)={\begin{cases}n&{\text{for }}x\in (0,1/n),\\0&{\text{otherwise.}}\end{cases}}$
• Example with uniform convergence: Let $S$ denote the set of all real numbers. Define
$f_{n}(x)={\begin{cases}{\frac {1}{n}}&{\text{for }}x\in [0,n],\\0&{\text{otherwise.}}\end{cases}}$
These sequences $(f_{n})_{n\in \mathbb {N} }$ converge on $S$ pointwise (respectively uniformly) to the zero function (with zero integral), but every $f_{n}$ has integral one.
The role of non-negativity
A suitable assumption concerning the negative parts of the sequence f1, f2, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let S denote the half line [0,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number n define
$f_{n}(x)={\begin{cases}-{\frac {1}{n}}&{\text{for }}x\in [n,2n],\\0&{\text{otherwise.}}\end{cases}}$
This sequence converges uniformly on S to the zero function and the limit, 0, is reached in a finite number of steps: for every x ≥ 0, if n > x, then fn(x) = 0. However, every function fn has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
As discussed in § Extensions and variations of Fatou's lemma below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
Reverse Fatou lemma
Let f1, f2, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a non-negative integrable function g on S such that fn ≤ g for all n, then
$\limsup _{n\to \infty }\int _{S}f_{n}\,d\mu \leq \int _{S}\limsup _{n\to \infty }f_{n}\,d\mu .$
Note: Here g integrable means that g is measurable and that $\textstyle \int _{S}g\,d\mu <\infty $.
Sketch of proof
We apply linearity of Lebesgue integral and Fatou's lemma to the sequence $g-f_{n}.$ Since $\textstyle \int _{S}g\,d\mu <+\infty ,$ this sequence is defined $\mu $-almost everywhere and non-negative.
Extensions and variations of Fatou's lemma
Integrable lower bound
Let f1, f2, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists an integrable function g on S such that fn ≥ −g for all n, then
$\int _{S}\liminf _{n\to \infty }f_{n}\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu .$
Proof
Apply Fatou's lemma to the non-negative sequence given by fn + g.
Pointwise convergence
If in the previous setting the sequence f1, f2, . . . converges pointwise to a function f μ-almost everywhere on S, then
$\int _{S}f\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu \,.$
Proof
Note that f has to agree with the limit inferior of the functions fn almost everywhere, and that the values of the integrand on a set of measure zero have no influence on the value of the integral.
Convergence in measure
The last assertion also holds, if the sequence f1, f2, . . . converges in measure to a function f.
Proof
There exists a subsequence such that
$\lim _{k\to \infty }\int _{S}f_{n_{k}}\,d\mu =\liminf _{n\to \infty }\int _{S}f_{n}\,d\mu .$
Since this subsequence also converges in measure to f, there exists a further subsequence, which converges pointwise to f almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence.
Fatou's Lemma with Varying Measures
In all of the above statements of Fatou's Lemma, the integration was carried out with respect to a single fixed measure μ. Suppose that μn is a sequence of measures on the measurable space (S,Σ) such that (see Convergence of measures)
$\mu _{n}(E)\to \mu (E),~\forall E\in {\mathcal {F}}.$
Then, with fn non-negative integrable functions and f being their pointwise limit inferior, we have
$\int _{S}f\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu _{n}.$
Proof
We will prove something a bit stronger here. Namely, we will allow fn to converge μ-almost everywhere on a subset E of S. We seek to show that
$\int _{E}f\,d\mu \leq \liminf _{n\to \infty }\int _{E}f_{n}\,d\mu _{n}\,.$
Let
$K=\{x\in E|f_{n}(x)\rightarrow f(x)\}$.
Then μ(E-K)=0 and
$\int _{E}f\,d\mu =\int _{E-K}f\,d\mu ,~~~\int _{E}f_{n}\,d\mu =\int _{E-K}f_{n}\,d\mu ~\forall n\in \mathbb {N} .$
Thus, replacing E by E-K we may assume that fn converge to f pointwise on E. Next, note that for any simple function φ we have
$\int _{E}\phi \,d\mu =\lim _{n\to \infty }\int _{E}\phi \,d\mu _{n}.$
Hence, by the definition of the Lebesgue Integral, it is enough to show that if φ is any non-negative simple function less than or equal to f, then
$\int _{E}\phi \,d\mu \leq \liminf _{n\rightarrow \infty }\int _{E}f_{n}\,d\mu _{n}$
Let a be the minimum non-negative value of φ. Define
$A=\{x\in E|\phi (x)>a\}$
We first consider the case when $\int _{E}\phi \,d\mu =\infty $. We must have that μ(A) is infinite since
$\int _{E}\phi \,d\mu \leq M\mu (A),$
where M is the (necessarily finite) maximum value of that φ attains.
Next, we define
$A_{n}=\{x\in E|f_{k}(x)>a~\forall k\geq n\}.$
We have that
$A\subseteq \bigcup _{n}A_{n}\Rightarrow \mu (\bigcup _{n}A_{n})=\infty .$
But An is a nested increasing sequence of functions and hence, by the continuity from below μ,
$\lim _{n\rightarrow \infty }\mu (A_{n})=\infty .$.
Thus,
$\lim _{n\to \infty }\mu _{n}(A_{n})=\mu (A_{n})=\infty .$.
At the same time,
$\int _{E}f_{n}\,d\mu _{n}\geq a\mu _{n}(A_{n})\Rightarrow \liminf _{n\to \infty }\int _{E}f_{n}\,d\mu _{n}=\infty =\int _{E}\phi \,d\mu ,$
proving the claim in this case.
The remaining case is when $\int _{E}\phi \,d\mu <\infty $. We must have that μ(A) is finite. Denote, as above, by M the maximum value of φ and fix ε>0. Define
$A_{n}=\{x\in E|f_{k}(x)>(1-\epsilon )\phi (x)~\forall k\geq n\}.$
Then An is a nested increasing sequence of sets whose union contains A. Thus, A-An is a decreasing sequence of sets with empty intersection. Since A has finite measure (this is why we needed to consider the two separate cases),
$\lim _{n\rightarrow \infty }\mu (A-A_{n})=0.$
Thus, there exists n such that
$\mu (A-A_{k})<\epsilon ,~\forall k\geq n.$
Therefore, since
$\lim _{n\to \infty }\mu _{n}(A-A_{k})=\mu (A-A_{k}),$
there exists N such that
$\mu _{k}(A-A_{k})<\epsilon ,~\forall k\geq N.$
Hence, for $k\geq N$
$\int _{E}f_{k}\,d\mu _{k}\geq \int _{A_{k}}f_{k}\,d\mu _{k}\geq (1-\epsilon )\int _{A_{k}}\phi \,d\mu _{k}.$
At the same time,
$\int _{E}\phi \,d\mu _{k}=\int _{A}\phi \,d\mu _{k}=\int _{A_{k}}\phi \,d\mu _{k}+\int _{A-A_{k}}\phi \,d\mu _{k}.$
Hence,
$(1-\epsilon )\int _{A_{k}}\phi \,d\mu _{k}\geq (1-\epsilon )\int _{E}\phi \,d\mu _{k}-\int _{A-A_{k}}\phi \,d\mu _{k}.$
Combining these inequalities gives that
$\int _{E}f_{k}\,d\mu _{k}\geq (1-\epsilon )\int _{E}\phi \,d\mu _{k}-\int _{A-A_{k}}\phi \,d\mu _{k}\geq \int _{E}\phi \,d\mu _{k}-\epsilon \left(\int _{E}\phi \,d\mu _{k}+M\right).$
Hence, sending ε to 0 and taking the liminf in n, we get that
$\liminf _{n\rightarrow \infty }\int _{E}f_{n}\,d\mu _{k}\geq \int _{E}\phi \,d\mu ,$
completing the proof.
Fatou's lemma for conditional expectations
In probability theory, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables X1, X2, . . . defined on a probability space $\scriptstyle (\Omega ,\,{\mathcal {F}},\,\mathbb {P} )$; the integrals turn into expectations. In addition, there is also a version for conditional expectations.
Standard version
Let X1, X2, . . . be a sequence of non-negative random variables on a probability space $\scriptstyle (\Omega ,{\mathcal {F}},\mathbb {P} )$ and let $\scriptstyle {\mathcal {G}}\,\subset \,{\mathcal {F}}$ be a sub-σ-algebra. Then
$\mathbb {E} {\Bigl [}\liminf _{n\to \infty }X_{n}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}\leq \liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}]$ almost surely.
Note: Conditional expectation for non-negative random variables is always well defined, finite expectation is not needed.
Proof
Besides a change of notation, the proof is very similar to the one for the standard version of Fatou's lemma above, however the monotone convergence theorem for conditional expectations has to be applied.
Let X denote the limit inferior of the Xn. For every natural number k define pointwise the random variable
$Y_{k}=\inf _{n\geq k}X_{n}.$
Then the sequence Y1, Y2, . . . is increasing and converges pointwise to X. For k ≤ n, we have Yk ≤ Xn, so that
$\mathbb {E} [Y_{k}|{\mathcal {G}}]\leq \mathbb {E} [X_{n}|{\mathcal {G}}]$ almost surely
by the monotonicity of conditional expectation, hence
$\mathbb {E} [Y_{k}|{\mathcal {G}}]\leq \inf _{n\geq k}\mathbb {E} [X_{n}|{\mathcal {G}}]$ almost surely,
because the countable union of the exceptional sets of probability zero is again a null set. Using the definition of X, its representation as pointwise limit of the Yk, the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely
${\begin{aligned}\mathbb {E} {\Bigl [}\liminf _{n\to \infty }X_{n}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}&=\mathbb {E} [X|{\mathcal {G}}]=\mathbb {E} {\Bigl [}\lim _{k\to \infty }Y_{k}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}=\lim _{k\to \infty }\mathbb {E} [Y_{k}|{\mathcal {G}}]\\&\leq \lim _{k\to \infty }\inf _{n\geq k}\mathbb {E} [X_{n}|{\mathcal {G}}]=\liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}].\end{aligned}}$
Extension to uniformly integrable negative parts
Let X1, X2, . . . be a sequence of random variables on a probability space $\scriptstyle (\Omega ,{\mathcal {F}},\mathbb {P} )$ and let $\scriptstyle {\mathcal {G}}\,\subset \,{\mathcal {F}}$ be a sub-σ-algebra. If the negative parts
$X_{n}^{-}:=\max\{-X_{n},0\},\qquad n\in {\mathbb {N} },$
are uniformly integrable with respect to the conditional expectation, in the sense that, for ε > 0 there exists a c > 0 such that
$\mathbb {E} {\bigl [}X_{n}^{-}1_{\{X_{n}^{-}>c\}}\,|\,{\mathcal {G}}{\bigr ]}<\varepsilon ,\qquad {\text{for all }}n\in \mathbb {N} ,\,{\text{almost surely}}$,
then
$\mathbb {E} {\Bigl [}\liminf _{n\to \infty }X_{n}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}\leq \liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}]$ almost surely.
Note: On the set where
$X:=\liminf _{n\to \infty }X_{n}$
satisfies
$\mathbb {E} [\max\{X,0\}\,|\,{\mathcal {G}}]=\infty ,$
the left-hand side of the inequality is considered to be plus infinity. The conditional expectation of the limit inferior might not be well defined on this set, because the conditional expectation of the negative part might also be plus infinity.
Proof
Let ε > 0. Due to uniform integrability with respect to the conditional expectation, there exists a c > 0 such that
$\mathbb {E} {\bigl [}X_{n}^{-}1_{\{X_{n}^{-}>c\}}\,|\,{\mathcal {G}}{\bigr ]}<\varepsilon \qquad {\text{for all }}n\in \mathbb {N} ,\,{\text{almost surely}}.$
Since
$X+c\leq \liminf _{n\to \infty }(X_{n}+c)^{+},$
where x+ := max{x,0} denotes the positive part of a real x, monotonicity of conditional expectation (or the above convention) and the standard version of Fatou's lemma for conditional expectations imply
$\mathbb {E} [X\,|\,{\mathcal {G}}]+c\leq \mathbb {E} {\Bigl [}\liminf _{n\to \infty }(X_{n}+c)^{+}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}\leq \liminf _{n\to \infty }\mathbb {E} [(X_{n}+c)^{+}\,|\,{\mathcal {G}}]$ almost surely.
Since
$(X_{n}+c)^{+}=(X_{n}+c)+(X_{n}+c)^{-}\leq X_{n}+c+X_{n}^{-}1_{\{X_{n}^{-}>c\}},$
we have
$\mathbb {E} [(X_{n}+c)^{+}\,|\,{\mathcal {G}}]\leq \mathbb {E} [X_{n}\,|\,{\mathcal {G}}]+c+\varepsilon $ almost surely,
hence
$\mathbb {E} [X\,|\,{\mathcal {G}}]\leq \liminf _{n\to \infty }\mathbb {E} [X_{n}\,|\,{\mathcal {G}}]+\varepsilon $ almost surely.
This implies the assertion.
References
• Carothers, N. L. (2000). Real Analysis. New York: Cambridge University Press. pp. 321–22. ISBN 0-521-49756-6.
• Royden, H. L. (1988). Real Analysis (3rd ed.). London: Collier Macmillan. ISBN 0-02-404151-3.
• Weir, Alan J. (1973). "The Convergence Theorems". Lebesgue Integration and Measure. Cambridge: Cambridge University Press. pp. 93–118. ISBN 0-521-08728-7.
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Wikipedia
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Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.
Hölder's inequality — Let (S, Σ, μ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1. Then for all measurable real- or complex-valued functions f and g on S,
$\|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}.$
If, in addition, p, q ∈ (1, ∞) and f ∈ Lp(μ) and g ∈ Lq(μ), then Hölder's inequality becomes an equality if and only if |f |p and |g|q are linearly dependent in L1(μ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α|f |p = β |g|q μ-almost everywhere.
The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality.[1] Hölder's inequality holds even if ‖fg‖1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in Lp(μ) and g is in Lq(μ), then the pointwise product fg is in L1(μ).
Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ), and also to establish that Lq(μ) is the dual space of Lp(μ) for p ∈ [1, ∞).
Hölder's inequality (in a slightly different form) was first found by Leonard James Rogers (1888). Inspired by Rogers' work, Hölder (1889) gave another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality,[2] which was in turn named for work of Johan Jensen building on Hölder's work.[3]
Remarks
Conventions
The brief statement of Hölder's inequality uses some conventions.
• In the definition of Hölder conjugates, 1/∞ means zero.
• If p, q ∈ [1, ∞), then ‖f ‖p and ‖g‖q stand for the (possibly infinite) expressions
${\begin{aligned}&\left(\int _{S}|f|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p}}\\&\left(\int _{S}|g|^{q}\,\mathrm {d} \mu \right)^{\frac {1}{q}}\end{aligned}}$
• If p = ∞, then ‖f ‖∞ stands for the essential supremum of |f |, similarly for ‖g‖∞.
• The notation ‖f ‖p with 1 ≤ p ≤ ∞ is a slight abuse, because in general it is only a norm of f if ‖f ‖p is finite and f is considered as equivalence class of μ-almost everywhere equal functions. If f ∈ Lp(μ) and g ∈ Lq(μ), then the notation is adequate.
• On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means 0. Multiplying a > 0 with ∞ gives ∞.
Estimates for integrable products
As above, let f and g denote measurable real- or complex-valued functions defined on S. If ‖fg‖1 is finite, then the pointwise products of f with g and its complex conjugate function are μ-integrable, the estimate
${\biggl |}\int _{S}f{\bar {g}}\,\mathrm {d} \mu {\biggr |}\leq \int _{S}|fg|\,\mathrm {d} \mu =\|fg\|_{1}$
and the similar one for fg hold, and Hölder's inequality can be applied to the right-hand side. In particular, if f and g are in the Hilbert space L2(μ), then Hölder's inequality for p = q = 2 implies
$|\langle f,g\rangle |\leq \|f\|_{2}\|g\|_{2},$
where the angle brackets refer to the inner product of L2(μ). This is also called Cauchy–Schwarz inequality, but requires for its statement that ‖f ‖2 and ‖g‖2 are finite to make sure that the inner product of f and g is well defined. We may recover the original inequality (for the case p = 2) by using the functions |f | and |g| in place of f and g.
Generalization for probability measures
If (S, Σ, μ) is a probability space, then p, q ∈ [1, ∞] just need to satisfy 1/p + 1/q ≤ 1, rather than being Hölder conjugates. A combination of Hölder's inequality and Jensen's inequality implies that
$\|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}$
for all measurable real- or complex-valued functions f and g on S.
Notable special cases
For the following cases assume that p and q are in the open interval (1,∞) with 1/p + 1/q = 1.
Counting measure
For the n-dimensional Euclidean space, when the set S is {1, ..., n} with the counting measure, we have
$\sum _{k=1}^{n}|x_{k}\,y_{k}|\leq {\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{\frac {1}{p}}{\biggl (}\sum _{k=1}^{n}|y_{k}|^{q}{\biggr )}^{\frac {1}{q}}{\text{ for all }}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})\in \mathbb {R} ^{n}{\text{ or }}\mathbb {C} ^{n}.$
Often the following practical form of this is used, for any $(r,s)\in \mathbb {R} _{+}$:
${\biggl (}\sum _{k=1}^{n}|x_{k}|^{r}\,|y_{k}|^{s}{\biggr )}^{r+s}\leq {\biggl (}\sum _{k=1}^{n}|x_{k}|^{r+s}{\biggr )}^{r}{\biggl (}\sum _{k=1}^{n}|y_{k}|^{r+s}{\biggr )}^{s}.$
For more than two sums, the following generalisation (Chen (2015)) holds, with real positive exponents $\lambda _{i}$ and $\lambda _{a}+\lambda _{b}+\cdots +\lambda _{z}=1$:
$\sum _{k=1}^{n}|a_{k}|^{\lambda _{a}}\,|b_{k}|^{\lambda _{b}}\cdots |z_{k}|^{\lambda _{z}}\leq {\biggl (}\sum _{k=1}^{n}|a_{k}|{\biggr )}^{\lambda _{a}}{\biggl (}\sum _{k=1}^{n}|b_{k}|{\biggr )}^{\lambda _{b}}\cdots {\biggl (}\sum _{k=1}^{n}|z_{k}|{\biggr )}^{\lambda _{z}}.$
Equality holds iff $|a_{1}|:|a_{2}|:\cdots :|a_{n}|=|b_{1}|:|b_{2}|:\cdots :|b_{n}|=\cdots =|z_{1}|:|z_{2}|:\cdots :|z_{n}|$ :|a_{n}|=|b_{1}|:|b_{2}|:\cdots :|b_{n}|=\cdots =|z_{1}|:|z_{2}|:\cdots :|z_{n}|} .
If S = N with the counting measure, then we get Hölder's inequality for sequence spaces:
$\sum _{k=1}^{\infty }|x_{k}\,y_{k}|\leq {\biggl (}\sum _{k=1}^{\infty }|x_{k}|^{p}{\biggr )}^{\frac {1}{p}}\left(\sum _{k=1}^{\infty }|y_{k}|^{q}\right)^{\frac {1}{q}}{\text{ for all }}(x_{k})_{k\in \mathbb {N} },(y_{k})_{k\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }{\text{ or }}\mathbb {C} ^{\mathbb {N} }.$
Lebesgue measure
If S is a measurable subset of Rn with the Lebesgue measure, and f and g are measurable real- or complex-valued functions on S, then Hölder inequality is
$\int _{S}{\bigl |}f(x)g(x){\bigr |}\,\mathrm {d} x\leq {\biggl (}\int _{S}|f(x)|^{p}\,\mathrm {d} x{\biggr )}^{\frac {1}{p}}{\biggl (}\int _{S}|g(x)|^{q}\,\mathrm {d} x{\biggr )}^{\frac {1}{q}}.$
Probability measure
For the probability space $(\Omega ,{\mathcal {F}},\mathbb {P} ),$ let $\mathbb {E} $ denote the expectation operator. For real- or complex-valued random variables $X$ and $Y$ on $\Omega ,$ Hölder's inequality reads
$\mathbb {E} [|XY|]\leqslant \left(\mathbb {E} {\bigl [}|X|^{p}{\bigr ]}\right)^{\frac {1}{p}}\left(\mathbb {E} {\bigl [}|Y|^{q}{\bigr ]}\right)^{\frac {1}{q}}.$
Let $1<r<s<\infty $ and define $p={\tfrac {s}{r}}.$ Then $q={\tfrac {p}{p-1}}$ is the Hölder conjugate of $p.$ Applying Hölder's inequality to the random variables $|X|^{r}$ and $1_{\Omega }$ we obtain
$\mathbb {E} {\bigl [}|X|^{r}{\bigr ]}\leqslant \left(\mathbb {E} {\bigl [}|X|^{s}{\bigr ]}\right)^{\frac {r}{s}}.$
In particular, if the sth absolute moment is finite, then the r th absolute moment is finite, too. (This also follows from Jensen's inequality.)
Product measure
For two σ-finite measure spaces (S1, Σ1, μ1) and (S2, Σ2, μ2) define the product measure space by
$S=S_{1}\times S_{2},\quad \Sigma =\Sigma _{1}\otimes \Sigma _{2},\quad \mu =\mu _{1}\otimes \mu _{2},$
where S is the Cartesian product of S1 and S2, the σ-algebra Σ arises as product σ-algebra of Σ1 and Σ2, and μ denotes the product measure of μ1 and μ2. Then Tonelli's theorem allows us to rewrite Hölder's inequality using iterated integrals: If f and g are Σ-measurable real- or complex-valued functions on the Cartesian product S, then
$\int _{S_{1}}\int _{S_{2}}|f(x,y)\,g(x,y)|\,\mu _{2}(\mathrm {d} y)\,\mu _{1}(\mathrm {d} x)\leq \left(\int _{S_{1}}\int _{S_{2}}|f(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\,\mu _{1}(\mathrm {d} x)\right)^{\frac {1}{p}}\left(\int _{S_{1}}\int _{S_{2}}|g(x,y)|^{q}\,\mu _{2}(\mathrm {d} y)\,\mu _{1}(\mathrm {d} x)\right)^{\frac {1}{q}}.$
This can be generalized to more than two σ-finite measure spaces.
Vector-valued functions
Let (S, Σ, μ) denote a σ-finite measure space and suppose that f = (f1, ..., fn) and g = (g1, ..., gn) are Σ-measurable functions on S, taking values in the n-dimensional real- or complex Euclidean space. By taking the product with the counting measure on {1, ..., n}, we can rewrite the above product measure version of Hölder's inequality in the form
$\int _{S}\sum _{k=1}^{n}|f_{k}(x)\,g_{k}(x)|\,\mu (\mathrm {d} x)\leq \left(\int _{S}\sum _{k=1}^{n}|f_{k}(x)|^{p}\,\mu (\mathrm {d} x)\right)^{\frac {1}{p}}\left(\int _{S}\sum _{k=1}^{n}|g_{k}(x)|^{q}\,\mu (\mathrm {d} x)\right)^{\frac {1}{q}}.$
If the two integrals on the right-hand side are finite, then equality holds if and only if there exist real numbers α, β ≥ 0, not both of them zero, such that
$\alpha \left(|f_{1}(x)|^{p},\ldots ,|f_{n}(x)|^{p}\right)=\beta \left(|g_{1}(x)|^{q},\ldots ,|g_{n}(x)|^{q}\right),$
for μ-almost all x in S.
This finite-dimensional version generalizes to functions f and g taking values in a normed space which could be for example a sequence space or an inner product space.
Proof of Hölder's inequality
There are several proofs of Hölder's inequality; the main idea in the following is Young's inequality for products.
Proof
If ‖f ‖p = 0, then f is zero μ-almost everywhere, and the product fg is zero μ-almost everywhere, hence the left-hand side of Hölder's inequality is zero. The same is true if ‖g‖q = 0. Therefore, we may assume ‖f ‖p > 0 and ‖g‖q > 0 in the following.
If ‖f ‖p = ∞ or ‖g‖q = ∞, then the right-hand side of Hölder's inequality is infinite. Therefore, we may assume that ‖f ‖p and ‖g‖q are in (0, ∞).
If p = ∞ and q = 1, then |fg| ≤ ‖f ‖∞ |g| almost everywhere and Hölder's inequality follows from the monotonicity of the Lebesgue integral. Similarly for p = 1 and q = ∞. Therefore, we may assume p, q ∈ (1,∞).
Dividing f and g by ‖f ‖p and ‖g‖q, respectively, we can assume that
$\|f\|_{p}=\|g\|_{q}=1.$
We now use Young's inequality for products, which states that whenever $p,q$ are in (1,∞) with ${\frac {1}{p}}+{\frac {1}{q}}=1$
$ab\leq {\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}$
for all nonnegative a and b, where equality is achieved if and only if ap = bq. Hence
$|f(s)g(s)|\leq {\frac {|f(s)|^{p}}{p}}+{\frac {|g(s)|^{q}}{q}},\qquad s\in S.$
Integrating both sides gives
$\|fg\|_{1}\leq {\frac {\|f\|_{p}^{p}}{p}}+{\frac {\|g\|_{q}^{q}}{q}}={\frac {1}{p}}+{\frac {1}{q}}=1,$
which proves the claim.
Under the assumptions p ∈ (1, ∞) and ‖f ‖p = ‖g‖q, equality holds if and only if |f |p = |g|q almost everywhere. More generally, if ‖f ‖p and ‖g‖q are in (0, ∞), then Hölder's inequality becomes an equality if and only if there exist real numbers α, β > 0, namely
$\alpha =\|g\|_{q}^{q},\qquad \beta =\|f\|_{p}^{p},$
such that
$\alpha |f|^{p}=\beta |g|^{q}$ μ-almost everywhere (*).
The case ‖f ‖p = 0 corresponds to β = 0 in (*). The case ‖g‖q = 0 corresponds to α = 0 in (*).
Alternative proof using Jensen's inequality:
Proof
The function $x\mapsto x^{p}$ on (0,∞) is convex because $p\geq 1$, so by Jensen's inequality,
$\int hd\nu \leq \left(\int h^{p}d\nu \right)^{\frac {1}{p}}$
where ν is any probability distribution and h any ν-measurable function. Let μ be any measure, and ν the distribution whose density w.r.t. μ is proportional to $g^{q}$, i.e.
$d\nu ={\frac {g^{q}}{\int g^{q}\,d\mu }}d\mu $
Hence we have, using ${\frac {1}{p}}+{\frac {1}{q}}=1$, hence $p(1-q)+q=0$, and letting $h=fg^{1-q}$,
$\int fg\,d\mu =\left(\int g^{q}\,d\mu \right)\int \underbrace {fg^{1-q}} _{h}\underbrace {{\frac {g^{q}}{\int g^{q}\,d\mu }}d\mu } _{d\nu }\leq \left(\int g^{q}d\mu \right)\left(\int \underbrace {f^{p}g^{p(1-q)}} _{h^{p}}\underbrace {{\frac {g^{q}}{\int g^{q}\,d\mu }}\,d\mu } _{d\nu }\right)^{\frac {1}{p}}=\left(\int g^{q}\,d\mu \right)\left(\int {\frac {f^{p}}{\int g^{q}\,d\mu }}\,d\mu \right)^{\frac {1}{p}}$
Finally, we get
$\int fg\,d\mu \leq \left(\int f^{p}\,d\mu \right)^{\frac {1}{p}}\left(\int g^{q}\,d\mu \right)^{\frac {1}{q}}$
This assumes that f, g are real and non-negative, but the extension to complex functions is straightforward (use the modulus of f, g). It also assumes that $\|f\|_{p},\|g\|_{q}$ are neither null nor infinity, and that $p,q>1$: all these assumptions can also be lifted as in the proof above.
Extremal equality
Statement
Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then for every f ∈ Lp(μ),
$\|f\|_{p}=\max \left\{\left|\int _{S}fg\,\mathrm {d} \mu \right|:g\in L^{q}(\mu ),\|g\|_{q}\leq 1\right\},$
where max indicates that there actually is a g maximizing the right-hand side. When p = ∞ and if each set A in the σ-field Σ with μ(A) = ∞ contains a subset B ∈ Σ with 0 < μ(B) < ∞ (which is true in particular when μ is σ-finite), then
$\|f\|_{\infty }=\sup \left\{\left|\int _{S}fg\,\mathrm {d} \mu \right|:g\in L^{1}(\mu ),\|g\|_{1}\leq 1\right\}.$
Proof of the extremal equality:
Proof
By Hölder's inequality, the integrals are well defined and, for 1 ≤ p ≤ ∞,
$\left|\int _{S}fg\,\mathrm {d} \mu \right|\leq \int _{S}|fg|\,\mathrm {d} \mu \leq \|f\|_{p},$
hence the left-hand side is always bounded above by the right-hand side.
Conversely, for 1 ≤ p ≤ ∞, observe first that the statement is obvious when ‖f ‖p = 0. Therefore, we assume ‖f ‖p > 0 in the following.
If 1 ≤ p < ∞, define g on S by
$g(x)={\begin{cases}\|f\|_{p}^{1-p}\,|f(x)|^{p}/f(x)&{\text{if }}f(x)\not =0,\\0&{\text{otherwise.}}\end{cases}}$
By checking the cases p = 1 and 1 < p < ∞ separately, we see that ‖g‖q = 1 and
$\int _{S}fg\,\mathrm {d} \mu =\|f\|_{p}.$
It remains to consider the case p = ∞. For ε ∈ (0, 1) define
$A=\left\{x\in S:|f(x)|>(1-\varepsilon )\|f\|_{\infty }\right\}.$
Since f is measurable, A ∈ Σ. By the definition of ‖f ‖∞ as the essential supremum of f and the assumption ‖f ‖∞ > 0, we have μ(A) > 0. Using the additional assumption on the σ-field Σ if necessary, there exists a subset B ∈ Σ of A with 0 < μ(B) < ∞. Define g on S by
$g(x)={\begin{cases}{\frac {1-\varepsilon }{\mu (B)}}{\frac {\|f\|_{\infty }}{f(x)}}&{\text{if }}x\in B,\\0&{\text{otherwise.}}\end{cases}}$
Then g is well-defined, measurable and |g(x)| ≤ 1/μ(B) for x ∈ B, hence ‖g‖1 ≤ 1. Furthermore,
$\left|\int _{S}fg\,\mathrm {d} \mu \right|=\int _{B}{\frac {1-\varepsilon }{\mu (B)}}\|f\|_{\infty }\,\mathrm {d} \mu =(1-\varepsilon )\|f\|_{\infty }.$
Remarks and examples
• The equality for $p=\infty $ fails whenever there exists a set $A$ of infinite measure in the $\sigma $-field $\Sigma $ with that has no subset $B\in \Sigma $ that satisfies: $0<\mu (B)<\infty .$ (the simplest example is the $\sigma $-field $\Sigma $ containing just the empty set and $S,$ and the measure $\mu $ with $\mu (S)=\infty .$) Then the indicator function $1_{A}$ satisfies $\|1_{A}\|_{\infty }=1,$ but every $g\in L^{1}(\mu )$ has to be $\mu $-almost everywhere constant on $A,$ because it is $\Sigma $-measurable, and this constant has to be zero, because $g$ is $\mu $-integrable. Therefore, the above supremum for the indicator function $1_{A}$ is zero and the extremal equality fails.
• For $p=\infty ,$ the supremum is in general not attained. As an example, let $S=\mathbb {N} ,\Sigma ={\mathcal {P}}(\mathbb {N} )$ and $\mu $ the counting measure. Define:
${\begin{cases}f:\mathbb {N} \to \mathbb {R} \\f(n)={\frac {n-1}{n}}\end{cases}}$
Then $\|f\|_{\infty }=1.$ For $g\in L^{1}(\mu ,\mathbb {N} )$ with $0<\|g\|_{1}\leqslant 1,$ let $m$ denote the smallest natural number with $g(m)\neq 0.$ Then
$\left|\int _{S}fg\,\mathrm {d} \mu \right|\leqslant {\frac {m-1}{m}}|g(m)|+\sum _{n=m+1}^{\infty }|g(n)|=\|g\|_{1}-{\frac {|g(m)|}{m}}<1.$
Applications
• The extremal equality is one of the ways for proving the triangle inequality ‖f1 + f2‖p ≤ ‖f1‖p + ‖f2‖p for all f1 and f2 in Lp(μ), see Minkowski inequality.
• Hölder's inequality implies that every f ∈ Lp(μ) defines a bounded (or continuous) linear functional κf on Lq(μ) by the formula
$\kappa _{f}(g)=\int _{S}fg\,\mathrm {d} \mu ,\qquad g\in L^{q}(\mu ).$
The extremal equality (when true) shows that the norm of this functional κf as element of the continuous dual space Lq(μ)* coincides with the norm of f in Lp(μ) (see also the Lp-space article).
Generalization with more than two functions
Statement
Assume that r ∈ (0, ∞] and p1, ..., pn ∈ (0, ∞] such that
$\sum _{k=1}^{n}{\frac {1}{p_{k}}}={\frac {1}{r}}$
where 1/∞ is interpreted as 0 in this equation. Then for all measurable real or complex-valued functions f1, ..., fn defined on S,
$\left\|\prod _{k=1}^{n}f_{k}\right\|_{r}\leq \prod _{k=1}^{n}\left\|f_{k}\right\|_{p_{k}}$
where we interpret any product with a factor of ∞ as ∞ if all factors are positive, but the product is 0 if any factor is 0.
In particular, if $f_{k}\in L^{p_{k}}(\mu )$ for all $k\in \{1,\ldots ,n\}$ then $\prod _{k=1}^{n}f_{k}\in L^{r}(\mu ).$
Note: For $r\in (0,1),$ contrary to the notation, ‖.‖r is in general not a norm because it doesn't satisfy the triangle inequality.
Proof of the generalization:
Proof
We use Hölder's inequality and mathematical induction. If $n=1$ then the result is immediate. Let us now pass from $n-1$ to $n.$ Without loss of generality assume that $p_{1}\leq \cdots \leq p_{n}.$
Case 1: If $p_{n}=\infty $ then
$\sum _{k=1}^{n-1}{\frac {1}{p_{k}}}={\frac {1}{r}}.$
Pulling out the essential supremum of |fn| and using the induction hypothesis, we get
${\begin{aligned}\left\|f_{1}\cdots f_{n}\right\|_{r}&\leq \left\|f_{1}\cdots f_{n-1}\right\|_{r}\left\|f_{n}\right\|_{\infty }\\&\leq \left\|f_{1}\right\|_{p_{1}}\cdots \left\|f_{n-1}\right\|_{p_{n-1}}\left\|f_{n}\right\|_{\infty }.\end{aligned}}$
Case 2: If $p_{n}<\infty $ then necessarily $r<\infty $ as well, and then
$p:={\frac {p_{n}}{p_{n}-r}},\qquad q:={\frac {p_{n}}{r}}$
are Hölder conjugates in (1, ∞). Application of Hölder's inequality gives
$\left\||f_{1}\cdots f_{n-1}|^{r}\,|f_{n}|^{r}\right\|_{1}\leq \left\||f_{1}\cdots f_{n-1}|^{r}\right\|_{p}\,\left\||f_{n}|^{r}\right\|_{q}.$
Raising to the power $1/r$ and rewriting,
$\|f_{1}\cdots f_{n}\|_{r}\leq \|f_{1}\cdots f_{n-1}\|_{pr}\|f_{n}\|_{qr}.$
Since $qr=p_{n}$ and
$\sum _{k=1}^{n-1}{\frac {1}{p_{k}}}={\frac {1}{r}}-{\frac {1}{p_{n}}}={\frac {p_{n}-r}{rp_{n}}}={\frac {1}{pr}},$
the claimed inequality now follows by using the induction hypothesis.
Interpolation
Let p1, ..., pn ∈ (0, ∞] and let θ1, ..., θn ∈ (0, 1) denote weights with θ1 + ... + θn = 1. Define $p$ as the weighted harmonic mean, that is,
${\frac {1}{p}}=\sum _{k=1}^{n}{\frac {\theta _{k}}{p_{k}}}.$
Given measurable real- or complex-valued functions $f_{k}$ on S, then the above generalization of Hölder's inequality gives
$\left\||f_{1}|^{\theta _{1}}\cdots |f_{n}|^{\theta _{n}}\right\|_{p}\leq \left\||f_{1}|^{\theta _{1}}\right\|_{\frac {p_{1}}{\theta _{1}}}\cdots \left\||f_{n}|^{\theta _{n}}\right\|_{\frac {p_{n}}{\theta _{n}}}=\|f_{1}\|_{p_{1}}^{\theta _{1}}\cdots \|f_{n}\|_{p_{n}}^{\theta _{n}}.$
In particular, taking $f_{1}=\cdots =f_{n}=:f$ gives
$\|f\|_{p}\leqslant \prod _{k=1}^{n}\|f\|_{p_{k}}^{\theta _{k}}.$
Specifying further θ1 = θ and θ2 = 1-θ, in the case $n=2,$ we obtain the interpolation result
Littlewood's inequality — For $\theta \in (0,1)$ and ${\frac {1}{p_{\theta }}}={\frac {\theta }{p_{1}}}+{\frac {1-\theta }{p_{0}}}$,
$\|f\|_{p_{\theta }}\leqslant \|f\|_{p_{1}}^{\theta }\cdot \|f\|_{p_{0}}^{1-\theta },$
An application of Hölder gives
Lyapunov's inequality — If $p=(1-\theta )p_{0}+\theta p_{1},\qquad \theta \in (0,1),$ then
$\left\||f_{0}|^{\frac {p_{0}(1-\theta )}{p}}\cdot |f_{1}|^{\frac {p_{1}\theta }{p}}\right\|_{p}^{p}\leq \|f_{0}\|_{p_{0}}^{p_{0}(1-\theta )}\|f_{1}\|_{p_{1}}^{p_{1}\theta }$
and in particular
$\|f\|_{p}^{p}\leqslant \|f\|_{p_{0}}^{p_{0}(1-\theta )}\cdot \|f\|_{p_{1}}^{p_{1}\theta }.$
Both Littlewood and Lyapunov imply that if $f\in L^{p_{0}}\cap L^{p_{1}}$ then $f\in L^{p}$ for all $p_{0}<p<p_{1}.$[4]
Reverse Hölder inequalities
Two functions
Assume that p ∈ (1, ∞) and that the measure space (S, Σ, μ) satisfies μ(S) > 0. Then for all measurable real- or complex-valued functions f and g on S such that g(s) ≠ 0 for μ-almost all s ∈ S,
$\|fg\|_{1}\geqslant \|f\|_{\frac {1}{p}}\,\|g\|_{\frac {-1}{p-1}}.$
If
$\|fg\|_{1}<\infty \quad {\text{and}}\quad \|g\|_{\frac {-1}{p-1}}>0,$
then the reverse Hölder inequality is an equality if and only if
$\exists \alpha \geqslant 0\quad |f|=\alpha |g|^{\frac {-p}{p-1}}\qquad \mu {\text{-almost everywhere}}.$
Note: The expressions:
$\|f\|_{\frac {1}{p}}$ and $\|g\|_{\frac {-1}{p-1}},$
are not norms, they are just compact notations for
$\left(\int _{S}|f|^{\frac {1}{p}}\,\mathrm {d} \mu \right)^{p}\quad {\text{and}}\quad \left(\int _{S}|g|^{\frac {-1}{p-1}}\,\mathrm {d} \mu \right)^{-(p-1)}.$
Proof of the reverse Hölder inequality (hidden, click show to reveal.)
Note that p and
$q:={\frac {p}{p-1}}\in (1,\infty )$
are Hölder conjugates. Application of Hölder's inequality gives
${\begin{aligned}\left\||f|^{\frac {1}{p}}\right\|_{1}&=\left\||fg|^{\frac {1}{p}}\,|g|^{-{\frac {1}{p}}}\right\|_{1}\\&\leqslant \left\||fg|^{\frac {1}{p}}\right\|_{p}\left\||g|^{-{\frac {1}{p}}}\right\|_{q}\\&=\|fg\|_{1}^{\frac {1}{p}}\left\||g|^{\frac {-1}{p-1}}\right\|_{1}^{\frac {p-1}{p}}\end{aligned}}$
Raising to the power p gives us:
$\left\||f|^{\frac {1}{p}}\right\|_{1}^{p}\leqslant \|fg\|_{1}\left\||g|^{\frac {-1}{p-1}}\right\|_{1}^{p-1}.$
Therefore:
$\left\||f|^{\frac {1}{p}}\right\|_{1}^{p}\left\||g|^{\frac {-1}{p-1}}\right\|_{1}^{-(p-1)}\leqslant \|fg\|_{1}.$
Now we just need to recall our notation.
Since g is not almost everywhere equal to the zero function, we can have equality if and only if there exists a constant α ≥ 0 such that |fg| = α |g|−q/p almost everywhere. Solving for the absolute value of f gives the claim.
Multiple functions
The Reverse Hölder inequality (above) can be generalized to the case of multiple functions if all but one conjugate is negative. That is,
Let $p_{1},...,p_{m-1}<0$ and $p_{m}\in \mathbb {R} $ be such that $\sum _{k=1}^{m}{\frac {1}{p_{k}}}=1$ (hence $0<p_{m}<1$). Let $f_{k}$ be measurable functions for $k=1,...,m$. Then
$\left\|\prod _{k=1}^{n}f_{k}\right\|_{1}\geq \prod _{k=1}^{n}\left\|f_{k}\right\|_{p_{k}}.$
This follows from the symmetric form of the Hölder inequality (see below).
Symmetric forms of Hölder inequality
It was observed by Aczél and Beckenbach[5] that Hölder's inequality can be put in a more symmetric form, at the price of introducing an extra vector (or function):
Let $f=(f(1),\dots ,f(m)),g=(g(1),\dots ,g(m)),h=(h(1),\dots ,h(m))$ be vectors with positive entries and such that $f(i)g(i)h(i)=1$ for all $i$. If $p,q,r$ are nonzero real numbers such that ${\frac {1}{p}}+{\frac {1}{q}}+{\frac {1}{r}}=0$, then:
• $\|f\|_{p}\|g\|_{q}\|h\|_{r}\geq 1$ if all but one of $p,q,r$ are positive;
• $\|f\|_{p}\|g\|_{q}\|h\|_{r}\leq 1$ if all but one of $p,q,r$ are negative.
The standard Hölder inequality follows immediately from this symmetric form (and in fact is easily seen to be equivalent to it). The symmetric statement also implies the reverse Hölder inequality (see above).
The result can be extended to multiple vectors:
Let $f_{1},\dots ,f_{n}$ be $n$ vectors in $\mathbb {R} ^{m}$ with positive entries and such that $f_{1}(i)\dots f_{n}(i)=1$ for all $i$. If $p_{1},\dots ,p_{n}$ are nonzero real numbers such that ${\frac {1}{p_{1}}}+\dots +{\frac {1}{p_{n}}}=0$, then:
• $\|f_{1}\|_{p_{1}}\dots \|f_{n}\|_{p_{n}}\geq 1$ if all but one of the numbers $p_{i}$ are positive;
• $\|f_{1}\|_{p_{1}}\dots \|f_{n}\|_{p_{n}}\leq 1$ if all but one of the numbers $p_{i}$ are negative.
As in the standard Hölder inequalities, there are corresponding statements for infinite sums and integrals.
Conditional Hölder inequality
Let (Ω, F, $\mathbb {P} $) be a probability space, G ⊂ F a sub-σ-algebra, and p, q ∈ (1, ∞) Hölder conjugates, meaning that 1/p + 1/q = 1. Then for all real- or complex-valued random variables X and Y on Ω,
$\mathbb {E} {\bigl [}|XY|{\big |}\,{\mathcal {G}}{\bigr ]}\leq {\bigl (}\mathbb {E} {\bigl [}|X|^{p}{\big |}\,{\mathcal {G}}{\bigr ]}{\bigr )}^{\frac {1}{p}}\,{\bigl (}\mathbb {E} {\bigl [}|Y|^{q}{\big |}\,{\mathcal {G}}{\bigr ]}{\bigr )}^{\frac {1}{q}}\qquad \mathbb {P} {\text{-almost surely.}}$
Remarks:
• If a non-negative random variable Z has infinite expected value, then its conditional expectation is defined by
$\mathbb {E} [Z|{\mathcal {G}}]=\sup _{n\in \mathbb {N} }\,\mathbb {E} [\min\{Z,n\}|{\mathcal {G}}]\quad {\text{a.s.}}$
• On the right-hand side of the conditional Hölder inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplying a > 0 with ∞ gives ∞.
Proof of the conditional Hölder inequality:
Proof
Define the random variables
$U={\bigl (}\mathbb {E} {\bigl [}|X|^{p}{\big |}\,{\mathcal {G}}{\bigr ]}{\bigr )}^{\frac {1}{p}},\qquad V={\bigl (}\mathbb {E} {\bigl [}|Y|^{q}{\big |}\,{\mathcal {G}}{\bigr ]}{\bigr )}^{\frac {1}{q}}$
and note that they are measurable with respect to the sub-σ-algebra. Since
$\mathbb {E} {\bigl [}|X|^{p}1_{\{U=0\}}{\bigr ]}=\mathbb {E} {\bigl [}1_{\{U=0\}}\underbrace {\mathbb {E} {\bigl [}|X|^{p}{\big |}\,{\mathcal {G}}{\bigr ]}} _{=\,U^{p}}{\bigr ]}=0,$
it follows that |X| = 0 a.s. on the set {U = 0}. Similarly, |Y| = 0 a.s. on the set {V = 0}, hence
$\mathbb {E} {\bigl [}|XY|{\big |}\,{\mathcal {G}}{\bigr ]}=0\qquad {\text{a.s. on }}\{U=0\}\cup \{V=0\}$
and the conditional Hölder inequality holds on this set. On the set
$\{U=\infty ,V>0\}\cup \{U>0,V=\infty \}$
the right-hand side is infinite and the conditional Hölder inequality holds, too. Dividing by the right-hand side, it therefore remains to show that
${\frac {\mathbb {E} {\bigl [}|XY|{\big |}\,{\mathcal {G}}{\bigr ]}}{UV}}\leq 1\qquad {\text{a.s. on the set }}H:=\{0<U<\infty ,\,0<V<\infty \}.$
This is done by verifying that the inequality holds after integration over an arbitrary
$G\in {\mathcal {G}},\quad G\subset H.$
Using the measurability of U, V, 1G with respect to the sub-σ-algebra, the rules for conditional expectations, Hölder's inequality and 1/p + 1/q = 1, we see that
${\begin{aligned}\mathbb {E} {\biggl [}{\frac {\mathbb {E} {\bigl [}|XY|{\big |}\,{\mathcal {G}}{\bigr ]}}{UV}}1_{G}{\biggr ]}&=\mathbb {E} {\biggl [}\mathbb {E} {\biggl [}{\frac {|XY|}{UV}}1_{G}{\bigg |}\,{\mathcal {G}}{\biggr ]}{\biggr ]}\\&=\mathbb {E} {\biggl [}{\frac {|X|}{U}}1_{G}\cdot {\frac {|Y|}{V}}1_{G}{\biggr ]}\\&\leq {\biggl (}\mathbb {E} {\biggl [}{\frac {|X|^{p}}{U^{p}}}1_{G}{\biggr ]}{\biggr )}^{\frac {1}{p}}{\biggl (}\mathbb {E} {\biggl [}{\frac {|Y|^{q}}{V^{q}}}1_{G}{\biggr ]}{\biggr )}^{\frac {1}{q}}\\&={\biggl (}\mathbb {E} {\biggl [}\underbrace {\frac {\mathbb {E} {\bigl [}|X|^{p}{\big |}\,{\mathcal {G}}{\bigr ]}}{U^{p}}} _{=\,1{\text{ a.s. on }}G}1_{G}{\biggr ]}{\biggr )}^{\frac {1}{p}}{\biggl (}\mathbb {E} {\biggl [}\underbrace {\frac {\mathbb {E} {\bigl [}|Y|^{q}{\big |}\,{\mathcal {G}}{\bigr ]}}{V^{p}}} _{=\,1{\text{ a.s. on }}G}1_{G}{\biggr ]}{\biggr )}^{\frac {1}{q}}\\&=\mathbb {E} {\bigl [}1_{G}{\bigr ]}.\end{aligned}}$
Hölder's inequality for increasing seminorms
Let S be a set and let $F(S,\mathbb {C} )$ be the space of all complex-valued functions on S. Let N be an increasing seminorm on $F(S,\mathbb {C} ),$ meaning that, for all real-valued functions $f,g\in F(S,\mathbb {C} )$ we have the following implication (the seminorm is also allowed to attain the value ∞):
$\forall s\in S\quad f(s)\geqslant g(s)\geqslant 0\qquad \Rightarrow \qquad N(f)\geqslant N(g).$
Then:
$\forall f,g\in F(S,\mathbb {C} )\qquad N(|fg|)\leqslant {\bigl (}N(|f|^{p}){\bigr )}^{\frac {1}{p}}{\bigl (}N(|g|^{q}){\bigr )}^{\frac {1}{q}},$
where the numbers $p$ and $q$ are Hölder conjugates.[6]
Remark: If (S, Σ, μ) is a measure space and $N(f)$ is the upper Lebesgue integral of $|f|$ then the restriction of N to all Σ-measurable functions gives the usual version of Hölder's inequality.
Distances based on Hölder inequality
Hölder inequality can be used to define statistical dissimilarity measures[7] between probability distributions. Those Hölder divergences are projective: They do not depend on the normalization factor of densities.
See also
• Cauchy–Schwarz inequality
• Minkowski inequality
• Jensen's inequality
• Young's inequality for products
• Clarkson's inequalities
• Brascamp–Lieb inequality
Citations
1. Roman 2008, p. 303 §12
2. Maligranda, Lech (1998), "Why Hölder's inequality should be called Rogers' inequality", Mathematical Inequalities & Applications, 1 (1): 69–83, doi:10.7153/mia-01-05, MR 1492911
3. Guessab, A.; Schmeisser, G. (2013), "Necessary and sufficient conditions for the validity of Jensen's inequality", Archiv der Mathematik, 100 (6): 561–570, doi:10.1007/s00013-013-0522-3, MR 3069109, S2CID 253600514, under the additional assumption that $\varphi ''$ exists, this inequality was already obtained by Hölder in 1889
4. Wojtaszczyk, P. (1991). Banach Spaces for Analysts. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. ISBN 978-0-521-56675-9.
5. Beckenbach, E. F. (1980). General inequalities 2. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d'Analyse Numérique. Vol. 47. Birkhäuser Basel. pp. 145–150. doi:10.1007/978-3-0348-6324-7. ISBN 978-3-7643-1056-1.
6. For a proof see (Trèves 1967, Lemma 20.1, pp. 205–206).
7. Nielsen, Frank; Sun, Ke; Marchand-Maillet, Stephane (2017). "On Hölder projective divergences". Entropy. 3 (19): 122. arXiv:1701.03916. Bibcode:2017Entrp..19..122N. doi:10.3390/e19030122.
References
• Grinshpan, A. Z. (2010), "Weighted inequalities and negative binomials", Advances in Applied Mathematics, 45 (4): 564–606, doi:10.1016/j.aam.2010.04.004
• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1934), Inequalities, Cambridge University Press, pp. XII+314, ISBN 0-521-35880-9, JFM 60.0169.01, Zbl 0010.10703.
• Hölder, O. (1889), "Ueber einen Mittelwertsatz", Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, Band (in German), 1889 (2): 38–47, JFM 21.0260.07. Available at Digi Zeitschriften.
• Kuptsov, L. P. (2001) [1994], "Hölder inequality", Encyclopedia of Mathematics, EMS Press.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rogers, L. J. (February 1888), "An extension of a certain theorem in inequalities", Messenger of Mathematics, New Series, XVII (10): 145–150, JFM 20.0254.02, archived from the original on August 21, 2007.
• Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5
• Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels, Pure and Applied Mathematics. A Series of Monographs and Textbooks, vol. 25, New York, London: Academic Press, MR 0225131, Zbl 0171.10402.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
External links
• Chen, Evan (2015), A Brief Introduction to Olympiad Inequalities (PDF).
• Kuttler, Kenneth (2007), An Introduction to Linear Algebra (PDF), Online e-book in PDF format, Brigham Young University.
• Lohwater, Arthur (1982), Introduction to Inequalities (PDF).
• Archived at Ghostarchive and the Wayback Machine: Tisdell, Chris (2012), Holder's Inequality, YouTube.
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Reverse Mathematics: Proofs from the Inside Out
Reverse Mathematics: Proofs from the Inside Out is a book by John Stillwell on reverse mathematics, the process of examining proofs in mathematics to determine which axioms are required by the proof. It was published in 2018 by the Princeton University Press (ISBN 978-0-691-17717-5).[1][2][3][4][5][6]
Topics
The book begins with a historical overview of the long struggles with the parallel postulate in Euclidean geometry,[3] and of the foundational crisis of the late 19th and early 20th centuries,[6] Then, after reviewing background material in real analysis and computability theory,[1] the book concentrates on the reverse mathematics of theorems in real analysis,[3] including the Bolzano–Weierstrass theorem, the Heine–Borel theorem, the intermediate value theorem and extreme value theorem, the Heine–Cantor theorem on uniform continuity,[6] the Hahn–Banach theorem, and the Riemann mapping theorem.[5] These theorems are analyzed with respect to three of the "big five" subsystems of second-order arithmetic, namely arithmetical comprehension, recursive comprehension, and the weak Kőnig's lemma.[1]
Audience
The book is aimed at a "general mathematical audience"[1] including undergraduate mathematics students with an introductory-level background in real analysis.[2] It is intended both to excite mathematicians, physicists, and computer scientists about the foundational issues in their fields,[6] and to provide an accessible introduction to the subject. However, it is not a textbook;[3][4] for instance, it has no exercises. One theme of the book is that many theorems in this area require axioms in second-order arithmetic that encompass infinite processes and uncomputable functions.[3]
Reception and related reading
Jeffry Hirst criticizes the book, writing that "if one is not too obsessive about the details, Proofs from the Inside Out is an interesting introduction," while finding details that he would prefer to be handled differently, in a topic for which details are important. In particular, in this area, there are multiple choices for how to build up the arithmetic on real numbers from simpler data types such as the natural numbers, and while Stillwell discusses three of them (decimal numerals, Dedekind cuts, and nested intervals), converting between them itself requires nontrivial axiomatic assumptions.[2]
However, James Case calls the book "very readable",[6] and Roman Kossak calls it "a stellar example of expository writing on mathematics".[5] Several other reviewers agree that this book could be helpful as a non-technical way to create interest in this topic in mathematicians who are not already familiar with it, and lead them to more in-depth material in this area.[1][2][3]
As additional reading on reverse mathematics in combinatorics, Hirst suggests Slicing the Truth by Denis Hirschfeldt.[2] Another book suggested by reviewer Reinhard Kahle is Stephen G. Simpson's Subsystems of Second Order Arithmetic.[1]
References
1. Kahle, Reinhard, "Review of Reverse Mathematics", Mathematical Reviews, MR 3729321
2. Hirst, Jeffry L. (June 2018), "Review of Reverse Mathematics", Bulletin of Symbolic Logic, 24 (2): 176–177, doi:10.1017/bsl.2018.19, JSTOR 26473950, S2CID 126256370
3. Cohen, Marion (October 2018), "Review of Reverse Mathematics", American Mathematical Monthly, 125 (9): 860–864, doi:10.1080/00029890.2018.1502995, S2CID 215791768
4. Bultheel, Adhemar (August 2018), "Review", EMS Reviews, European Mathematical Society
5. Kossak, Roman (November 2018), "Review of Reverse Mathematics", The Mathematical Intelligencer, 41 (1): 81–82, doi:10.1007/s00283-018-9841-3, S2CID 125295465
6. Case, James (March 2019), "A new mathematical field answers old questions", SIAM News
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Transpose graph
In the mathematical and algorithmic study of graph theory, the converse,[1] transpose[2] or reverse[3] of a directed graph G is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of the corresponding edges in G. That is, if G contains an edge (u, v) then the converse/transpose/reverse of G contains an edge (v, u) and vice versa.
Notation
The name converse arises because the reversal of arrows corresponds to taking the converse of an implication in logic. The name transpose is because the adjacency matrix of the transpose directed graph is the transpose of the adjacency matrix of the original directed graph.
There is no general agreement on preferred terminology.
The converse is denoted symbolically as G', GT, GR, or other notations, depending on which terminology is used and which book or article is the source for the notation.
Applications
Although there is little difference mathematically between a graph and its transpose, the difference may be larger in computer science, depending on how a given graph is represented. For instance, for the web graph, it is easy to determine the outgoing links of a vertex, but hard to determine the incoming links, while in the reversal of this graph the opposite is true. In graph algorithms, therefore, it may sometimes be useful to construct an explicit representation of the reversal of a graph, in order to put the graph into a form which is more suitable for the operations being performed on it. An example of this is Kosaraju's algorithm for strongly connected components, which applies depth-first search twice, once to the given graph and a second time to its reversal.
Related concepts
A skew-symmetric graph is a graph that is isomorphic to its own transpose graph, via a special kind of isomorphism that pairs up all of the vertices.
The converse relation of a binary relation is the relation that reverses the ordering of each pair of related objects. If the relation is interpreted as a directed graph, this is the same thing as the transpose of the graph. In particular, the dual order of a partial order can be interpreted in this way as the transposition of a transitively-closed directed acyclic graph.
See also
• Converse relation – Reversal of the order of elements of a binary relation
References
1. Harary, Frank; Norman, Robert Z.; Cartwright, Dorwin (1965), Structural Models: An Introduction to the Theory of Directed Graphs, New York: Wiley
2. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. Introduction to Algorithms. MIT Press and McGraw-Hill., ex. 22.1–3, p. 530.
3. Essam, John W.; Fisher, Michael E. (1970), "Some basic definitions in graph theory", Reviews of Modern Physics, 42 (2): 275, Bibcode:1970RvMP...42..271E, doi:10.1103/RevModPhys.42.271, entry 2.24
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Reverse perspective
Reverse perspective, also called inverse perspective,[1] inverted perspective,[2] divergent perspective,[3][4] or Byzantine perspective,[5] is a form of perspective drawing in which the objects depicted in a scene are placed between the projective point and the viewing plane. Objects farther away from the viewing plane are drawn as larger, and closer objects are drawn as smaller, in contrast to the more conventional linear perspective for which closer objects appear larger.[3] Lines that are parallel in three-dimensional space are drawn as diverging against the horizon, rather than converging as they do in linear perspective.[1] Technically, the vanishing points are placed outside the painting with the illusion that they are "in front of" the painting.
The name Byzantine perspective comes from the use of this perspective in Byzantine and Russian Orthodox icons; it is also found in the art of many pre-Renaissance cultures, and was sometimes used in Cubism and other movements of modern art, as well as in children's drawings.[3][4] The reasons for the convention are still debated among art historians;[6] since the artists involved in forming the convention did not have access to the more realistic linear perspective convention, it is not clear how deliberate the effects achieved were.[7]
References
1. Hopkins, Robert (1998), Picture, Image and Experience: A Philosophical Inquiry, Cambridge University Press, p. 157, ISBN 9780521582599.
2. Rauschenbach, Boris V. (1983), "On my concept of perceptual perspective that accounts for parallel and inverted perspective in pictorial art", Leonardo, 16 (1): 28–30, doi:10.2307/1575038, JSTOR 1575038, S2CID 192987663.
3. Kulvicki, John V. (2006), On Images : Their Structure and Content, Oxford University Press, pp. 102–105, ISBN 9780191537455.
4. Howard, Ian P.; Allison, Robert S. (2011), "Drawing with divergent perspective, ancient and modern" (PDF), Perception, 40 (9): 1017–1033, doi:10.1068/p6876, PMID 22208125, S2CID 11085186.
5. Deregowski, Jan B.; Parker, Denis M.; Massironi, Manfredo (1994), "The perception of spatial structure with oblique viewing: an explanation for Byzantine perspective?", Perception, 23 (1): 5–13, doi:10.1068/p230005, PMID 7936976, S2CID 16046480.
6. Antonova, Clemena (2010), "On the problem of "reverse perspective": definitions east and west", Leonardo, 43 (5): 464–469, doi:10.1162/LEON_a_00039, S2CID 57559265, The author ... identifies six distinct views on reverse perspective, some of which are mutually exclusive.
7. Antonova, Clemena (2010), Space, Time, and Presence in the Icon: Seeing the World with the Eyes of God, Ashgate studies in the history of philosophical theology, Ashgate Publishing, Ltd., p. 54, ISBN 9780754667988, In the case of "reverse perspective", on the other hand, there is no evidence that icon-painters had recourse to mathematically correct systems of measurement to enable them to represent vanishing point systems".
External links
• Video demonstrating consistent reverse perspective
• Example of computer generated reverse perspectives
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Reversed compound agent theorem
In probability theory, the reversed compound agent theorem (RCAT) is a set of sufficient conditions for a stochastic process expressed in any formalism to have a product form stationary distribution[1] (assuming that the process is stationary[2][1]). The theorem shows that product form solutions in Jackson's theorem,[1] the BCMP theorem[3] and G-networks are based on the same fundamental mechanisms.[4]
The theorem identifies a reversed process using Kelly's lemma, from which the stationary distribution can be computed.[1]
Notes
1. Harrison, P. G. (2003). "Turning back time in Markovian process algebra". Theoretical Computer Science. 290 (3): 1947–2013. doi:10.1016/S0304-3975(02)00375-4.
2. Harrison, P. G. (2006). "Process Algebraic Non-product-forms". Electronic Notes in Theoretical Computer Science. 151 (3): 61–76. doi:10.1016/j.entcs.2006.03.012.
3. Harrison, P. G. (2004). "Reversed processes, product forms and a non-product form". Linear Algebra and Its Applications. 386: 359–381. doi:10.1016/j.laa.2004.02.020.
4. Hillston, J. (2005). "Process Algebras for Quantitative Analysis" (PDF). 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05). pp. 239–248. doi:10.1109/LICS.2005.35. ISBN 0-7695-2266-1. S2CID 1236394.
References
• Bradley, Jeremy T. (28 February 2008). "RCAT: From PEPA to product form" (PDF). Archived from the original on 3 March 2016. Retrieved 10 December 2022. {{cite journal}}: Cite journal requires |journal= (help)
A short introduction to RCAT.
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Reversible cellular automaton
A reversible cellular automaton is a cellular automaton in which every configuration has a unique predecessor. That is, it is a regular grid of cells, each containing a state drawn from a finite set of states, with a rule for updating all cells simultaneously based on the states of their neighbors, such that the previous state of any cell before an update can be determined uniquely from the updated states of all the cells. The time-reversed dynamics of a reversible cellular automaton can always be described by another cellular automaton rule, possibly on a much larger neighborhood.
Several methods are known for defining cellular automata rules that are reversible; these include the block cellular automaton method, in which each update partitions the cells into blocks and applies an invertible function separately to each block, and the second-order cellular automaton method, in which the update rule combines states from two previous steps of the automaton. When an automaton is not defined by one of these methods, but is instead given as a rule table, the problem of testing whether it is reversible is solvable for block cellular automata and for one-dimensional cellular automata, but is undecidable for other types of cellular automata.
Reversible cellular automata form a natural model of reversible computing, a technology that could lead to ultra-low-power computing devices. Quantum cellular automata, one way of performing computations using the principles of quantum mechanics, are often required to be reversible. Additionally, many problems in physical modeling, such as the motion of particles in an ideal gas or the Ising model of alignment of magnetic charges, are naturally reversible and can be simulated by reversible cellular automata.
Properties related to reversibility may also be used to study cellular automata that are not reversible on their entire configuration space, but that have a subset of the configuration space as an attractor that all initially random configurations converge towards. As Stephen Wolfram writes, "once on an attractor, any system—even if it does not have reversible underlying rules—must in some sense show approximate reversibility."[1]
Examples
One-dimensional automata
A cellular automaton is defined by its cells (often a one- or two-dimensional array), a finite set of values or states that can go into each cell, a neighborhood associating each cell with a finite set of nearby cells, and an update rule according to which the values of all cells are updated, simultaneously, as a function of the values of their neighboring cells. The simplest possible cellular automata have a one-dimensional array of cells, each of which can hold a binary value (either 0 or 1), with each cell having a neighborhood consisting only of it and its two nearest cells on either side; these are called the elementary cellular automata.[2] If the update rule for such an automaton causes each cell to always remain in the same state, then the automaton is reversible: the previous state of all cells can be recovered from their current states, because for each cell the previous and current states are the same. Similarly, if the update rule causes every cell to change its state from 0 to 1 and vice versa, or if it causes a cell to copy the state from a fixed neighboring cell, or if it causes it to copy a state and then reverse its value, it is necessarily reversible.[3] Toffoli & Margolus (1990) call these types of reversible cellular automata, in which the state of each cell depends only on the previous state of one neighboring cell, "trivial". Despite its simplicity, the update rule that causes each cell to copy the state of a neighboring cell is important in the theory of symbolic dynamics, where it is known as the shift map.[4]
A little less trivially, suppose that the cells again form a one-dimensional array, but that each state is an ordered pair (l,r) consisting of a left part l and a right part r, each drawn from a finite set of possible values. Define a transition function that sets the left part of a cell to be the left part of its left neighbor and the right part of a cell to be the right part of its right neighbor. That is, if the left neighbor's state is (a,b) and the right neighbor's state is (c,d), the new state of a cell is the result of combining these states using a pairwise operation × defined by the equation (a,b) × (c,d) = (a,d). An example of this construction is given in the illustration, in which the left part is represented graphically as a shape and the right part is represented as a color; in this example, each cell is updated with the shape of its left neighbor and the color of its right neighbor. Then this automaton is reversible: the values on the left side of each pair migrate rightwards and the values on the right side migrate leftwards, so the prior state of each cell can be recovered by looking for these values in neighboring cells. The operation × used to combine pairs of states in this automaton forms an algebraic structure known as a rectangular band.[5]
Multiplication of decimal numbers by two or by five can be performed by a one-dimensional reversible cellular automaton with ten states per cell (the ten decimal digits). Each digit of the product depends only on a neighborhood of two digits in the given number: the digit in the same position and the digit one position to the right. More generally, multiplication or division of doubly infinite digit sequences in any radix b, by a multiplier or divisor x all of whose prime factors are also prime factors of b, is an operation that forms a cellular automaton because it depends only on a bounded number of nearby digits, and is reversible because of the existence of multiplicative inverses.[6] Multiplication by other values (for instance, multiplication of decimal numbers by three) remains reversible, but does not define a cellular automaton, because there is no fixed bound on the number of digits in the initial value that are needed to determine a single digit in the result.
There are no nontrivial reversible elementary cellular automata.[7] However, a near-miss is provided by Rule 90 and other elementary cellular automata based on the exclusive or function. In Rule 90, the state of each cell is the exclusive or of the previous states of its two neighbors. This use of the exclusive or makes the transition rule locally invertible, in the sense that any contiguous subsequence of states can be generated by this rule. Rule 90 is not a reversible cellular automaton rule, because in Rule 90 every assignment of states to the complete array of cells has exactly four possible predecessors, whereas reversible rules are required to have exactly one predecessor per configuration.[8]
Critters' rule
Conway's Game of Life, one of the most famous cellular automaton rules, is not reversible: for instance, it has many patterns that die out completely, so the configuration in which all cells are dead has many predecessors, and it also has Garden of Eden patterns with no predecessors. However, another rule called "Critters" by its inventors, Tommaso Toffoli and Norman Margolus, is reversible and has similar dynamic behavior to Life.[9]
The Critters rule is a block cellular automaton in which, at each step, the cells of the automaton are partitioned into 2×2 blocks and each block is updated independently of the other blocks. Its transition function flips the state of every cell in a block that does not have exactly two live cells, and in addition rotates by 180° blocks with exactly three live cells. Because this function is invertible, the automaton defined by these rules is a reversible cellular automaton.[9]
When started with a smaller field of random cells centered within a larger region of dead cells, many small patterns similar to Life's glider escape from the central random area and interact with each other. The Critters rule can also support more complex spaceships of varying speeds as well as oscillators with infinitely many different periods.[9]
Constructions
Several general methods are known for constructing cellular automaton rules that are automatically reversible.
Block cellular automata
A block cellular automaton is an automaton at which, in each time step, the cells of the automaton are partitioned into congruent subsets (called blocks), and the same transformation is applied independently to each block. Typically, such an automaton will use more than one partition into blocks, and will rotate between these partitions at different time steps of the system.[10] In a frequently used form of this design, called the Margolus neighborhood, the cells of the automaton form a square grid and are partitioned into larger 2 × 2 square blocks at each step. The center of a block at one time step becomes the corner of four blocks at the next time step, and vice versa; in this way, the four cells in each 2 × 2 belong to four different 2 × 2 squares of the previous partition.[11] The Critters rule discussed above is an example of this type of automaton.
Designing reversible rules for block cellular automata, and determining whether a given rule is reversible, is easy: for a block cellular automaton to be reversible it is necessary and sufficient that the transformation applied to the individual blocks at each step of the automaton is itself reversible. When a block cellular automaton is reversible, the time-reversed version of its dynamics can also be described as a block cellular automaton with the same block structure, using a time-reversed sequence of partitions of cells into blocks, and with the transition function for each block being the inverse function of the original rule.[10]
Simulation of irreversible automata
Toffoli (1977) showed how to embed any irreversible d-dimensional cellular automaton rule into a reversible (d + 1)-dimensional rule. Each d-dimensional slice of the new reversible rule simulates a single time step of the original rule. In this way, Toffoli showed that many features of irreversible cellular automata, such as the ability to simulate arbitrary Turing machines, could also be extended to reversible cellular automata.
As Toffoli conjectured and Hertling (1998) proved, the increase in dimension incurred by Toffoli's method is a necessary payment for its generality: under mild assumptions (such as the translation-invariance of the embedding), any embedding of a cellular automaton that has a Garden of Eden into a reversible cellular automaton must increase the dimension.
Morita (1995) describes another type of simulation that does not obey Hertling's assumptions and does not change the dimension. Morita's method can simulate the finite configurations of any irreversible automaton in which there is a "quiescent" or "dead" state, such that if a cell and all its neighbors are quiescent then the cell remains quiescent in the next step. The simulation uses a reversible block cellular automaton of the same dimension as the original irreversible automaton. The information that would be destroyed by the irreversible steps of the simulated automaton is instead sent away from the configuration into the infinite quiescent region of the simulating automaton. This simulation does not update all cells of the simulated automaton simultaneously; rather, the time to simulate a single step is proportional to the size of the configuration being simulated. Nevertheless, the simulation accurately preserves the behavior of the simulated automaton, as if all of its cells were being updated simultaneously. Using this method it is possible to show that even one-dimensional reversible cellular automata are capable of universal computation.[12]
Second-order cellular automata
The second-order cellular automaton technique is a method of transforming any cellular automaton into a reversible cellular automaton, invented by Edward Fredkin and first published by several other authors in 1984.[13] In this technique, the state of each cell in the automaton at time t is a function both of its neighborhood at time t − 1 and of its own state at time t − 2. Specifically, the transition function of the automaton maps each neighborhood at time t − 1 to a permutation on the set of states, and then applies that permutation to the state at time t − 2. The reverse dynamics of the automaton may be computed by mapping each neighborhood to the inverse permutation and proceeding in the same way.[14]
In the case of automata with binary-valued states (zero or one), there are only two possible permutations on the states (the identity permutation and the permutation that swaps the two states), which may themselves be represented as the exclusive or of a state with a binary value. In this way, any conventional two-valued cellular automaton may be converted to a second-order cellular automaton rule by using the conventional automaton's transition function on the states at time t − 1, and then computing the exclusive or of these states with the states at time t − 2 to determine the states at time t. However, the behavior of the reversible cellular automaton determined in this way may not bear any resemblance to the behavior of the cellular automaton from which it was defined.[14]
Any second-order automaton may be transformed into a conventional cellular automaton, in which the transition function depends only on the single previous time step, by combining pairs of states from consecutive time steps of the second-order automaton into single states of a conventional cellular automaton.[14]
Conserved landscape
A one-dimensional cellular automaton found by Patt (1971) uses a neighborhood consisting of four contiguous cells. In this automaton, a cell flips its state whenever it occupies the "?" position in the pattern "0?10". No two such patterns can overlap, so the same "landscape" surrounding the flipped cell continues to be present after the transition. In the next step, the cell in the same "?" position will flip again, back to its original state. Therefore, this automaton is its own inverse, and is reversible. Patt performed a brute force search of all two-state one-dimensional cellular automata with small neighborhoods; this search led to the discovery of this automaton, and showed that it was the simplest possible nontrivial one-dimensional two-state reversible cellular automaton. There are no nontrivial reversible two-state automata with three-cell neighborhoods, and all two-state reversible automata with four-cell neighborhoods are simple variants on Patt's automaton.[15]
Patt's automaton can be viewed in retrospect as an instance of the "conserved landscape" technique for designing reversible cellular automata. In this technique, a change to the state of a cell is triggered by a pattern among a set of neighbors that do not themselves change states. In this way, the existence of the same pattern can be used to trigger the inverse change in the time-reversed dynamics of the automaton. Patt's automaton has very simple dynamics (all cyclic sequences of configurations have length two), but automata using the same conserved landscape technique with more than one triggering pattern are capable of more complex behavior. In particular they can simulate any second-order cellular automaton.[15]
The SALT model of Miller & Fredkin (2005) is a special case of the conserved landscape technique. In this model, the cells of an integer grid are split into even and odd subsets. In each time step certain pairs of cells of one parity are swapped, based on the configuration of nearby cells of the other parity. Rules using this model can simulate the billiard ball computer,[16] or support long strings of live cells that can move at many different speeds or vibrate at many different frequencies.[17]
Theory
A cellular automaton consists of an array of cells, each one of which has a finite number of possible states, together with a rule for updating all cells simultaneously based only on the states of neighboring cells. A configuration of a cellular automaton is an assignment of a state to every cell of the automaton; the update rule of a cellular automaton forms a function from configurations to configurations, with the requirement that the updated value of any cell depends only on some finite neighborhood of the cell, and that the function is invariant under translations of the input array.
With these definitions, a cellular automaton is reversible when it satisfies any one of the following conditions, all of which are mathematically equivalent to each other:[18]
1. Every configuration of the automaton has a unique predecessor that is mapped to it by the update rule.
2. The update rule of the automaton is a bijection; that is, a function that is both one-to-one and onto.
3. The update rule is an injective function, that is, there are no two configurations that both map to the same common configuration. This condition is obviously implied by the assumption that the update rule is a bijection. In the other direction, the Garden of Eden theorem for cellular automata implies that every injective update rule is bijective.[19]
4. The time-reversed dynamics of the automaton can be described by another cellular automaton. Clearly, for this to be possible, the update rule must be bijective. In the other direction, if the update rule is bijective, then it has an inverse function that is also bijective. This inverse function must be a cellular automaton rule. The proof of this fact uses the Curtis–Hedlund–Lyndon theorem, a topological characterization of cellular automata rules as the translation-invariant functions that are continuous with respect to the Cantor topology on the space of configurations.[20]
5. The update rule of the automaton is an automorphism of the shift dynamical system defined by the state space and the translations of the lattice of cells. That is, it is a homeomorphism that commutes with the shift map, as the Curtis–Hedlund–Lyndon theorem implies.[21]
Di Gregorio & Trautteur (1975) analyze several alternative definitions of reversibility for cellular automata. Most of these turn out to be equivalent either to injectivity or to surjectivity of the transition function of the automaton; however, there is one more alternative that does not match either of these two definitions. It applies to automata such as the Game of Life that have a quiescent or dead state. In such an automaton, one can define a configuration to be "finite" if it has only finitely many non-quiescent cells, and one can consider the class of automata for which every finite configuration has at least one finite predecessor. This class turns out to be distinct from both the surjective and injective automata, and in some subsequent research, automata with this property have been called invertible finite automata.[22]
Testing reversibility
It was first shown by Amoroso & Patt (1972) that the problem of testing reversibility of a given one-dimensional cellular automaton has an algorithmic solution. Alternative algorithms based on automata theory and de Bruijn graphs were given by Culik (1987) and Sutner (1991), respectively.
• Culik begins with the observation that a cellular automaton has an injective transition function if and only if the transition function is injective on the subsets of configurations that are periodic (repeating the same substring infinitely often in both directions). He defines a nondeterministic finite-state transducer that performs the transition rule of the automaton on periodic strings. This transducer works by remembering the neighborhood of the automaton at the start of the string and entering an accepting state when that neighborhood concatenated to the end of the input would cause its nondeterministically chosen transitions to be correct. Culik then swaps the input and output of the transducer. The transducer resulting from this swap simulates the inverse dynamics of the given automaton. Finally, Culik applies previously known algorithms to test whether the resulting swapped transducer maps each input to a single output.[23]
• Sutner defines a directed graph (a type of de Bruijn graph) in which each vertex represents a pair of assignments of states for the cells in a contiguous sequence of cells. The length of this sequence is chosen to be one less than the neighborhood size of the automaton. An edge in Sutner's graph represents a pair of sequences of cells that overlap in all but one cell, so that the union of the sequences is a full neighborhood in the cellular automaton. Each such edge is directed from the overlapping subsequence on the left to the subsequence on the right. Edges are only included in the graph when they represent compatible state assignments on the overlapping parts of their cell sequences, and when the automaton rule (applied to the neighborhood determined by the potential edge) would give the same results for both assignments of states. By performing a linear-time strong connectivity analysis of this graph, it is possible to determine which of its vertices belong to cycles. The transition rule is non-injective if and only if this graph contains a directed cycle in which at least one vertex has two differing state assignments.[8]
These methods take polynomial time, proportional to the square of the size of the state transition table of the input automaton.[24] A related algorithm of Hillman (1991) determines whether a given rule is surjective when applied to finite-length arrays of cells with periodic boundary conditions, and if so, for which lengths.
For a block cellular automaton, testing reversibility is also easy: the automaton is reversible if and only if the transition function on the blocks of the automaton is invertible, and in this case the reverse automaton has the same block structure with the inverse transition function.[10]
However, for cellular automata with other neighborhoods in two or more dimensions, the problem of testing reversibility is undecidable, meaning that there cannot exist an algorithm that always halts and always correctly answers the problem. The proof of this fact by Kari (1990) is based on the previously known undecidability of tiling the plane by Wang tiles, sets of square tiles with markings on their edges that constrain which pairs of tiles can fit edge-to-edge. Kari defines a cellular automaton from a set of Wang tiles, such that the automaton fails to be injective if and only if the given tile set can tile the entire plane. His construction uses the von Neumann neighborhood, and cells with large numbers of states. In the same paper, Kari also showed that it is undecidable to test whether a given cellular automaton rule of two or more dimensions is surjective (that is, whether it has a Garden of Eden).
Reverse neighborhood size
In a one-dimensional reversible cellular automaton with n states per cell, in which the neighborhood of a cell is an interval of m cells, the automaton representing the reverse dynamics has neighborhoods that consist of at most nm − 1 − m + 1 cells. This bound is known to be tight for m = 2: there exist n-state reversible cellular automata with two-cell neighborhoods whose time-reversed dynamics forms a cellular automaton with neighborhood size exactly n − 1.[25]
For any integer m there are only finitely many two-dimensional reversible m-state cellular automata with the von Neumann neighborhood. Therefore, there is a well-defined function f(m) such that all reverses of m-state cellular automata with the von Neumann neighborhood use a neighborhood with radius at most f(m): simply let f(m) be the maximum, among all of the finitely many reversible m-state cellular automata, of the neighborhood size needed to represent the time-reversed dynamics of the automaton. However, because of Kari's undecidability result, there is no algorithm for computing f(m) and the values of this function must grow very quickly, more quickly than any computable function.[12]
Wolfram's classification
A well-known classification of cellular automata by Stephen Wolfram studies their behavior on random initial conditions. For a reversible cellular automaton, if the initial configuration is chosen uniformly at random among all possible configurations, then that same uniform randomness continues to hold for all subsequent states. Thus it would appear that most reversible cellular automata are of Wolfram's Class 3: automata in which almost all initial configurations evolve pseudo-randomly or chaotically. However, it is still possible to distinguish among different reversible cellular automata by analyzing the effect of local perturbations on the behavior of the automaton. Making a change to the initial state of a reversible cellular automaton may cause changes to later states to remain only within a bounded region, to propagate irregularly but unboundedly, or to spread quickly, and Wolfram (1984) lists one-dimensional reversible cellular automaton rules exhibiting all three of these types of behavior.
Later work by Wolfram identifies the one-dimensional Rule 37R automaton as being particularly interesting in this respect. When run on a finite array of cells with periodic boundary conditions, starting from a small seed of random cells centered within a larger empty neighborhood, it tends to fluctuate between ordered and chaotic states. However, with the same initial conditions on an unbounded set of cells its configurations tend to organize themselves into several types of simple moving particles.[26]
Abstract algebra
Another way to formalize reversible cellular automata involves abstract algebra, and this formalization has been useful in developing computerized searches for reversible cellular automaton rules. Boykett (2004) defines a semicentral bigroupoid to be an algebraic structure consisting of a set S of elements and two operations → and ← on pairs of elements, satisfying the two equational axioms:
• for all elements a, b, and c in S, (a → b) ← (b → c) = b, and
• for all elements a, b, and c in S, (a ← b) → (b ← c) = b.
For instance, this is true for the two operations in which operation → returns its right argument and operation ← returns its left argument.
As Boykett argues, any one-dimensional reversible cellular automaton is equivalent to an automaton in rectangular form, in which the cells are offset a half unit at each time step, and in which both the forward and reverse evolution of the automaton have neighborhoods with just two cells, the cells a half unit away in each direction. If a reversible automaton has neighborhoods larger than two cells, it can be simulated by a reversible automaton with smaller neighborhoods and more states per cell, in which each cell of the simulating automaton simulates a contiguous block of cells in the simulated automaton. The two axioms of a semicentral bigroupoid are exactly the conditions required on the forward and reverse transition functions of these two-cell neighborhoods to be the reverses of each other. That is, every semicentral bigroupoid defines a reversible cellular automaton in rectangular form, in which the transition function of the automaton uses the → operation to combine the two cells of its neighborhood, and in which the ← operation similarly defines the reverse dynamics of the automaton. Every one-dimensional reversible cellular automaton is equivalent to one in this form.[5]
Boykett used this algebraic formulation as the basis for algorithms that exhaustively list all possible inequivalent reversible cellular automata.[27]
Conservation laws
When researchers design reversible cellular automata to simulate physical systems, they typically incorporate into the design the conservation laws of the system; for instance, a cellular automaton that simulates an ideal gas should conserve the number of gas particles and their total momentum, for otherwise it would not provide an accurate simulation. However, there has also been some research on the conservation laws that reversible cellular automata can have, independent of any intentional design. The typical type of conserved quantity measured in these studies takes the form of a sum, over all contiguous subsets of k cells of the automaton, of some numerical function of the states of the cells in each subset. Such a quantity is conserved if, whenever it takes a finite value, that value automatically remains constant through each time step of the automaton, and in this case it is called a kth-order invariant of the automaton.[28]
For instance, recall the one-dimensional cellular automaton defined as an example from a rectangular band, in which the cell states are pairs of values (l,r) drawn from sets L and R of left values and right values, the left value of each cell moves rightwards at each time step, and the right value of each cell moves leftwards. In this case, for each left or right value x of the band, one can define a conserved quantity, the total number of cells that have that value. If there are λ left values and ρ right values, then there are λ + ρ − 2 independent first-order-invariants, and any first-order invariant can be represented as a linear combination of these fundamental ones. The conserved quantities associated with left values flow uniformly to the right at a constant rate: that is, if the number of left values equal to x within some region C of the line takes a certain value at time 0, then it will take the same value for the shifted region C + t/2 at time t. Similarly, the conserved quantities associated with right values flow uniformly to the left.[29]
Any one-dimensional reversible cellular automaton may be placed into rectangular form, after which its transition rule may be factored into the action of an idempotent semicentral bigroupoid (a reversible rule for which regions of cells with a single state value change only at their boundaries) together with a permutation on the set of states. The first-order invariants for the idempotent lifting of the automaton rule (the modified rule formed by omitting the permutation) necessarily behave like the ones for a rectangular band: they have a basis of invariants that flow either leftwards or rightwards at a constant rate without interaction. The first-order invariants for the overall automaton are then exactly the invariants for the idempotent lifting that give equal weight to every pair of states that belong to the same orbit of the permutation. However, the permutation of states in the rule may cause these invariants to behave differently from in the idempotent lifting, flowing non-uniformly and with interactions.[29]
In physical systems, Noether's theorem provides an equivalence between conservation laws and symmetries of the system. However, for cellular automata this theorem does not directly apply, because instead of being governed by the energy of the system the behavior of the automaton is encoded into its rules, and the automaton is guaranteed to obey certain symmetries (translation invariance in both space and time) regardless of any conservation laws it might obey. Nevertheless, the conserved quantities of certain reversible systems behave similarly to energy in some respects. For instance, if different regions of the automaton have different average values of some conserved quantity, the automaton's rules may cause this quantity to dissipate, so that the distribution of the quantity is more uniform in later states. Using these conserved quantities as a stand-in for the energy of the system can allow it to be analyzed using methods from classical physics.[30]
Applications
Lattice gas automata
A lattice gas automaton is a cellular automaton designed to simulate the motion of particles in a fluid or an ideal gas. In such a system, gas particles move on straight lines with constant velocity, until undergoing elastic collision with other particles. Lattice gas automata simplify these models by only allowing a constant number of velocities (typically, only one speed and either four or six directions of motion) and by simplifying the types of collision that are possible.[31]
Specifically, the HPP lattice gas model consists of particles moving at unit velocity in the four axis-parallel directions. When two particles meet on the same line in opposite directions, they collide and are sent outwards from the collision point on the perpendicular line. This system obeys the conservation laws of physical gases, and produces simulations whose appearance resembles the behavior of physical gases. However, it was found to obey unrealistic additional conservation laws. For instance, the total momentum within any single line is conserved. As well, the differences between axis-parallel and non-axis-parallel directions in this model (its anisotropy) is undesirably high. The FHP lattice gas model improves the HPP model by having particles moving in six different directions, at 60 degree angles to each other, instead of only four directions. In any head-on collision, the two outgoing particles are deflected at 60 degree angles from the two incoming particles. Three-way collisions are also possible in the FHP model and are handled in a way that both preserves total momentum and avoids the unphysical added conservation laws of the HPP model.[31]
Because the motion of the particles in these systems is reversible, they are typically implemented with reversible cellular automata. In particular, both the HPP and FHP lattice gas automata can be implemented with a two-state block cellular automaton using the Margolus neighborhood.[31]
Ising model
The Ising model is used to model the behavior of magnetic systems. It consists of an array of cells, the state of each of which represents a spin, either up or down. The energy of the system is measured by a function that depends on the number of neighboring pairs of cells that have the same spin as each other. Therefore, if a cell has equal numbers of neighbors in the two states, it may flip its own state without changing the total energy. However, such a flip is energy-conserving only if no two adjacent cells flip at the same time.[32]
Cellular automaton models of this system divide the square lattice into two alternating subsets, and perform updates on one of the two subsets at a time. In each update, every cell that can flip does so. This defines a reversible cellular automaton which can be used to investigate the Ising model.[32]
Billiard-ball computation and low-power computing
Fredkin & Toffoli (1982) proposed the billiard-ball computer as part of their investigations into reversible computing. A billiard-ball computer consists of a system of synchronized particles (the billiard balls) moving in tracks and guided by a fixed set of obstacles. When the particles collide with each other or with the obstacles, they undergo an elastic collision much as real billiard balls would do. The input to the computer is encoded using the presence or absence of particles on certain input tracks, and its output is similarly encoded using the presence or absence of particles on output tracks. The tracks themselves may be envisioned as wires, and the particles as being Boolean signals transported on those wires. When a particle hits an obstacle, it reflects from it. This reflection may be interpreted as a change in direction of the wire the particle is following. Two particles on different tracks may collide, forming a logic gate at their collision point.[33]
As Margolus (1984) showed, billiard-ball computers may be simulated using a two-state reversible block cellular automaton with the Margolus neighborhood. In this automaton's update rule, blocks with exactly one live cell rotate by 180°, blocks with two diagonally opposite live cells rotate by 90°, and all other blocks remain unchanged. These rules cause isolated live cells to behave like billiard balls, moving on diagonal trajectories. Connected groups of more than one live cell behave instead like the fixed obstacles of the billiard-ball computer. In an appendix, Margolus also showed that a three-state second-order cellular automaton using the two-dimensional Moore neighborhood could simulate billiard-ball computers.
Unsolved problem in mathematics:
Is every three-dimensional reversible cellular automaton locally reversible?
(more unsolved problems in mathematics)
One reason to study reversible universal models of computation such as the billiard-ball model is that they could theoretically lead to actual computer systems that consume very low quantities of energy. According to Landauer's principle, irreversible computational steps require a certain minimal amount of energy per step, but reversible steps can be performed with an amount of energy per step that is arbitrarily close to zero.[34] However, in order to perform computation using less energy than Landauer's bound, it is not good enough for a cellular automaton to have a transition function that is globally reversible: what is required is that the local computation of the transition function also be done in a reversible way. For instance, reversible block cellular automata are always locally reversible: the behavior of each individual block involves the application of an invertible function with finitely many inputs and outputs. Toffoli & Margolus (1990) were the first to ask whether every reversible cellular automaton has a locally reversible update rule. Kari (1996) showed that for one- and two-dimensional automata the answer is positive, and Durand-Lose (2001) showed that any reversible cellular automaton could be simulated by a (possibly different) locally reversible cellular automaton. However, the question of whether every reversible transition function is locally reversible remains open for dimensions higher than two.[35]
Synchronization
The "Tron" rule of Toffoli and Margolus is a reversible block cellular rule with the Margolus neighborhood. When a 2 × 2 block of cells all have the same state, all cells of the block change state; in all other cases, the cells of the block remain unchanged. As Toffoli and Margolus argue, the evolution of patterns generated by this rule can be used as a clock to synchronize any other rule on the Margolus neighborhood. A cellular automaton synchronized in this way obeys the same dynamics as the standard Margolus-neighborhood rule while running on an asynchronous cellular automaton.[36]
Encryption
Kari (1990) proposed using multidimensional reversible cellular automata as an encryption system. In Kari's proposal, the cellular automaton rule would be the encryption key. Encryption would be performed by running the rule forward one step, and decryption would be performed by running it backward one step. Kari suggests that a system such as this may be used as a public-key cryptosystem. In principle, an attacker could not algorithmically determine the decryption key (the reverse rule) from a given encryption key (forward rule) because of the undecidability of testing reversibility, so the forward rule could be made public without compromising the security of the system. However, Kari did not specify which types of reversible cellular automaton should be used for such a system, or show how a cryptosystem using this approach would be able to generate encryption/decryption key pairs.
Chai, Cao & Zhou (2005) have proposed an alternative encryption system. In their system, the encryption key determines the local rule for each cell of a one-dimensional cellular automaton. A second-order automaton based on that rule is run for several rounds on an input to transform it into an encrypted output. The reversibility property of the automaton ensures that any encrypted message can be decrypted by running the same system in reverse. In this system, keys must be kept secret, because the same key is used both for encryption and decryption.
Quantum computing
Quantum cellular automata are arrays of automata whose states and state transitions obey the laws of quantum dynamics. Quantum cellular automata were suggested as a model of computation by Feynman (1982) and first formalized by Watrous (1995). Several competing notions of these automata remain under research, many of which require that the automata constructed in this way be reversible.[37]
Physical universality
Janzing (2010) asked whether it was possible for a cellular automaton to be physically universal, meaning that, for any bounded region of the automaton's cells, it should be possible to surround that region with cells whose states form an appropriate support scaffolding that causes the automaton to implement any arbitrary transformation on sets of states within the region. Such an automaton must be reversible, or at least locally injective, because automata without this property have Garden of Eden patterns, and it is not possible to implement a transformation that creates a Garden of Eden.
Schaeffer (2015) constructed a reversible cellular automaton that is physically universal in this sense. Schaeffer's automaton is a block cellular automaton with two states and the Margolis neighborhood, closely related to the automata for the billiard ball model and for the HPP lattice gas. However, the billiard ball model is not physically universal, as it can be used to construct impenetrable walls preventing the state within some region from being read and transformed. In Schaeffer's model, every pattern eventually decomposes into particles moving diagonally in four directions. Thus, his automaton is not Turing complete. However, Schaeffer showed that it is possible to surround any finite configuration by scaffolding that decays more slowly than it. After the configuration decomposes into particles, the scaffolding intercepts those particles, and uses them as the input to a system of Boolean circuits constructed within the scaffolding. These circuits can be used to compute arbitrary functions of the initial configuration. The scaffolding then translates the output of the circuits back into a system of moving particles, which converge on the initial region and collide with each other to build a copy of the transformed state. In this way, Schaeffer's system can be used to apply any function to any bounded region of the state space, showing that this automaton rule is physically universal.[38]
Notes
1. Wolfram (2002), p. 1018.
2. Schiff (2008), p. 44.
3. Toffoli & Margolus (1990).
4. Blanchard, Devaney & Keen (2004), p. 38: "The shift map is without doubt the fundamental object in symbolic dynamics."
5. Boykett (2004).
6. Wolfram (2002), p. 1093.
7. Patt (1971).
8. Sutner (1991).
9. Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005).
10. Toffoli & Margolus (1987), Section 14.5, "Partitioning technique", pp. 150–153; Schiff (2008), Section 4.2.1, "Partitioning Cellular Automata", pp. 115–116.
11. Toffoli & Margolus (1987), Chapter 12, "The Margolus Neighborhood", pp. 119–138.
12. Kari (2005).
13. Margolus (1984); Vichniac (1984); Wolfram (1984).
14. Toffoli & Margolus (1987), Section 14.2, "Second-order technique", pp. 147–149. Wolfram (2002), pp. 437ff. McIntosh (2009).
15. Toffoli & Margolus (1990), section 5.3, "Conserved-landscape permutations", pp. 237–238.
16. Miller & Fredkin (2005).
17. Miller & Fredkin (2012).
18. In the one-dimensional case, several of these equivalences were already presented, in the language of dynamical systems rather than cellular automata, by Hedlund (1969), Theorem 4.1. For higher dimensions, see Richardson (1972) and Di Gregorio & Trautteur (1975).
19. Myhill (1963).
20. Richardson (1972).
21. Hedlund (1969).
22. Moraal (2000).
23. Culik cites a 1979 automata theory textbook for this result, but see Béal et al. (2003) for more recent developments on efficiently testing whether a transducer defines a function.
24. Neither Amoroso & Patt (1972) nor Culik (1987) state their algorithms' time complexities explicitly, but Sutner (1991) does, and this bound can also be found e.g. in Czeizler & Kari (2007).
25. Kari (1992); Czeizler (2004); Czeizler & Kari (2007).
26. Wolfram (2002), pp. 454–457.
27. Boykett (2004). See Hillman (1991) and Seck Tuoh Mora et al. (2005) for closely related work on the enumeration of width-2 reversible cellular automata.
28. Hattori & Takesue (1991); Fukś (2007).
29. Boykett, Kari & Taati (2008).
30. Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011).
31. Toffoli & Margolus (1987), Chapter 16, "Fluid dynamics", pp. 172–184.
32. Toffoli & Margolus (1987), Chapter 17.2, "Ising systems", pp. 186–190.
33. Durand-Lose (2002).
34. Fredkin & Toffoli (1982).
35. Kari (2005, 2009)
36. Toffoli & Margolus (1987), Section 12.8.3, "Asynchronous computation", pp. 134–136.
37. Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008).
38. See also "A Physically Universal Cellular Automaton", Shtetl-Optimized, Scott Aaronson, June 26, 2014.
References
• Amoroso, S.; Patt, Y. N. (1972), "Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures", Journal of Computer and System Sciences, 6 (5): 448–464, doi:10.1016/S0022-0000(72)80013-8, MR 0317852.
• Béal, Marie-Pierre; Carton, Olivier; Prieur, Christophe; Sakarovitch, Jacques (2003), "Squaring transducers: an efficient procedure for deciding functionality and sequentiality", Theoretical Computer Science, 292 (1): 45–63, doi:10.1016/S0304-3975(01)00214-6, MR 1964625.
• Blanchard, Paul; Devaney, Robert L.; Keen, Linda (2004), "Complex dynamics and symbolic dynamics", in Williams, Susan G. (ed.), Symbolic Dynamics and its Applications, Proceedings of Symposia in Applied Mathematics, vol. 60, Providence, RI: American Mathematical Society, pp. 37–60, doi:10.1090/psapm/060/2078845, MR 2078845.
• Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738.
• Boykett, Tim; Kari, Jarkko; Taati, Siamak (2008), "Conservation laws in rectangular CA" (PDF), Journal of Cellular Automata, 3 (2): 115–122, MR 2394641, archived from the original (PDF) on 2015-09-30.
• Capobianco, Silvio; Toffoli, Tommaso (2011), "Can anything from Noether's theorem be salvaged for discrete dynamical systems?", Proceedings of the 10th International Conference on Unconventional Computation (UC 2011), Lecture Notes in Computer Science, vol. 6714, Springer-Verlag, pp. 77–88, arXiv:1103.4785, doi:10.1007/978-3-642-21341-0_13, S2CID 42541816.
• Chai, Zhenchuan; Cao, Zhenfu; Zhou, Yuan (2005), "Encryption based on reversible second-order cellular automata", Parallel and Distributed Processing and Applications (ISPA 2005 Workshops), Lecture Notes in Computer Science, vol. 3759, Springer-Verlag, pp. 350–358, doi:10.1007/11576259_39.
• Culik, Karel, II (1987), "On invertible cellular automata" (PDF), Complex Systems, 1 (6): 1035–1044, MR 0931401{{citation}}: CS1 maint: multiple names: authors list (link).
• Czeizler, Eugen (2004), "On the size of the inverse neighborhoods for one-dimensional reversible cellular automata", Theoretical Computer Science, 325 (2): 273–284, doi:10.1016/j.tcs.2004.06.009, MR 2086740.
• Czeizler, Eugen; Kari, Jarkko (2007), "A tight linear bound on the synchronization delay of bijective automata", Theoretical Computer Science, 380 (1–2): 23–36, doi:10.1016/j.tcs.2007.02.052, MR 2330639.
• Di Gregorio, S.; Trautteur, G. (1975), "On reversibility in cellular automata", Journal of Computer and System Sciences, 11 (3): 382–391, doi:10.1016/S0022-0000(75)80059-6, MR 0392201.
• Durand-Lose, Jérôme (2001), "Representing reversible cellular automata with reversible block cellular automata", Discrete models: combinatorics, computation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrèt. (MIMD), Paris, pp. 145–154, MR 1888769.
• Durand-Lose, Jérôme (2002), "Computing inside the billiard ball model", in Adamatzky, Andrew (ed.), Collision-Based Computing, Springer-Verlag, pp. 135–160.
• Feynman, Richard P. (1982), "Simulating physics with computers", International Journal of Theoretical Physics, 21 (6–7): 467–488, Bibcode:1982IJTP...21..467F, doi:10.1007/BF02650179, MR 0658311, S2CID 124545445.
• Fredkin, Edward; Toffoli, Tommaso (1982), "Conservative logic", International Journal of Theoretical Physics, 21 (3–4): 219–253, Bibcode:1982IJTP...21..219F, doi:10.1007/BF01857727, MR 0657156, S2CID 37305161. Reprinted in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 47–82.
• Fukś, Henryk (2007), "Remarks on the critical behavior of second order additive invariants in elementary cellular automata", Fundamenta Informaticae, 78 (3): 329–341, arXiv:nlin/0502037, Bibcode:2005nlin......2037F, MR 2346870.
• Hattori, Tetsuya; Takesue, Shinji (1991), "Additive conserved quantities in discrete-time lattice dynamical systems", Physica D: Nonlinear Phenomena, 49 (3): 295–322, Bibcode:1991PhyD...49..295H, doi:10.1016/0167-2789(91)90150-8, MR 1115865.
• Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927.
• Hertling, Peter (1998), "Embedding cellular automata into reversible ones", Unconventional models of computation (Auckland, 1998), Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer-Verlag, pp. 243–256, MR 1653663.
• Hillman, David (1991), "The structure of reversible one-dimensional cellular automata", Physica D: Nonlinear Phenomena, 52 (2–3): 277–292, Bibcode:1991PhyD...52..277H, doi:10.1016/0167-2789(91)90128-V, MR 1128996.
• Janzing, Dominik (2010), Is there a physically universal cellular automaton or Hamiltonian?, arXiv:1009.1720, Bibcode:2010arXiv1009.1720J.
• Kari, Jarkko (1990), "Reversibility of 2D cellular automata is undecidable", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 379–385, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882.
• Kari, Jarkko (1992), "On the inverse neighborhoods of reversible cellular automata", Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology, Springer-Verlag, pp. 477–495, doi:10.1007/978-3-642-58117-5_29, MR 1226709.
• Kari, Jarkko (1996), "Representation of reversible cellular automata with block permutations", Mathematical Systems Theory, 29 (1): 47–61, doi:10.1007/BF01201813, MR 1360196, S2CID 31986003.
• Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250.
• Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7–11, 2009, Proceedings, Lecture Notes in Computer Science, vol. 5715, Springer-Verlag, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, MR 2539690.
• Margolus, Norman (1984), "Physics-like models of computation", Physica D: Nonlinear Phenomena, 10 (1–2): 81–95, Bibcode:1984PhyD...10...81M, doi:10.1016/0167-2789(84)90252-5, MR 0762656. Reprinted in Wolfram, Stephen (1986), Theory and Applications of Cellular Automata, Advanced series on complex systems, vol. 1, World Scientific, pp. 232–246, Bibcode:1986taca.book.....W, and in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 83–104.
• Margolus, Norman (1999), "Crystalline computation", in Hey, Anthony J. G. (ed.), Feynman and Computation, Perseus Books, pp. 267–305, arXiv:comp-gas/9811002, Bibcode:1998comp.gas.11002M.
• Marotta, Sebastian M. (2005), "Living in Critters' world", Revista Ciências Exatas e Naturais, 7 (1), archived from the original on 2012-03-19.
• McIntosh, Harold V. (2009), "12. Reversible cellular automata", One Dimensional Cellular Automata, Luniver Press, pp. 205–246.
• Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940.
• Miller, Daniel B.; Fredkin, Edward (2005), "Two-state, reversible, universal cellular automata in three dimensions", Proceedings of the 2nd Conference on Computing Frontiers (CF '05), New York, NY, USA: ACM, pp. 45–51, arXiv:nlin/0501022, doi:10.1145/1062261.1062271, ISBN 1-59593-019-1, S2CID 14082792.
• Miller, Daniel B.; Fredkin, Edward (2012), Circular motion of strings in cellular automata, and other surprises, arXiv:1206.2060, Bibcode:2012arXiv1206.2060M.
• Moraal, Hendrik (2000), "Graph-theoretical characterization of invertible cellular automata", Physica D: Nonlinear Phenomena, 141 (1–2): 1–18, Bibcode:2000PhyD..141....1M, doi:10.1016/S0167-2789(00)00020-8, MR 1764165.
• Morita, Kenichi (1995), "Reversible simulation of one-dimensional irreversible cellular automata", Theoretical Computer Science, 148 (1): 157–163, doi:10.1016/0304-3975(95)00038-X, MR 1347674.
• Myhill, John (1963), "The converse of Moore's Garden-of-Eden theorem", Proceedings of the American Mathematical Society, 14 (4): 685–686, doi:10.2307/2034301, JSTOR 2034301, MR 0155764. Reprinted in Burks, Arthur W. (1970), Essays on Cellular Automata, University of Illinois Press, pp. 204–205.
• Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.
• Patt, Y. N. (1971), Injections of neighborhood size three and four on the set of configurations from the infinite one-dimensional tessellation automata of two-state cells, Technical Report ECON-N1-P-1, Ft. Monmouth, NJ 07703. As cited by Amoroso & Patt (1972) and Toffoli & Margolus (1990).
• Pomeau, Y. (1984), "Invariants in cellular automata", Journal of Physics A: Mathematical and General, 17 (8): L415–L418, Bibcode:1984JPhA...17L.415P, doi:10.1088/0305-4470/17/8/004, MR 0750565.
• Richardson, D. (1972), "Tessellations with local transformations", Journal of Computer and System Sciences, 6 (5): 373–388, doi:10.1016/S0022-0000(72)80009-6, MR 0319678.
• Schaeffer, Luke (2015), "A physically universal cellular automaton", Proceedings of the 6th Innovations in Theoretical Computer Science Conference (ITCS 2015), Association for Computing Machinery, pp. 237–246, doi:10.1145/2688073.2688107, S2CID 16903144, ECCC TR14-084.
• Schiff, Joel L. (2008), Cellular Automata: A Discrete View of the World, Wiley, ISBN 978-0-470-16879-0.
• Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S.
• Seck Tuoh Mora, Juan Carlos; Chapa Vergara, Sergio V.; Juárez Martínez, Genaro; McIntosh, Harold V. (2005), "Procedures for calculating reversible one-dimensional cellular automata" (PDF), Physica D: Nonlinear Phenomena, 202 (1–2): 134–141, Bibcode:2005PhyD..202..134S, doi:10.1016/j.physd.2005.01.018, MR 2131890.
• Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768.
• Sutner, Klaus (1991), "De Bruijn graphs and linear cellular automata" (PDF), Complex Systems, 5: 19–30, MR 1116419.
• Takesue, Shinji (1990), "Relaxation properties of elementary reversible cellular automata", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 278–284, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882.
• Toffoli, Tommaso (1977), "Computation and construction universality of reversible cellular automata", Journal of Computer and System Sciences, 15 (2): 213–231, doi:10.1016/S0022-0000(77)80007-X, MR 0462816.
• Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 9780262200608.
• Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
• Vichniac, Gérard Y. (1984), "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7, MR 0762657.
• Watrous, John (1995), "On one-dimensional quantum cellular automata", Proceedings of the 36th Annual Symposium on Foundations of Computer Science (Milwaukee, WI, 1995), Los Alamitos, CA: IEEE Computer Society Press, pp. 528–537, doi:10.1109/SFCS.1995.492583, MR 1619103, S2CID 7441203.
• Wolfram, Stephen (1984), "Cellular automata as models of complexity" (PDF), Nature, 311 (5985): 419–424, Bibcode:1984Natur.311..419W, doi:10.1038/311419a0, S2CID 4237923.
• Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418
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Reversible diffusion
In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov.
Kolmogorov's characterization of reversible diffusions
Let B denote a d-dimensional standard Brownian motion; let b : Rd → Rd be a Lipschitz continuous vector field. Let X : [0, +∞) × Ω → Rd be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation
$\mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\mathrm {d} B_{t}$
with square-integrable initial condition, i.e. X0 ∈ L2(Ω, Σ, P; Rd). Then the following are equivalent:
• The process X is reversible with stationary distribution μ on Rd.
• There exists a scalar potential Φ : Rd → R such that b = −∇Φ, μ has Radon–Nikodym derivative
${\frac {\mathrm {d} \mu (x)}{\mathrm {d} x}}=\exp \left(-2\Phi (x)\right)$
and
$\int _{\mathbf {R} ^{d}}\exp \left(-2\Phi (x)\right)\,\mathrm {d} x=1.$
(Of course, the condition that b be the negative of the gradient of Φ only determines Φ up to an additive constant; this constant may be chosen so that exp(−2Φ(·)) is a probability density function with integral 1.)
References
• Voß, Jochen (2004). Some large deviation results for diffusion processes (Thesis). Universität Kaiserslautern: PhD thesis. (See theorem 1.4)
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Revised simplex method
In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming.
The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constraints adjusted to a set of basic variables, it maintains a representation of a basis of the matrix representing the constraints. The matrix-oriented approach allows for greater computational efficiency by enabling sparse matrix operations.[1]
Problem formulation
For the rest of the discussion, it is assumed that a linear programming problem has been converted into the following standard form:
${\begin{array}{rl}{\text{maximize}}&{\boldsymbol {c}}^{\mathrm {T} }{\boldsymbol {x}}\\{\text{subject to}}&{\boldsymbol {Ax}}={\boldsymbol {b}},{\boldsymbol {x}}\geq {\boldsymbol {0}}\end{array}}$
where A ∈ ℝm×n. Without loss of generality, it is assumed that the constraint matrix A has full row rank and that the problem is feasible, i.e., there is at least one x ≥ 0 such that Ax = b. If A is rank-deficient, either there are redundant constraints, or the problem is infeasible. Both situations can be handled by a presolve step.
Algorithmic description
Optimality conditions
For linear programming, the Karush–Kuhn–Tucker conditions are both necessary and sufficient for optimality. The KKT conditions of a linear programming problem in the standard form is
${\begin{aligned}{\boldsymbol {Ax}}&={\boldsymbol {b}},\\{\boldsymbol {A}}^{\mathrm {T} }{\boldsymbol {\lambda }}+{\boldsymbol {s}}&={\boldsymbol {c}},\\{\boldsymbol {x}}&\geq {\boldsymbol {0}},\\{\boldsymbol {s}}&\geq {\boldsymbol {0}},\\{\boldsymbol {s}}^{\mathrm {T} }{\boldsymbol {x}}&=0\end{aligned}}$
where λ and s are the Lagrange multipliers associated with the constraints Ax = b and x ≥ 0, respectively.[2] The last condition, which is equivalent to sixi = 0 for all 1 < i < n, is called the complementary slackness condition.
By what is sometimes known as the fundamental theorem of linear programming, a vertex x of the feasible polytope can be identified by being a basis B of A chosen from the latter's columns.[lower-alpha 1] Since A has full rank, B is nonsingular. Without loss of generality, assume that A = [B N]. Then x is given by
${\boldsymbol {x}}={\begin{bmatrix}{\boldsymbol {x_{B}}}\\{\boldsymbol {x_{N}}}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {B}}^{-1}{\boldsymbol {b}}\\{\boldsymbol {0}}\end{bmatrix}}$
where xB ≥ 0. Partition c and s accordingly into
${\begin{aligned}{\boldsymbol {c}}&={\begin{bmatrix}{\boldsymbol {c_{B}}}\\{\boldsymbol {c_{N}}}\end{bmatrix}},\\{\boldsymbol {s}}&={\begin{bmatrix}{\boldsymbol {s_{B}}}\\{\boldsymbol {s_{N}}}\end{bmatrix}}.\end{aligned}}$
To satisfy the complementary slackness condition, let sB = 0. It follows that
${\begin{aligned}{\boldsymbol {B}}^{\mathrm {T} }{\boldsymbol {\lambda }}&={\boldsymbol {c_{B}}},\\{\boldsymbol {N}}^{\mathrm {T} }{\boldsymbol {\lambda }}+{\boldsymbol {s_{N}}}&={\boldsymbol {c_{N}}},\end{aligned}}$
which implies that
${\begin{aligned}{\boldsymbol {\lambda }}&=({\boldsymbol {B}}^{\mathrm {T} })^{-1}{\boldsymbol {c_{B}}},\\{\boldsymbol {s_{N}}}&={\boldsymbol {c_{N}}}-{\boldsymbol {N}}^{\mathrm {T} }{\boldsymbol {\lambda }}.\end{aligned}}$
If sN ≥ 0 at this point, the KKT conditions are satisfied, and thus x is optimal.
Pivot operation
If the KKT conditions are violated, a pivot operation consisting of introducing a column of N into the basis at the expense of an existing column in B is performed. In the absence of degeneracy, a pivot operation always results in a strict decrease in cTx. Therefore, if the problem is bounded, the revised simplex method must terminate at an optimal vertex after repeated pivot operations because there are only a finite number of vertices.[4]
Select an index m < q ≤ n such that sq < 0 as the entering index. The corresponding column of A, Aq, will be moved into the basis, and xq will be allowed to increase from zero. It can be shown that
${\frac {\partial ({\boldsymbol {c}}^{\mathrm {T} }{\boldsymbol {x}})}{\partial x_{q}}}=s_{q},$
i.e., every unit increase in xq results in a decrease by −sq in cTx.[5] Since
${\boldsymbol {Bx_{B}}}+{\boldsymbol {A}}_{q}x_{q}={\boldsymbol {b}},$
xB must be correspondingly decreased by ΔxB = B−1Aqxq subject to xB − ΔxB ≥ 0. Let d = B−1Aq. If d ≤ 0, no matter how much xq is increased, xB − ΔxB will stay nonnegative. Hence, cTx can be arbitrarily decreased, and thus the problem is unbounded. Otherwise, select an index p = argmin1≤i≤m {xi/di | di > 0} as the leaving index. This choice effectively increases xq from zero until xp is reduced to zero while maintaining feasibility. The pivot operation concludes with replacing Ap with Aq in the basis.
Numerical example
See also: Simplex method § Example
Consider a linear program where
${\begin{aligned}{\boldsymbol {c}}&={\begin{bmatrix}-2&-3&-4&0&0\end{bmatrix}}^{\mathrm {T} },\\{\boldsymbol {A}}&={\begin{bmatrix}3&2&1&1&0\\2&5&3&0&1\end{bmatrix}},\\{\boldsymbol {b}}&={\begin{bmatrix}10\\15\end{bmatrix}}.\end{aligned}}$
Let
${\begin{aligned}{\boldsymbol {B}}&={\begin{bmatrix}{\boldsymbol {A}}_{4}&{\boldsymbol {A}}_{5}\end{bmatrix}},\\{\boldsymbol {N}}&={\begin{bmatrix}{\boldsymbol {A}}_{1}&{\boldsymbol {A}}_{2}&{\boldsymbol {A}}_{3}\end{bmatrix}}\end{aligned}}$
initially, which corresponds to a feasible vertex x = [0 0 0 10 15]T. At this moment,
${\begin{aligned}{\boldsymbol {\lambda }}&={\begin{bmatrix}0&0\end{bmatrix}}^{\mathrm {T} },\\{\boldsymbol {s_{N}}}&={\begin{bmatrix}-2&-3&-4\end{bmatrix}}^{\mathrm {T} }.\end{aligned}}$
Choose q = 3 as the entering index. Then d = [1 3]T, which means a unit increase in x3 results in x4 and x5 being decreased by 1 and 3, respectively. Therefore, x3 is increased to 5, at which point x5 is reduced to zero, and p = 5 becomes the leaving index.
After the pivot operation,
${\begin{aligned}{\boldsymbol {B}}&={\begin{bmatrix}{\boldsymbol {A}}_{3}&{\boldsymbol {A}}_{4}\end{bmatrix}},\\{\boldsymbol {N}}&={\begin{bmatrix}{\boldsymbol {A}}_{1}&{\boldsymbol {A}}_{2}&{\boldsymbol {A}}_{5}\end{bmatrix}}.\end{aligned}}$
Correspondingly,
${\begin{aligned}{\boldsymbol {x}}&={\begin{bmatrix}0&0&5&5&0\end{bmatrix}}^{\mathrm {T} },\\{\boldsymbol {\lambda }}&={\begin{bmatrix}0&-4/3\end{bmatrix}}^{\mathrm {T} },\\{\boldsymbol {s_{N}}}&={\begin{bmatrix}2/3&11/3&4/3\end{bmatrix}}^{\mathrm {T} }.\end{aligned}}$
A positive sN indicates that x is now optimal.
Practical issues
Degeneracy
See also: Simplex method § Degeneracy: stalling and cycling
Because the revised simplex method is mathematically equivalent to the simplex method, it also suffers from degeneracy, where a pivot operation does not result in a decrease in cTx, and a chain of pivot operations causes the basis to cycle. A perturbation or lexicographic strategy can be used to prevent cycling and guarantee termination.[6]
Basis representation
Two types of linear systems involving B are present in the revised simplex method:
${\begin{aligned}{\boldsymbol {Bz}}&={\boldsymbol {y}},\\{\boldsymbol {B}}^{\mathrm {T} }{\boldsymbol {z}}&={\boldsymbol {y}}.\end{aligned}}$
Instead of refactorizing B, usually an LU factorization is directly updated after each pivot operation, for which purpose there exist several strategies such as the Forrest−Tomlin and Bartels−Golub methods. However, the amount of data representing the updates as well as numerical errors builds up over time and makes periodic refactorization necessary.[1][7]
Notes and references
Notes
1. The same theorem also states that the feasible polytope has at least one vertex and that there is at least one vertex which is optimal.[3]
References
1. Morgan 1997, §2.
2. Nocedal & Wright 2006, p. 358, Eq. 13.4.
3. Nocedal & Wright 2006, p. 363, Theorem 13.2.
4. Nocedal & Wright 2006, p. 370, Theorem 13.4.
5. Nocedal & Wright 2006, p. 369, Eq. 13.24.
6. Nocedal & Wright 2006, p. 381, §13.5.
7. Nocedal & Wright 2006, p. 372, §13.4.
Bibliography
• Morgan, S. S. (1997). A Comparison of Simplex Method Algorithms (MSc thesis). University of Florida. Archived from the original on 7 August 2011.
• Nocedal, J.; Wright, S. J. (2006). Mikosch, T. V.; Resnick, S. I.; Robinson, S. M. (eds.). Numerical Optimization. Springer Series in Operations Research and Financial Engineering (2nd ed.). New York, NY, USA: Springer. ISBN 978-0-387-30303-1.
Optimization: Algorithms, methods, and heuristics
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Revolutions in Mathematics
Revolutions in Mathematics is a 1992 collection of essays in the history and philosophy of mathematics.
Contents
• Michael J. Crowe, Ten "laws" concerning patterns of change in the history of mathematics (1975) (15–20);
• Herbert Mehrtens, T. S. Kuhn's theories and mathematics: a discussion paper on the "new historiography" of mathematics (1976) (21–41);
• Herbert Mehrtens, Appendix (1992): revolutions reconsidered (42–48);
• Joseph Dauben, Conceptual revolutions and the history of mathematics: two studies in the growth of knowledge (1984) (49–71);
• Joseph Dauben, Appendix (1992): revolutions revisited (72–82);
• Paolo Mancosu, Descartes's Géométrie and revolutions in mathematics (83–116);
• Emily Grosholz, Was Leibniz a mathematical revolutionary? (117–133);
• Giulio Giorello, The "fine structure" of mathematical revolutions: metaphysics, legitimacy, and rigour. The case of the calculus from Newton to Berkeley and Maclaurin (134–168);
• Yu Xin Zheng, Non-Euclidean geometry and revolutions in mathematics (169–182);
• Luciano Boi, The "revolution" in the geometrical vision of space in the nineteenth century, and the hermeneutical epistemology of mathematics (183–208);
• Caroline Dunmore, Meta-level revolutions in mathematics (209–225);
• Jeremy Gray, The nineteenth-century revolution in mathematical ontology (226–248);
• Herbert Breger, A restoration that failed: Paul Finsler's theory of sets (249–264);
• Donald A. Gillies, The Fregean revolution in logic (265–305);
• Michael Crowe, Afterword (1992): a revolution in the historiography of mathematics? (306–316).
Reviews
The book was reviewed by Pierre Kerszberg for Mathematical Reviews and by Michael S. Mahoney for American Mathematical Monthly. Mahoney says "The title should have a question mark." He sets the context by referring to paradigm shifts that characterize scientific revolutions as described by Thomas Kuhn in his book The Structure of Scientific Revolutions. According to Michael Crowe in chapter one, revolutions never occur in mathematics. Mahoney explains how mathematics grows upon itself and does not discard earlier gains in understanding with new ones, such as happens in biology, physics, or other sciences. A nuanced version of revolution in mathematics is described by Caroline Dunmore who sees change at the level of "meta-mathematical values of the community that define the telos and methods of the subject, and encapsulate general beliefs about its value." On the other hand, reaction to innovation in mathematics is noted, resulting in "clashes of intellectual and social values".
Editions
• Gillies, Donald (1992) Revolutions in Mathematics, Oxford Science Publications, The Clarendon Press, Oxford University Press.
References
• Pierre Kerszberg (1994, 2009) Review of Revolutions in Mathematics in Mathematical Reviews.
• Michael S. Mahoney (1994) "Review of Revolutions in Mathematics", American Mathematical Monthly 101(3):283–7.
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Theodor Reye
Karl Theodor Reye (born 20 June 1838 in Ritzebüttel, Germany and died 2 July 1919 in Würzburg, Germany) was a German mathematician. He contributed to geometry, particularly projective geometry and synthetic geometry. He is best known for his introduction of configurations in the second edition of his book, Geometrie der Lage (Geometry of Position, 1876).[1] The Reye configuration of 12 points, 12 planes, and 16 lines is named after him.
Reye also developed a novel solution to the following three-dimensional extension of the problem of Apollonius: Construct all possible spheres that are simultaneously tangent to four given spheres.[2]
Life
Reye obtained his Ph.D. from the University of Göttingen in 1861. His dissertation was entitled "Die mechanische Wärme-Theorie und das Spannungsgesetz der Gase" (The mechanical theory of heat and the potential law of gases).
Mathematical work
Reye worked on conic sections, quadrics and projective geometry.
Reye's work on linear manifolds of projective plane pencils and of bundles on spheres influenced later work by Corrado Segre on manifolds. He introduced Reye congruences, the earliest examples of Enriques surfaces.
References
1. Scott, Charlotte Angas (1899). "Reye's Geometrie der Lage". Bull. Amer. Math. Soc. 5 (4): 175–181. doi:10.1090/S0002-9904-1899-00580-9.
2. Reye T (1879). Synthetische Geometrie der Kugeln (PDF) (in German). Leipzig: B. G. Teubner.
Further reading
• Reye, Karl Theodor (1860) [1859-11-08]. Written at Zürich. Bornemann, K. R. (ed.). "Zur Theorie der Zapfenreibung" [On the theory of pivot friction]. Der Civilingenieur - Zeitschrift für das Ingenieurwesen. Neue Folge (NF) (in German). Freiberg: Buchhandlung J. G. Engelhardt. 6: 235–255. Retrieved 2018-05-25. (NB. Theodor Reye was a polytechnician in Zürich in 1860, but later became a professor in Straßburg. This paper established Reye's hypothesis and laid the foundation to what is known as Reye–Archard–Khrushchov wear law today.
External links
• Works by Theodor Reye at Project Gutenberg
• Works by or about Theodor Reye at Internet Archive
• Theodor Reye (1892) Die Geometrie der Lage from archive.org.
• Theodor Reye at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Theodor Reye", MacTutor History of Mathematics Archive, University of St Andrews
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Enriques surface
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0. Over fields of characteristic not 2 they are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces were first studied in detail by Enriques (1896) as an answer to a question discussed by Castelnuovo (1895) about whether a surface with q = pg = 0 is necessarily rational, though some of the Reye congruences introduced earlier by Reye (1882) are also examples of Enriques surfaces.
Enriques surfaces can also be defined over other fields. Over fields of characteristic other than 2, Artin (1960) showed that the theory is similar to that over the complex numbers. Over fields of characteristic 2 the definition is modified, and there are two new families, called singular and supersingular Enriques surfaces, described by Bombieri & Mumford (1976). These two extra families are related to the two non-discrete algebraic group schemes of order 2 in characteristic 2.
Invariants of complex Enriques surfaces
The plurigenera Pn are 1 if n is even and 0 if n is odd. The fundamental group has order 2. The second cohomology group H2(X, Z) is isomorphic to the sum of the unique even unimodular lattice II1,9 of dimension 10 and signature -8 and a group of order 2.
Hodge diamond:
1
00
0100
00
1
Marked Enriques surfaces form a connected 10-dimensional family, which Kondo (1994) showed is rational.
Characteristic 2
In characteristic 2 there are some new families of Enriques surfaces, sometimes called quasi Enriques surfaces or non-classical Enriques surfaces or (super)singular Enriques surfaces. (The term "singular" does not mean that the surface has singularities, but means that the surface is "special" in some way.) In characteristic 2 the definition of Enriques surfaces is modified: they are defined to be minimal surfaces whose canonical class K is numerically equivalent to 0 and whose second Betti number is 10. (In characteristics other than 2 this is equivalent to the usual definition.) There are now 3 families of Enriques surfaces:
• Classical: dim(H1(O)) = 0. This implies 2K = 0 but K is nonzero, and Picτ is Z/2Z. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme μ2.
• Singular: dim(H1(O)) = 1 and is acted on non-trivially by the Frobenius endomorphism. This implies K = 0, and Picτ is μ2. The surface is a quotient of a K3 surface by the group scheme Z/2Z.
• Supersingular: dim(H1(O)) = 1 and is acted on trivially by the Frobenius endomorphism. This implies K = 0, and Picτ is α2. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme α2.
All Enriques surfaces are elliptic or quasi elliptic.
Examples
• A Reye congruence is the family of lines contained in at least 2 quadrics of a given 3-dimensional linear system of quadrics in P3. If the linear system is generic then the Reye congruence is an Enriques surface. These were found by Reye (1882), and may be the earliest examples of Enriques surfaces.
• Take a surface of degree 6 in 3 dimensional projective space with double lines along the edges of a tetrahedron, such as
$w^{2}x^{2}y^{2}+w^{2}x^{2}z^{2}+w^{2}y^{2}z^{2}+x^{2}y^{2}z^{2}+wxyzQ(w,x,y,z)=0$
for some general homogeneous polynomial Q of degree 2. Then its normalization is an Enriques surface. This is the family of examples found by Enriques (1896).
• The quotient of a K3 surface by a fixed point free involution is an Enriques surface, and all Enriques surfaces in characteristic other than 2 can be constructed like this. For example, if S is the K3 surface w4 + x4 + y4 + z4 = 0 and T is the order 4 automorphism taking (w,x,y,z) to (w,ix,–y,–iz) then T2 has eight fixed points. Blowing up these eight points and taking the quotient by T2 gives a K3 surface with a fixed-point-free involution T, and the quotient of this by T is an Enriques surface. Alternatively, the Enriques surface can be constructed by taking the quotient of the original surface by the order 4 automorphism T and resolving the eight singular points of the quotient. Another example is given by taking the intersection of 3 quadrics of the form Pi(u,v,w) + Qi(x,y,z) = 0 and taking the quotient by the involution taking (u:v:w:x:y:z) to (–x:–y:–z:u:v:w). For generic quadrics this involution is a fixed-point-free involution of a K3 surface so the quotient is an Enriques surface.
See also
• List of algebraic surfaces
• Enriques–Kodaira classification
• Supersingular variety
References
• Artin, Michael (1960), On Enriques surfaces, PhD thesis, Harvard
• Compact Complex Surfaces by Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven ISBN 3-540-00832-2 This is the standard reference book for compact complex surfaces.
• Bombieri, Enrico; Mumford, David (1976), "Enriques' classification of surfaces in char. p. III." (PDF), Inventiones Mathematicae, 35 (1): 197–232, Bibcode:1976InMat..35..197B, doi:10.1007/BF01390138, ISSN 0020-9910, MR 0491720, S2CID 122816845
• Castelnuovo, G. (1895), "Sulle superficie di genere zero", Mem. Delle Soc. Ital. Delle Scienze, Série III, 10: 103–123
• Cossec, François R.; Dolgachev, Igor V. (1989), Enriques surfaces. I, Progress in Mathematics, vol. 76, Boston: Birkhäuser Boston, ISBN 978-0-8176-3417-9, MR 0986969
• Dolgachev, Igor V. (2016), A brief introduction to Enriques surfaces (PDF)
• Enriques, Federigo (1896), "Introduzione alla geometria sopra le superficie algebriche.", Mem. Soc. Ital. Delle Scienze, 10: 1–81
• Enriques, Federigo (1949), Le Superficie Algebriche, Nicola Zanichelli, Bologna, MR 0031770
• Kondo, Shigeyuki (1994), "The rationality of the moduli space of Enriques surfaces", Compositio Mathematica, 91 (2): 159–173
• Reye, T. (1882), Die Geometrie der Lage, Leipzig: Baumgärtnerś Buchhandlung
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Reynolds decomposition
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity $u$ the decomposition would be
$u(x,y,z,t)={\overline {u(x,y,z)}}+u'(x,y,z,t)$
where ${\overline {u}}$ denotes the expectation value of $u$, (often called the steady component/time, spatial or ensemble average), and $u'$, are the deviations from the expectation value (or fluctuations). The fluctuations are defined as the expectation value subtracted from quantity $u$ such that their time average equals zero. [1][2]
The expected value, ${\overline {u}}$, is often found from an ensemble average which is an average taken over multiple experiments under identical conditions. The expected value is also sometime denoted $\langle u\rangle $, but it is also seen often with the over-bar notation.[3]
Direct numerical simulation, or resolution of the Navier–Stokes equations completely in $(x,y,z,t)$, is only possible on extremely fine computational grids and small time steps even when Reynolds numbers are low, and becomes prohibitively computationally expensive at high Reynolds' numbers. Due to computational constraints, simplifications of the Navier-Stokes equations are useful to parameterize turbulence that are smaller than the computational grid, allowing larger computational domains.[4]
Reynolds decomposition allows the simplification of the Navier–Stokes equations by substituting in the sum of the steady component and perturbations to the velocity profile and taking the mean value. The resulting equation contains a nonlinear term known as the Reynolds stresses which gives rise to turbulence.
See also
• Reynolds-averaged Navier–Stokes equations
References
1. Müller, Peter (2006). The Equations of Oceanic Motions. p. 112.
2. Adrian, R (2000). "Analysis and Interpretation of instantaneous turbulent velocity fields". Experiments in Fluids. 29 (3): 275–290. Bibcode:2000ExFl...29..275A. doi:10.1007/s003489900087. S2CID 122145330.
3. Kundu, Pijush (27 March 2015). Fluid Mechanics. Academic Press. p. 609. ISBN 978-0-12-405935-1.
4. Mukerji, Sudip (1997-01-01). "Turbulence Computations with 3-D Small-Scale Additive Turbulent Decomposition and Data-Fitting Using Chaotic Map Combinations". doi:10.2172/666048. OSTI 666048. {{cite journal}}: Cite journal requires |journal= (help)
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Reynolds operator
In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountered in models of turbulent flows, particularly the Reynolds-averaged Navier–Stokes equations, where the average is typically taken over the fluid flow under the group of time translations. In invariant theory the average is often taken over a compact group or reductive algebraic group acting on a commutative algebra, such as a ring of polynomials. Reynolds operators were introduced into fluid dynamics by Osbourne Reynolds (1895) and named by J. Kampé de Fériet (1934, 1935, 1949).
Definition
Reynolds operators are used in fluid dynamics, functional analysis, and invariant theory, and the notation and definitions in these areas differ slightly. A Reynolds operator acting on φ is sometimes denoted by $R(\phi ),P(\phi ),\rho (\phi ),\langle \phi \rangle $ or ${\overline {\phi }}$. Reynolds operators are usually linear operators acting on some algebra of functions, satisfying the identity
$R(R(\phi )\psi )=R(\phi )R(\psi )\quad {\text{ for all }}\phi ,\psi $
and sometimes some other conditions, such as commuting with various group actions.
Invariant theory
In invariant theory a Reynolds operator R is usually a linear operator satisfying
$R(R(\phi )\psi )=R(\phi )R(\psi )\quad {\text{ for all }}\phi ,\psi $
and
$R(1)=1$
Together these conditions imply that R is idempotent: R2 = R. The Reynolds operator will also usually commute with some group action, and project onto the invariant elements of this group action.
Functional analysis
In functional analysis a Reynolds operator is a linear operator R acting on some algebra of functions φ, satisfying the Reynolds identity
$ R(\phi \psi )=R(\phi )R(\psi )+R\left(\left(\phi -R(\phi )\right)\left(\psi -R(\psi )\right)\right)\quad {\text{ for all }}\phi ,\psi $
The operator R is called an averaging operator if it is linear and satisfies
$R(R(\phi )\psi )=R(\phi )R(\psi )\quad {\text{ for all }}\phi ,\psi $
If R(R(φ)) = R(φ) for all φ then R is an averaging operator if and only if it is a Reynolds operator. Sometimes the R(R(φ)) = R(φ) condition is added to the definition of Reynolds operators.
Fluid dynamics
Let $\phi $ and $\psi $ be two random variables, and $a$ be an arbitrary constant. Then the properties satisfied by Reynolds operators, for an operator $\langle \rangle ,$ include linearity and the averaging property:
$\langle \phi +\psi \rangle =\langle \phi \rangle +\langle \psi \rangle ,\,$
$\langle a\phi \rangle =a\langle \phi \rangle ,\,$
$\langle \langle \phi \rangle \psi \rangle =\langle \phi \rangle \langle \psi \rangle ,\,$ which implies $\langle \langle \phi \rangle \rangle =\langle \phi \rangle .\,$
In addition the Reynolds operator is often assumed to commute with space and time translations:
$\left\langle {\frac {\partial \phi }{\partial t}}\right\rangle ={\frac {\partial \langle \phi \rangle }{\partial t}},\qquad \left\langle {\frac {\partial \phi }{\partial x}}\right\rangle ={\frac {\partial \langle \phi \rangle }{\partial x}},$
$\left\langle \int \phi ({\boldsymbol {x}},t)\,d{\boldsymbol {x}}\,dt\right\rangle =\int \langle \phi ({\boldsymbol {x}},t)\rangle \,d{\boldsymbol {x}}\,dt.$
Any operator satisfying these properties is a Reynolds operator.[1]
Examples
Reynolds operators are often given by projecting onto an invariant subspace of a group action.
• The "Reynolds operator" considered by Reynolds (1895) was essentially the projection of a fluid flow to the "average" fluid flow, which can be thought of as projection to time-invariant flows. Here the group action is given by the action of the group of time-translations.
• Suppose that G is a reductive algebraic group or a compact group, and V is a finite-dimensional representation of G. Then G also acts on the symmetric algebra SV of polynomials. The Reynolds operator R is the G-invariant projection from SV to the subring SVG of elements fixed by G.
References
1. Sagaut, Pierre (2006). Large Eddy Simulation for Incompressible Flows (Third ed.). Springer. ISBN 3-540-26344-6.
• Kampé de Fériet, J. (1934), "L'état actuel du problème de la turbulence I", La Science Aérienne, 3: 9–34
• Kampé de Fériet, J. (1935), "L'état actuel du problème de la turbulence II", La Science Aérienne, 4: 12–52
• Kampé de Fériet, J. (1949), "Sur un problème d'algèbre abstraite posé par la définition de la moyenne dans la théorie de la turbulence", Annales de la Société Scientifique de Bruxelles. Série I. Sciences Mathématiques, Astronomiques et Physiques, 63: 165–180, ISSN 0037-959X, MR 0032718
• Reynolds, O. (1895), "On the dynamical theory of incompressible viscous fluids and the determination of the criterion", Philosophical Transactions of the Royal Society A, 186: 123–164, Bibcode:1895RSPTA.186..123R, doi:10.1098/rsta.1895.0004, JSTOR 90643
• Rota, Gian-Carlo (2003), Gian-Carlo Rota on analysis and probability, Contemporary Mathematicians, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4275-4, MR 1944526 Reprints several of Rota's papers on Reynolds operators, with commentary.
• Rota, Gian-Carlo (1964), "Reynolds operators", Proc. Sympos. Appl. Math., vol. XVI, Providence, R.I.: Amer. Math. Soc., pp. 70–83, MR 0161140
• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-7091-4368-1, ISBN 978-3-211-82445-0, MR 1255980
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Reza Sadeghi (mathematician)
Reza Sadeghi (Persian: رضا صادقی;(April 20, 1977 – March 17, 1998) was an Iranian mathematician. He is an alumnus of National Organization for Development of Exceptional Talents (NODET) Mashad, Iran (Hasheminejad highschool).[1]
Reza Sadeghi
BornApril 20, 1977
Mashad, Iran
NationalityIranian
Alma materSharif University of Technology
Scientific career
FieldsMathematician
Prizes
Reza Sadeghi won the silver medal in the International Mathematical Olympiad in Hong Kong in 1994. He won the gold medal in the International Mathematical Olympiad in Canada in 1995.[2]
References
1. http://www.hn1.sampadrazavi.ir/view.aspx?kh=2&cod=12%5B%5D
2. "International Mathematical Olympiad". www.imo-official.org.
Mathematics in Iran
Mathematicians
Before
20th Century
• Abu al-Wafa' Buzjani
• Jamshīd al-Kāshī (al-Kashi's theorem)
• Omar Khayyam (Khayyam-Pascal's triangle, Khayyam-Saccheri quadrilateral, Khayyam's Solution of Cubic Equations)
• Al-Mahani
• Muhammad Baqir Yazdi
• Nizam al-Din al-Nisapuri
• Al-Nayrizi
• Kushyar Gilani
• Ayn al-Quzat Hamadani
• Al-Isfahani
• Al-Isfizari
• Al-Khwarizmi (Al-jabr)
• Najm al-Din al-Qazwini al-Katibi
• Nasir al-Din al-Tusi
• Al-Biruni
Modern
• Maryam Mirzakhani
• Caucher Birkar
• Sara Zahedi
• Farideh Firoozbakht (Firoozbakht's conjecture)
• S. L. Hakimi (Havel–Hakimi algorithm)
• Siamak Yassemi
• Freydoon Shahidi (Langlands–Shahidi method)
• Hamid Naderi Yeganeh
• Esmail Babolian
• Ramin Takloo-Bighash
• Lotfi A. Zadeh (Fuzzy mathematics, Fuzzy set, Fuzzy logic)
• Ebadollah S. Mahmoodian
• Reza Sarhangi (The Bridges Organization)
• Siavash Shahshahani
• Gholamhossein Mosaheb
• Amin Shokrollahi
• Reza Sadeghi
• Mohammad Mehdi Zahedi
• Mohsen Hashtroodi
• Hossein Zakeri
• Amir Ali Ahmadi
Prize Recipients
Fields Medal
• Maryam Mirzakhani (2014)
• Caucher Birkar (2018)
EMS Prize
• Sara Zahedi (2016)
Satter Prize
• Maryam Mirzakhani (2013)
Organizations
• Iranian Mathematical Society
Institutions
• Institute for Research in Fundamental Sciences
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Rho calculus
There are two different calculi that use the name rho-calculus:
• The first is a formalism intended to combine the higher-order facilities of lambda calculus with the pattern matching of term rewriting.
• The second is a reflective higher-order variant[1] of the asynchronous polyadic pi calculus.
References
1. Meredith, L. G.; Radestock, Mattias (22 December 2005). "A Reflective Higher-Order Calculus". Electronic Notes in Theoretical Computer Science. 141 (5): 49–67. doi:10.1016/j.entcs.2005.05.016.
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Rhode Island Math League
The Rhode Island Mathematics League (RIML) competition consists of four meets spanning the entire year. It culminates at the state championship held at Bishop Hendricken High School. Top schools from the state championship are invited to the New England Association of Math Leagues (NEAML) championship.
Format
Each meet consists of five rounds and a team round. Each team consists of five students, and each school may have as many as six teams. However, each team may have a maximum of two seniors and four sophomores/juniors. At least one sophomore or freshman must be on each team (or the team may compete with an empty slot). Three students from each team participate in a round. Therefore, each student participates in three rounds and the team round. The first five rounds consist of three questions each. Beginning in 2007, one of the five rounds is designated as "calculator-free", in 2008, this number was increased to two, and in 2018, calculators were banned from all meets. The first question in each round is worth one point, the second two points, and the third three points. Each student works on the questions independently in the ten minutes allotted. All answers must be presented in simplified and rationalized form unless specified otherwise. After the completion of the first five rounds, there is a team round. All five players from each team collaborate on five questions worth two points each. The maximum score for one team is 100 points, and the maximum score for one student is 18 points.
Rounds
At the first meet the rounds are as follows:
Round 1: Arithmetic, Number Theory, and Matrices
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the second meet the rounds are as follows:
Round 1: Arithmetic, Number Theory, and Matrices
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the third meet the rounds are as follows:
Round 1: Statistics and Probability
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the fourth meet the rounds are as follows:
Round 1: Statistics and Probability
Round 2: Algebra I
Round 3: Geometry
Round 4: Algebra II
Round 5: Miscellaneous Math
Team Round
At the playoff meet the rounds are as follows:
Round 1: Arithmetic, Number Theory, and Matrices
Round 2: Statistics and Probability
Round 3: Algebra I
Round 4: Geometry
Round 5: Algebra II
Round 6: Miscellaneous Math
At the end of the six rounds, a relay round will occur, where four people from a team of six will participate. In this round, four questions are given, and each student after the first must use the answer given to them from the previous question to answer the next one.
At the end of the relay round, there will be a team round, where four people from the team will compete to answer five questions together.
Miscellaneous Math
As of the 2019-2020 year, certain rounds were replaced with a round called Miscellaneous Math, which tests anything from the first four (or five at the playoff meet) rounds, plus Trigonometry, Analytical Geometry, and Conics.
Calculator Usage
As of the 2018-2019 year, calculators have been prohibited on all rounds, including the team rounds.
Current events
The 2007-08, and 2008-09 league champion was Wheeler School. Wheeler School ranked third among small schools at the 2006[1] and 2007[2] New England championships. In 2009 two seniors from Wheeler, Matthew Halpern and Karan Takhar, tied for first in the state. Barrington High School ranked first among medium schools at the 2006 New England championships.[1]
Barrington High School won first place for both the normal season and the playoff in the 2017-2018 year.
Wheeler School won first place for both the normal season and the playoff in the 2018-2019 year.
Due to the rise of the COVID-19 pandemic, the playoffs for 2020 were cancelled, and only the four normal meets counted towards school and individual scores. The top three schools were Wheeler School with 340 points, East Greenwich High School with 306 points, and Barrington High School with 292 points. The top three individuals scorers were Wheeler's Eric Tang with a perfect 72 points, The Prout School's Robert Hoffert with 68 points, and Andrew Babb and Jessica Liang from East Greenwich and Barrington, respectively, who tied for third with 65 points. Narragansett High School was dubbed to be the "most improved school" of the year due to having the largest percent change of scores of any school in the state.[3]
Practice Problems
A link to previous competition problems can be found here.
References
1. 2006 team stats Archived 2009-01-08 at the Wayback Machine
2. 2007 team stats Archived 2009-01-08 at the Wayback Machine
3. Morin, Ray (March 10, 2020). "RIML Meet 4 Results" (PDF). Retrieved January 1, 2021.{{cite web}}: CS1 maint: url-status (link)
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Rhombic enneacontahedron
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.
Rhombic enneacontahedron
Typezonohedron
Faces90 rhombi (60 wide, 30 narrow)
Edges180 (60+120)
Vertices92 (12+20+60)
Vertex configuration43, 45, 46
Schläfli symbolrt{3,5}
Conway notationjtI = dakD[1]
Symmetry groupIh, [5,3], *532
Dual polyhedronRectified truncated icosahedron
Propertiesconvex
Net
Construction
It can also be seen as a nonuniform truncated icosahedron with pyramids augmented to the pentagonal and hexagonal faces with heights adjusted until the dihedral angles are zero, and the two pyramid type side edges are equal length. This construction is expressed in the Conway polyhedron notation jtI with join operator j. Without the equal edge constraint, the wide rhombi are kites if limited only by the icosahedral symmetry.
The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2. The face angles of these rhombi are approximately 70.528° and 109.471°. The thirty slim rhombic faces have face vertex angles of 41.810° and 138.189°; the diagonals are in ratio of 1 to φ2.
It is also called a rhombic enenicontahedron in Lloyd Kahn's Domebook 2.
Close-packing density
The optimal packing fraction of rhombic enneacontahedra is given by
$\eta =16-{\frac {34}{\sqrt {5}}}\approx 0.7947377530014315$.
It was noticed that this optimal value is obtained in a Bravais lattice by de Graaf (2011). Since the rhombic enneacontahedron is contained in a rhombic dodecahedron whose inscribed sphere is identical to its own inscribed sphere, the value of the optimal packing fraction is a corollary of the Kepler conjecture: it can be achieved by putting a rhombicuboctahedron in each cell of the rhombic dodecahedral honeycomb, and it cannot be surpassed, since otherwise the optimal packing density of spheres could be surpassed by putting a sphere in each rhombicuboctahedron of the hypothetical packing which surpasses it.
References
1. "PolyHédronisme".
• Weisstein, Eric W. "Rhombic enneacontahedron". MathWorld.
• VRML model: George Hart,
• George Hart's Conway Generator Try dakD
• Domebook2 by Kahn, Lloyd (Editor); Easton, Bob; Calthorpe, Peter; et al., Pacific Domes, Los Gatos, CA (1971), page 102
• de Graaf, J.; van Roij, R.; Dijkstra, M. (2011), "Dense Regular Packings of Irregular Nonconvex Particles", Phys. Rev. Lett., 107: 155501, arXiv:1107.0603, Bibcode:2011PhRvL.107o5501D, doi:10.1103/PhysRevLett.107.155501, PMID 22107298
• Torquato, S.; Jiao, Y. (2009), "Dense packings of the Platonic and Archimedean solids", Nature, 460: 876, arXiv:0908.4107, Bibcode:2009Natur.460..876T, doi:10.1038/nature08239, PMID 19675649
• Hales, Thomas C. (2005), "A proof of the Kepler conjecture", Annals of Mathematics, 162: 1065, arXiv:math/9811078, doi:10.4007/annals.2005.162.1065
External links
• Weisstein, Eric W. "Rhombic enneacontahedron". MathWorld.
• The Rhombic Enneacontahedron and relations
• Rhombic Enneacontahedron
• George Hart
• A Color-Matching Dissection of the Rhombic Enneacontahedron
• Color-Matching Dissection of the Rhombic Enneacontahedron
• VRML model
Polyhedra
Listed by number of faces and type
1–10 faces
• Monohedron
• Dihedron
• Trihedron
• Tetrahedron
• Pentahedron
• Hexahedron
• Heptahedron
• Octahedron
• Enneahedron
• Decahedron
11–20 faces
• Hendecahedron
• Dodecahedron
• Tridecahedron
• Tetradecahedron
• Pentadecahedron
• Hexadecahedron
• Heptadecahedron
• Octadecahedron
• Enneadecahedron
• Icosahedron
>20 faces
• Icositetrahedron (24)
• Triacontahedron (30)
• Hexecontahedron (60)
• Enneacontahedron (90)
• Hectotriadiohedron (132)
• Apeirohedron (∞)
elemental things
• face
• edge
• vertex
• uniform polyhedron (two infinite groups and 75)
• regular polyhedron (9)
• quasiregular polyhedron (7)
• semiregular polyhedron (two infinite groups and 59)
convex polyhedron
• Platonic solid (5)
• Archimedean solid (13)
• Catalan solid (13)
• Johnson solid (92)
non-convex polyhedron
• Kepler–Poinsot polyhedron (4)
• Star polyhedron (infinite)
• Uniform star polyhedron (57)
prismatoids
• prism
• antiprism
• frustum
• cupola
• wedge
• pyramid
• parallelepiped
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Disphenoid
In geometry, a disphenoid (from Greek sphenoeides 'wedgelike') is a tetrahedron whose four faces are congruent acute-angled triangles.[1] It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are isotetrahedron,[2] sphenoid,[3] bisphenoid,[3] isosceles tetrahedron,[4] equifacial tetrahedron,[5] almost regular tetrahedron,[6] and tetramonohedron.[7]
The tetragonal and digonal disphenoids can be positioned inside a cuboid bisecting two opposite faces. Both have four equal edges going around the sides. The digonal has two pairs of congruent isosceles triangle faces, while the tetragonal has four congruent isosceles triangle faces.
A rhombic disphenoid has congruent scalene triangle faces, and can fit diagonally inside of a cuboid. It has three sets of edge lengths, existing as opposite pairs.
All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles. However, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, and its edges have three different lengths.
Special cases and generalizations
Further information: Tetrahedron § Isometries of irregular tetrahedra
If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with Td tetrahedral symmetry, although this is not normally called a disphenoid. When the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid. In this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D2 dihedral symmetry. Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral.[8] Both tetragonal disphenoids and rhombic disphenoids are isohedra: as well as being congruent to each other, all of their faces are symmetric to each other.
It is not possible to construct a disphenoid with right triangle or obtuse triangle faces.[4] When right triangles are glued together in the pattern of a disphenoid, they form a flat figure (a doubly-covered rectangle) that does not enclose any volume.[8] When obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid (by Alexandrov's uniqueness theorem) but one with acute triangle faces and with edges that in general do not lie along the edges of the given obtuse triangles.
Two more types of tetrahedron generalize the disphenoid and have similar names. The digonal disphenoid has faces with two different shapes, both isosceles triangles, with two faces of each shape. The phyllic disphenoid similarly has faces with two shapes of scalene triangles.
Disphenoids can also be seen as digonal antiprisms or as alternated quadrilateral prisms.
Characterizations
A tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right-angled.[9]
We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide.[10]
Another characterization states that if d1, d2 and d3 are the common perpendiculars of AB and CD; AC and BD; and AD and BC respectively in a tetrahedron ABCD, then the tetrahedron is a disphenoid if and only if d1, d2 and d3 are pairwise perpendicular.[9]
The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting.[11]
The disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area,[10] and the tetrahedra in which the angular defects of all four vertices equal π. They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints.[6]
Metric formulas
The volume of a disphenoid with opposite edges of length l, m and n is given by[12]
$V={\sqrt {\frac {(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}}.$
The circumscribed sphere has radius[12] (the circumradius)
$R={\sqrt {\frac {l^{2}+m^{2}+n^{2}}{8}}}$
and the inscribed sphere has radius[12]
$r={\frac {3V}{4T}}$
where V is the volume of the disphenoid and T is the area of any face, which is given by Heron's formula. There is also the following interesting relation connecting the volume and the circumradius:[12]
$\displaystyle 16T^{2}R^{2}=l^{2}m^{2}n^{2}+9V^{2}.$
The squares of the lengths of the bimedians are[12]
${\tfrac {1}{2}}(l^{2}+m^{2}-n^{2}),\quad {\tfrac {1}{2}}(l^{2}-m^{2}+n^{2}),\quad {\tfrac {1}{2}}(-l^{2}+m^{2}+n^{2}).$
Other properties
If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid.[10]
If the four faces of a tetrahedron have the same area, then it is a disphenoid.[9][10]
The centers in the circumscribed and inscribed spheres coincide with the centroid of the disphenoid.[12]
The bimedians are perpendicular to the edges they connect and to each other.[12]
Honeycombs and crystals
Some tetragonal disphenoids will form honeycombs. The disphenoid whose four vertices are (-1, 0, 0), (1, 0, 0), (0, 1, 1), and (0, 1, -1) is such a disphenoid.[13][14] Each of its four faces is an isosceles triangle with edges of lengths √3, √3, and 2. It can tessellate space to form the disphenoid tetrahedral honeycomb. As Gibb (1990) describes, it can be folded without cutting or overlaps from a single sheet of a4 paper.[15]
"Disphenoid" is also used to describe two forms of crystal:
• A wedge-shaped crystal form of the tetragonal or orthorhombic system. It has four triangular faces that are alike and that correspond in position to alternate faces of the tetragonal or orthorhombic dipyramid. It is symmetrical about each of three mutually perpendicular diad axes of symmetry in all classes except the tetragonal-disphenoidal, in which the form is generated by an inverse tetrad axis of symmetry.
• A crystal form bounded by eight scalene triangles arranged in pairs, constituting a tetragonal scalenohedron.
Other uses
Six tetragonal disphenoids attached end-to-end in a ring construct a kaleidocycle, a paper toy that can rotate on 4 sets of faces in a hexagon.
See also
• Irregular tetrahedra
• Orthocentric tetrahedron
• Snub disphenoid - A Johnson solid with 12 equilateral triangle faces and D2d symmetry.
• Trirectangular tetrahedron
References
1. Coxeter, H. S. M. (1973), Regular Polytopes (3rd ed.), Dover Publications, p. 15, ISBN 0-486-61480-8
2. Akiyama, Jin; Matsunaga, Kiyoko (2020), "An Algorithm for Folding a Conway Tile into an Isotetrahedron or a Rectangle Dihedron", Journal of Information Processing, 28 (28): 750–758, doi:10.2197/ipsjjip.28.750, S2CID 230108666.
3. Whittaker, E. J. W. (2013), Crystallography: An Introduction for Earth Science (and other Solid State) Students, Elsevier, p. 89, ISBN 9781483285566.
4. Leech, John (1950), "Some properties of the isosceles tetrahedron", The Mathematical Gazette, 34 (310): 269–271, doi:10.2307/3611029, JSTOR 3611029, MR 0038667, S2CID 125145099.
5. Hajja, Mowaffaq; Walker, Peter (2001), "Equifacial tetrahedra", International Journal of Mathematical Education in Science and Technology, 32 (4): 501–508, doi:10.1080/00207390110038231, MR 1847966, S2CID 218495301.
6. Akiyama, Jin (2007), "Tile-makers and semi-tile-makers", American Mathematical Monthly, 114 (7): 602–609, doi:10.1080/00029890.2007.11920450, JSTOR 27642275, MR 2341323, S2CID 32897155.
7. Demaine, Erik; O'Rourke, Joseph (2007), Geometric Folding Algorithms, Cambridge University Press, p. 424, ISBN 978-0-521-71522-5.
8. Petitjean, Michel (2015), "The most chiral disphenoid" (PDF), MATCH Communications in Mathematical and in Computer Chemistry, 73 (2): 375–384, MR 3242747.
9. Andreescu, Titu; Gelca, Razvan (2009), Mathematical Olympiad Challenges (2nd ed.), Birkhäuser, pp. 30–31.
10. Brown, B. H. (April 1926), "Theorem of Bang. Isosceles tetrahedra", Undergraduate Mathematics Clubs: Club Topics, American Mathematical Monthly, 33 (4): 224–226, doi:10.1080/00029890.1926.11986564, JSTOR 2299548.
11. Fuchs, Dmitry [in German]; Fuchs, Ekaterina (2007), "Closed geodesics on regular polyhedra" (PDF), Moscow Mathematical Journal, 7 (2): 265–279, 350, doi:10.17323/1609-4514-2007-7-2-265-279, MR 2337883.
12. Leech, John (1950), "Some properties of the isosceles tetrahedron", Mathematical Gazette, 34 (310): 269–271, doi:10.2307/3611029, JSTOR 3611029, S2CID 125145099.
13. Coxeter (1973, pp. 71–72).
14. Senechal, Marjorie (1981), "Which tetrahedra fill space?", Mathematics Magazine, 54 (5): 227–243, doi:10.2307/2689983, JSTOR 2689983, MR 0644075
15. Gibb, William (1990), "Paper patterns: solid shapes from metric paper", Mathematics in School, 19 (3): 2–4 Reprinted in Pritchard, Chris, ed. (2003), The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, Cambridge University Press, pp. 363–366, ISBN 0-521-53162-4
External links
• Mathematical Analysis of Disphenoid by H C Rajpoot from Academia.edu
• Weisstein, Eric W. "Disphenoid". MathWorld.
• Weisstein, Eric W. "Isosceles tetrahedron". MathWorld.
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Wikipedia
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Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Rhombic dodecahedron
(Click here for rotating model)
TypeCatalan solid
Coxeter diagram
Conway notationjC
Face typeV3.4.3.4
rhombus
Faces12
Edges24
Vertices14
Vertices by type8{3}+6{4}
Symmetry groupOh, B3, [4,3], (*432)
Rotation groupO, [4,3]+, (432)
Dihedral angle120°
Propertiesconvex, face-transitive isohedral, isotoxal, parallelohedron
Cuboctahedron
(dual polyhedron)
Net
Properties
The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly √2 times the short face-diagonal length; thus, the acute angles on each face measure arccos(1/3), or approximately 70.53°.
Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces. In elementary terms, this means that for any two faces A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.
The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the octahedron, while the 8 vertices where 3 rhombi meet correspond to the vertices of the cube.
The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic triacontahedron.
The rhombic dodecahedron can be used to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane.
This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice. It is the Brillouin zone of body centered cubic (bcc) crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. As Johannes Kepler noted in his 1611 book on snowflakes (Strena seu de Nive Sexangula), honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.[1]
The graph of the rhombic dodecahedron is nonhamiltonian.
A rhombic dodecahedron can be dissected into 4 obtuse trigonal trapezohedra around its center. These rhombohedra are the cells of a trigonal trapezohedral honeycomb. Analogy: a regular hexagon can be dissected into 3 rhombi around its center. These rhombi are the tiles of a rhombille.
The collections of the Louvre include a die in the shape of a rhombic dodecahedron dating from Ptolemaic Egypt. The faces are inscribed with Greek letters representing the numbers 1 through 12: Α Β Γ Δ Ε Ϛ Z Η Θ Ι ΙΑ ΙΒ. The function of the die is unknown.[2]
• Rhombic dodecahedron dissected into 4 rhombohedra
• Hexagon dissected into 3 rhombi
• A garnet crystal
• This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the center-face-pyramids of a cube.
Dimensions
Denoting by a the edge length of a rhombic dodecahedron,
• the radius of its inscribed sphere (tangent to each of the rhombic dodecahedron's faces) is
$r_{\mathrm {i} }={\frac {\sqrt {6}}{3}}~a\approx 0.816\,496\,5809~a\quad $ (OEIS: A157697),
• the radius of its midsphere is
$r_{\mathrm {m} }={\frac {2{\sqrt {2}}}{3}}~a\approx 0.942\,809\,041\,58~a\quad $ (OEIS: A179587),
• the radius of the sphere passing through the six order 4 vertices, but not through the eight order 3 vertices, is
$r_{\mathrm {o} }={\frac {2{\sqrt {3}}}{3}}~a\approx 1.154\,700\,538~a\quad $ (OEIS: A020832),
• the radius of the sphere passing through the eight order 3 vertices is exactly equal to the length of the sides
$r_{\mathrm {t} }=a$
Area and volume
The surface area A and the volume V of the rhombic dodecahedron with edge length a are:
$A=8{\sqrt {2}}~a^{2}\approx 11.313\,7085~a^{2}$
$V={\frac {16{\sqrt {3}}}{9}}~a^{3}\approx 3.079\,201\,44~a^{3}$
Orthogonal projections
The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, and the two types of vertex, threefold and fourfold. The last two correspond to the B2 and A2 Coxeter planes.
Orthogonal projections
Projective
symmetry
[4] [6] [2] [2]
Rhombic
dodecahedron
Cuboctahedron
(dual)
Cartesian coordinates
Pyritohedron variations between a cube and rhombic dodecahedron
Expansion of a rhombic dodecahedron
For edge length √3, the eight vertices where three faces meet at their obtuse angles have Cartesian coordinates:
(±1, ±1, ±1)
The coordinates of the six vertices where four faces meet at their acute angles are:
(±2, 0, 0), (0, ±2, 0) and (0, 0, ±2)
The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates (±1, ±1, ±1) and (0, 1 + h, 1 − h2) with parameter h = 1.
Topologically equivalent forms
Parallelohedron
The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron, dodecahedrille, being the dual to the tetroctahedrille or half cubic honeycomb, and described by two Coxeter diagrams: and . With D3d symmetry, it can be seen as an elongated trigonal trapezohedron.
The rhombic dodecahedron can tessellate space by translational copies of itself, as can the stellated rhombic dodecahedron.
The rhombic dodecahedron can be constructed with 4 sets of 6 parallel edges.
Dihedral rhombic dodecahedron
Other symmetry constructions of the rhombic dodecahedron are also space-filling, and as parallelotopes they are similar to variations of space-filling truncated octahedra.[3]
For example, with 4 square faces, and 60-degree rhombic faces, and D4h dihedral symmetry, order 16. It can be seen as a cuboctahedron with square pyramids augmented on the top and bottom.
Net
Coordinates
(0, 0, ±2)
(±1, ±1, 0)
(±1, 0, ±1)
(0, ±1, ±1)
Bilinski dodecahedron
Main article: Bilinski dodecahedron
Bilinski dodecahedron with edges and front faces colored by their symmetry positions.
Bilinski dodecahedron colored by parallel edges
In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron. It has the same topology but different geometry. The rhombic faces in this form have the golden ratio.[4][5]
Faces
First form Second form
√2:1 √5 + 1/2:1
Deltoidal dodecahedron
Drawing and crystal model of deltoidal dodecahedron
Another topologically equivalent variation, sometimes called a deltoidal dodecahedron,[6] is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites (deltoids). It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by (a,b), where b and a depend on each other such that the tetrahedron defined by the four vertices of a face has volume zero, i.e. is a planar face. (1,1) is the rhombic solution. As a approaches 1/2, b approaches infinity. It always holds that 1/a + 1/b = 2, with a, b > 1/2.
(±2, 0, 0), (0, ±2, 0), (0, 0, ±2)
(a, a, a), (−a, −a, a), (−a, a, −a), (a, −a, −a)
(−b, −b, −b), (−b, b, b), (b, −b, b), (b, b, −b)
(1,1) (7/8,7/6) (3/4,3/2) (2/3,2) (5/8,5/2) (9/16,9/2)
Related polyhedra
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
When projected onto a sphere (see right), it can be seen that the edges make up the edges of two tetrahedra arranged in their dual positions (the stella octangula). This trend continues on with the deltoidal icositetrahedron and deltoidal hexecontahedron for the dual pairings of the other regular polyhedra (alongside the triangular bipyramid if improper tilings are to be considered), giving this shape the alternative systematic name of deltoidal dodecahedron.
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4
This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.
Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32 Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Tiling
Conf. V(3.3)2 V(3.4)2 V(3.5)2 V(3.6)2 V(3.7)2 V(3.8)2 V(3.∞)2
*n42 symmetry mutations of quasiregular dual tilings: V(4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[iπ/λ,4]
Tiling
Conf.
V4.3.4.3
V4.4.4.4
V4.5.4.5
V4.6.4.6
V4.7.4.7
V4.8.4.8
V4.∞.4.∞
V4.∞.4.∞
Similarly it relates to the infinite series of tilings with the face configurations V3.2n.3.2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.
V3.4.3.4
(Drawn as a net)
V3.6.3.6
Euclidean plane tiling
Rhombille tiling
V3.8.3.8
Hyperbolic plane tiling
(Drawn in a Poincaré disk model)
Stellations
Like many convex polyhedra, the rhombic dodecahedron can be stellated by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke.[7]
The first stellation, often simply called the stellated rhombic dodecahedron, is well known. It can be seen as a rhombic dodecahedron with each face augmented by attaching a rhombic-based pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces:
• The first stellation of the rhombic dodecahedron
• 3D model of decomposition into 12 pyramids and 4 half-cubes
Luke describes four more stellations: the second and third stellations (expanding outwards), one formed by removing the second from the third, and another by adding the original rhombic dodecahedron back to the previous one.
SecondThird
Great stellated rhombic dodecahedron
Stellated rhombic dodecahedron
Related polytopes
The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent rhombohedra, giving eight possible rhombohedra as projections of the tesseracts 8 cubic cells. One set of projective vectors are: u = (1,1,−1,−1), v = (−1,1,−1,1), w = (1,−1,−1,1).
The rhombic dodecahedron forms the maximal cross-section of a 24-cell, and also forms the hull of its vertex-first parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of six pairs of the 24-cell's octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The non-regularity of these images are due to projective distortion; the facets of the 24-cell are regular octahedra in 4-space.
This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a cube into six congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses. The 24-cell may also be constructed in an analogous way using two tesseracts.[8]
Practical usage
In spacecraft reaction wheel layout, a tetrahedral configuration of four wheels is commonly used. For wheels that perform equally (from a peak torque and max angular momentum standpoint) in both spin directions and across all four wheels, the maximum torque and maximum momentum envelopes for the 3-axis attitude control system (considering idealized actuators) are given by projecting the tesseract representing the limits of each wheel's torque or momentum into 3D space via the 3 × 4 matrix of wheel axes; the resulting 3D polyhedron is a rhombic dodecahedron.[9] Such an arrangement of reaction wheels is not the only possible configuration (a simpler arrangement consists of three wheels mounted to spin about orthogonal axes), but it is advantageous in providing redundancy to mitigate the failure of one of the four wheels (with degraded overall performance available from the remaining three active wheels) and in providing a more convex envelope than a cube, which leads to less agility dependence on axis direction (from an actuator/plant standpoint). Spacecraft mass properties influence overall system momentum and agility, so decreased variance in envelope boundary does not necessarily lead to increased uniformity in preferred axis biases (that is, even with a perfectly distributed performance limit within the actuator subsystem, preferred rotation axes are not necessarily arbitrary at the system level).
See also
• Dodecahedron
• Rhombic triacontahedron
• Trapezo-rhombic dodecahedron
• Truncated rhombic dodecahedron
• 24-cell – 4D analog of rhombic dodecahedron
• Archimede construction systems
• Fully truncated rhombic dodecahedron
References
1. Dodecahedral Crystal Habit Archived 2009-04-12 at the Wayback Machine. khulsey.com
2. Perdrizet, Paul. (1930). "Le jeu alexandrin de l'icosaèdre". Bulletin de l'Institut français d'archéologie orientale. 30: 1–16.
3. Order in Space: A design source book, Keith Critchlow, p.56–57
4. Branko Grünbaum (2010). "The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra" (PDF). 32 (4): 5–15. Archived from the original (PDF) on 2015-04-02. {{cite journal}}: Cite journal requires |journal= (help)
5. H.S.M Coxeter, "Regular polytopes", Dover publications, 1973.
6. Economic Mineralogy: A Practical Guide to the Study of Useful Minerals, p.8
7. Luke, D. (1957). "Stellations of the rhombic dodecahedron". The Mathematical Gazette. 41 (337): 189–194. doi:10.2307/3609190. JSTOR 3609190. S2CID 126103579.
8. Archived at Ghostarchive and the Wayback Machine: "There are SIX Platonic Solids". YouTube.
9. Markley, F. Landis (September 2010). "Maximum Torque and Momentum Envelopes for Reaction-Wheel Arrays". ntrs.nasa.gov. Retrieved 2020-08-20.
Further reading
• 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)
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. doi:10.1017/CBO9780511569371. ISBN 978-0-521-54325-5. MR 0730208. (The thirteen semiregular convex polyhedra and their duals, Page 19, Rhombic dodecahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, p. 285, Rhombic dodecahedron )
External links
• Weisstein, Eric W. "Rhombic dodecahedron". MathWorld.
• Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
Computer models
• Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron, Rhombic Dodecahedron 5-Compound and Rhombic Dodecahedron 5-Compound by Sándor Kabai, The Wolfram Demonstrations Project.
Paper projects
• Rhombic Dodecahedron Calendar – make a rhombic dodecahedron calendar without glue
• Another Rhombic Dodecahedron Calendar – made by plaiting paper strips
Practical applications
• Archimede Institute Examples of actual housing construction projects using this geometry
Catalan solids
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)
Archimedean duals
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)
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.
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Cuboctahedral pyramid
In 4-dimensional geometry, the cuboctahedral pyramid is bounded by one cuboctahedron on the base, 6 square pyramid, and 8 triangular pyramid cells which meet at the apex. It has 38 faces: 32 triangles and 6 squares. It has 32 edges, and 13 vertices.
Cuboctahedral pyramid
Schlegel diagram
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ r{4,3}
Cells 15 1 cuboctahedron
6 square pyramids
8 triangular pyramids
Faces 38: 8+24 triangles
6 squares
Edges 36
Vertices 13
Dual rhombic dodecahedral pyramid
Symmetry group B3, [4,3,1], order 48
Properties convex
Since a cuboctahedron's circumradius is equal to its edge length,[1] the triangles must be taller than equilateral to create a positive height.
The dual to the cuboctahedral pyramid is a rhombic dodecahedral pyramid, seen as a rhombic dodecahedral base, and 12 rhombic pyramids meeting at an apex.
References
1. Klitzing, Richard. "3D convex uniform polyhedra o3x4o - co".
External links
• Olshevsky, George. "Pyramid". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra
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Hemi-cuboctahedron
A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron.
Hemi-cuboctahedron
Schlegel diagram
Typeabstract polyhedron
globally projective polyhedron
Faces7:
4 triangles
3 squares
Edges12
Vertices6
Vertex configuration3.4.3.4
Schläfli symbolr{3,4}/2 or r{3,4}3
Symmetry groupS4, order 24
Propertiesnon-orientable
Euler characteristic 1
It has 4 triangular faces and 3 square faces, 12 edges, and 6 vertices. It can be seen as a rectified hemi-octahedron or rectified hemi-cube.
It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 triangles and 3 square), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected.
Dual
Its dual polyhedron is a rhombic hemi-dodecahedron which has 7 vertices (1-7), 12 edges (a-l), and 6 rhombic faces (A-F).
Related polyhedra
It has a real presentation as a uniform star polyhedron, the tetrahemihexahedron.
See also
• Hemi-dodecahedron
• Hemi-icosahedron
References
• McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective Regular Polytopes", Abstract Regular Polytopes (1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0
External links
• The hemicubeoctahedron
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Rhombic hexecontahedron
In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach.[1]
Rhombic hexecontahedron
TypeStellation of rhombic triacontahedron
Vertices62 (12+20+30)
Edges120 (60+60)
Faces60 golden rhombi
SymmetryIh, [5,3], (*532)
Propertiesnon-convex, zonohedron
It is topologically identical to the convex deltoidal hexecontahedron which has kite faces.
Dissection
The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length a to be $V=(10+2{\sqrt {5}})a^{3}$ and the area to be $A=(24{\sqrt {5}})a^{2}$.
Construction
A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of (ϕ+1)/2 ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to (3-ϕ)/2 ≈ 0.691 of its original value, and the 30 edge centers are left unchanged, then a rhombic hexecontahedron is formed. (The circumradius is increased by 30.9% and the inradius is decreased by the same 30.9%.) Scaling the points by different amounts results in hexecontahedra with kite-shaped faces or other polyhedra.
Every golden rhombic face has a face center, a vertex, and two edge centers of the original dodecahedron, with the edge centers forming the short diagonal. Each edge center is connected to two vertices and two face centers. Each face center is connected to five edge centers, and each vertex is connected to three edge centers.
Stellation
The rhombic hexecontahedron is one of 227 self-supporting stellations of the rhombic triacontahedron. Its stellation diagram looks like this, with the original rhombic triacontahedron faces as the central rhombus.
Related polyhedra
The great rhombic triacontahedron contains the 30 larger intersecting rhombic faces:
In popular culture
In Brazilian culture, handcrafted rhombic hexecontahedra used to be made from colored fabric and cardboard, called giramundos ("world turners" in Portuguese) or happiness stars, sewn by mothers and given as wedding gifts to their daughters. The custom got lost with the urbanization of Brazil, though the technique for producing the handicrafts was still taught in Brazilian rural schools up until the first half of the twentieth century.[2]
The logo of the WolframAlpha website is a red rhombic hexecontahedron and was inspired by the logo of the related Mathematica software.[3]
References
1. Grünbaum (1996b)
2. Artesanato se antecipou à descoberta de poliedro [Handicraft anticipated the discovery of a polyhedron] (in Portuguese), IMPA, retrieved 2019-01-08
3. "What's in the Logo? That Which We Call a Rhombic Hexecontahedron—Wolfram|Alpha Blog".
Bibliography
• Unkelbach, H. "Die kantensymmetrischen, gleichkantigen Polyeder. Deutsche Math. 5, 306-316, 1940.
• Grünbaum, B. (1996a). "A New Rhombic Hexecontahedron". Geombinatorics: 15–18.
• Grünbaum, B. (1996b). "A New Rhombic Hexecontahedron—Once More". Geombinatorics: 55–59.
• Grünbaum, B. (1997). "Still More Rhombic Hexecontahedra". Geombinatorics: 140–142.
• Grünbaum, B. Parallelogram-Faced Isohedra with Edges in Mirror-Planes. Discrete Math. 221, 93–100, 2000.
External links
• Weisstein, Eric W. "Rhombic hexecontahedron". MathWorld.
• http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html
• The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra. Branko Grünbaum
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Rhombic icosahedron
The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi;[1] 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on its axis of 5-fold symmetry, which is perpendicular to 5 axes of 2-fold symmetry through the midpoints of opposite equatorial edges (example on top figure: most left-hand and most right-hand mid-edges). Its other 10 faces follow its equator, 5 above and 5 below it; each of these 10 rhombi has 2 of its 4 sides lying on this zig-zag skew decagon equator. The rhombic icosahedron has 22 vertices. It has D5d, [2+,10], (2*5) symmetry group, of order 20; thus it has a center of symmetry (since 5 is odd).
Not to be confused with Rhombicosahedron.
Rhombic icosahedron
Typezonohedron
Faces20 congruent golden rhombi
Edges40
Vertices22
Faces per vertex3, 4, or 5
Dual polyhedronirregular-faced pentagonal gyrobicupola
SymmetryD5d = D5v, [2+,10], (2*5)
Propertiesconvex, zonohedron
Even though all its faces are congruent, the rhombic icosahedron is not face-transitive, since one can distinguish whether a particular face is near the equator or near a pole by examining the types of vertices surrounding this face.
Zonohedron
The rhombic icosahedron is a zonohedron, that is dual to a pentagonal gyrobicupola with regular triangular, regular pentagonal, but irregular quadrilateral faces.
The rhombic icosahedron has 5 sets of 8 parallel edges, described as 85 belts.
The edges of the rhombic icosahedron can be grouped in 5 parallel-sets, seen in this wireframe orthogonal projection.
The rhombic icosahedron forms the convex hull of the vertex-first projection of a 5-cube to 3 dimensions. The 32 vertices of a 5-cube map into the 22 exterior vertices of the rhombic icosahedron, with the remaining 10 interior vertices forming a pentagonal antiprism.
In the same way, one can obtain a Bilinski dodecahedron from a 4-cube, and a rhombic triacontahedron from a 6-cube.
Related polyhedra
The rhombic icosahedron can be derived from the rhombic triacontahedron by removing a belt of 10 middle faces.
A rhombic triacontahedron can be seen as an elongated rhombic icosahedron.
The rhombic icosahedron and the rhombic triacontahedron have the same 10-fold symmetric orthogonal projection. (*)
(*) (For example, on the left-hand figure):
The orthogonal projection of the (vertical) belt of 10 middle faces of the rhombic triacontahedron is just the (horizontal) exterior regular decagon of the common orthogonal projection.
References
1. Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-20.
External links
• Weisstein, Eric W. "Rhombic icosahedron". MathWorld.
• http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html
• VRML Model
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Axiality and rhombicity
In physics and mathematics, axiality and rhombicity are two characteristics of a symmetric second-rank tensor in three-dimensional Euclidean space, describing its directional asymmetry.
Let A denote a second-rank tensor in R3, which can be represented by a 3-by-3 matrix. We assume that A is symmetric. This implies that A has three real eigenvalues, which we denote by $A_{xx}$, $A_{yy}$ and $A_{zz}$. We assume that they are ordered such that
$A_{xx}\leq A_{yy}\leq A_{zz}.$
The axiality of A is defined by
$\Delta A=2A_{zz}-(A_{xx}+A_{yy}).\,$
The rhombicity is the difference between the smallest and the second-smallest eigenvalue:
$\delta A=A_{yy}-A_{xx}.\,$
Other definitions of axiality and rhombicity differ from the ones given above by constant factors which depend on the context. For example, when using them as parameters in the irreducible spherical tensor expansion, it is most convenient to divide the above definition of axiality by ${\sqrt {6}}$ and that of rhombicity by ${2}$.
Applications
The description of physical interactions in terms of axiality and rhombicity is frequently encountered in spin dynamics and, in particular, in spin relaxation theory, where many traceless bilinear interaction Hamiltonians, having the (eigenframe) form
${\hat {H}}={\hat {\vec {\mathbf {a} }}}\cdot \mathbf {A} \cdot {\hat {\vec {\mathbf {b} }}}=A_{xx}{\hat {a}}_{x}{\hat {b}}_{x}+A_{yy}{\hat {a}}_{y}{\hat {b}}_{y}+A_{zz}{\hat {a}}_{z}{\hat {b}}_{z}$
(hats denote spin projection operators) may be conveniently rotated using rank 2 irreducible spherical tensor operators:
${\hat {\vec {\mathbf {a} }}}\cdot \mathbf {A} \cdot {\hat {\vec {\mathbf {b} }}}={\frac {\delta A}{2}}{\hat {T}}_{2,-2}+{\frac {\delta A}{2}}{\hat {T}}_{2,2}+{\frac {\Delta A}{\sqrt {6}}}{\hat {T}}_{2,-2}$
${\hat {\hat {R}}}_{\alpha ,\beta ,\gamma }({\hat {T}}_{l,m})=\sum _{k=-2}^{2}{\hat {T}}_{l,k}{\mathfrak {D}}_{k,m}^{(l)}(\alpha ,\beta ,\gamma )$
where ${\mathfrak {D}}_{k,m}^{(l)}(\alpha ,\beta ,\gamma )$ are Wigner functions, $(\alpha ,\beta ,\gamma )$ are Euler angles, and the expressions for the rank 2 irreducible spherical tensor operators are:
${\hat {T}}_{2,2}=+{\frac {1}{2}}{\hat {a}}_{+}{\hat {b}}_{+}$
${\hat {T}}_{2,1}=-{\frac {1}{2}}({\hat {a}}_{z}{\hat {b}}_{+}+{\hat {a}}_{+}{\hat {b}}_{z})$
${\hat {T}}_{2,0}=+{\sqrt {\frac {2}{3}}}({\hat {a}}_{z}{\hat {b}}_{z}-{\frac {1}{4}}({\hat {a}}_{+}{\hat {b}}_{-}+{\hat {a}}_{-}{\hat {b}}_{+}))$
${\hat {T}}_{2,-1}=+{\frac {1}{2}}({\hat {a}}_{z}{\hat {b}}_{-}+{\hat {a}}_{-}{\hat {b}}_{z})$
${\hat {T}}_{2,-2}=+{\frac {1}{2}}{\hat {a}}_{-}{\hat {b}}_{-}$
Defining Hamiltonian rotations in this way (axiality, rhombicity, three angles) significantly simplifies calculations, since the properties of Wigner functions are well understood.
References
D.M. Brink and G.R. Satchler, Angular momentum, 3rd edition, 1993, Oxford: Clarendon Press.
D.A. Varshalovich, A.N. Moskalev, V.K. Khersonski, Quantum theory of angular momentum: irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols, 1988, Singapore: World Scientific Publications.
I. Kuprov, N. Wagner-Rundell, P.J. Hore, J. Magn. Reson., 2007 (184) 196-206. Article
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Rhombicosacron
In geometry, the rhombicosacron (or midly dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.
Rhombicosacron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 50 (χ = −10)
Symmetry groupIh, [5,3], *532
Index referencesDU56
dual polyhedronRhombicosahedron
Proportions
Each face has two angles of $\arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }$ and two angles of $\arccos(-{\frac {1}{6}})\approx 99.594\,068\,226\,86^{\circ }$. The diagonals of each antiparallelogram intersect at an angle of $\arccos({\frac {1}{8}}+{\frac {7{\sqrt {5}}}{24}})\approx 38.996\,309\,663\,87^{\circ }$. The dihedral angle equals $\arccos(-{\frac {5}{7}})\approx 135.584\,691\,402\,81^{\circ }$. The ratio between the lengths of the long edges and the short ones equals ${\frac {3}{2}}+{\frac {1}{2}}{\sqrt {5}}$, which is the square of the golden ratio.
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Rhombicosacron". 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
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Rhombidodecadodecahedron
In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices.[1] It is given a Schläfli symbol t0,2{5⁄2,5}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.
Rhombidodecadodecahedron
TypeUniform star polyhedron
ElementsF = 54, E = 120
V = 60 (χ = −6)
Faces by sides30{4}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol5/2 5 | 2
Symmetry groupIh, [5,3], *532
Index referencesU38, C48, W76
Dual polyhedronMedial deltoidal hexecontahedron
Vertex figure
4.5/2.4.5
Bowers acronymRaded
Cartesian coordinates
Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of
(±1/τ2, 0, ±τ2)
(±1, ±1, ±√5)
(±2, ±1/τ, ±τ)
where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
Related polyhedra
It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the icosidodecadodecahedron (having the pentagonal and pentagrammic faces in common) and the rhombicosahedron (having the square faces in common).
convex hull
Rhombidodecadodecahedron
Icosidodecadodecahedron
Rhombicosahedron
Compound of ten triangular prisms
Compound of twenty triangular prisms
Medial deltoidal hexecontahedron
Medial deltoidal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 54 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU38
dual polyhedronRhombidodecadodecahedron
The medial deltoidal hexecontahedron (or midly lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "38: rhombidodecadodecahedron". 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. "Rhombidodecadodecahedron". MathWorld.
• Weisstein, Eric W. "Medial deltoidal hexecontahedron". 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
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Rhombitrihexagonal tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.
Rhombitrihexagonal tiling
TypeSemiregular tiling
Vertex configuration
3.4.6.4
Schläfli symbolrr{6,3} or $r{\begin{Bmatrix}6\\3\end{Bmatrix}}$
Wythoff symbol3 | 6 2
Coxeter diagram
Symmetryp6m, [6,3], (*632)
Rotation symmetryp6, [6,3]+, (632)
Bowers acronymRothat
DualDeltoidal trihexagonal tiling
PropertiesVertex-transitive
John Conway calls it a rhombihexadeltille.[1] It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language.
There are three regular and eight semiregular tilings in the plane.
Uniform colorings
There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)
With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t{3} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,6}. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling, .
Symmetry [6,3], (*632) [6,3+], (3*3)
Name Rhombitrihexagonal Cantic snub triangular Snub triangular
Image
Uniform face coloring
Uniform edge coloring
Nonuniform geometry
Limit
Schläfli
symbol
rr{3,6} s2{3,6} s{3,6}
Coxeter
diagram
Examples
From The Grammar of Ornament (1856)
The game Kensington
Floor tiling, Archeological Museum of Seville, Sevilla, Spain
The Temple of Diana in Nîmes, France
Roman floor mosaic in Castel di Guido
Related tilings
There is one related 2-uniform tiling, having hexagons dissected into six triangles.[3][4] The rhombitrihexagonal tiling is also related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:
1-uniform Dissection 2-uniform dissections
3.4.6.4
3.3.4.3.4 & 36
to CH
Dual Tilings
3.4.6.4
4.6.12
to 3
Circle packing
The rhombitrihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (kissing number).[5] The translational lattice domain (red rhombus) contains six distinct circles.
Wythoff construction
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 eight forms, seven topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)
[6,3+]
(3*3)
{6,3} t{6,3} r{6,3} t{3,6} {3,6} rr{6,3} tr{6,3} sr{6,3} s{3,6}
63 3.122 (3.6)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6 3.3.3.3.3.3
Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.6.4 V.4.6.12 V34.6 V36
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paracomp.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4
Deltoidal trihexagonal tiling
Deltoidal trihexagonal tiling
TypeDual semiregular tiling
Faceskite
Coxeter diagram
Symmetry groupp6m, [6,3], (*632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronRhombitrihexagonal tiling
Face configurationV3.4.6.4
Propertiesface-transitive
The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway calls it a tetrille.[1] The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.[6]
The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling.[7] Its faces are deltoids or kites.
Related polyhedra and tilings
It is one of seven 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
This tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with three mirrors meeting at a point, and threefold rotation points.[8]
Isohedral variations
Symmetry p6m, [6,3], (*632) p31m, [6,3+], (3*3)
Form
Faces Kite Half regular hexagon Quadrilaterals
This tiling is related to the trihexagonal tiling by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.
The deltoidal trihexagonal tiling is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.
Symmetry mutations
This tiling is topologically related as a part of sequence of tilings with face configurations V3.4.n.4, and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4
Other deltoidal (kite) tiling
Other deltoidal tilings are possible.
Point symmetry allows the plane to be filled by growing kites, with the topology as a square tiling, V4.4.4.4, and can be created by crossing string of a dream catcher. Below is an example with dihedral hexagonal symmetry.
Another face transitive tiling with kite faces, also a topological variation of a square tiling and with face configuration V4.4.4.4. It is also vertex transitive, with every vertex containing all orientations of the kite face.
Symmetry D6, [6], (*66) pmg, [∞,(2,∞)+], (22*) p6m, [6,3], (*632)
Tiling
Configuration V4.4.4.4 V6.4.3.4
See also
Wikimedia Commons has media related to Uniform tiling 3-4-6-4 (rhombitrihexagonal tiling).
• Tilings of regular polygons
• List of uniform tilings
Notes
1. Conway, 2008, p288 table
2. Ring Cycles a Jacks Chain variation
3. 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.
4. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
5. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern B
6. 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.
7. Weisstein, Eric W. "Dual tessellation". MathWorld. (See comparative overlay of this tiling and its dual)
8. Tilings and Patterns
References
• 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. ISBN 0-486-23729-X. p40
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.
• Weisstein, Eric W. "Uniform tessellation". MathWorld.
• Weisstein, Eric W. "Semiregular tessellation". MathWorld.
• Klitzing, Richard. "2D Euclidean tilings x3o6x - rothat - O8".
• Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2
• Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 116
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
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Tetrahexagonal tiling
In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.
Tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.6)2
Schläfli symbolr{6,4} or ${\begin{Bmatrix}6\\4\end{Bmatrix}}$
rr{6,6}
r(4,4,3)
t0,1,2,3(∞,3,∞,3)
Wythoff symbol2 | 6 4
Coxeter diagram or
or
Symmetry group[6,4], (*642)
[6,6], (*662)
[(4,4,3)], (*443)
[(∞,3,∞,3)], (*3232)
DualOrder-6-4 quasiregular rhombic tiling
PropertiesVertex-transitive edge-transitive
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1+], gives [6,6], (*662). Removing the first mirror [1+,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1+,6,4,1+], leaving [(3,∞,3,∞)] (*3232).
Four uniform constructions of 4.6.4.6
Uniform
Coloring
Fundamental
Domains
Schläfli r{6,4} r{4,6}1⁄2 r{6,4}1⁄2 r{6,4}1⁄4
Symmetry [6,4]
(*642)
[6,6] = [6,4,1+]
(*662)
[(4,4,3)] = [1+,6,4]
(*443)
[(∞,3,∞,3)] = [1+,6,4,1+]
(*3232)
or
Symbol r{6,4} rr{6,6} r(4,3,4) t0,1,2,3(∞,3,∞,3)
Coxeter
diagram
= = =
or
Symmetry
The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.
Related polyhedra and tiling
*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Symmetry mutation of quasiregular tilings: 6.n.6.n
Symmetry
*6n2
[n,6]
Euclidean Compact hyperbolic Paracompact Noncompact
*632
[3,6]
*642
[4,6]
*652
[5,6]
*662
[6,6]
*762
[7,6]
*862
[8,6]...
*∞62
[∞,6]
[iπ/λ,6]
Quasiregular
figures
configuration
6.3.6.3
6.4.6.4
6.5.6.5
6.6.6.6
6.7.6.7
6.8.6.8
6.∞.6.∞
6.∞.6.∞
Dual figures
Rhombic
figures
configuration
V6.3.6.3
V6.4.6.4
V6.5.6.5
V6.6.6.6
V6.7.6.7
V6.8.6.8
V6.∞.6.∞
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}
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
Similar H2 tilings in *3232 symmetry
Coxeter
diagrams
Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image
Dual
See also
Wikimedia Commons has media related to Uniform tiling 4-6-4-6.
• 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
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Compound of five small cubicuboctahedra
This uniform polyhedron compound is a composition of 5 small cubicuboctahedra, in the same vertex arrangement as the compound of 5 small rhombicuboctahedra.
Compound of five small cubicuboctahedra
TypeUniform compound
IndexUC64
Polyhedra5 small cubicuboctahedra
Faces40 triangles, 30 squares, 30 octagons
Edges240
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Rhombille tiling
In geometry, the rhombille tiling,[1] also known as tumbling blocks,[2] reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.
Rhombille tiling
TypeLaves tiling
Faces60°–120° rhombus
Coxeter diagram
Symmetry groupp6m, [6,3], *632
p3m1, [3[3]], *333
Rotation groupp6, [6,3]+, (632)
p3, [3[3]]+, (333)
Dual polyhedronTrihexagonal tiling
Face configurationV3.6.3.6
Propertiesedge-transitive, face-transitive
Properties
The rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling. It can also be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi.
The diagonals of each rhomb are in the ratio 1:√3. This is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the face configuration for monohedral tilings it is denoted [3.6.3.6].[4]
It is also one of 56 possible isohedral tilings by quadrilaterals,[5] and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.[6]
It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points (x,y,z) with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, and more strongly such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube.[7]
Artistic and decorative applications
The rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion.[8]
In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, and Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms.[9] In another of his works, Cycle (1938), Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements (drawn isometrically) and an upstairs patio tiled with the rhombille tiling. A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so.[10] These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more generally, and includes a depiction of the reversible cubes illusion on a flag within the scene.[11]
The rhombille tiling is also used as a design for parquetry[12] and for floor or wall tiling, sometimes with variations in the shapes of its rhombi.[13] It appears in ancient Greek floor mosaics from Delos[14] and from Italian floor tilings from the 11th century,[15] although the tiles with this pattern in Siena Cathedral are of a more recent vintage.[16] In quilting, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation.[2][15][17] As a quilting pattern it also has many other names including cubework, heavenly stairs, and Pandora's box.[17] It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape. See Quilts of the Underground Railroad.[18] In these decorative applications, the rhombi may appear in multiple colors, but are typically given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms.[19]
Other applications
The rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field. The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers.[20]
In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice. It is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals,[21] and it has also been studied in percolation theory.[22]
Related polyhedra and tilings
Combinatorially equivalent tilings by parallelograms
The rhombille tiling is the dual of the trihexagonal tiling. It is one of many different ways of tiling the plane by congruent rhombi. Others include a diagonally flattened variation of the square tiling (with translational symmetry on all four sides of the rhombi), the tiling used by the Miura-ori folding pattern (alternating between translational and reflectional symmetry), and the Penrose tiling which uses two kinds of rhombi with 36° and 72° acute angles aperiodically. When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry.
Tilings combinatorially equivalent to the rhombille tiling can also be realized by parallelograms, and interpreted as axonometric projections of three dimensional cubic steps.
There are only eight edge tessellations, tilings of the plane with the property that reflecting any tile across any one of its edges produces another tile; one of them is the rhombille tiling.[6]
Examples
• The rhombille tiling overlaid on its dual, the trihexagonal tiling.
• Rhombille tiling floor mosaic in Delos
• Rhombille tiling pattern on the floor of Siena Cathedral
• Rhombille tiling in cruise terminal in Tallinn, Estonia
See also
Wikimedia Commons has media related to Rhombille tiling.
• Tiling by regular polygons
References
1. Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5.
2. Smith, Barbara (2002), Tumbling Blocks: New Quilts from an Old Favorite, Collector Books, ISBN 9781574327892.
3. Richard K. Guy & Robert E. Woodrow, The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, 1996, p.79, Figure 10
4. Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Section 2.7, Tilings with regular vertices, pp. 95–98.
5. Grünbaum & Shephard (1987), Figure 9.1.2, Tiling P4-42, p. 477.
6. 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.
7. Deza, Michel; Grishukhin, Viatcheslav; Shtogrin, Mikhail (2004), Scale-isometric polytopal graphs in hypercubes and cubic lattices: Polytopes in hypercubes and $\mathbb {Z} _{n}$, London: Imperial College Press, p. 150, doi:10.1142/9781860945489, ISBN 1-86094-421-3, MR 2051396.
8. Warren, Howard Crosby (1919), Human psychology, Houghton Mifflin, p. 262.
9. Kaplan, Craig S. (2008), "Metamorphosis in Escher's art", Bridges 2008: Mathematical Connections in Art, Music and Science (PDF), pp. 39–46.
10. Escher, Maurits Cornelis (2001), M.C. Escher, the Graphic Work, Taschen, pp. 29–30, ISBN 9783822858646.
11. De May, Jos (2003), "Painting after M. C. Escher", in Schattschneider, D.; Emmer, M. (eds.), M. C. Escher's Legacy: A Centennial Celebration, Springer, pp. 130–141.
12. Schleining, Lon; O'Rourke, Randy (2003), "Tricking the eyes with tumbling blocks", Treasure Chests: The Legacy of Extraordinary Boxes, Taunton Press, p. 58, ISBN 9781561586516.
13. Tessellation Tango, The Mathematical Tourist, Drexel University, retrieved 2012-05-23.
14. Dunbabin, Katherine M. D. (1999), Mosaics of the Greek and Roman World, Cambridge University Press, p. 32, ISBN 9780521002301.
15. Tatem, Mary (2010), "Tumbling Blocks", Quilt of Joy: Stories of Hope from the Patchwork Life, Revell, p. 115, ISBN 9780800733643.
16. Wallis, Henry (1902), Italian ceramic art, Bernard Quaritch, p. xxv.
17. Fowler, Earlene (2008), Tumbling Blocks, Benni Harper Mysteries, Penguin, ISBN 9780425221235. This is a mystery novel, but it also includes a brief description of the tumbling blocks quilt pattern in its front matter.
18. Tobin, Jacqueline L.; Dobard, Raymond G. (2000), Hidden in Plain View: A Secret Story of Quilts and the Underground Railroad, Random House Digital, Inc., p. 81, ISBN 9780385497671.
19. Aux armes: symbolism, Symbolism in arms, Pleiade, retrieved 2013-04-17.
20. The Q*Bert neighbourhood, Tim Tyler, accessed 2012-05-23.
21. Fisher, Michael E. (1959), "Transformations of Ising models", Physical Review, 113 (4): 969–981, Bibcode:1959PhRv..113..969F, doi:10.1103/PhysRev.113.969.
22. Yonezawa, Fumiko; Sakamoto, Shoichi; Hori, Motoo (1989), "Percolation in two-dimensional lattices. I. A technique for the estimation of thresholds", Phys. Rev. B, 40 (1): 636–649, Bibcode:1989PhRvB..40..636Y, doi:10.1103/PhysRevB.40.636.
Further reading
• Keith Critchlow, Order in Space: A design source book, 1970, pp. 77–76, pattern 1
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
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Rhombipentahexagonal tiling
In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{6,5}.
Rhombipentahexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration5.4.6.4
Schläfli symbolrr{6,5} or $r{\begin{Bmatrix}6\\5\end{Bmatrix}}$
Wythoff symbol5 | 6 2
Coxeter diagram
Symmetry group[6,5], (*652)
DualDeltoidal pentahexagonal tiling
PropertiesVertex-transitive
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
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 4-5-4-6.
• 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
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Tetrapentagonal tiling
In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}.
Tetrapentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.5)2
Schläfli symbolr{5,4} or ${\begin{Bmatrix}5\\4\end{Bmatrix}}$
rr{5,5} or $r{\begin{Bmatrix}5\\5\end{Bmatrix}}$
Wythoff symbol2 | 5 4
5 5 | 2
Coxeter diagram or
or
Symmetry group[5,4], (*542)
[5,5], (*552)
DualOrder-5-4 rhombille tiling
PropertiesVertex-transitive edge-transitive
Symmetry
A half symmetry [1+,4,5] = [5,5] construction exists, which can be seen as two colors of pentagons. This coloring can be called a rhombipentapentagonal tiling.
Dual tiling
The dual tiling is made of rhombic faces and has a face configuration V4.5.4.5:
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
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
*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
*5n2 symmetry mutations of quasiregular tilings: (5.n)2
Symmetry
*5n2
[n,5]
Spherical Hyperbolic Paracompact Noncompact
*352
[3,5]
*452
[4,5]
*552
[5,5]
*652
[6,5]
*752
[7,5]
*852
[8,5]...
*∞52
[∞,5]
[ni,5]
Figures
Config. (5.3)2 (5.4)2 (5.5)2 (5.6)2 (5.7)2 (5.8)2 (5.∞)2 (5.ni)2
Rhombic
figures
Config. V(5.3)2 V(5.4)2 V(5.5)2 V(5.6)2 V(5.7)2 V(5.8)2 V(5.∞)2 V(5.∞)2
See also
Wikimedia Commons has media related to Uniform tiling 4-5-4-5.
• 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
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Compound of five great cubicuboctahedra
This uniform polyhedron compound is a composition of 5 great cubicuboctahedra, in the same arrangement as the compound of 5 truncated cubes.
Compound of five great cubicuboctahedra
TypeUniform compound
IndexUC65
Polyhedra5 great cubicuboctahedra
Faces40 triangles, 30 squares, 30 octagrams
Edges240
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Expanded icosidodecahedron
The expanded icosidodecahedron is a polyhedron, constructed as an expanded icosidodecahedron. It has 122 faces: 20 triangles, 60 squares, 12 pentagons, and 30 rhombs. The 120 vertices exist at two sets of 60, with a slightly different distance from its center.
Expanded icosidodecahedron
Schläfli symbolrr${\begin{Bmatrix}5\\3\end{Bmatrix}}$ = rrr{5,3}
Conway notationedaD = aaaD
Faces122:
20 {3}
60 {4}
12 {5}
30 rhombs
Edges240
Vertices120
Symmetry groupIh, [5,3], (*532) order 120
Rotation groupI, [5,3]+, (532), order 60
Dual polyhedronDeltoidal hecatonicosahedron
Propertiesconvex
Net
It can also be constructed as a rectified rhombicosidodecahedron.
Other names
• Expanded rhombic triacontahedron
• Rectified rhombicosidodecahedron
• Rectified small rhombicosidodecahedron
• Rhombirhombicosidodecahedron
Expansion
The expansion operation from the rhombic triacontahedron can be seen in this animation:
Dissection
This polyhedron can be dissected into a central rhombic triacontahedron surrounded by: 30 rhombic prisms, 20 tetrahedra, 12 pentagonal pyramids, 60 triangular prisms.
If the central rhombic triacontahedron and the 30 rhombic prisms are removed, you can create a toroidal polyhedron with all regular polygon faces.
Related polyhedra
Name Dodeca-
hedron
Icosidodeca-
hedron
Rhomb-
icosidodeca-
hedron
Expanded
icosidodeca-
hedron
Coxeter[1] D ID rID rrID
Conway aD aaD = eD aaaD = eaD
Image
Conway dD = I daD = jD deD = oD deaD = oaD
Dual
See also
• Rhombicosidodecahedron (expanded dodecahedron)
• Truncated rhombicosidodecahedron
• Expanded cuboctahedron
References
1. "Uniform Polyhedron".
• 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, VRML model
• Convex Polyhedra containing Golden Rhombi: 2. Expanded RTC ('XRTC') and related polyhedral
• Variations on a Rhombic Theme
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Compound of six cubes with rotational freedom
This uniform polyhedron compound is a symmetric arrangement of 6 cubes, considered as square prisms. It can be constructed by superimposing six identical cubes, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each cube is rotated by an equal (and opposite, within a pair) angle θ.
Compound of six cubes with rotational freedom
TypeUniform compound
IndexUC7
Polyhedra6 cubes
Faces12+24 squares
Edges72
Vertices48
Symmetry groupoctahedral (Oh)
Subgroup restricting to one constituent4-fold rotational (C4h)
When θ = 0, all six cubes coincide. When θ is 45 degrees, the cubes coincide in pairs yielding (two superimposed copies of) the compound of three cubes.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the permutations of
$(\pm (\cos(\theta )+\sin(\theta )),\pm (\cos(\theta )-\sin(\theta )),\pm 1).$
Gallery
• Compounds of six cubes with rotational freedom
• θ = 0°
• θ = 5°
• θ = 10°
• θ = 15°
• θ = 20°
• θ = 25°
• θ = 30°
• θ = 35°
• θ = 40°
• θ = 45°
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.
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Compound of five small rhombihexahedra
This uniform polyhedron compound is a composition of 5 small rhombihexahedra, in the same vertex and edge arrangement as the compound of 5 small rhombicuboctahedra.
Compound of five small rhombihexahedra
TypeUniform compound
IndexUC63
Polyhedra5 small rhombihexahedra
Faces60 squares, 30 octagons
Edges240
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Compound of five great rhombihexahedra
This uniform polyhedron compound is a composition of 5 great rhombihexahedra, in the same vertex arrangement as the compound of 5 truncated cubes.
Compound of five great rhombihexahedra
TypeUniform compound
IndexUC66
Polyhedra5 great rhombihexahedra
Faces60 squares, 30 octagrams
Edges240
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
Filling
There is some controversy on how to colour the faces of this polyhedron compound. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in). In addition, overlapping regions of coplanar faces can cancel each other out.[1]
Traditional filling
"Neo filling"
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.
1. "Uniform Polyhedra".
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Compound of five rhombicuboctahedra
This uniform polyhedron compound is a composition of 5 rhombicuboctahedra, in the same vertex arrangement (i.e. sharing vertices with) the compound of 5 stellated truncated hexahedra.
Compound of five rhombicuboctahedra
TypeUniform compound
IndexUC62
Polyhedra5 rhombicuboctahedra
Faces40 triangles, 30+60 squares
Edges240
Vertices120
Symmetry groupicosahedral (Ih)
Subgroup restricting to one constituentpyritohedral (Th)
References
• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.
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Rhombitetraapeirogonal tiling
In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.
Rhombitetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.∞.4
Schläfli symbolrr{∞,4} or $r{\begin{Bmatrix}\infty \\4\end{Bmatrix}}$
Wythoff symbol4 | ∞ 2
Coxeter diagram or
Symmetry group[∞,4], (*∞42)
DualDeltoidal tetraapeirogonal tiling
PropertiesVertex-transitive
Constructions
There are two uniform constructions of this tiling, one from [∞,4] or (*∞42) symmetry, and secondly removing the mirror middle, [∞,1+,4], gives a rectangular fundamental domain [∞,∞,∞], (*∞222).
Two uniform constructions of 4.4.4.∞
Name Rhombitetrahexagonal tiling
Image
Symmetry [∞,4]
(*∞42)
[∞,∞,∞] = [∞,1+,4]
(*∞222)
Schläfli symbol rr{∞,4} t0,1,2,3{∞,∞,∞}
Coxeter diagram
Symmetry
The dual of this tiling, called a deltoidal tetraapeirogonal tiling represents the fundamental domains of (*∞222) orbifold symmetry. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.
Related polyhedra and tiling
*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
V3.4.4.4
V4.4.4.4
V5.4.4.4
V6.4.4.4
V7.4.4.4
V8.4.4.4
V∞.4.4.4
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 4-4-4-i.
• 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
|
Rhombitetraheptagonal tiling
In geometry, the rhombitetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{4,7}. It can be seen as constructed as a rectified tetraheptagonal tiling, r{7,4}, as well as an expanded order-4 heptagonal tiling or expanded order-7 square tiling.
Rhombitetraheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.4.7.4
Schläfli symbolrr{7,4} or $r{\begin{Bmatrix}7\\4\end{Bmatrix}}$
Wythoff symbol4 | 7 2
Coxeter diagram
Symmetry group[7,4], (*742)
DualDeltoidal tetraheptagonal tiling
PropertiesVertex-transitive
Dual tiling
The dual is called the deltoidal tetraheptagonal tiling with face configuration V.4.4.4.7.
Related polyhedra and tiling
*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
V3.4.4.4
V4.4.4.4
V5.4.4.4
V6.4.4.4
V7.4.4.4
V8.4.4.4
V∞.4.4.4
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 4-4-4-7.
• 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
|
Rhombitriapeirogonal tiling
In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
Rhombitriapeirogonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.4.∞.4
Schläfli symbolrr{∞,3} or $r{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$
s2{3,∞}
Wythoff symbol3 | ∞ 2
Coxeter diagram or
Symmetry group[∞,3], (*∞32)
[∞,3+], (3*∞)
DualDeltoidal triapeirogonal tiling
PropertiesVertex-transitive
Symmetry
This tiling has [∞,3], (*∞32) symmetry. There is only one uniform coloring.
Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*∞) orbifold notation. The apeireogons can be considered as truncated, t{∞} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,∞}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an infinite-order triangular tiling results, constructed as a snub triapeirotrigonal tiling, .
Related polyhedra and tiling
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.∞
Symmetry mutations
This hyperbolic tiling is topologically related as a part of sequence of uniform cantellated polyhedra with vertex configurations (3.4.n.4), and [n,3] Coxeter group symmetry.
*n42 symmetry mutation of expanded tilings: 3.4.n.4
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]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4
See also
Wikimedia Commons has media related to Uniform tiling 3-4-i-4.
• 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
|
Rhombitriheptagonal tiling
In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.
Rhombitriheptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.4.7.4
Schläfli symbolrr{7,3} or $r{\begin{Bmatrix}7\\3\end{Bmatrix}}$
Wythoff symbol3 | 7 2
Coxeter diagram or
Symmetry group[7,3], (*732)
DualDeltoidal triheptagonal tiling
PropertiesVertex-transitive
Dual tiling
The dual tiling is called a deltoidal triheptagonal tiling, and consists of congruent kites. It is formed by overlaying an order-3 heptagonal tiling and an order-7 triangular tiling.
Related polyhedra and tilings
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 8 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
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4
See also
Wikimedia Commons has media related to Uniform tiling 3-4-7-4.
• Rhombitrihexagonal tiling
• Order-3 heptagonal tiling
• Tilings of regular polygons
• List of uniform tilings
• Kagome lattice
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
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Rhombitrioctagonal tiling
In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.
Rhombitrioctagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.4.8.4
Schläfli symbolrr{8,3} or $r{\begin{Bmatrix}8\\3\end{Bmatrix}}$
s2{3,8}
Wythoff symbol3 | 8 2
Coxeter diagram or
Symmetry group[8,3], (*832)
[8,3+], (3*4)
DualDeltoidal trioctagonal tiling
PropertiesVertex-transitive
Symmetry
This tiling has [8,3], (*832) symmetry. There is only one uniform coloring.
Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*4) orbifold notation. The octagons can be considered as truncated squares, t{4} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,8}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an order-8 triangular tiling results, constructed as a snub tritetratrigonal tiling, .
Related polyhedra and tilings
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
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
*n42 symmetry mutation of expanded tilings: 3.4.n.4
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]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4
See also
Wikimedia Commons has media related to Uniform tiling 3-4-8-4.
• Rhombitrihexagonal tiling
• Order-3 octagonal tiling
• Tilings of regular polygons
• List of uniform tilings
• Kagome lattice
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
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Truncated icosidodecahedron
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,[1] great rhombicosidodecahedron,[2][3] omnitruncated dodecahedron or omnitruncated icosahedron[4] is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
Truncated icosidodecahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 62, E = 180, V = 120 (χ = 2)
Faces by sides30{4}+20{6}+12{10}
Conway notationbD or taD
Schläfli symbolstr{5,3} or $t{\begin{Bmatrix}5\\3\end{Bmatrix}}$
t0,1,2{5,3}
Wythoff symbol2 3 5 |
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle6-10: 142.62°
4-10: 148.28°
4-6: 159.095°
ReferencesU28, C31, W16
PropertiesSemiregular convex zonohedron
Colored faces
4.6.10
(Vertex figure)
Disdyakis triacontahedron
(dual polyhedron)
Net
It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a 15-zonohedron.
Names
The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of an icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid.
Alternate interchangeable names are:
• Truncated icosidodecahedron (Johannes Kepler)
• Rhombitruncated icosidodecahedron (Magnus Wenninger[1])
• Great rhombicosidodecahedron (Robert Williams,[2] Peter Cromwell[3])
• Omnitruncated dodecahedron or icosahedron (Norman Johnson[4])
Icosidodecahedron and its truncation
The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.
Area and volume
The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:
${\begin{aligned}A&=30\left(1+{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 174.292\,0303a^{2}.\\V&=\left(95+50{\sqrt {5}}\right)a^{3}&&\approx 206.803\,399a^{3}.\end{aligned}}$
If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:[5]
(±1/φ, ±1/φ, ±(3 + φ)),
(±2/φ, ±φ, ±(1 + 2φ)),
(±1/φ, ±φ2, ±(−1 + 3φ)),
(±(2φ − 1), ±2, ±(2 + φ)) and
(±φ, ±3, ±2φ),
where φ = 1 + √5/2 is the golden ratio.
Dissection
The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares, whose height to base ratio is φ. The rest of its space can be dissected into nonuniform cupolas, namely 12 between inner pentagons and outer decagons and 20 between inner triangles and outer hexagons.
An alternative dissection also has a rhombicosidodecahedral core. It has 12 pentagonal rotundae between inner pentagons and outer decagons. The remaining part is a toroidal polyhedron.
dissection images
These images show the rhombicosidodecahedron (violet) and the truncated icosidodecahedron (green). If their edge lengths are 1, the distance between corresponding squares is φ.
Orthogonal projections
The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge
4-6
Edge
4-10
Edge
6-10
Face
square
Face
hexagon
Face
decagon
Solid
Wireframe
Projective
symmetry
[2]+ [2] [2] [2] [2] [6] [10]
Dual
image
Spherical tilings and Schlegel diagrams
The truncated icosidodecahedron 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 Hexagon-centered Square-centered
Geometric variations
Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces. The truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases.
Truncated icosidodecahedral graph
Truncated icosidodecahedral graph
5-fold symmetry
Vertices120
Edges180
Radius15
Diameter15
Girth4
Automorphisms120 (A5×2)
Chromatic number2
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters
In the mathematical field of graph theory, a truncated icosidodecahedral graph (or great rhombicosidodecahedral graph) is the graph of vertices and edges of the truncated icosidodecahedron, one of the Archimedean solids. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph.[6]
Schlegel diagram graphs
3-fold symmetry
2-fold symmetry
Related polyhedra and tilings
Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.[7]
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 can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*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]
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
Notes
1. Wenninger Model Number 16
2. Williams (Section 3-9, p. 94)
3. Cromwell (p. 82)
4. Norman Woodason Johnson, "The Theory of Uniform Polytopes and Honeycombs", 1966
5. Weisstein, Eric W. "Icosahedral group". MathWorld.
6. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
7. Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
References
• Wenninger, Magnus (1974), Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR 0467493
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
• Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
• Eric W. Weisstein, GreatRhombicosidodecahedron (Archimedean solid) at MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x5x - grid".
External links
• Weisstein, Eric W. "Great rhombicosidodecahedron". MathWorld.
• * Weisstein, Eric W. "Great rhombicosidodecahedral graph". MathWorld.
• Editable printable net of a truncated icosidodecahedron 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.
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Truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron.
Truncated octahedron
(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 14, E = 36, V = 24 (χ = 2)
Faces by sides6{4}+8{6}
Conway notationtO
bT
Schläfli symbolst{3,4}
tr{3,3} or $t{\begin{Bmatrix}3\\3\end{Bmatrix}}$
t0,1{3,4} or t0,1,2{3,3}
Wythoff symbol2 4 | 3
3 3 2 |
Coxeter diagram
Symmetry groupOh, B3, [4,3], (*432), order 48
Th, [3,3] and (*332), order 24
Rotation groupO, [4,3]+, (432), order 24
Dihedral angle4-6: arccos(−1/√3) = 125°15′51″
6-6: arccos(−1/3) = 109°28′16″
ReferencesU08, C20, W7
PropertiesSemiregular convex parallelohedron
permutohedron
zonohedron
Colored faces
4.6.6
(Vertex figure)
Tetrakis hexahedron
(dual polyhedron)
Net
The truncated octahedron was called the "mecon" by Buckminster Fuller.[1]
Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths 9/8√2 and 3/2√2.
Construction
A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point. These pyramids have both base side length (a) and lateral side length (e) of a, to form equilateral triangles. The base area is then a2. Note that this shape is exactly similar to half an octahedron or Johnson solid J1.
From the properties of square pyramids, we can now find the slant height, s, and the height, h, of the pyramid:
${\begin{aligned}h&={\sqrt {e^{2}-{\tfrac {1}{2}}a^{2}}}&&={\tfrac {1}{\sqrt {2}}}a\\s&={\sqrt {h^{2}+{\tfrac {1}{4}}a^{2}}}&&={\sqrt {{\tfrac {1}{2}}a^{2}+{\tfrac {1}{4}}a^{2}}}&&={\tfrac {\sqrt {3}}{2}}a\end{aligned}}$
The volume, V, of the pyramid is given by:
$V={\tfrac {1}{3}}a^{2}h={\tfrac {\sqrt {2}}{6}}a^{3}$
Because six pyramids are removed by truncation, there is a total lost volume of √2a3.
Orthogonal projections
The truncated octahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B2 and A2 Coxeter planes.
Orthogonal projections
Centered by Vertex Edge
4-6
Edge
6-6
Face
Square
Face
Hexagon
Solid
Wireframe
Dual
Projective
symmetry
[2] [2] [2] [4] [6]
Spherical tiling
The truncated octahedron 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.
square-centered
hexagon-centered
Orthographic projection Stereographic projections
Coordinates
Orthogonal projection in bounding box
(±2,±2,±2)
Truncated octahedron with hexagons replaced by 6 coplanar triangles. There are 8 new vertices at: (±1,±1,±1). Truncated octahedron subdivided into as a topological rhombic triacontahedron
All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.
The edge vectors have Cartesian coordinates (0, ±1, ±1) and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are (0, 0, ±1), (0, ±1, 0) and (±1, 0, 0). The face normals of the 8 hexagonal faces are (±1/√3, ±1/√3, ±1/√3). The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. The dihedral angle is approximately 1.910633 radians (109.471° OEIS: A156546) at edges shared by two hexagons or 2.186276 radians (125.263° OEIS: A195698) at edges shared by a hexagon and a square.
Dissection
The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupolae on each face, and 6 square pyramids above the vertices.[2]
Removing the central octahedron and 2 or 4 triangular cupolae creates two Stewart toroids, with dihedral and tetrahedral symmetry:
Genus 2 Genus 3
D3d, [2+,6], (2*3), order 12 Td, [3,3], (*332), order 24
Permutohedron
The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1, 2, 3, 4) form the vertices of a truncated octahedron in the three-dimensional subspace x + y + z + w = 10. Therefore, the truncated octahedron is the permutohedron of order 4: each vertex corresponds to a permutation of (1, 2, 3, 4) and each edge represents a single pairwise swap of two elements.
Area and volume
The surface area S and the volume V of a truncated octahedron of edge length a are:
${\begin{aligned}S&=\left(6+12{\sqrt {3}}\right)a^{2}&&\approx 26.784\,6097a^{2}\\V&=8{\sqrt {2}}a^{3}&&\approx 11.313\,7085a^{3}.\end{aligned}}$
Uniform colorings
There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a truncated triangular antiprism. The constructional names are given for each. Their Conway polyhedron notation is given in parentheses.
1-uniform 2-uniform
Oh, [4,3], (*432)
Order 48
Td, [3,3], (*332)
Order 24
D4h, [4,2], (*422)
Order 16
D3d, [2+,6], (2*3)
Order 12
122 coloring
123 coloring
122 & 322 colorings
122 & 123 colorings
Truncated octahedron
(tO)
Bevelled tetrahedron
(bT)
Truncated square bipyramid
(tdP4)
Truncated triangular antiprism
(tA3)
Chemistry
The truncated octahedron exists in the structure of the faujasite crystals.
Data hiding
The truncated octahedron (in fact, the generalized truncated octahedron) appears in the error analysis of quantization index modulation (QIM) in conjunction with repetition coding.[3]
Related polyhedra
The truncated octahedron 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
It also exists as the omnitruncate of the tetrahedron family:
Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3
Symmetry mutations
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*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]
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
*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.∞.∞
This polyhedron is a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter–Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures n.6.6, extending into the hyperbolic plane:
*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
The truncated octahedron 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.∞.∞
Related polytopes
The truncated octahedron (bitruncated cube), is first 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}
It is possible to slice a tesseract by a hyperplane so that its sliced cross-section is a truncated octahedron.[4]
Tessellations
The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations):
Bitruncated cubic Cantitruncated cubic Truncated alternated cubic
The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centered cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedra.
Objects
Jungle gym nets often include truncated octahedra.
• ancient Chinese die
• sculpture in Bonn
• Rubik's Cube variant
• model made with Polydron construction set
• Pyrite crystal
Truncated octahedral graph
Truncated octahedral graph
3-fold symmetric Schlegel diagram
Vertices24
Edges36
Automorphisms48
Chromatic number2
Book thickness3
Queue number2
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters
In the mathematical field of graph theory, a truncated octahedral graph is the graph of vertices and edges of the truncated octahedron. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[5] It has book thickness 3 and queue number 2.[6]
As a Hamiltonian cubic graph, it can be represented by LCF notation in multiple ways: [3, −7, 7, −3]6, [5, −11, 11, 7, 5, −5, −7, −11, 11, −5, −7, 7]2, and [−11, 5, −3, −7, −9, 3, −5, 5, −3, 9, 7, 3, −5, 11, −3, 7, 5, −7, −9, 9, 7, −5, −7, 3].[7]
References
1. "Truncated Octahedron". Wolfram Mathworld.
2. Doskey, Alex. "Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1". www.doskey.com.
3. Perez-Gonzalez, F.; Balado, F.; Martin, J.R.H. (2003). "Performance analysis of existing and new methods for data hiding with known-host information in additive channels". IEEE Transactions on Signal Processing. 51 (4): 960–980. Bibcode:2003ITSP...51..960P. doi:10.1109/TSP.2003.809368.
4. Borovik, Alexandre V.; Borovik, Anna (2010), "Exercise 14.4", Mirrors and Reflections, Universitext, New York: Springer, p. 109, doi:10.1007/978-0-387-79066-4, ISBN 978-0-387-79065-7, MR 2561378
5. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
6. Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
7. Weisstein, Eric W. "Truncated octahedral graph". MathWorld.
• 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)
• Freitas, Robert A. Jr. "Uniform space-filling using only truncated octahedra". Figure 5.5 of Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999. Retrieved 2006-09-08. {{cite web}}: External link in |publisher= (help)
• Gaiha, P. & Guha, S.K. (1977). "Adjacent vertices on a permutohedron". SIAM Journal on Applied Mathematics. 32 (2): 323–327. doi:10.1137/0132025.
• Hart, George W. "VRML model of truncated octahedron". Virtual Polyhedra: The Encyclopedia of Polyhedra. Retrieved 2006-09-08. {{cite web}}: External link in |publisher= (help)
• Mäder, Roman. "The Uniform Polyhedra: Truncated Octahedron". Retrieved 2006-09-08.
• Alexandrov, A.D. (1958). Konvexe Polyeder. Berlin: Springer. p. 539. ISBN 3-540-23158-7.
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
Wikimedia Commons has media related to Truncated octahedron.
• Eric W. Weisstein, Truncated octahedron (Archimedean solid) at MathWorld.
• Weisstein, Eric W. "Truncated octahedral graph". MathWorld.
• Weisstein, Eric W. "Permutohedron". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x4o - toe".
• Editable printable net of a truncated octahedron with interactive 3D view
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.
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Wikipedia
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Rhombohedron
In geometry, a rhombohedron (also called a rhombic hexahedron[1] or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.
For the crystal system, see Rhombohedral crystal system.
Rhombohedron
Typeprism
Faces6 rhombi
Edges12
Vertices8
Symmetry groupCi , [2+,2+], (×), order 2
Propertiesconvex, equilateral, zonohedron, parallelohedron
In general a rhombohedron can have up to three types of rhombic faces in congruent opposite pairs, Ci symmetry, order 2.
Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.[2]
Rhombohedral lattice system
Main article: Rhombohedral lattice system
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron:
Special cases by symmetry
Form Cube Trigonal trapezohedron Right rhombic prism Oblique rhombic prism
Angle
constraints
$\alpha =\beta =\gamma =90^{\circ }$ $\alpha =\beta =\gamma $ $\alpha =\beta =90^{\circ }$ $\alpha =\beta $
Symmetry Oh
order 48
D3d
order 12
D2h
order 8
C2h
order 4
Faces 6 squares 6 congruent rhombi 2 rhombi, 4 squares 6 rhombi
• Cube: with Oh symmetry, order 48. All faces are squares.
• Trigonal trapezohedron (also called isohedral rhombohedron):[3] with D3d symmetry, order 12. All non-obtuse internal angles of the faces are equal (all faces are congruent rhombi). This can be seen by stretching a cube on its body-diagonal axis. For example, a regular octahedron with two regular tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron.
• Right rhombic prism: with D2h symmetry, order 8. It is constructed by two rhombi and four squares. This can be seen by stretching a cube on its face-diagonal axis. For example, two right prisms with regular triangular bases attached together makes a 60 degree right rhombic prism.
• Oblique rhombic prism: with C2h symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces.
Solid geometry
For a unit (i.e.: with side length 1) isohedral rhombohedron,[3] with rhombic acute angle $\theta ~$, with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are
e1 : ${\biggl (}1,0,0{\biggr )},$
e2 : ${\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},$
e3 : ${\biggl (}\cos \theta ,{\cos \theta -\cos ^{2}\theta \over \sin \theta },{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }{\biggr )}.$
The other coordinates can be obtained from vector addition[4] of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .
The volume $V$ of an isohedral rhombohedron, in terms of its side length $a$ and its rhombic acute angle $\theta ~$, is a simplification of the volume of a parallelepiped, and is given by
$V=a^{3}(1-\cos \theta ){\sqrt {1+2\cos \theta }}=a^{3}{\sqrt {(1-\cos \theta )^{2}(1+2\cos \theta )}}=a^{3}{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }}~.$
We can express the volume $V$ another way :
$V=2{\sqrt {3}}~a^{3}\sin ^{2}\left({\frac {\theta }{2}}\right){\sqrt {1-{\frac {4}{3}}\sin ^{2}\left({\frac {\theta }{2}}\right)}}~.$
As the area of the (rhombic) base is given by $a^{2}\sin \theta ~$, and as the height of a rhombohedron is given by its volume divided by the area of its base, the height $h$ of an isohedral rhombohedron in terms of its side length $a$ and its rhombic acute angle $\theta $ is given by
$h=a~{(1-\cos \theta ){\sqrt {1+2\cos \theta }} \over \sin \theta }=a~{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }~.$
Note:
$h=a~z$3 , where $z$3 is the third coordinate of e3 .
The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.
See also
• Lists of shapes
References
1. "David Mitchell's Origami Heaven - Rhombic Polyhedra".
2. Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly, 41 (8): 499–502, doi:10.2307/2300415, JSTOR 2300415.
3. Lines, L (1965). Solid geometry: with chapters on space-lattices, sphere-packs and crystals. Dover Publications.
4. "Vector Addition". Wolfram. 17 May 2016. Retrieved 17 May 2016.
External links
• Weisstein, Eric W. "Rhombohedron". MathWorld.
• Volume Calculator https://rechneronline.de/pi/rhombohedron.php
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.
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Wikipedia
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Hexagonal crystal family
In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section crystal systems below).[1] In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-quartz).
Not to be confused with Hexagonal lattice.
Crystal system Trigonal Hexagonal
Lattice system
Rhombohedral
Hexagonal
Example
Dolomite
α-Quartz
Beryl
The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.[2] There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral.
Lattice systems
The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice.
Hexagonal crystal family
Bravais lattice Hexagonal Rhombohedral
Pearson symbol hP hR
Hexagonal
unit cell
Rhombohedral
unit cell
In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes (a by a), an included angle of 120° (γ) and a height (c, which can be different from a) perpendicular to the two base axes.
The hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell. There are two ways to do this, which can be thought of as two notations which represent the same structure. In the usual so-called obverse setting, the additional lattice points are at coordinates (2⁄3, 1⁄3, 1⁄3) and (1⁄3, 2⁄3, 2⁄3), whereas in the alternative reverse setting they are at the coordinates (1⁄3,2⁄3,1⁄3) and (2⁄3,1⁄3,2⁄3).[3] In either case, there are 3 lattice points per unit cell in total and the lattice is non-primitive.
The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes.[4] The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice). This is a unit cell with parameters a = b = c; α = β = γ ≠ 90°.[5] In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are often shown (for the rhombohedral lattice) in textbooks because this cell reveals the 3m symmetry of the crystal lattice.
The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered[1] cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates (1⁄3, 1⁄3, 1⁄3) and (2⁄3, 2⁄3, 2⁄3). However, such a description is rarely used.
Crystal systems
Crystal system Required symmetries of point group Point groups Space groups Bravais lattices Lattice system
Trigonal 1 threefold axis of rotation 5 7 1 Rhombohedral
18 1 Hexagonal
Hexagonal 1 sixfold axis of rotation 7 27
The hexagonal crystal family consists of two crystal systems: trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system (see table in Crystal system#Crystal classes).
The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis, which includes space groups 143 to 167. These 5 point groups have 7 corresponding space groups (denoted by R) assigned to the rhombohedral lattice system and 18 corresponding space groups (denoted by P) assigned to the hexagonal lattice system. Hence, the trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups.
The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis. These 7 point groups have 27 space groups (168 to 194), all of which are assigned to the hexagonal lattice system.
Trigonal crystal system
The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals.[6][7][8]
Space group no. Point group Type Examples Space groups
Name[1] Intl Schoen. Orb. Cox. Hexagonal Rhombohedral
143–146 Trigonal pyramidal 3 C3 33 [3]+ enantiomorphic polar carlinite, jarosite P3, P31, P32 R3
147–148 Rhombohedral 3 C3i (S6) 3× [2+,6+] centrosymmetric dolomite, ilmenite P3 R3
149–155 Trigonal trapezohedral 32 D3 223 [2,3]+ enantiomorphic abhurite, alpha-quartz (152, 154), cinnabar P312, P321, P3112, P3121, P3212, P3221 R32
156–161 Ditrigonal pyramidal 3m C3v *33 [3] polar schorl, cerite, tourmaline, alunite, lithium tantalate P3m1, P31m, P3c1, P31c R3m, R3c
162–167 Ditrigonal scalenohedral 3m D3d 2*3 [2+,6] centrosymmetric antimony, hematite, corundum, calcite, bismuth P31m, P31c, P3m1, P3c1 R3m, R3c
Hexagonal crystal system
The 7 point groups (crystal classes) in this crystal system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation, and mineral examples, if they exist.[2][9]
Space group no. Point group Type Examples Space groups
Name[1] Intl Schoen. Orb. Cox.
168–173 Hexagonal pyramidal 6 C6 66 [6]+ enantiomorphic polar nepheline, cancrinite P6, P61, P65, P62, P64, P63
174 Trigonal dipyramidal 6 C3h 3* [2,3+] laurelite and boric acid P6
175–176 Hexagonal dipyramidal 6/m C6h 6* [2,6+] centrosymmetric apatite, vanadinite P6/m, P63/m
177–182 Hexagonal trapezohedral 622 D6 226 [2,6]+ enantiomorphic kalsilite and high quartz P622, P6122, P6522, P6222, P6422, P6322
183–186 Dihexagonal pyramidal 6mm C6v *66 [6] polar greenockite, wurtzite[10] P6mm, P6cc, P63cm, P63mc
187–190 Ditrigonal dipyramidal 6m2 D3h *223 [2,3] benitoite P6m2, P6c2, P62m, P62c
191–194 Dihexagonal dipyramidal 6/mmm D6h *226 [2,6] centrosymmetric beryl P6/mmm, P6/mcc, P63/mcm, P63/mmc
The unit cell volume is given by a2c•sin(60°)
Hexagonal close packed
Main article: Close-packing of equal spheres
Hexagonal close packed (hcp) is one of the two simple types of atomic packing with the highest density, the other being the face-centered cubic (fcc). However, unlike the fcc, it is not a Bravais lattice, as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two-atom motif (the additional atom at about (2⁄3, 1⁄3, 1⁄2)) associated with each lattice point.[11]
Multi-element structures
Compounds that consist of more than one element (e.g. binary compounds) often have crystal structures based on the hexagonal crystal family. Some of the more common ones are listed here. These structures can be viewed as two or more interpenetrating sublattices where each sublattice occupies the interstitial sites of the others.
Wurtzite structure
The wurtzite crystal structure is referred to by the Strukturbericht designation B4 and the Pearson symbol hP4. The corresponding space group is No. 186 (in International Union of Crystallography classification) or P63mc (in Hermann–Mauguin notation). The Hermann-Mauguin symbols in P63mc can be read as follows:[13]
• 63.. : a six fold screw rotation around the c-axis
• .m. : a mirror plane with normal {100}
• ..c : glide plane in the c-directions with normal {120}.
Among the compounds that can take the wurtzite structure are wurtzite itself (ZnS with up to 8% iron instead of zinc), silver iodide (AgI), zinc oxide (ZnO), cadmium sulfide (CdS), cadmium selenide (CdSe), silicon carbide (α-SiC), gallium nitride (GaN), aluminium nitride (AlN), boron nitride (w-BN) and other semiconductors.[14] In most of these compounds, wurtzite is not the favored form of the bulk crystal, but the structure can be favored in some nanocrystal forms of the material.
In materials with more than one crystal structure, the prefix "w-" is sometimes added to the empirical formula to denote the wurtzite crystal structure, as in w-BN.
Each of the two individual atom types forms a sublattice which is hexagonal close-packed (HCP-type). When viewed all together, the atomic positions are the same as in lonsdaleite (hexagonal diamond). Each atom is tetrahedrally coordinated. The structure can also be described as an HCP lattice of zinc with sulfur atoms occupying half of the tetrahedral voids or vice versa.
The wurtzite structure is non-centrosymmetric (i.e., lacks inversion symmetry). Due to this, wurtzite crystals can (and generally do) have properties such as piezoelectricity and pyroelectricity, which centrosymmetric crystals lack.
Nickel arsenide structure
The nickel arsenide structure consists of two interpenetrating sublattices: a primitive hexagonal nickel sublattice and a hexagonal close-packed arsenic sublattice. Each nickel atom is octahedrally coordinated to six arsenic atoms, while each arsenic atom is trigonal prismatically coordinated to six nickel atoms.[15] The structure can also be described as an HCP lattice of arsenic with nickel occupying each octahedral void.
Compounds adopting the NiAs structure are generally the chalcogenides, arsenides, antimonides and bismuthides of transition metals.
The following are the members of the nickeline group:[16]
• Achavalite: FeSe
• Breithauptite: NiSb
• Freboldite: CoSe
• Kotulskite: Pd(Te,Bi)
• Langistite: (Co,Ni)As
• Nickeline: NiAs
• Sobolevskite: Pd(Bi,Te)
• Sudburyite: (Pd,Ni)Sb
In two dimensions
Main article: Hexagonal lattice
There is only one hexagonal Bravais lattice in two dimensions: the hexagonal lattice.
Bravais lattice Hexagonal
Pearson symbol hp
Unit cell
See also
• Close-packing
• Crystal structure
References
1. Hahn, Theo, ed. (2005). International tables for crystallography (5th ed.). Dordrecht, Netherlands: Published for the International Union of Crystallography by Springer. ISBN 978-0-7923-6590-7.
2. Dana, James Dwight; Hurlbut, Cornelius Searle (1959). Dana's Manual of Mineralogy (17th ed.). New York: Chapman Hall. pp. 78–89.
3. Edward Prince (2004). Mathematical Techniques in Crystallography and Materials Science. Springer Science & Business Media. p. 41.
4. "Medium-Resolution Space Group Diagrams and Tables". img.chem.ucl.ac.uk.
5. Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics (1st ed.). p. 119. ISBN 0-03-083993-9.
6. Pough, Frederick H.; Peterson, Roger Tory (1998). A Field Guide to Rocks and Minerals. Houghton Mifflin Harcourt. p. 62. ISBN 0-395-91096-X.
7. Hurlbut, Cornelius S.; Klein, Cornelis (1985). Manual of Mineralogy (20th ed.). pp. 78–89. ISBN 0-471-80580-7.
8. "Crystallography and Minerals Arranged by Crystal Form". Webmineral.
9. "Crystallography". Webmineral.com. Retrieved 2014-08-03.
10. "Minerals in the Hexagonal crystal system, Dihexagonal Pyramidal class (6mm)". Mindat.org. Retrieved 2014-08-03.
11. Jaswon, Maurice Aaron (1965-01-01). An introduction to mathematical crystallography. American Elsevier Pub. Co.
12. De Graef, Marc; McHenry, Michael E. (2012). Structure of Materials; An introduction to Crystallography, Diffraction and Symmetry (PDF). Cambridge University Press. p. 16.
13. Hitchcock, Peter B (1988). International tables for crystallography volume A.
14. Togo, Atsushi; Chaput, Laurent; Tanaka, Isao (2015-03-20). "Distributions of phonon lifetimes in Brillouin zones". Physical Review B. 91 (9): 094306. arXiv:1501.00691. Bibcode:2015PhRvB..91i4306T. doi:10.1103/PhysRevB.91.094306. S2CID 118851924.
15. Inorganic Chemistry by Duward Shriver and Peter Atkins, 3rd Edition, W.H. Freeman and Company, 1999, pp.47,48.
16. http://www.mindat.org/min-2901.html Mindat.org
External links
• Media related to Hexagonal lattices at Wikimedia Commons
• Mineralogy database
Crystal systems
• Bravais lattice
• Crystallographic point group
Seven 3D systems
• triclinic (anorthic)
• monoclinic
• orthorhombic
• tetragonal
• trigonal & hexagonal
• cubic (isometric)
Four 2D systems
• oblique
• rectangular
• square
• hexagonal
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Rhomboid
Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. The terms rhomboid and parallelogram are often erroneously conflated with each other (i.e, when most people refer to a "parallelogram" they almost always mean a rhomboid, a specific subtype of parallelogram), however while all rhomboids are parallelograms, not all parallelograms are rhomboids.
Rhomboid
A rhomboid is a parallelogram with two edge lengths and no right angles
Typequadrilateral, trapezium
Edges and vertices4
Symmetry groupC2, [2]+,
Areab × h (base × height);
ab sin θ (product of adjacent sides and sine of the vertex angle determined by them)
Propertiesconvex
A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboid.
A parallelogram with right angled corners is a rectangle but not a rhomboid.
The term rhomboid is now more often used for a rhombohedron or a more general parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in three-dimensional rhomboids. This solid is also sometimes called a rhombic prism. The term occurs frequently in science terminology referring to both its two- and three-dimensional meaning.
History
Euclid introduced the term in his Elements in Book I, Definition 22,
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
— Translation from the page of D.E. Joyce, Dept. Math. & Comp. Sci., Clark University
Euclid never used the definition of rhomboid again and introduced the word parallelogram in Proposition 34 of Book I; "In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas." Heath suggests that rhomboid was an older term already in use.
Symmetries
The rhomboid has no line of symmetry, but it has rotational symmetry of order 2.
In biology
In biology, rhomboid may describe a geometric rhomboid (e.g. the rhomboid muscles) or a bilaterally-symmetrical kite-shaped or diamond-shaped outline, as in leaves or cephalopod fins.[1]
In medicine
In a type of arthritis called pseudogout, crystals of calcium pyrophosphate dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope.
References
1. "Decapodiform Fin Shapes".
External links
• Weisstein, Eric W. "Rhomboid". MathWorld.
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Rhonda Hughes
Rhonda Jo Hughes (born Rhonda Weisberg September 28, 1947)[1] is an American mathematician, the Helen Herrmann Professor Emeritus of Mathematics at Bryn Mawr College.[2]
Rhonda Jo Hughes
Born
Rhonda Weisberg
(1947-09-28) September 28, 1947
NationalityAmerican
Alma materUniversity of Illinois at Chicago
Known forFounding the EDGE Program
Awards
• Fellow of the Association for Women in Mathematics (2017)
• Gweneth Humphreys Award (2010)
Scientific career
FieldsMathematics, Functional analysis
InstitutionsTufts University, Bryn Mawr College
ThesisSemi-Groups of Unbounded Linear Operators in Banach Space (1975)
Doctoral advisorShmuel Kantorovitz
Education
Hughes grew up on the South Side of Chicago. She attended Gage Park High School, where she was a cheerleader and valedictorian of her class. She studied engineering at the University of Illinois at Urbana–Champaign for one and a half years, then left school and worked for six months before resuming her education at the University of Illinois at Chicago on an Illinois State Scholarship studying mathematics. There, she came under the mentorship of Yoram Sagher, who encouraged her to pursue graduate studies in mathematics.[1] She earned a Ph.D. from the same university in 1975, under the supervision of Shmuel Kantorovitz, with a dissertation entitled Semi-Groups of Unbounded Linear Operators in Banach Space.[1][3]
Career
She began her teaching career at Tufts University then spent a year as a fellow at the Bunting Institute of Radcliffe College. She moved to Bryn Mawr College in 1980,[1] where she served as department chair for six years. She is the Helen Herrmann professor emeritus of mathematics at Bryn Mawr, and retired in 2011.[4]
She was president of the Association for Women in Mathematics (AWM) 1987–1988.[1][5] She has served on the Commission on Physical Science, Mathematics, and Applications of the United States National Research Council.[1] She served as an American Mathematical Society (AMS) Council member at large from 1988 to 1990.[6]
She and Sylvia Bozeman organized the Spelman-Bryn Mawr Summer Mathematics Program for female undergraduate students from 1992 to 1994.[1] In 1998, they founded the EDGE Program (Enhancing Diversity in Graduate Education), a transition program for women entering graduate programs in the mathematical sciences. The program is now in its twentieth year.
Her most recent research involves ill-posed problems.[2]
Honors
Hughes received a Distinguished Teaching Award from the Mathematical Association of America in 1997. In 2004 she received the AAAS Mentor Award for Lifetime Achievement, in 2010 the Gweneth Humphreys Award for Mentorship of Undergraduate Women in Mathematics of the Association for Women in Mathematics,[7] and in 2013 she received the Elizabeth Bingham Award of the Philadelphia Chapter of the Association for Women in Science.[4]
In 2017, she was selected as a fellow of the Association for Women in Mathematics in the inaugural class.[8]
References
1. Morrow, Charlene; Perl, Teri, eds. (1998), "Rhonda Hughes (1947–)", Notable Women in Mathematics, a Biographical Dictionary, Greenwood Press, pp. 85–89.
2. Faculty profile, Bryn Mawr College, retrieved 2017-08-15
3. Rhonda Hughes at the Mathematics Genealogy Project
4. "Professor Emeritus Rhonda Hughes Awarded Elizabeth Bingham Award", Inside Bryn Mawr, Bryn Mawr College, March 28, 2013, retrieved 2016-02-04.
5. Blum, Lenore (1991), "A brief history of the Association for Women in Mathematics: the Presidents' perspectives", Notices of the American Mathematical Society, 38 (7): 738–754, MR 1125380, archived from the original on 7 March 2018, retrieved 4 May 2020.
6. "AMS Committees". American Mathematical Society. Retrieved 2023-03-29.
7. Rhonda Hughes to Receive First AWM Humphreys Award, Edge for Women, December 15, 2010, retrieved 2016-02-04.
8. "2018 Inaugural Class of AWM Fellows". awm-math.org/awards/awm-fellows/. Association for Women in Mathematics. Retrieved 4 May 2020.
External links
• A Tribute to the Work of Professor Emeritus Rhonda Hughes
Presidents of the Association for Women in Mathematics
1971–1990
• Mary W. Gray (1971–1973)
• Alice T. Schafer (1973–1975)
• Lenore Blum (1975–1979)
• Judith Roitman (1979–1981)
• Bhama Srinivasan (1981–1983)
• Linda Preiss Rothschild (1983–1985)
• Linda Keen (1985–1987)
• Rhonda Hughes (1987–1989)
• Jill P. Mesirov (1989–1991)
1991–2010
• Carol S. Wood (1991–1993)
• Cora Sadosky (1993–1995)
• Chuu-Lian Terng (1995–1997)
• Sylvia M. Wiegand (1997–1999)
• Jean E. Taylor (1999–2001)
• Suzanne Lenhart (2001–2003)
• Carolyn S. Gordon (2003–2005)
• Barbara Keyfitz (2005–2007)
• Cathy Kessel (2007–2009)
• Georgia Benkart (2009–2011)
2011–0000
• Jill Pipher (2011–2013)
• Ruth Charney (2013–2015)
• Kristin Lauter (2015–2017)
• Ami Radunskaya (2017–2019)
• Ruth Haas (2019–2021)
• Kathryn Leonard (2021–2023)
• Talitha Washington (2023–2025)
Authority control
International
• VIAF
National
• Poland
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Rhombicosahedron
In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. It has 50 faces (30 squares and 20 hexagons), 120 edges and 60 vertices.[1] Its vertex figure is an antiparallelogram.
Not to be confused with Rhombic icosahedron.
Rhombicosahedron
TypeUniform star polyhedron
ElementsF = 50, E = 120
V = 60 (χ = −10)
Faces by sides30{4}+20{6}
Coxeter diagram (with extra double-covered pentagrams)
(with extra double-covered pentagons)
Wythoff symbol2 3 (5/4 5/2) |
Symmetry groupIh, [5,3], *532
Index referencesU56, C72, W96
Dual polyhedronRhombicosacron
Vertex figure
4.6.4/3.6/5
Bowers acronymRi
Related polyhedra
A rhombicosahedron shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the rhombidodecadodecahedron (having the square faces in common) and the icosidodecadodecahedron (having the hexagonal faces in common).
Convex hull
Rhombidodecadodecahedron
Icosidodecadodecahedron
Rhombicosahedron
Compound of ten triangular prisms
Compound of twenty triangular prisms
Rhombicosacron
Rhombicosacron
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 50 (χ = −10)
Symmetry groupIh, [5,3], *532
Index referencesDU56
dual polyhedronRhombicosahedron
The rhombicosacron is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.
References
1. Maeder, Roman. "56: rhombicosahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Rhombicosacron". MathWorld.
• Weisstein, Eric W. "Rhombicosahedron". 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
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Ribbon Hopf algebra
A ribbon Hopf algebra $(A,\nabla ,\eta ,\Delta ,\varepsilon ,S,{\mathcal {R}},\nu )$ is a quasitriangular Hopf algebra which possess an invertible central element $\nu $ more commonly known as the ribbon element, such that the following conditions hold:
$\nu ^{2}=uS(u),\;S(\nu )=\nu ,\;\varepsilon (\nu )=1$
$\Delta (\nu )=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-1}(\nu \otimes \nu )$
where $u=\nabla (S\otimes {\text{id}})({\mathcal {R}}_{21})$. Note that the element u exists for any quasitriangular Hopf algebra, and $uS(u)$ must always be central and satisfies $S(uS(u))=uS(u),\varepsilon (uS(u))=1,\Delta (uS(u))=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-2}(uS(u)\otimes uS(u))$, so that all that is required is that it have a central square root with the above properties.
Here
$A$ is a vector space
$\nabla $ is the multiplication map $\nabla :A\otimes A\rightarrow A$
$\Delta $ is the co-product map $\Delta :A\rightarrow A\otimes A$
$\eta $ is the unit operator $\eta :\mathbb {C} \rightarrow A$ :\mathbb {C} \rightarrow A}
$\varepsilon $ is the co-unit operator $\varepsilon :A\rightarrow \mathbb {C} $
$S$ is the antipode $S:A\rightarrow A$
${\mathcal {R}}$ is a universal R matrix
We assume that the underlying field $K$ is $\mathbb {C} $
If $A$ is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if $A$ is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.
See also
• Quasitriangular Hopf algebra
• Quasi-triangular quasi-Hopf algebra
References
• Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150: 83–107. arXiv:hep-th/9202047. Bibcode:1992CMaPh.150...83A. doi:10.1007/bf02096567.
• Chari, V. C.; Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press. ISBN 0-521-55884-0.
• Drinfeld, Vladimir (1989). "Quasi-Hopf algebras". Leningrad Math J. 1: 1419–1457.
• Majid, Shahn (1995). Foundations of Quantum Group Theory. Cambridge University Press.
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Ribenboim Prize
The Ribenboim Prize, named in honour of Paulo Ribenboim, is awarded by the Canadian Number Theory Association for distinguished research in number theory by a mathematician who is Canadian or has close connections to Canadian mathematics.[1] Normally the winner will have received their Ph.D. in the last 12 years. The winner is expected to give a plenary talk at the award ceremony.[2]
Winners
YearNameUniversity
1999Andrew Granville[3] University of Georgia
2002Henri Darmon[4]McGill University
2004Michael A. Bennett[4]University of British Columbia
2006Vinayak Vatsal[4]University of British Columbia
2008Adrian Iovita[4]Concordia University
2010Valentin Blomer[5]University of Toronto, Universität Göttingen
2012Dragos Ghioca[2]University of British Columbia
2014Florian Herzig[6]University of Toronto
2016Jacob Tsimerman[7]University of Toronto
2018Maksym Radziwill[2]McGill University
See also
• List of mathematics awards
References
1. "Ribenboim Prize – Canadian Number Theory Association CNTA XIV – University of Calgary". www.ucalgary.ca.
2. "Canadian Number Theory Association CNTA XV". archimede.mat.ulaval.ca.
3. "The Valuation Theory Home Page: Ribenboim Prize for Andrew Granville". math.usask.ca.
4. "Ribenboim Prize". www-history.mcs.st-andrews.ac.uk.
5. "CANADIAN NUMBER THEORY DAY XII June 17-22, 2012 Lethbridge". www.cs.uleth.ca.
6. "Fields Institute – CNTA XIII". www.fields.utoronto.ca.
7. "CNTA XIV – Ribenboim Prize".
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Ribet's theorem
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true.
In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation).[1]
Statement
Let f be a weight 2 newform on Γ0(qN) – i.e. of level qN where q does not divide N – with absolutely irreducible 2-dimensional mod p Galois representation ρf,p unramified at q if q ≠ p and finite flat at q = p. Then there exists a weight 2 newform g of level N such that
$\rho _{f,p}\simeq \rho _{g,p}.$
In particular, if E is an elliptic curve over $\mathbb {Q} $ with conductor qN, then the modularity theorem guarantees that there exists a weight 2 newform f of level qN such that the 2-dimensional mod p Galois representation ρf, p of f is isomorphic to the 2-dimensional mod p Galois representation ρE, p of E. To apply Ribet's Theorem to ρE, p, it suffices to check the irreducibility and ramification of ρE, p. Using the theory of the Tate curve, one can prove that ρE, p is unramified at q ≠ p and finite flat at q = p if p divides the power to which q appears in the minimal discriminant ΔE. Then Ribet's theorem implies that there exists a weight 2 newform g of level N such that ρg, p ≈ ρE, p.
Level lowering
Ribet's theorem states that beginning with an elliptic curve E of conductor qN does not guarantee the existence of an elliptic curve E′ of level N such that ρE, p ≈ ρE′, p. The newform g of level N may not have rational Fourier coefficients, and hence may be associated to a higher-dimensional abelian variety, not an elliptic curve. For example, elliptic curve 4171a1 in the Cremona database given by the equation
$E:y^{2}+xy+y=x^{3}-663204x+206441595$
with conductor 43 × 97 and discriminant 437 × 973 does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod p Galois representation is isomorphic to the mod p Galois representation of an irrational newform g of level 97.
However, for p large enough compared to the level N of the level-lowered newform, a rational newform (e.g. an elliptic curve) must level-lower to another rational newform (e.g. elliptic curve). In particular for p ≫ NN1+ε, the mod p Galois representation of a rational newform cannot be isomorphic to an irrational newform of level N.[2]
Similarly, the Frey-Mazur conjecture predicts that for large enough p (independent of the conductor N), elliptic curves with isomorphic mod p Galois representations are in fact isogenous, and hence have the same conductor. Thus non-trivial level-lowering between rational newforms is not predicted to occur for large p (p > 17).
History
In his thesis, Yves Hellegouarch originated the idea of associating solutions (a,b,c) of Fermat's equation with a different mathematical object: an elliptic curve.[3] If p is an odd prime and a, b, and c are positive integers such that
$a^{p}+b^{p}=c^{p},$
then a corresponding Frey curve is an algebraic curve given by the equation
$y^{2}=x(x-a^{p})(x+b^{p}).$
This is a nonsingular algebraic curve of genus one defined over $\mathbb {Q} $, and its projective completion is an elliptic curve over $\mathbb {Q} $.
In 1982 Gerhard Frey called attention to the unusual properties of the same curve, now called a Frey curve.[4] This provided a bridge between Fermat and Taniyama by showing that a counterexample to FLT would create a curve that would not be modular. The conjecture attracted considerable interest when Frey suggested that the Taniyama–Shimura–Weil conjecture implies FLT. However, his argument was not complete.[5] In 1985 Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof.[6][7] This showed that a proof of the semistable case of the Taniyama–Shimura conjecture would imply FLT. Serre did not provide a complete proof and the missing bit became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied FLT.[8]
Implications
Suppose that the Fermat equation with exponent p ≥ 5[8] had a solution in non-zero integers a, b, c. The corresponding Frey curve Eap,bp,cp is an elliptic curve whose minimal discriminant Δ is equal to 2−8 (abc)2p and whose conductor N is the radical of abc, i.e. the product of all distinct primes dividing abc. An elementary consideration of the equation ap + bp = cp, makes it clear that one of a, b, c is even and hence so is N. By the Taniyama–Shimura conjecture, E is a modular elliptic curve. Since all odd primes dividing a, b, c in N appear to a pth power in the minimal discriminant Δ, by Ribet's theorem repetitive level descent modulo p strips all odd primes from the conductor. However, no newforms of level 2 remain because the genus of the modular curve X0(2) is zero (and newforms of level N are differentials on X0(N)).
See also
• ABC conjecture
• Wiles' proof of Fermat's Last Theorem
Notes
1. "The Proof of Fermat's Last Theorem". 2008-12-10. Archived from the original on 2008-12-10.
2. Silliman, Jesse; Vogt, Isabel (2015). "Powers in Lucas Sequences via Galois Representations". Proceedings of the American Mathematical Society. 143 (3): 1027–1041. arXiv:1307.5078. CiteSeerX 10.1.1.742.7591. doi:10.1090/S0002-9939-2014-12316-1. MR 3293720. S2CID 16892383.
3. Hellegouarch, Yves (1972). "Courbes elliptiques et equation de Fermat". Doctoral Dissertation. BNF 359121326.
4. Frey, Gerhard (1982), "Rationale Punkte auf Fermatkurven und getwisteten Modulkurven" [Rational points on Fermat curves and twisted modular curves], J. Reine Angew. Math. (in German), 1982 (331): 185–191, doi:10.1515/crll.1982.331.185, MR 0647382, S2CID 118263144
5. Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, MR 0853387
6. Serre, J.-P. (1987), "Lettre à J.-F. Mestre [Letter to J.-F. Mestre]", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemporary Mathematics (in French), vol. 67, Providence, RI: American Mathematical Society, pp. 263–268, doi:10.1090/conm/067/902597, ISBN 9780821850749, MR 0902597
7. Serre, Jean-Pierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Duke Mathematical Journal, 54 (1): 179–230, doi:10.1215/S0012-7094-87-05413-5, ISSN 0012-7094, MR 0885783
8. Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones Mathematicae. 100 (2): 431–476. Bibcode:1990InMat.100..431R. doi:10.1007/BF01231195. MR 1047143. S2CID 120614740.
References
• Kenneth Ribet, From the Taniyama-Shimura conjecture to Fermat's last theorem. Annales de la faculté des sciences de Toulouse Sér. 5, 11 no. 1 (1990), p. 116–139.
• Andrew Wiles (May 1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559.
• Richard Taylor and Andrew Wiles (May 1995). "Ring-theoretic properties of certain Hecke algebras" (PDF). Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. ISSN 0003-486X. JSTOR 2118560. OCLC 37032255. Zbl 0823.11030.
• Frey Curve and Ribet's Theorem
External links
• Ken Ribet and Fermat's Last Theorem by Kevin Buzzard June 28, 2008
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Wikipedia
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Ricardo Baeza Rodríguez
Ricardo Baeza Rodríguez is a Chilean mathematician who works as a professor at the University of Talca.[1][2] He earned his Ph.D. in 1970 from Saarland University, under the joint supervision of Robert W. Berger and Manfred Knebusch.[2][3] His research interest is in number theory.[4]
Baeza became a member of the Chilean Academy of Sciences in 1983.[1] He was the 2009 winner of the Chilean National Prize for Exact Sciences.[2][4] In 2012, he became one of the inaugural fellows of the American Mathematical Society, the only Chilean to be so honored.[2][5]
References
1. Member profile, Chilean Academy of Sciences, retrieved 2015-01-12.
2. "Miembro de Excelencia: Ricardo Baeza será distinguido por la American Mathematical Society", Sala de Prensa, University of Talca, November 16, 2012, retrieved 2015-01-12.
3. Ricardo Baeza Rodríguez at the Mathematics Genealogy Project.
4. "Ricardo Baeza gana Premio Nacional de Ciencias Exactas", Nacion.cl, August 27, 2009.
5. List of Fellows of the American Mathematical Society, retrieved 2015-01-12.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
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Wikipedia
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Ricardo Cortez (mathematician)
Ricardo Cortez is an American mathematician and currently the Pendergraft William Larkin Duren Professor at Tulane University.
Professional career
Ricardo Cortez earned a BS in mechanical engineering in 1986 and a BA in applied mathematics in 1988 from Arizona State University.[1] In 1995, he earned his applied mathematics PhD from the University of California at Berkeley.[2] He was an instructor at the Courant Institute of Mathematical Sciences, which is New York University's mathematics research school, from 1995-1998.[1] Since 1998 he has been: Assistant Professor, Mathematics Department, Tulane University (1998-2001), Associate Professor, Mathematics Department, Tulane University 2001-2007. As of 2007 he is a Professor, Mathematics Department, at Tulane University.[1]
Awards and honors
Cortez was awarded the Blackwell-Tapia Prize in 2012.[3] He was named to the 2021 class of fellows of the American Mathematical Society "for contributions in numerical methods for fluid dynamics and leadership in promoting opportunities in mathematical sciences for underrepresented groups".[4]
References
1. "Ricardo Cortez, Ph.D." School of Science & Engineering. 2019-02-16. Retrieved 2020-08-04.
2. "Ricardo Cortez | Math Alliance: The National Alliance for Doctoral Studies in the Mathematical Sciences". Retrieved 2020-08-04.
3. "Grad alum Ricardo Cortez, now at Tulane, wins the prestigious Blackwell-Tapia Prize | Berkeley Graduate Division". grad.berkeley.edu. June 2012. Retrieved 2020-09-08.
4. 2021 Class of Fellows of the AMS, American Mathematical Society, retrieved 2020-11-02
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
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Wikipedia
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Riccati equation
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form
$y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)$
where $q_{0}(x)\neq 0$ and $q_{2}(x)\neq 0$. If $q_{0}(x)=0$ the equation reduces to a Bernoulli equation, while if $q_{2}(x)=0$ the equation becomes a first order linear ordinary differential equation.
The equation is named after Jacopo Riccati (1676–1754).[1]
More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation.
Conversion to a second order linear equation
The non-linear Riccati equation can always be converted to a second order linear ordinary differential equation (ODE):[2] If
$y'=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}\!$
then, wherever $q_{2}$ is non-zero and differentiable, $v=yq_{2}$ satisfies a Riccati equation of the form
$v'=v^{2}+R(x)v+S(x),\!$
where $S=q_{2}q_{0}$ and $R=q_{1}+{\frac {q_{2}'}{q_{2}}}$, because
$v'=(yq_{2})'=y'q_{2}+yq_{2}'=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+v{\frac {q_{2}'}{q_{2}}}=q_{0}q_{2}+\left(q_{1}+{\frac {q_{2}'}{q_{2}}}\right)v+v^{2}.\!$
Substituting $v=-u'/u$, it follows that $u$ satisfies the linear 2nd order ODE
$u''-R(x)u'+S(x)u=0\!$
since
$v'=-(u'/u)'=-(u''/u)+(u'/u)^{2}=-(u''/u)+v^{2}\!$
so that
$u''/u=v^{2}-v'=-S-Rv=-S+Ru'/u\!$
and hence
$u''-Ru'+Su=0.\!$
A solution of this equation will lead to a solution $y=-u'/(q_{2}u)$ of the original Riccati equation.
Application to the Schwarzian equation
An important application of the Riccati equation is to the 3rd order Schwarzian differential equation
$S(w):=(w''/w')'-(w''/w')^{2}/2=f$
which occurs in the theory of conformal mapping and univalent functions. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative $S(w)$ has the remarkable property that it is invariant under Möbius transformations, i.e. $S((aw+b)/(cw+d))=S(w)$ whenever $ad-bc$ is non-zero.) The function $y=w''/w'$ satisfies the Riccati equation
$y'=y^{2}/2+f.$
By the above $y=-2u'/u$ where $u$ is a solution of the linear ODE
$u''+(1/2)fu=0.$
Since $w''/w'=-2u'/u$, integration gives $w'=C/u^{2}$ for some constant $C$. On the other hand any other independent solution $U$ of the linear ODE has constant non-zero Wronskian $U'u-Uu'$ which can be taken to be $C$ after scaling. Thus
$w'=(U'u-Uu')/u^{2}=(U/u)'$
so that the Schwarzian equation has solution $w=U/u.$
Obtaining solutions by quadrature
The correspondence between Riccati equations and second-order linear ODEs has other consequences. For example, if one solution of a 2nd order ODE is known, then it is known that another solution can be obtained by quadrature, i.e., a simple integration. The same holds true for the Riccati equation. In fact, if one particular solution $y_{1}$ can be found, the general solution is obtained as
$y=y_{1}+u$
Substituting
$y_{1}+u$
in the Riccati equation yields
$y_{1}'+u'=q_{0}+q_{1}\cdot (y_{1}+u)+q_{2}\cdot (y_{1}+u)^{2},$
and since
$y_{1}'=q_{0}+q_{1}\,y_{1}+q_{2}\,y_{1}^{2},$
it follows that
$u'=q_{1}\,u+2\,q_{2}\,y_{1}\,u+q_{2}\,u^{2}$
or
$u'-(q_{1}+2\,q_{2}\,y_{1})\,u=q_{2}\,u^{2},$
which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is
$z={\frac {1}{u}}$
Substituting
$y=y_{1}+{\frac {1}{z}}$
directly into the Riccati equation yields the linear equation
$z'+(q_{1}+2\,q_{2}\,y_{1})\,z=-q_{2}$
A set of solutions to the Riccati equation is then given by
$y=y_{1}+{\frac {1}{z}}$
where z is the general solution to the aforementioned linear equation.
See also
• Linear-quadratic regulator
• Algebraic Riccati equation
• Linear-quadratic-Gaussian control
References
1. Riccati, Jacopo (1724) "Animadversiones in aequationes differentiales secundi gradus" (Observations regarding differential equations of the second order), Actorum Eruditorum, quae Lipsiae publicantur, Supplementa, 8 : 66-73. Translation of the original Latin into English by Ian Bruce.
2. Ince, E. L. (1956) [1926], Ordinary Differential Equations, New York: Dover Publications, pp. 23–25
Further reading
• Hille, Einar (1997) [1976], Ordinary Differential Equations in the Complex Domain, New York: Dover Publications, ISBN 0-486-69620-0
• Nehari, Zeev (1975) [1952], Conformal Mapping, New York: Dover Publications, ISBN 0-486-61137-X
• Polyanin, Andrei D.; Zaitsev, Valentin F. (2003), Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.), Boca Raton, Fla.: Chapman & Hall/CRC, ISBN 1-58488-297-2
• Zelikin, Mikhail I. (2000), Homogeneous Spaces and the Riccati Equation in the Calculus of Variations, Berlin: Springer-Verlag
• Reid, William T. (1972), Riccati Differential Equations, London: Academic Press
External links
• "Riccati equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Riccati Equation at EqWorld: The World of Mathematical Equations.
• Riccati Differential Equation at Mathworld
• MATLAB function for solving continuous-time algebraic Riccati equation.
• SciPy has functions for solving the continuous algebraic Riccati equation and the discrete algebraic Riccati equation.
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Wikipedia
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Ricci-flat manifold
In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a (pseudo-)Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in vacuum with vanishing cosmological constant.
In Lorentzian geometry, a number of Ricci-flat metrics are known from works of Karl Schwarzschild, Roy Kerr, and Yvonne Choquet-Bruhat. In Riemannian geometry, Shing-Tung Yau's resolution of the Calabi conjecture produced a number of Ricci-flat metrics on Kähler manifolds.
Definition
A pseudo-Riemannian manifold is said to be Ricci-flat if its Ricci curvature is zero.[1] It is direct to verify that, except in dimension two, a metric is Ricci-flat if and only if its Einstein tensor is zero.[2] Ricci-flat manifolds are one of three special types of Einstein manifold, arising as the special case of scalar curvature equaling zero.
From the definition of the Weyl curvature tensor, it is direct to see that any Ricci-flat metric has Weyl curvature equal to Riemann curvature tensor. By taking traces, it is straightforward to see that the converse also holds. This may also be phrased as saying that Ricci-flatness is characterized by the vanishing of the two non-Weyl parts of the Ricci decomposition.
Since the Weyl curvature vanishes in two or three dimensions, every Ricci-flat metric in these dimensions is flat. Conversely, it is automatic from the definitions that any flat metric is Ricci-flat. The study of flat metrics is usually considered as a topic unto itself. As such, the study of Ricci-flat metrics is only a distinct topic in dimension four and above.
Examples
As noted above, any flat metric is Ricci-flat. However it is nontrivial to identify Ricci-flat manifolds whose full curvature is nonzero.
In 1916, Karl Schwarzschild found the Schwarzschild metrics, which are Ricci-flat Lorentzian manifolds of nonzero curvature.[3] Roy Kerr later found the Kerr metrics, a two-parameter family containing the Schwarzschild metrics as a special case.[4] These metrics are fully explicit and are of fundamental interest in the mathematics and physics of black holes. More generally, in general relativity, Ricci-flat Lorentzian manifolds represent the vacuum solutions of Einstein's field equations with vanishing cosmological constant.[5]
Many pseudo-Riemannian manifolds are constructed as homogeneous spaces. However, these constructions are not directly helpful for Ricci-flat Riemannian metrics, in the sense that any homogeneous Riemannian manifold which is Ricci-flat must be flat.[6] However, there are homogeneous (and even symmetric) Lorentzian manifolds which are Ricci-flat but not flat, as follows from an explicit construction and computation of Lie algebras.[7]
Until Shing-Tung Yau's resolution of the Calabi conjecture in the 1970s, it was not known whether every Ricci-flat Riemannian metric on a closed manifold is flat.[8] His work, using techniques of partial differential equations, established a comprehensive existence theory for Ricci-flat metrics in the special case of Kähler metrics on closed complex manifolds. Due to his analytical techniques, the metrics are non-explicit even in the simplest cases. Such Riemannian manifolds are often called Calabi–Yau manifolds, although various authors use this name in slightly different ways.[9]
Analytical character
Relative to harmonic coordinates, the condition of Ricci-flatness for a Riemannian metric can be interpreted as a system of elliptic partial differential equations. It is a straightforward consequence of standard elliptic regularity results that any Ricci-flat Riemannian metric on a smooth manifold is analytic, in the sense that harmonic coordinates define a compatible analytic structure, and the local representation of the metric is real-analytic. This also holds in the broader setting of Einstein Riemannian metrics.[10]
Analogously, relative to harmonic coordinates, Ricci-flatness of a Lorentzian metric can be interpreted as a system of hyperbolic partial differential equations. Based on this perspective, Yvonne Choquet-Bruhat developed the well-posedness of the Ricci-flatness condition. She reached a definitive result in collaboration with Robert Geroch in the 1960s, establishing how a certain class of maximally extended Ricci-flat Lorentzian metrics are prescribed and constructed by certain Riemannian data. These are known as maximal globally hyperbolic developments. In general relativity, this is typically interpreted as an initial value formulation of Einstein's field equations for gravitation.[11]
The study of Ricci-flatness in the Riemannian and Lorentzian cases are quite distinct. This is already indicated by the fundamental distinction between the geodesically complete metrics which are typical of Riemannian geometry and the maximal globally hyperbolic developments which arise from Choquet-Bruhat and Geroch's work. Moreover, the analyticity and corresponding unique continuation of a Ricci-flat Riemannian metric has a fundamentally different character than Ricci-flat Lorentzian metrics, which have finite speeds of propagation and fully localizable phenomena. This can be viewed as a nonlinear geometric analogue of the difference between the Laplace equation and the wave equation.
Topology of Ricci-flat Riemannian manifolds
Yau's existence theorem for Ricci-flat Kähler metrics established the precise topological condition under which such a metric exists on a given closed complex manifold: the first Chern class of the holomorphic tangent bundle must be zero. The necessity of this condition was previously known by Chern–Weil theory.
Beyond Kähler geometry, the situation is not as well understood. A four-dimensional closed and oriented manifold supporting any Einstein Riemannian metric must satisfy the Hitchin–Thorpe inequality on its topological data. As particular cases of well-known theorems on Riemannian manifolds of nonnegative Ricci curvature, any manifold with a complete Ricci-flat Riemannian metric must:[12]
• have first Betti number less than or equal to the dimension, whenever the manifold is closed
• have fundamental group of polynomial growth.
Mikhael Gromov and Blaine Lawson introduced the notion of enlargeability of a closed manifold. The class of enlargeable manifolds is closed under homotopy equivalence, the taking of products, and under the connected sum with an arbitrary closed manifold. Every Ricci-flat Riemannian manifold in this class is flat, which is a corollary of Cheeger and Gromoll's splitting theorem.[13]
Ricci-flatness and holonomy
On a simply-connected Kähler manifold, a Kähler metric is Ricci-flat if and only if the holonomy group is contained in the special unitary group. On a general Kähler manifold, the if direction still holds, but only the restricted holonomy group of a Ricci-flat Kähler metric is necessarily contained in the special unitary group.[14]
A hyperkähler manifold is a Riemannian manifold whose holonomy group is contained in the symplectic group. This condition on a Riemannian manifold may also be characterized (roughly speaking) by the existence of a 2-sphere of complex structures which are all parallel. This says in particular that every hyperkähler metric is Kähler; furthermore, via the Ambrose–Singer theorem, every such metric is Ricci-flat. The Calabi–Yau theorem specializes to this context, giving a general existence and uniqueness theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact spaces had earlier been obtained by Eugenio Calabi. The Eguchi–Hanson space, discovered at the same time, is a special case of his construction.[15]
A quaternion-Kähler manifold is a Riemannian manifold whose holonomy group is contained in the Lie group Sp(n)·Sp(1). Marcel Berger showed that any such metric must be Einstein. Furthermore, any Ricci-flat quaternion-Kähler manifold must be locally hyperkähler, meaning that the restricted holonomy group is contained in the symplectic group.[16]
A G2 manifold or Spin(7) manifold is a Riemannian manifold whose holonomy group is contained in the Lie groups Spin(7) or G2. The Ambrose–Singer theorem implies that any such manifold is Ricci-flat.[17] The existence of closed manifolds of this type was established by Dominic Joyce in the 1990s.[18]
Marcel Berger commented that all known examples of irreducible Ricci-flat Riemannian metrics on simply-connected closed manifolds have special holonomy groups, according to the above possibilities. It is not known whether this suggests an unknown general theorem or simply a limitation of known techniques. For this reason, Berger considered Ricci-flat manifolds to be "extremely mysterious."[19]
References
Notes.
1. O'Neill 1983, p. 87.
2. O'Neill 1983, p. 336.
3. Besse 1987, Section 3F; Misner, Thorne & Wheeler 1973, Chapter 31; O'Neill 1983, Chapter 13; Schwarzschild 1916.
4. Kerr 1963; Misner, Thorne & Wheeler 1973, Chapter 33.
5. Besse 1987, Section 3C.
6. Besse 1987, Theorem 7.61.
7. Besse 1987, Theorem 7.118.
8. Besse 1987, Paragraph 0.30.
9. Besse 1987, Sections 11B–C; Yau 1978.
10. Besse 1987, Section 5F.
11. Hawking & Ellis 1973, Sections 7.5–7.6.
12. Besse 1987, Sections 6D–E.
13. Lawson & Michelsohn 1989, Section IV.5.
14. Besse 1987, Proposition 10.29.
15. Besse 1987, Sections 14A–C.
16. Besse 1987, Section 14D.
17. Besse 1987, Section 10F.
18. Berger 2003, Section 13.5.1; Joyce 2000.
19. Berger 2003, Section 11.4.6.
Sources.
• Berger, Marcel (2003). A panoramic view of Riemannian geometry. Berlin: Springer-Verlag. doi:10.1007/978-3-642-18245-7. ISBN 3-540-65317-1. MR 2002701. Zbl 1038.53002.
• Besse, Arthur L. (1987). Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 10. Reprinted in 2008. Berlin: Springer-Verlag. doi:10.1007/978-3-540-74311-8. ISBN 3-540-15279-2. MR 0867684. Zbl 0613.53001.
• Einstein, A. (1916). Translated by Perrett, W.; Jeffery, G. B. "Die Grundlage der allgemeinen Relativitätstheorie" [The foundation of the general theory of relativity]. Annalen der Physik. 354 (7): 769–822. JFM 46.1293.01.
• Hawking, S. W.; Ellis, G. F. R. (1973). The large scale structure of space-time. Cambridge Monographs on Mathematical Physics. Vol. 1. London−New York: Cambridge University Press. doi:10.1017/CBO9780511524646. ISBN 0-521-20016-4. MR 0424186. Zbl 0265.53054.
• Joyce, Dominic D. (2000). Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford: Oxford University Press. ISBN 0-19-850601-5. MR 1787733. Zbl 1027.53052.
• Kerr, Roy P. (1963). "Gravitational field of a spinning mass as an example of algebraically special metrics". Physical Review Letters. 11 (5): 237–238. doi:10.1103/PhysRevLett.11.237. MR 0156674. Zbl 0112.21904.
• Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise (1989). Spin geometry. Princeton Mathematical Series. Vol. 38. Princeton, NJ: Princeton University Press. ISBN 0-691-08542-0. MR 1031992. Zbl 0688.57001.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco, CA: W. H. Freeman and Company. ISBN 0-7503-0948-2. MR 0418833. Zbl 1375.83002.
• O'Neill, Barrett (1983). Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. Vol. 103. New York: Academic Press, Inc. doi:10.1016/s0079-8169(08)x6002-7. ISBN 0-12-526740-1. MR 0719023. Zbl 0531.53051.
• Schwarzschild, K. (1916). "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, Physikalisch-Mathematische Klasse: 189–196. JFM 46.1296.02.
Schwarzschild, K. (2003). Translated by Antoci, S.; Loinger, A. "On the gravitational field of a mass point according to Einstein's theory". General Relativity and Gravitation. 35 (5): 951–959. doi:10.1023/A:1022971926521. MR 1982197. Zbl 1020.83005.
• Yau, Shing Tung (1978). "On the Ricci curvature of a compact Kähler manifold and the complex Monge−Ampère equation. I". Communications on Pure and Applied Mathematics. 31 (3): 339–411. doi:10.1002/cpa.3160310304. MR 0480350. Zbl 0369.53059.
String theory
Background
• Strings
• Cosmic strings
• History of string theory
• First superstring revolution
• Second superstring revolution
• String theory landscape
Theory
• Nambu–Goto action
• Polyakov action
• Bosonic string theory
• Superstring theory
• Type I string
• Type II string
• Type IIA string
• Type IIB string
• Heterotic string
• N=2 superstring
• F-theory
• String field theory
• Matrix string theory
• Non-critical string theory
• Non-linear sigma model
• Tachyon condensation
• RNS formalism
• GS formalism
String duality
• T-duality
• S-duality
• U-duality
• Montonen–Olive duality
Particles and fields
• Graviton
• Dilaton
• Tachyon
• Ramond–Ramond field
• Kalb–Ramond field
• Magnetic monopole
• Dual graviton
• Dual photon
Branes
• D-brane
• NS5-brane
• M2-brane
• M5-brane
• S-brane
• Black brane
• Black holes
• Black string
• Brane cosmology
• Quiver diagram
• Hanany–Witten transition
Conformal field theory
• Virasoro algebra
• Mirror symmetry
• Conformal anomaly
• Conformal algebra
• Superconformal algebra
• Vertex operator algebra
• Loop algebra
• Kac–Moody algebra
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Wikipedia
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Ricci decomposition
In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry.
Definition of the decomposition
Let (M,g) be a Riemannian or pseudo-Riemannian n-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention
$R_{ijkl}=g_{lp}{\Big (}\partial _{i}\Gamma _{jk}^{p}-\partial _{j}\Gamma _{ik}^{p}+\Gamma _{iq}^{p}\Gamma _{jk}^{q}-\Gamma _{jq}^{p}\Gamma _{ik}^{q}{\Big )};$
written multilinearly, this is the convention
$\operatorname {Rm} (W,X,Y,Z)=g{\Big (}\nabla _{W}\nabla _{X}Y-\nabla _{X}\nabla _{W}Y-\nabla _{[W,X]}Y,Z{\Big )}.$
With this convention, the Ricci tensor is a (0,2)-tensor field defined by Rjk=gilRijkl and the scalar curvature is defined by R=gjkRjk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci tensor with the opposite sign. Under that more common convention, the signs of the Ricci tensor and scalar must be changed in the equations below.) Define the traceless Ricci tensor
$Z_{jk}=R_{jk}-{\frac {1}{n}}Rg_{jk},$
and then define three (0,4)-tensor fields S, E, and W by
${\begin{aligned}S_{ijkl}&={\frac {R}{n(n-1)}}{\big (}g_{il}g_{jk}-g_{ik}g_{jl}{\big )}\\E_{ijkl}&={\frac {1}{n-2}}{\big (}Z_{il}g_{jk}-Z_{jl}g_{ik}-Z_{ik}g_{jl}+Z_{jk}g_{il}{\big )}\\W_{ijkl}&=R_{ijkl}-S_{ijkl}-E_{ijkl}.\end{aligned}}$
The "Ricci decomposition" is the statement
$R_{ijkl}=S_{ijkl}+E_{ijkl}+W_{ijkl}.$
As stated, this is vacuous since it is just a reorganization of the definition of W. The importance of the decomposition is in the properties of the three new tensors S, E, and W.
Terminological note. The tensor W is called the Weyl tensor. The notation W is standard in mathematics literature, while C is more common in physics literature. The notation R is standard in both, while there is no standardized notation for S, Z, and E.
Basic properties
Properties of the pieces
Each of the tensors S, E, and W has the same algebraic symmetries as the Riemann tensor. That is:
${\begin{aligned}S_{ijkl}&=-S_{jikl}=-S_{ijlk}=S_{klij}\\E_{ijkl}&=-E_{jikl}=-E_{ijlk}=E_{klij}\\W_{ijkl}&=-W_{jikl}=-W_{ijlk}=W_{klij}\end{aligned}}$
together with
${\begin{aligned}S_{ijkl}+S_{jkil}+S_{kijl}&=0\\E_{ijkl}+E_{jkil}+E_{kijl}&=0\\W_{ijkl}+W_{jkil}+W_{kijl}&=0.\end{aligned}}$
The Weyl tensor has the additional symmetry that it is completely traceless:
$g^{il}W_{ijkl}=0.$
Hermann Weyl showed that W has the remarkable property of measuring the deviation of a Riemannian or pseudo-Riemannian manifold from local conformal flatness; if it is zero, then M can be covered by charts relative to which g has the form gij=efδij for some function f defined chart by chart.
Properties of the decomposition
One may check that the Ricci decomposition is orthogonal in the sense that
$S_{ijkl}E^{ijkl}=S_{ijkl}W^{ijkl}=E_{ijkl}W^{ijkl}=0,$
recalling the general definition $T^{ijkl}=g^{ip}g^{jq}g^{kr}g^{ls}T_{pqrs}.$ This has the consequence, which could be proved directly, that
$R_{ijkl}R^{ijkl}=S_{ijkl}S^{ijkl}+E_{ijkl}E^{ijkl}+W_{ijkl}W^{ijkl}.$
Terminological note. It would be symbolically clean to present this orthogonality as saying
$\langle S,E\rangle _{g}=\langle S,W\rangle _{g}=\langle E,W\rangle _{g}=0,$
together with
$|\operatorname {Rm} |_{g}^{2}=|S|_{g}^{2}+|E|_{g}^{2}+|W|_{g}^{2}.$
However, there is an unavoidable ambiguity with such notation depending on whether one views $\operatorname {Rm} ,S,E,W$ as multilinear maps $T_{p}M\times T_{p}M\times T_{p}M\times T_{p}M\to \mathbb {R} $ or as linear maps $\wedge ^{2}T_{p}M\to \wedge ^{2}T_{p}M,$ in which case the corresponding norms and inner products would differ by a constant factor. Although this would not lead to any inconsistencies in the above equations, since all terms would be changed by the same factor, it can lead to confusion in more involved contexts. For this reason, the index notation can often be easier to understand.
Related formulas
One can compute the "norm formulas"
${\begin{aligned}S_{ijkl}S^{ijkl}&={\frac {2R^{2}}{n(n-1)}}\\E_{ijkl}E^{ijkl}&={\frac {4R_{ij}R^{ij}}{n-2}}-{\frac {4R^{2}}{n(n-2)}}\\W_{ijkl}W^{ijkl}&=R_{ijkl}R^{ijkl}-{\frac {4R_{ij}R^{ij}}{n-2}}+{\frac {2R^{2}}{(n-1)(n-2)}}\end{aligned}}$
and the "trace formulas"
${\begin{aligned}g^{il}S_{ijkl}&={\frac {1}{n}}Rg_{jk}\\g^{il}E_{ijkl}&=R_{jk}-{\frac {1}{n}}Rg_{jk}\\g^{il}W_{ijkl}&=0.\end{aligned}}$
Mathematical explanation of the decomposition
Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group (Besse 1987, Chapter 1, §G). Let V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature). Here V is modeled on the cotangent space at a point, so that a curvature tensor R (with all indices lowered) is an element of the tensor product V⊗V⊗V⊗V. The curvature tensor is skew symmetric in its first and last two entries:
$R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)\,$
and obeys the interchange symmetry
$R(x,y,z,w)=R(z,w,x,y),\,$
for all x,y,z,w ∈ V∗. As a result, R is an element of the subspace $S^{2}\Lambda ^{2}V$, the second symmetric power of the second exterior power of V. A curvature tensor must also satisfy the Bianchi identity, meaning that it is in the kernel of the linear map $b:S^{2}\Lambda ^{2}V\to \Lambda ^{4}V$ given by
$b(R)(x,y,z,w)=R(x,y,z,w)+R(y,z,x,w)+R(z,x,y,w).\,$
The space RV = ker b in S2Λ2V is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping
$c:S^{2}\Lambda ^{2}V\to S^{2}V$
is given by
$c(R)(x,y)=\operatorname {tr} R(x,\cdot ,y,\cdot ).$
This associates a symmetric 2-form to an algebraic curvature tensor. Conversely, given a pair of symmetric 2-forms h and k, the Kulkarni–Nomizu product of h and k
$(h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k)(x,y,z,w)=h(x,z)k(y,w)+h(y,w)k(x,z)-h(x,w)k(y,z)-h(y,z)k(x,w)$
produces an algebraic curvature tensor.
If n ≥ 4, then there is an orthogonal decomposition into (unique) irreducible subspaces
RV = SV ⊕ EV ⊕ CV
where
$\mathbf {S} V=\mathbb {R} g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}g$, where $\mathbb {R} $ is the space of real scalars
$\mathbf {E} V=g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}S_{0}^{2}V$, where S2
0
V is the space of trace-free symmetric 2-forms
$\mathbf {C} V=\ker c\cap \ker b.$
The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors. In particular,
$R=S+E+C$
is an orthogonal decomposition in the sense that
$|R|^{2}=|S|^{2}+|E|^{2}+|C|^{2}.$
This decomposition expresses the space of tensors with Riemann symmetries as a direct sum of the scalar submodule, the Ricci submodule, and Weyl submodule, respectively. Each of these modules is an irreducible representation for the orthogonal group (Singer & Thorpe 1969), and thus the Ricci decomposition is a special case of the splitting of a module for a semisimple Lie group into its irreducible factors. In dimension 4, the Weyl module decomposes further into a pair of irreducible factors for the special orthogonal group: the self-dual and antiself-dual parts W+ and W−.
Physical interpretation
The Ricci decomposition can be interpreted physically in Einstein's theory of general relativity, where it is sometimes called the Géhéniau-Debever decomposition. In this theory, the Einstein field equation
$G_{ab}=8\pi \,T_{ab}$
where $T_{ab}$ is the stress–energy tensor describing the amount and motion of all matter and all nongravitational field energy and momentum, states that the Ricci tensor—or equivalently, the Einstein tensor—represents that part of the gravitational field which is due to the immediate presence of nongravitational energy and momentum. The Weyl tensor represents the part of the gravitational field which can propagate as a gravitational wave through a region containing no matter or nongravitational fields. Regions of spacetime in which the Weyl tensor vanishes contain no gravitational radiation and are also conformally flat.
See also
• Bel decomposition of the Riemann tensor
• Conformal geometry
• Petrov classification
• Plebanski tensor
• Ricci calculus
• Schouten tensor
• Trace-free Ricci tensor
References
• Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8.
• Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9. Section 6.1 discusses the decomposition. Versions of the decomposition also enter into the discussion of conformal and projective geometries, in chapters 7 and 8.
• Singer, I.M.; Thorpe, J.A. (1969), "The curvature of 4-dimensional Einstein spaces", Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, pp. 355–365.
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Wikipedia
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Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard Hamilton, who used it through the 1980s to prove striking new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen.
Following Shing-Tung Yau's suggestion that the singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture, Hamilton produced a number of results in the 1990s which were directed towards the conjecture's resolution. In 2002 and 2003, Grigori Perelman presented a number of fundamental new results about the Ricci flow, including a novel variant of some technical aspects of Hamilton's program. Hamilton and Perelman's works are now widely regarded as forming a proof of the Thurston conjecture, including as a special case the Poincaré conjecture, which had been a well-known open problem in the field of geometric topology since 1904. Their results are considered as a milestone in the fields of geometry and topology.
Mathematical definition
On a smooth manifold M, a smooth Riemannian metric g automatically determines the Ricci tensor Ricg. For each element p of M, by definition gp is a positive-definite inner product on the tangent space TpM at p. If given a one-parameter family of Riemannian metrics gt, one may then consider the derivative ∂/∂t gt, which then assigns to each particular value of t and p a symmetric bilinear form on TpM. Since the Ricci tensor of a Riemannian metric also assigns to each p a symmetric bilinear form on TpM, the following definition is meaningful.
• Given a smooth manifold M and an open real interval (a, b), a Ricci flow assigns, to each t in the interval (a,b), a Riemannian metric gt on M such that ∂/∂t gt = −2 Ricgt.
The Ricci tensor is often thought of as an average value of the sectional curvatures, or as an algebraic trace of the Riemann curvature tensor. However, for the analysis of existence and uniqueness of Ricci flows, it is extremely significant that the Ricci tensor can be defined, in local coordinates, by a formula involving the first and second derivatives of the metric tensor. This makes the Ricci flow into a geometrically-defined partial differential equation. The analysis of the ellipticity of the local coordinate formula provides the foundation for the existence of Ricci flows; see the following section for the corresponding result.
Let k be a nonzero number. Given a Ricci flow gt on an interval (a,b), consider Gt = gkt for t between a/k and b/k. Then ∂/∂t Gt = −2k RicGt. So, with this very trivial change of parameters, the number −2 appearing in the definition of the Ricci flow could be replaced by any other nonzero number. For this reason, the use of −2 can be regarded as an arbitrary convention, albeit one which essentially every paper and exposition on Ricci flow follows. The only significant difference is that if −2 were replaced by a positive number, then the existence theorem discussed in the following section would become a theorem which produces a Ricci flow that moves backwards (rather than forwards) in parameter values from initial data.
The parameter t is usually called time, although this is only as part of standard informal terminology in the mathematical field of partial differential equations. It is not physically meaningful terminology. In fact, in the standard quantum field theoretic interpretation of the Ricci flow in terms of the renormalization group, the parameter t corresponds to length or energy, rather than time.[1]
Normalized Ricci flow
Suppose that M is a compact smooth manifold, and let gt be a Ricci flow for t in the interval (a, b). Define Ψ:(a, b) → (0, ∞) so that each of the Riemannian metrics Ψ(t)gt has volume 1; this is possible since M is compact. (More generally, it would be possible if each Riemannian metric gt had finite volume.) Then define F:(a, b) → (0, ∞) to be the antiderivative of Ψ which vanishes at a. Since Ψ is positive-valued, F is a bijection onto its image (0, S). Now the Riemannian metrics Gs = Ψ(F −1(s))gF −1(s), defined for parameters s ∈ (0, S), satisfy
${\frac {\partial }{\partial s}}G_{s}=-2\operatorname {Ric} ^{G_{s}}+{\frac {2}{n}}{\frac {\int _{M}R^{G_{s}}\,d\mu _{G_{s}}}{\int _{M}d\mu _{G_{s}}}}G_{s}.$
Here R denotes scalar curvature. This is called the normalized Ricci flow equation. Thus, with an explicitly defined change of scale Ψ and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The converse also holds, by reversing the above calculations.
The primary reason for considering the normalized Ricci flow is that it allows a convenient statement of the major convergence theorems for Ricci flow. However, it is not essential to do so, and for virtually all purposes it suffices to consider Ricci flow in its standard form. Moreover, the normalized Ricci flow is not generally meaningful on noncompact manifolds.
Existence and uniqueness
Let $M$ be a smooth closed manifold, and let $g_{0}$ be any smooth Riemannian metric on $M$. Making use of the Nash–Moser implicit function theorem, Hamilton (1982) showed the following existence theorem:
• There exists a positive number $T$ and a Ricci flow $g_{t}$ parametrized by $t\in (0,T)$ such that $g_{t}$ converges to $g_{0}$ in the $C^{\infty }$ topology as $t$ decreases to 0.
He showed the following uniqueness theorem:
• If $\{g_{t}:t\in (0,T)\}$ and $\{{\widetilde {g}}_{t}:t\in (0,{\widetilde {T}})\}$ are two Ricci flows as in the above existence theorem, then $g_{t}={\widetilde {g}}_{t}$ for all $t\in (0,\min\{T,{\widetilde {T}}\}).$
The existence theorem provides a one-parameter family of smooth Riemannian metrics. In fact, any such one-parameter family also depends smoothly on the parameter. Precisely, this says that relative to any smooth coordinate chart $(U,\phi )$ on $M$, the function $g_{ij}:U\times (0,T)\to \mathbb {R} $ is smooth for any $i,j=1,\dots ,n$.
Dennis DeTurck subsequently gave a proof of the above results which uses the Banach implicit function theorem instead.[2] His work is essentially a simpler Riemannian version of Yvonne Choquet-Bruhat's well-known proof and interpretation of well-posedness for the Einstein equations in Lorentzian geometry.
As a consequence of Hamilton's existence and uniqueness theorem, when given the data $(M,g_{0})$, one may speak unambiguously of the Ricci flow on $M$ with initial data $g_{0}$, and one may select $T$ to take on its maximal possible value, which could be infinite. The principle behind virtually all major applications of Ricci flow, in particular in the proof of the Poincaré conjecture and geometrization conjecture, is that, as $t$ approaches this maximal value, the behavior of the metrics $g_{t}$ can reveal and reflect deep information about $M$.
Convergence theorems
Complete expositions of the following convergence theorems are given in Andrews & Hopper (2011) and Brendle (2010).
Let (M, g0) be a smooth closed Riemannian manifold. Under any of the following three conditions:
• M is two-dimensional
• M is three-dimensional and g0 has positive Ricci curvature
• M has dimension greater than three and the product metric on (M, g0) × ℝ has positive isotropic curvature
the normalized Ricci flow with initial data g0 exists for all positive time and converges smoothly, as t goes to infinity, to a metric of constant curvature.
The three-dimensional result is due to Hamilton (1982). Hamilton's proof, inspired by and loosely modeled upon James Eells and Joseph Sampson's epochal 1964 paper on convergence of the harmonic map heat flow,[3] included many novel features, such as an extension of the maximum principle to the setting of symmetric 2-tensors. His paper (together with that of Eells−Sampson) is among the most widely cited in the field of differential geometry. There is an exposition of his result in Chow, Lu & Ni (2006, Chapter 3).
In terms of the proof, the two-dimensional case is properly viewed as a collection of three different results, one for each of the cases in which the Euler characteristic of M is positive, zero, or negative. As demonstrated by Hamilton (1988), the negative case is handled by the maximum principle, while the zero case is handled by integral estimates; the positive case is more subtle, and Hamilton dealt with the subcase in which g0 has positive curvature by combining a straightforward adaptation of Peter Li and Shing-Tung Yau's gradient estimate to the Ricci flow together with an innovative "entropy estimate". The full positive case was demonstrated by Bennett Chow (1991), in an extension of Hamilton's techniques. Since any Ricci flow on a two-dimensional manifold is confined to a single conformal class, it can be recast as a partial differential equation for a scalar function on the fixed Riemannian manifold (M, g0). As such, the Ricci flow in this setting can also be studied by purely analytic methods; correspondingly, there are alternative non-geometric proofs of the two-dimensional convergence theorem.
The higher-dimensional case has a longer history. Soon after Hamilton's breakthrough result, Gerhard Huisken extended his methods to higher dimensions, showing that if g0 almost has constant positive curvature (in the sense of smallness of certain components of the Ricci decomposition), then the normalized Ricci flow converges smoothly to constant curvature. Hamilton (1986) found a novel formulation of the maximum principle in terms of trapping by convex sets, which led to a general criterion relating convergence of the Ricci flow of positively curved metrics to the existence of "pinching sets" for a certain multidimensional ordinary differential equation. As a consequence, he was able to settle the case in which M is four-dimensional and g0 has positive curvature operator. Twenty years later, Christoph Böhm and Burkhard Wilking found a new algebraic method of constructing "pinching sets", thereby removing the assumption of four-dimensionality from Hamilton's result (Böhm & Wilking 2008). Simon Brendle and Richard Schoen showed that positivity of the isotropic curvature is preserved by the Ricci flow on a closed manifold; by applying Böhm and Wilking's method, they were able to derive a new Ricci flow convergence theorem (Brendle & Schoen 2009). Their convergence theorem included as a special case the resolution of the differentiable sphere theorem, which at the time had been a long-standing conjecture. The convergence theorem given above is due to Brendle (2008), which subsumes the earlier higher-dimensional convergence results of Huisken, Hamilton, Böhm & Wilking, and Brendle & Schoen.
Corollaries
The results in dimensions three and higher show that any smooth closed manifold M which admits a metric g0 of the given type must be a space form of positive curvature. Since these space forms are largely understood by work of Élie Cartan and others, one may draw corollaries such as
• Suppose that M is a smooth closed 3-dimensional manifold which admits a smooth Riemannian metric of positive Ricci curvature. If M is simply-connected then it must be diffeomorphic to the 3-sphere.
So if one could show directly that any smooth closed simply-connected 3-dimensional manifold admits a smooth Riemannian metric of positive Ricci curvature, then the Poincaré conjecture would immediately follow. However, as matters are understood at present, this result is only known as a (trivial) corollary of the Poincaré conjecture, rather than vice versa.
Possible extensions
Given any n larger than two, there exist many closed n-dimensional smooth manifolds which do not have any smooth Riemannian metrics of constant curvature. So one cannot hope to be able to simply drop the curvature conditions from the above convergence theorems. It could be possible to replace the curvature conditions by some alternatives, but the existence of compact manifolds such as complex projective space, which has a metric of nonnegative curvature operator (the Fubini-Study metric) but no metric of constant curvature, makes it unclear how much these conditions could be pushed. Likewise, the possibility of formulating analogous convergence results for negatively curved Riemannian metrics is complicated by the existence of closed Riemannian manifolds whose curvature is arbitrarily close to constant and yet admit no metrics of constant curvature.[4]
Li–Yau inequalities
Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, Hamilton (1993a) proved the following "Li–Yau inequality".[5]
• Let $M$ be a smooth manifold, and let $g_{t}$ be a solution of the Ricci flow with $t\in (0,T)$ such that each $g_{t}$ is complete with bounded curvature. Furthermore, suppose that each $g_{t}$ has nonnegative curvature operator. Then, for any curve $\gamma :[t_{1},t_{2}]\to M$ :[t_{1},t_{2}]\to M} with $[t_{1},t_{2}]\subset (0,T)$, one has
${\frac {d}{dt}}{\big (}R^{g(t)}(\gamma (t)){\big )}+{\frac {R^{g(t)}(\gamma (t))}{t}}+{\frac {1}{2}}\operatorname {Ric} ^{g(t)}(\gamma '(t),\gamma '(t))\geq 0.$
Perelman (2002) showed the following alternative Li–Yau inequality.
• Let $M$ be a smooth closed $n$-manifold, and let $g_{t}$ be a solution of the Ricci flow. Consider the backwards heat equation for $n$-forms, i.e. ${\tfrac {\partial }{\partial t}}\omega +\Delta ^{g(t)}\omega =0$; given $p\in M$ and $t_{0}\in (0,T)$, consider the particular solution which, upon integration, converges weakly to the Dirac delta measure as $t$ increases to $t_{0}$. Then, for any curve $\gamma :[t_{1},t_{2}]\to M$ :[t_{1},t_{2}]\to M} with $[t_{1},t_{2}]\subset (0,T)$, one has
${\frac {d}{dt}}{\big (}f(\gamma (t),t){\big )}+{\frac {f{\big (}\gamma (t),t{\big )}}{2(t_{0}-t)}}\leq {\frac {R^{g(t)}(\gamma (t))+|\gamma '(t)|_{g(t)}^{2}}{2}}.$
where $\omega =(4\pi (t_{0}-t))^{-n/2}e^{-f}{\text{d}}\mu _{g(t)}$.
Both of these remarkable inequalities are of profound importance for the proof of the Poincaré conjecture and geometrization conjecture. The terms on the right hand side of Perelman's Li–Yau inequality motivates the definition of his "reduced length" functional, the analysis of which leads to his "noncollapsing theorem". The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton 1995) to construct "singularity models", which are Ricci flows on new three-dimensional manifolds. Owing to the Hamilton–Ivey estimate, these new Ricci flows have nonnegative curvature. Hamilton's Li–Yau inequality can then be applied to see that the scalar curvature is, at each point, a nondecreasing (nonnegative) function of time. This is a powerful result that allows many further arguments to go through. In the end, Perelman shows that any of his singularity models is asymptotically like a complete gradient shrinking Ricci soliton, which are completely classified; see the previous section.
See Chow, Lu & Ni (2006, Chapters 10 and 11) for details on Hamilton's Li–Yau inequality; the books Chow et al. (2008) and Müller (2006) contain expositions of both inequalities above.
Examples
Constant-curvature and Einstein metrics
Let $(M,g)$ be a Riemannian manifold which is Einstein, meaning that there is a number $\lambda $ such that ${\text{Ric}}^{g}=\lambda g$. Then $g_{t}=(1-2\lambda t)g$ is a Ricci flow with $g_{0}=g$, since then
${\frac {\partial }{\partial t}}g_{t}=-2\lambda g=-2\operatorname {Ric} ^{g}=-2\operatorname {Ric} ^{g_{t}}.$
If $M$ is closed, then according to Hamilton's uniqueness theorem above, this is the only Ricci flow with initial data $g$. One sees, in particular, that:
• if $\lambda $ is positive, then the Ricci flow "contracts" $g$ since the scale factor $1-2\lambda t$ is less than 1 for positive $t$; furthermore, one sees that $t$ can only be less than $1/2\lambda $, in order that $g_{t}$ is a Riemannian metric. This is the simplest examples of a "finite-time singularity".
• if $\lambda $ is zero, which is synonymous with $g$ being Ricci-flat, then $g_{t}$ is independent of time, and so the maximal interval of existence is the entire real line.
• if $\lambda $ is negative, then the Ricci flow "expands" $g$ since the scale factor $1-2\lambda t$ is greater than 1 for all positive $t$; furthermore one sees that $t$ can be taken arbitrarily large. One says that the Ricci flow, for this initial metric, is "immortal".
In each case, since the Riemannian metrics assigned to different values of $t$ differ only by a constant scale factor, one can see that the normalized Ricci flow $G_{s}$ exists for all time and is constant in $s$; in particular, it converges smoothly (to its constant value) as $s\to \infty $.
The Einstein condition has as a special case that of constant curvature; hence the particular examples of the sphere (with its standard metric) and hyperbolic space appear as special cases of the above.
Ricci solitons
Ricci solitons are Ricci flows that may change their size but not their shape up to diffeomorphisms.
• Cylinders Sk × Rl (for k ≥ 2) shrink self similarly under the Ricci flow up to diffeomorphisms
• A significant 2-dimensional example is the cigar soliton, which is given by the metric (dx2 + dy2)/(e4t + x2 + y2) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons.
• An example of a 3-dimensional steady Ricci soliton is the Bryant soliton, which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations. A similar construction works in arbitrary dimension.
• There exist numerous families of Kähler manifolds, invariant under a U(n) action and birational to Cn, which are Ricci solitons. These examples were constructed by Cao and Feldman-Ilmanen-Knopf. (Chow-Knopf 2004)
• A 4-dimensional example exhibiting only torus symmetry was recently discovered by Bamler-Cifarelli-Conlon-Deruelle.
A gradient shrinking Ricci soliton consists of a smooth Riemannian manifold (M,g) and f ∈ C∞(M) such that
$\operatorname {Ric} ^{g}+\operatorname {Hess} ^{g}f={\frac {1}{2}}g.$
One of the major achievements of Perelman (2002) was to show that, if M is a closed three-dimensional smooth manifold, then finite-time singularities of the Ricci flow on M are modeled on complete gradient shrinking Ricci solitons (possibly on underlying manifolds distinct from M). In 2008, Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu completed the classification of these solitons, showing:
• Suppose (M,g,f) is a complete gradient shrinking Ricci soliton with dim(M) = 3. If M is simply-connected then the Riemannian manifold (M,g) is isometric to $\mathbb {R} ^{3}$, $S^{3}$, or $S^{2}\times \mathbb {R} $, each with their standard Riemannian metrics. This was originally shown by Perelman (2003a) with some extra conditional assumptions. Note that if M is not simply-connected, then one may consider the universal cover $\pi :M'\to M,$ and then the above theorem applies to $(M',\pi ^{\ast }g,f\circ \pi ).$
There is not yet a good understanding of gradient shrinking Ricci solitons in any higher dimensions.
Relationship to uniformization and geometrization
Hamilton's first work on Ricci flow was published at the same time as William Thurston's geometrization conjecture, which concerns the topological classification of three-dimensional smooth manifolds.[6] Hamilton's idea was to define a kind of nonlinear diffusion equation which would tend to smooth out irregularities in the metric. Suitable canonical forms had already been identified by Thurston; the possibilities, called Thurston model geometries, include the three-sphere S3, three-dimensional Euclidean space E3, three-dimensional hyperbolic space H3, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. (This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes.)
Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of positive Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow, renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time.) However, this doesn't prove the full geometrization conjecture, because of the restrictive assumption on curvature.
Indeed, a triumph of nineteenth-century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.
Note that the term "uniformization" suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" suggests placing a geometry on a smooth manifold. Geometry is being used here in a precise manner akin to Klein's notion of geometry (see Geometrization conjecture for further details). In particular, the result of geometrization may be a geometry that is not isotropic. In most cases including the cases of constant curvature, the geometry is unique. An important theme in this area is the interplay between real and complex formulations. In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.
Singularities
Hamilton showed that a compact Riemannian manifold always admits a short-time Ricci flow solution. Later Shi generalized the short-time existence result to complete manifolds of bounded curvature.[7] In general, however, due to the highly non-linear nature of the Ricci flow equation, singularities form in finite time. These singularities are curvature singularities, which means that as one approaches the singular time the norm of the curvature tensor $|\operatorname {Rm} |$ blows up to infinity in the region of the singularity. A fundamental problem in Ricci flow is to understand all the possible geometries of singularities. When successful, this can lead to insights into the topology of manifolds. For instance, analyzing the geometry of singular regions that may develop in 3d Ricci flow, is the crucial ingredient in Perelman's proof the Poincare and Geometrization Conjectures.
Blow-up limits of singularities
To study the formation of singularities it is useful, as in the study of other non-linear differential equations, to consider blow-ups limits. Intuitively speaking, one zooms into the singular region of the Ricci flow by rescaling time and space. Under certain assumptions, the zoomed in flow tends to a limiting Ricci flow $(M_{\infty },g_{\infty }(t)),t\in (-\infty ,0]$, called a singularity model. Singularity models are ancient Ricci flows, i.e. they can be extended infinitely into the past. Understanding the possible singularity models in Ricci flow is an active research endeavor.
Below, we sketch the blow-up procedure in more detail: Let $(M,g_{t}),\,t\in [0,T),$ be a Ricci flow that develops a singularity as $t\rightarrow T$. Let $(p_{i},t_{i})\in M\times [0,T)$ be a sequence of points in spacetime such that
$K_{i}:=\left|\operatorname {Rm} (g_{t_{i}})\right|(p_{i})\rightarrow \infty $
as $i\rightarrow \infty $. Then one considers the parabolically rescaled metrics
$g_{i}(t)=K_{i}g\left(t_{i}+{\frac {t}{K_{i}}}\right),\quad t\in [-K_{i}t_{i},0]$
Due to the symmetry of the Ricci flow equation under parabolic dilations, the metrics $g_{i}(t)$ are also solutions to the Ricci flow equation. In the case that
$|Rm|\leq K_{i}{\text{ on }}M\times [0,t_{i}],$
i.e. up to time $t_{i}$ the maximum of the curvature is attained at $p_{i}$, then the pointed sequence of Ricci flows $(M,g_{i}(t),p_{i})$ subsequentially converges smoothly to a limiting ancient Ricci flow $(M_{\infty },g_{\infty }(t),p_{\infty })$. Note that in general $M_{\infty }$ is not diffeomorphic to $M$.
Type I and Type II singularities
Hamilton distinguishes between Type I and Type II singularities in Ricci flow. In particular, one says a Ricci flow $(M,g_{t}),\,t\in [0,T)$, encountering a singularity a time $T$ is of Type I if
$\sup _{t<T}(T-t)|Rm|<\infty $.
Otherwise the singularity is of Type II. It is known that the blow-up limits of Type I singularities are gradient shrinking Ricci solitons.[8] In the Type II case it is an open question whether the singularity model must be a steady Ricci soliton—so far all known examples are.
Singularities in 3d Ricci flow
In 3d the possible blow-up limits of Ricci flow singularities are well-understood. By Hamilton, Perelman and recent work by Brendle, blowing up at points of maximum curvature leads to one of the following three singularity models:
• The shrinking round spherical space form $S^{3}/\Gamma $
• The shrinking round cylinder $S^{2}\times \mathbb {R} $
• The Bryant soliton
The first two singularity models arise from Type I singularities, whereas the last one arises from a Type II singularity.
Singularities in 4d Ricci flow
In four dimensions very little is known about the possible singularities, other than that the possibilities are far more numerous than in three dimensions. To date the following singularity models are known
• $S^{3}\times \mathbb {R} $
• $S^{2}\times \mathbb {R} ^{2}$
• The 4d Bryant soliton
• Compact Einstein manifold of positive scalar curvature
• Compact gradient Kahler–Ricci shrinking soliton
• The FIK shrinker [9]
• The BCCD shrinker [10]
Note that the first three examples are generalizations of 3d singularity models. The FIK shrinker models the collapse of an embedded sphere with self-intersection number −1.
Relation to diffusion
To see why the evolution equation defining the Ricci flow is indeed a kind of nonlinear diffusion equation, we can consider the special case of (real) two-manifolds in more detail. Any metric tensor on a two-manifold can be written with respect to an exponential isothermal coordinate chart in the form
$ds^{2}=\exp(2\,p(x,y))\,\left(dx^{2}+dy^{2}\right).$
(These coordinates provide an example of a conformal coordinate chart, because angles, but not distances, are correctly represented.)
The easiest way to compute the Ricci tensor and Laplace-Beltrami operator for our Riemannian two-manifold is to use the differential forms method of Élie Cartan. Take the coframe field
$\sigma ^{1}=\exp(p)\,dx,\;\;\sigma ^{2}=\exp(p)\,dy$
so that metric tensor becomes
$\sigma ^{1}\otimes \sigma ^{1}+\sigma ^{2}\otimes \sigma ^{2}=\exp(2p)\,\left(dx\otimes dx+dy\otimes dy\right).$
Next, given an arbitrary smooth function $h(x,y)$, compute the exterior derivative
$dh=h_{x}dx+h_{y}dy=\exp(-p)h_{x}\,\sigma ^{1}+\exp(-p)h_{y}\,\sigma ^{2}.$
Take the Hodge dual
$\star dh=-\exp(-p)h_{y}\,\sigma ^{1}+\exp(-p)h_{x}\,\sigma ^{2}=-h_{y}\,dx+h_{x}\,dy.$
Take another exterior derivative
$d\star dh=-h_{yy}\,dy\wedge dx+h_{xx}\,dx\wedge dy=\left(h_{xx}+h_{yy}\right)\,dx\wedge dy$
(where we used the anti-commutative property of the exterior product). That is,
$d\star dh=\exp(-2p)\,\left(h_{xx}+h_{yy}\right)\,\sigma ^{1}\wedge \sigma ^{2}.$
Taking another Hodge dual gives
$\Delta h=\star d\star dh=\exp(-2p)\,\left(h_{xx}+h_{yy}\right)$
which gives the desired expression for the Laplace/Beltrami operator
$\Delta =\exp(-2\,p(x,y))\left(D_{x}^{2}+D_{y}^{2}\right).$
To compute the curvature tensor, we take the exterior derivative of the covector fields making up our coframe:
$d\sigma ^{1}=p_{y}\exp(p)dy\wedge dx=-\left(p_{y}dx\right)\wedge \sigma ^{2}=-{\omega ^{1}}_{2}\wedge \sigma ^{2}$
$d\sigma ^{2}=p_{x}\exp(p)dx\wedge dy=-\left(p_{x}dy\right)\wedge \sigma ^{1}=-{\omega ^{2}}_{1}\wedge \sigma ^{1}.$
From these expressions, we can read off the only independent Spin connection one-form
${\omega ^{1}}_{2}=p_{y}dx-p_{x}dy,$
where we have taken advantage of the anti-symmetric property of the connection (${\omega ^{2}}_{1}=-{\omega ^{1}}_{2}$). Take another exterior derivative
$d{\omega ^{1}}_{2}=p_{yy}dy\wedge dx-p_{xx}dx\wedge dy=-\left(p_{xx}+p_{yy}\right)\,dx\wedge dy.$
This gives the curvature two-form
${\Omega ^{1}}_{2}=-\exp(-2p)\left(p_{xx}+p_{yy}\right)\,\sigma ^{1}\wedge \sigma ^{2}=-\Delta p\,\sigma ^{1}\wedge \sigma ^{2}$
from which we can read off the only linearly independent component of the Riemann tensor using
${\Omega ^{1}}_{2}={R^{1}}_{212}\,\sigma ^{1}\wedge \sigma ^{2}.$
Namely
${R^{1}}_{212}=-\Delta p$
from which the only nonzero components of the Ricci tensor are
$R_{22}=R_{11}=-\Delta p.$
From this, we find components with respect to the coordinate cobasis, namely
$R_{xx}=R_{yy}=-\left(p_{xx}+p_{yy}\right).$
But the metric tensor is also diagonal, with
$g_{xx}=g_{yy}=\exp(2p)$
and after some elementary manipulation, we obtain an elegant expression for the Ricci flow:
${\frac {\partial p}{\partial t}}=\Delta p.$
This is manifestly analogous to the best known of all diffusion equations, the heat equation
${\frac {\partial u}{\partial t}}=\Delta u$
where now $\Delta =D_{x}^{2}+D_{y}^{2}$ is the usual Laplacian on the Euclidean plane. The reader may object that the heat equation is of course a linear partial differential equation—where is the promised nonlinearity in the p.d.e. defining the Ricci flow?
The answer is that nonlinearity enters because the Laplace-Beltrami operator depends upon the same function p which we used to define the metric. But notice that the flat Euclidean plane is given by taking $p(x,y)=0$. So if $p$ is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation. This computation suggests that, just as (according to the heat equation) an irregular temperature distribution in a hot plate tends to become more homogeneous over time, so too (according to the Ricci flow) an almost flat Riemannian manifold will tend to flatten out the same way that heat can be carried off "to infinity" in an infinite flat plate. But if our hot plate is finite in size, and has no boundary where heat can be carried off, we can expect to homogenize the temperature, but clearly we cannot expect to reduce it to zero. In the same way, we expect that the Ricci flow, applied to a distorted round sphere, will tend to round out the geometry over time, but not to turn it into a flat Euclidean geometry.
Recent developments
The Ricci flow has been intensively studied since 1981. Some recent work has focused on the question of precisely how higher-dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving $n$-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the flow approaches some characteristic time $t_{0}$. In certain cases, such neckpinches will produce manifolds called Ricci solitons.
For a 3-dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.
Kähler metrics remain Kähler under Ricci flow, and so Ricci flow has also been studied in this setting, where it is called Kähler–Ricci flow.
Notes
1. Friedan, D. (1980). "Nonlinear models in 2+ε dimensions". Physical Review Letters (Submitted manuscript). 45 (13): 1057–1060. Bibcode:1980PhRvL..45.1057F. doi:10.1103/PhysRevLett.45.1057.
2. DeTurck, Dennis M. (1983). "Deforming metrics in the direction of their Ricci tensors". J. Differential Geom. 18 (1): 157–162. doi:10.4310/jdg/1214509286.
3. Eells, James Jr.; Sampson, J.H. (1964). "Harmonic mappings of Riemannian manifolds". Amer. J. Math. 86 (1): 109–160. doi:10.2307/2373037. JSTOR 2373037.
4. Gromov, M.; Thurston, W. (1987). "Pinching constants for hyperbolic manifolds". Invent. Math. 89 (1): 1–12. doi:10.1007/BF01404671. S2CID 119850633.
5. Li, Peter; Yau, Shing-Tung (1986). "On the parabolic kernel of the Schrödinger operator". Acta Math. 156 (3–4): 153–201. doi:10.1007/BF02399203. S2CID 120354778.
6. Weeks, Jeffrey R. (1985). The Shape of Space: how to visualize surfaces and three-dimensional manifolds. New York: Marcel Dekker. ISBN 978-0-8247-7437-0.. A popular book that explains the background for the Thurston classification program.
7. Shi, W.-X. (1989). "Deforming the metric on complete Riemannian manifolds". Journal of Differential Geometry. 30: 223–301. doi:10.4310/jdg/1214443292.
8. Enders, J.; Mueller, R.; Topping, P. (2011). "On Type I Singularities in Ricci flow". Communications in Analysis and Geometry. 19 (5): 905–922. arXiv:1005.1624. doi:10.4310/CAG.2011.v19.n5.a4. S2CID 968534.
9. Maximo, D. (2014). "On the blow-up of four-dimensional Ricci flow singularities". J. Reine Angew. Math. 2014 (692): 153–171. arXiv:1204.5967. doi:10.1515/crelle-2012-0080. S2CID 17651053.
10. Bamler, R.; Cifarelli, C.; Conlon, R.; Deruelle, A. (2022). "A new complete two-dimensional shrinking gradient Kähler-Ricci soliton". arXiv:2206.10785 [math.DG].
References
Articles for a popular mathematical audience.
• Anderson, Michael T. (2004). "Geometrization of 3-manifolds via the Ricci flow" (PDF). Notices Amer. Math. Soc. 51 (2): 184–193. MR 2026939.
• Milnor, John (2003). "Towards the Poincaré Conjecture and the classification of 3-manifolds" (PDF). Notices Amer. Math. Soc. 50 (10): 1226–1233. MR 2009455.
• Morgan, John W. (2005). "Recent progress on the Poincaré conjecture and the classification of 3-manifolds". Bull. Amer. Math. Soc. (N.S.). 42 (1): 57–78. doi:10.1090/S0273-0979-04-01045-6. MR 2115067.
• Tao, T. (2008). "Ricci flow" (PDF). In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics. Princeton University Press. pp. 279–281. ISBN 978-0-691-11880-2.
Research articles.
• Böhm, Christoph; Wilking, Burkhard (2008). "Manifolds with positive curvature operators are space forms". Ann. of Math. (2). 167 (3): 1079–1097. arXiv:math/0606187. doi:10.4007/annals.2008.167.1079. JSTOR 40345372. MR 2415394. S2CID 15521923.
• Brendle, Simon (2008). "A general convergence result for the Ricci flow in higher dimensions". Duke Math. J. 145 (3): 585–601. arXiv:0706.1218. doi:10.1215/00127094-2008-059. MR 2462114. S2CID 438716. Zbl 1161.53052.
• Brendle, Simon; Schoen, Richard (2009). "Manifolds with 1/4-pinched curvature are space forms". J. Amer. Math. Soc. 22 (1): 287–307. arXiv:0705.0766. Bibcode:2009JAMS...22..287B. doi:10.1090/S0894-0347-08-00613-9. JSTOR 40587231. MR 2449060. S2CID 2901565.
• Cao, Huai-Dong; Xi-Ping Zhu (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures — application of the Hamilton-Perelman theory of the Ricci flow" (PDF). Asian Journal of Mathematics. 10 (2). MR 2488948. Erratum.
• Revised version: Huai-Dong Cao; Xi-Ping Zhu (2006). "Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". arXiv:math.DG/0612069.
• Chow, Bennett (1991). "The Ricci flow on the 2-sphere". J. Differential Geom. 33 (2): 325–334. doi:10.4310/jdg/1214446319. MR 1094458. Zbl 0734.53033.
• Colding, Tobias H.; Minicozzi, William P., II (2005). "Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman" (PDF). J. Amer. Math. Soc. 18 (3): 561–569. arXiv:math/0308090. doi:10.1090/S0894-0347-05-00486-8. JSTOR 20161247. MR 2138137. S2CID 2810043.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Hamilton, Richard S. (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry. 17 (2): 255–306. doi:10.4310/jdg/1214436922. MR 0664497. Zbl 0504.53034.
• Hamilton, Richard S. (1986). "Four-manifolds with positive curvature operator". J. Differential Geom. 24 (2): 153–179. doi:10.4310/jdg/1214440433. MR 0862046. Zbl 0628.53042.
• Hamilton, Richard S. (1988). "The Ricci flow on surfaces". Mathematics and general relativity (Santa Cruz, CA, 1986). Contemp. Math. Vol. 71. Amer. Math. Soc., Providence, RI. pp. 237–262. doi:10.1090/conm/071/954419. MR 0954419.
• Hamilton, Richard S. (1993a). "The Harnack estimate for the Ricci flow". J. Differential Geom. 37 (1): 225–243. doi:10.4310/jdg/1214453430. MR 1198607. Zbl 0804.53023.
• Hamilton, Richard S. (1993b). "Eternal solutions to the Ricci flow". J. Differential Geom. 38 (1): 1–11. doi:10.4310/jdg/1214454093. MR 1231700. Zbl 0792.53041.
• Hamilton, Richard S. (1995a). "A compactness property for solutions of the Ricci flow". Amer. J. Math. 117 (3): 545–572. doi:10.2307/2375080. JSTOR 2375080. MR 1333936.
• Hamilton, Richard S. (1995b). "The formation of singularities in the Ricci flow". Surveys in differential geometry, Vol. II (Cambridge, MA, 1993). Int. Press, Cambridge, MA. pp. 7–136. doi:10.4310/SDG.1993.v2.n1.a2. MR 1375255.
• Hamilton, Richard S. (1997). "Four-manifolds with positive isotropic curvature". Comm. Anal. Geom. 5 (1): 1–92. doi:10.4310/CAG.1997.v5.n1.a1. MR 1456308. Zbl 0892.53018.
• Hamilton, Richard S. (1999). "Non-singular solutions of the Ricci flow on three-manifolds". Comm. Anal. Geom. 7 (4): 695–729. doi:10.4310/CAG.1999.v7.n4.a2. MR 1714939.
• Bruce Kleiner; John Lott (2008). "Notes on Perelman's papers". Geometry & Topology. 12 (5): 2587–2855. arXiv:math.DG/0605667. doi:10.2140/gt.2008.12.2587. MR 2460872. S2CID 119133773.
• Perelman, Grisha (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math/0211159.
• Perelman, Grisha (2003a). "Ricci flow with surgery on three-manifolds". arXiv:math/0303109.
• Perelman, Grisha (2003b). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math/0307245.
Textbooks
• Andrews, Ben; Hopper, Christopher (2011). The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem. Lecture Notes in Mathematics. Vol. 2011. Heidelberg: Springer. doi:10.1007/978-3-642-16286-2. ISBN 978-3-642-16285-5.
• Brendle, Simon (2010). Ricci Flow and the Sphere Theorem. Graduate Studies in Mathematics. Vol. 111. Providence, RI: American Mathematical Society. doi:10.1090/gsm/111. ISBN 978-0-8218-4938-5.
• Cao, H.D.; Chow, B.; Chu, S.C.; Yau, S.T., eds. (2003). Collected Papers on Ricci Flow. Series in Geometry and Topology. Vol. 37. Somerville, MA: International Press. ISBN 1-57146-110-8.
• Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2007). The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects. Mathematical Surveys and Monographs. Vol. 135. Providence, RI: American Mathematical Society. doi:10.1090/surv/135. ISBN 978-0-8218-3946-1.
• Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2008). The Ricci Flow: Techniques and Applications. Part II. Analytic Aspects. Mathematical Surveys and Monographs. Vol. 144. Providence, RI: American Mathematical Society. doi:10.1090/surv/144. ISBN 978-0-8218-4429-8.
• Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2010). The Ricci Flow: Techniques and Applications. Part III. Geometric-Analytic Aspects. Mathematical Surveys and Monographs. Vol. 163. Providence, RI: American Mathematical Society. doi:10.1090/surv/163. ISBN 978-0-8218-4661-2.
• Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2015). The Ricci Flow: Techniques and Applications. Part IV. Long-Time Solutions and Related Topics. Mathematical Surveys and Monographs. Vol. 206. Providence, RI: American Mathematical Society. doi:10.1090/surv/206. ISBN 978-0-8218-4991-0.
• Chow, Bennett; Knopf, Dan (2004). The Ricci Flow: An Introduction. Mathematical Surveys and Monographs. Vol. 110. Providence, RI: American Mathematical Society. doi:10.1090/surv/110. ISBN 0-8218-3515-7.
• Chow, Bennett; Lu, Peng; Ni, Lei (2006). Hamilton's Ricci Flow. Graduate Studies in Mathematics. Vol. 77. Beijing, New York: American Mathematical Society, Providence, RI; Science Press. doi:10.1090/gsm/077. ISBN 978-0-8218-4231-7.
• Morgan, John W.; Fong, Frederick Tsz-Ho (2010). Ricci Flow and Geometrization of 3-Manifolds. University Lecture Series. Vol. 53. Providence, RI: American Mathematical Society. doi:10.1090/ulect/053. ISBN 978-0-8218-4963-7.
• Morgan, John; Tian, Gang (2007). Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs. Vol. 3. Providence, RI and Cambridge, MA: American Mathematical Society and Clay Mathematics Institute. ISBN 978-0-8218-4328-4.
• Müller, Reto (2006). Differential Harnack inequalities and the Ricci flow. EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS). doi:10.4171/030. hdl:2318/1701023. ISBN 978-3-03719-030-2.
• Topping, Peter (2006). Lectures on the Ricci Flow. London Mathematical Society Lecture Note Series. Vol. 325. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511721465. ISBN 0-521-68947-3.
• Zhang, Qi S. (2011). Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture. Boca Raton, FL: CRC Press. ISBN 978-1-4398-3459-6.
External links
• Isenberg, James A. "Ricci Flow" (video). Brady Haran. Archived from the original on 2021-12-12. Retrieved 23 April 2014.
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Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe.
Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form (Besse 1987, p. 43).[1] Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function. However, there are other ways to draw the same analogy.
In three-dimensional topology, the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor. In part, this simplicity allows for the application of many geometric and analytic tools, which led to the solution of the Poincaré conjecture through the work of Richard S. Hamilton and Grigory Perelman.
In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. This is since lower bounds on the Ricci tensor can be successfully used in studying the length functional in Riemannian geometry, as first shown in 1941 via Myers's theorem.
One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the Bochner formula, which is used ubiquitously in Riemannian geometry. For example, this formula explains why the gradient estimates due to Shing-Tung Yau (and their developments such as the Cheng-Yau and Li-Yau inequalities) nearly always depend on a lower bound for the Ricci curvature.
In 2007, John Lott, Karl-Theodor Sturm, and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form.[2] This established a deep link between Ricci curvature and Wasserstein geometry and optimal transport, which is presently the subject of much research.
Definition
Suppose that $\left(M,g\right)$ is an $n$-dimensional Riemannian or pseudo-Riemannian manifold, equipped with its Levi-Civita connection $\nabla $. The Riemann curvature of $M$ is a map which takes smooth vector fields $X$, $Y$, and $Z$,
and returns the vector field
$R(X,Y)Z:=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z$
on vector fields $X,Y,Z$. Since $R$ is a tensor field, for each point $p\in M$, it gives rise to a (multilinear) map:
$\operatorname {R} _{p}:T_{p}M\times T_{p}M\times T_{p}M\to T_{p}M.$
Define for each point $p\in M$ the map $\operatorname {Ric} _{p}:T_{p}M\times T_{p}M\to \mathbb {R} $ by
$\operatorname {Ric} _{p}(Y,Z):=\operatorname {tr} {\big (}X\mapsto \operatorname {R} _{p}(X,Y)Z{\big )}.$
That is, having fixed $Y$ and $Z$, then for any orthonormal basis $v_{1},\ldots ,v_{n}$ of the vector space $T_{p}M$, one has
$\operatorname {Ric} _{p}(Y,Z)=\sum _{i=1}\langle \operatorname {R} _{p}(v_{i},Y)Z,v_{i}\rangle .$
It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis $v_{1},\ldots ,v_{n}$.
In abstract index notation,
$\mathrm {Ric} _{ab}=\mathrm {R} ^{c}{}_{bca}=\mathrm {R} ^{c}{}_{acb}.$
Sign conventions. Note that some sources define $R(X,Y)Z$ to be what would here be called $-R(X,Y)Z;$ they would then define $\operatorname {Ric} _{p}$ as $-\operatorname {tr} (X\mapsto \operatorname {R} _{p}(X,Y)Z).$ Although sign conventions differ about the Riemann tensor, they do not differ about the Ricci tensor.
Definition via local coordinates on a smooth manifold
Let $\left(M,g\right)$ be a smooth Riemannian or pseudo-Riemannian $n$-manifold. Given a smooth chart $\left(U,\varphi \right)$ one then has functions $g_{ij}:\varphi (U)\rightarrow \mathbb {R} $ and $g^{ij}:\varphi (U)\rightarrow \mathbb {R} $ for each $i,j=1,\ldots ,n$ which satisfy
$\sum _{k=1}^{n}g^{ik}(x)g_{kj}(x)=\delta _{j}^{i}={\begin{cases}1&i=j\\0&i\neq j\end{cases}}$
for all $x\in \varphi (U)$. The latter shows that, expressed as matrices, $g^{ij}(x)=(g^{-1})_{ij}(x)$. The functions $g_{ij}$ are defined by evaluating $g$ on coordinate vector fields, while the functions $g^{ij}$ are defined so that, as a matrix-valued function, they provide an inverse to the matrix-valued function $x\mapsto g_{ij}(x)$.
Now define, for each $a$, $b$, $c$, $i$, and $j$ between 1 and $n$, the functions
${\begin{aligned}\Gamma _{ab}^{c}&:={\frac {1}{2}}\sum _{d=1}^{n}\left({\frac {\partial g_{bd}}{\partial x^{a}}}+{\frac {\partial g_{ad}}{\partial x^{b}}}-{\frac {\partial g_{ab}}{\partial x^{d}}}\right)g^{cd}\\R_{ij}&:=\sum _{a=1}^{n}{\frac {\partial \Gamma _{ij}^{a}}{\partial x^{a}}}-\sum _{a=1}^{n}{\frac {\partial \Gamma _{ai}^{a}}{\partial x^{j}}}+\sum _{a=1}^{n}\sum _{b=1}^{n}\left(\Gamma _{ab}^{a}\Gamma _{ij}^{b}-\Gamma _{ib}^{a}\Gamma _{aj}^{b}\right)\end{aligned}}$
as maps $\varphi :U\rightarrow \mathbb {R} $.
Now let $\left(U,\varphi \right)$ and $\left(V,\psi \right)$ be two smooth charts with $U\cap V\neq \emptyset $. Let $R_{ij}:\varphi (U)\rightarrow \mathbb {R} $ be the functions computed as above via the chart $\left(U,\varphi \right)$ and let $r_{ij}:\psi (V)\rightarrow \mathbb {R} $ be the functions computed as above via the chart $\left(V,\psi \right)$. Then one can check by a calculation with the chain rule and the product rule that
$R_{ij}(x)=\sum _{k,l=1}^{n}r_{kl}\left(\psi \circ \varphi ^{-1}(x)\right)D_{i}{\Big |}_{x}\left(\psi \circ \varphi ^{-1}\right)^{k}D_{j}{\Big |}_{x}\left(\psi \circ \varphi ^{-1}\right)^{l}.$
where $D_{i}$ is the first derivative along $i$th direction of $\mathbb {R} ^{n}$. This shows that the following definition does not depend on the choice of $\left(U,\varphi \right)$. For any $p\in U$, define a bilinear map $\operatorname {Ric} _{p}:T_{p}M\times T_{p}M\rightarrow \mathbb {R} $ by
$(X,Y)\in T_{p}M\times T_{p}M\mapsto \operatorname {Ric} _{p}(X,Y)=\sum _{i,j=1}^{n}R_{ij}(\varphi (x))X^{i}(p)Y^{j}(p),$
where $X^{1},\ldots ,X^{n}$ and $Y^{1},\ldots ,Y^{n}$ are the components of the tangent vectors at $p$ in $X$ and $Y$ relative to the coordinate vector fields of $\left(U,\varphi \right)$.
It is common to abbreviate the above formal presentation in the following style:
Let $M$ be a smooth manifold, and let g be a Riemannian or pseudo-Riemannian metric. In local smooth coordinates, define the Christoffel symbols
${\begin{aligned}\Gamma _{ij}^{k}&:={\frac {1}{2}}g^{kl}\left(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij}\right)\\R_{jk}&:=\partial _{i}\Gamma _{jk}^{i}-\partial _{j}\Gamma _{ki}^{i}+\Gamma _{ip}^{i}\Gamma _{jk}^{p}-\Gamma _{jp}^{i}\Gamma _{ik}^{p}.\end{aligned}}$
It can be directly checked that
$R_{jk}={\widetilde {R}}_{ab}{\frac {\partial {\widetilde {x}}^{a}}{\partial x^{j}}}{\frac {\partial {\widetilde {x}}^{b}}{\partial x^{k}}},$
so that $R_{ij}$ define a (0,2)-tensor field on $M$. In particular, if $X$ and $Y$ are vector fields on $M$, then relative to any smooth coordinates one has
${\begin{aligned}R_{jk}X^{j}Y^{k}&=\left({\widetilde {R}}_{ab}{\frac {\partial {\widetilde {x}}^{a}}{\partial x^{j}}}{\frac {\partial {\widetilde {x}}^{b}}{\partial x^{k}}}\right)\left({\widetilde {X}}^{c}{\frac {\partial x^{j}}{\partial {\widetilde {x}}^{c}}}\right)\left({\widetilde {Y}}^{d}{\frac {\partial x^{k}}{\partial {\widetilde {x}}^{d}}}\right)\\&={\widetilde {R}}_{ab}{\widetilde {X}}^{c}{\widetilde {Y}}^{d}\left({\frac {\partial {\widetilde {x}}^{a}}{\partial x^{j}}}{\frac {\partial x^{j}}{\partial {\widetilde {x}}^{c}}}\right)\left({\frac {\partial {\widetilde {x}}^{b}}{\partial x^{k}}}{\frac {\partial x^{k}}{\partial {\widetilde {x}}^{d}}}\right)\\&={\widetilde {R}}_{ab}{\widetilde {X}}^{c}{\widetilde {Y}}^{d}\delta _{c}^{a}\delta _{d}^{b}\\&={\widetilde {R}}_{ab}{\widetilde {X}}^{a}{\widetilde {Y}}^{b}.\end{aligned}}$
The final line includes the demonstration that the bilinear map Ric is well-defined, which is much easier to write out with the informal notation.
Comparison of the definitions
The two above definitions are identical. The formulas defining $\Gamma _{ij}^{k}$ and $R_{ij}$ in the coordinate approach have an exact parallel in the formulas defining the Levi-Civita connection, and the Riemann curvature via the Levi-Civita connection. Arguably, the definitions directly using local coordinates are preferable, since the "crucial property" of the Riemann tensor mentioned above requires $M$ to be Hausdorff in order to hold. By contrast, the local coordinate approach only requires a smooth atlas. It is also somewhat easier to connect the "invariance" philosophy underlying the local approach with the methods of constructing more exotic geometric objects, such as spinor fields.
The complicated formula defining $R_{ij}$ in the introductory section is the same as that in the following section. The only difference is that terms have been grouped so that it is easy to see that $R_{ij}=R_{ji}.$
Properties
As can be seen from the Bianchi identities, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that
$\operatorname {Ric} (X,Y)=\operatorname {Ric} (Y,X)$
for all $X,Y\in T_{p}M.$
It thus follows linear-algebraically that the Ricci tensor is completely determined by knowing the quantity $\operatorname {Ric} (X,X)$ for all vectors $X$ of unit length. This function on the set of unit tangent vectors is often also called the Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor.
The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information. Indeed, if $\xi $ is a vector of unit length on a Riemannian $n$-manifold, then $\operatorname {Ric} (\xi ,\xi )$ is precisely $(n-1)$ times the average value of the sectional curvature, taken over all the 2-planes containing $\xi $. There is an $(n-2)$-dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. The second fundamental form, which determines the full curvature via the Gauss–Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor was introduced by Ricci for this reason.
As can be seen from the second Bianchi identity, one has
$\operatorname {div} \operatorname {Ric} ={\frac {1}{2}}dR,$
where $R$ is the scalar curvature, defined in local coordinates as $g^{ij}R_{ij}.$ This is often called the contracted second Bianchi identity.
Informal properties
The Ricci curvature is sometimes thought of as (a negative multiple of) the Laplacian of the metric tensor (Chow & Knopf 2004, Lemma 3.32).[3] Specifically, in harmonic local coordinates the components satisfy
$R_{ij}=-{\frac {1}{2}}\Delta \left(g_{ij}\right)+{\text{lower-order terms}},$
where $\Delta =\nabla \cdot \nabla $ is the Laplace–Beltrami operator, here regarded as acting on the locally-defined functions $g_{ij}$. This fact motivates, for instance, the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric. Alternatively, in a normal coordinate system based at $p$,
$R_{ij}=-{\frac {2}{3}}\Delta \left(g_{ij}\right).$
Direct geometric meaning
Near any point $p$ in a Riemannian manifold $\left(M,g\right)$, one can define preferred local coordinates, called geodesic normal coordinates. These are adapted to the metric so that geodesics through $p$ correspond to straight lines through the origin, in such a manner that the geodesic distance from $p$ corresponds to the Euclidean distance from the origin. In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense that
$g_{ij}=\delta _{ij}+O\left(|x|^{2}\right).$
In fact, by taking the Taylor expansion of the metric applied to a Jacobi field along a radial geodesic in the normal coordinate system, one has
$g_{ij}=\delta _{ij}-{\frac {1}{3}}R_{ikjl}x^{k}x^{l}+O\left(|x|^{3}\right).$
In these coordinates, the metric volume element then has the following expansion at p:
$d\mu _{g}=\left[1-{\frac {1}{6}}R_{jk}x^{j}x^{k}+O\left(|x|^{3}\right)\right]d\mu _{\text{Euclidean}},$
which follows by expanding the square root of the determinant of the metric.
Thus, if the Ricci curvature $\operatorname {Ric} (\xi ,\xi )$ is positive in the direction of a vector $\xi $, the conical region in $M$ swept out by a tightly focused family of geodesic segments of length $\varepsilon $ emanating from $p$, with initial velocity inside a small cone about $\xi $, will have smaller volume than the corresponding conical region in Euclidean space, at least provided that $\varepsilon $ is sufficiently small. Similarly, if the Ricci curvature is negative in the direction of a given vector $\xi $, such a conical region in the manifold will instead have larger volume than it would in Euclidean space.
The Ricci curvature is essentially an average of curvatures in the planes including $\xi $. Thus if a cone emitted with an initially circular (or spherical) cross-section becomes distorted into an ellipse (ellipsoid), it is possible for the volume distortion to vanish if the distortions along the principal axes counteract one another. The Ricci curvature would then vanish along $\xi $. In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of worldlines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.
Applications
Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
Ricci curvature also appears in the Ricci flow equation, where certain one-parameter families of Riemannian metrics are singled out as solutions of a geometrically-defined partial differential equation. This system of equations can be thought of as a geometric analog of the heat equation, and was first introduced by Richard S. Hamilton in 1982. Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, if one is given a manifold, the Ricci flow may be hoped to produce an 'equilibrium' Riemannian metric which is Einstein or of constant curvature. However, such a clean "convergence" picture cannot be achieved since many manifolds cannot support such metrics. A detailed study of the nature of solutions of the Ricci flow, due principally to Hamilton and Grigori Perelman, shows that the types of "singularities" that occur along a Ricci flow, corresponding to the failure of convergence, encodes deep information about 3-dimensional topology. The culmination of this work was a proof of the geometrization conjecture first proposed by William Thurston in the 1970s, which can be thought of as a classification of compact 3-manifolds.
On a Kähler manifold, the Ricci curvature determines the first Chern class of the manifold (mod torsion). However, the Ricci curvature has no analogous topological interpretation on a generic Riemannian manifold.
Global geometry and topology
Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry. Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has no topological implications. (The Ricci curvature is said to be positive if the Ricci curvature function $\operatorname {Ric} (\xi ,\xi )$ is positive on the set of non-zero tangent vectors $\xi $.) Some results are also known for pseudo-Riemannian manifolds.
1. Myers' theorem (1941) states that if the Ricci curvature is bounded from below on a complete Riemannian n-manifold by $(n-1)k>0$, then the manifold has diameter $\leq \pi /{\sqrt {k}}$. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group. Cheng (1975) showed that, in this setting, equality in the diameter inequality occurs if only if the manifold is isometric to a sphere of a constant curvature $k$.
2. The Bishop–Gromov inequality states that if a complete $n$-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a geodesic ball is less than or equal to the volume of a geodesic ball of the same radius in Euclidean $n$-space. Moreover, if $v_{p}(R)$ denotes the volume of the ball with center $p$ and radius $R$ in the manifold and $V(R)=c_{n}R^{n}$ denotes the volume of the ball of radius $R$ in Euclidean $n$-space then the function $v_{p}(R)/V(R)$ is nonincreasing. This can be generalized to any lower bound on the Ricci curvature (not just nonnegativity), and is the key point in the proof of Gromov's compactness theorem.)
3. The Cheeger–Gromoll splitting theorem states that if a complete Riemannian manifold $\left(M,g\right)$ with $\operatorname {Ric} \geq 0$ contains a line, meaning a geodesic $\gamma :\mathbb {R} \to M$ :\mathbb {R} \to M} such that $d(\gamma (u),\gamma (v))=\left|u-v\right|$ for all $u,v\in \mathbb {R} $, then it is isometric to a product space $\mathbb {R} \times L$. Consequently, a complete manifold of positive Ricci curvature can have at most one topological end. The theorem is also true under some additional hypotheses for complete Lorentzian manifolds (of metric signature $\left(+--\ldots \right)$) with non-negative Ricci tensor (Galloway 2000).
4. Hamilton's first convergence theorem for Ricci flow has, as a corollary, that the only compact 3-manifolds which have Riemannian metrics of positive Ricci curvature are the quotients of the 3-sphere by discrete subgroups of SO(4) which act properly discontinuously. He later extended this to allow for nonnegative Ricci curvature. In particular, the only simply-connected possibility is the 3-sphere itself.
These results, particularly Myers' and Hamilton's, show that positive Ricci curvature has strong topological consequences. By contrast, excluding the case of surfaces, negative Ricci curvature is now known to have no topological implications; Lohkamp (1994) has shown that any manifold of dimension greater than two admits a complete Riemannian metric of negative Ricci curvature. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear topological implications. There are very few two-dimensional manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.
Behavior under conformal rescaling
If the metric $g$ is changed by multiplying it by a conformal factor $e^{2f}$, the Ricci tensor of the new, conformally-related metric ${\tilde {g}}=e^{2f}g$ is given (Besse 1987, p. 59) by
${\widetilde {\operatorname {Ric} }}=\operatorname {Ric} +(2-n)\left[\nabla df-df\otimes df\right]+\left[\Delta f-(n-2)\|df\|^{2}\right]g,$
where $\Delta =*d*d$ is the (positive spectrum) Hodge Laplacian, i.e., the opposite of the usual trace of the Hessian.
In particular, given a point $p$ in a Riemannian manifold, it is always possible to find metrics conformal to the given metric $g$ for which the Ricci tensor vanishes at $p$. Note, however, that this is only pointwise assertion; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.
For two dimensional manifolds, the above formula shows that if $f$ is a harmonic function, then the conformal scaling $g\mapsto e^{2f}g$ does not change the Ricci tensor (although it still changes its trace with respect to the metric unless $f=0$.
Trace-free Ricci tensor
In Riemannian geometry and pseudo-Riemannian geometry, the trace-free Ricci tensor (also called traceless Ricci tensor) of a Riemannian or pseudo-Riemannian $n$-manifold $\left(M,g\right)$ is the tensor defined by
$Z=\operatorname {Ric} -{\frac {1}{n}}Rg,$
where $\operatorname {Ric} $ and $R$ denote the Ricci curvature and scalar curvature of $g$. The name of this object reflects the fact that its trace automatically vanishes: $\operatorname {tr} _{g}Z\equiv g^{ab}Z_{ab}=0.$ However, it is quite an important tensor since it reflects an "orthogonal decomposition" of the Ricci tensor.
The orthogonal decomposition of the Ricci tensor
The following, not so trivial, property is
$\operatorname {Ric} =Z+{\frac {1}{n}}Rg.$
It is less immediately obvious that the two terms on the right hand side are orthogonal to each other:
$\left\langle Z,{\frac {1}{n}}Rg\right\rangle _{g}\equiv g^{ab}\left(R_{ab}-{\frac {1}{n}}Rg_{ab}\right)=0.$
An identity which is intimately connected with this (but which could be proved directly) is that
$\left|\operatorname {Ric} \right|_{g}^{2}=|Z|_{g}^{2}+{\frac {1}{n}}R^{2}.$
The trace-free Ricci tensor and Einstein metrics
By taking a divergence, and using the contracted Bianchi identity, one sees that $Z=0$ implies $ {\frac {1}{2}}dR-{\frac {1}{n}}dR=0$. So, provided that n ≥ 3 and $M$ is connected, the vanishing of $Z$ implies that the scalar curvature is constant. One can then see that the following are equivalent:
• $Z=0$
• $\operatorname {Ric} =\lambda g$ for some number $\lambda $
• $\operatorname {Ric} ={\frac {1}{n}}Rg$
In the Riemannian setting, the above orthogonal decomposition shows that $R^{2}=n|\operatorname {Ric} |^{2}$ is also equivalent to these conditions. In the pseudo-Riemmannian setting, by contrast, the condition $|Z|_{g}^{2}=0$ does not necessarily imply $Z=0,$ so the most that one can say is that these conditions imply $R^{2}=n\left|\operatorname {Ric} \right|_{g}^{2}.$
In particular, the vanishing of trace-free Ricci tensor characterizes Einstein manifolds, as defined by the condition $\operatorname {Ric} =\lambda g$ for a number $\lambda .$ In general relativity, this equation states that $\left(M,g\right)$ is a solution of Einstein's vacuum field equations with cosmological constant.
Kähler manifolds
On a Kähler manifold $X$, the Ricci curvature determines the curvature form of the canonical line bundle (Moroianu 2007, Chapter 12). The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler differentials:
$\kappa = \bigwedge }^{n}~\Omega _{X}.$
The Levi-Civita connection corresponding to the metric on $X$ gives rise to a connection on $\kappa $. The curvature of this connection is the 2-form defined by
$\rho (X,Y)\;{\stackrel {\text{def}}{=}}\;\operatorname {Ric} (JX,Y)$
where $J$ is the complex structure map on the tangent bundle determined by the structure of the Kähler manifold. The Ricci form is a closed 2-form. Its cohomology class is, up to a real constant factor, the first Chern class of the canonical bundle, and is therefore a topological invariant of $X$ (for compact $X$) in the sense that it depends only on the topology of $X$ and the homotopy class of the complex structure.
Conversely, the Ricci form determines the Ricci tensor by
$\operatorname {Ric} (X,Y)=\rho (X,JY).$
In local holomorphic coordinates $z^{\alpha }$, the Ricci form is given by
$\rho =-i\partial {\overline {\partial }}\log \det \left(g_{\alpha {\overline {\beta }}}\right)$
where ∂ is the Dolbeault operator and
$g_{\alpha {\overline {\beta }}}=g\left({\frac {\partial }{\partial z^{\alpha }}},{\frac {\partial }{\partial {\overline {z}}^{\beta }}}\right).$
If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group $SL(n;\mathbb {C} )$. However, Kähler manifolds already possess holonomy in $U(n)$, and so the (restricted) holonomy of a Ricci-flat Kähler manifold is contained in $SU(n)$. Conversely, if the (restricted) holonomy of a 2$n$-dimensional Riemannian manifold is contained in $SU(n)$, then the manifold is a Ricci-flat Kähler manifold (Kobayashi & Nomizu 1996, IX, §4).
Generalization to affine connections
The Ricci tensor can also be generalized to arbitrary affine connections, where it is an invariant that plays an especially important role in the study of projective geometry (geometry associated to unparameterized geodesics) (Nomizu & Sasaki 1994). If $\nabla $ denotes an affine connection, then the curvature tensor $R$ is the (1,3)-tensor defined by
$R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z$
for any vector fields $X,Y,Z$. The Ricci tensor is defined to be the trace:
$\operatorname {ric} (X,Y)=\operatorname {tr} {\big (}Z\mapsto R(Z,X)Y{\big )}.$
In this more general situation, the Ricci tensor is symmetric if and only if there exists locally a parallel volume form for the connection.
Discrete Ricci curvature
Notions of Ricci curvature on discrete manifolds have been defined on graphs and networks, where they quantify local divergence properties of edges. Ollivier's Ricci curvature is defined using optimal transport theory.[4] A different (and earlier) notion, Forman's Ricci curvature, is based on topological arguments.[5]
See also
• Curvature of Riemannian manifolds
• Scalar curvature
• Ricci calculus
• Ricci decomposition
• Ricci-flat manifold
• Christoffel symbols
• Introduction to the mathematics of general relativity
Footnotes
1. Here it is assumed that the manifold carries its unique Levi-Civita connection. For a general affine connection, the Ricci tensor need not be symmetric.
2. Lott, John; Villani, Cedric (2006-06-23). "Ricci curvature for metric-measure spaces via optimal transport". arXiv:math/0412127.
3. Chow, Bennett (2004). The Ricci flow : an introduction. Dan Knopf. Providence, R.I.: American Mathematical Society. ISBN 0-8218-3515-7. OCLC 54692148.
4. Ollivier, Yann (2009-02-01). "Ricci curvature of Markov chains on metric spaces". Journal of Functional Analysis. 256 (3): 810–864. doi:10.1016/j.jfa.2008.11.001. ISSN 0022-1236. S2CID 14316364.
5. Forman (2003-02-01). "Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature". Discrete & Computational Geometry. 29 (3): 323–374. doi:10.1007/s00454-002-0743-x. ISSN 1432-0444. S2CID 9584267.
References
• Besse, A.L. (1987), Einstein manifolds, Springer, ISBN 978-3-540-15279-8.
• Chow, Bennet & Knopf, Dan (2004), The Ricci Flow: an introduction, American Mathematical Society, ISBN 0-8218-3515-7.
• Eisenhart, L.P. (1949), Riemannian geometry, Princeton Univ. Press.
• Forman (2003), "Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature", Discrete & Computational Geometry, 29 (3): 323–374. doi:10.1007/s00454-002-0743-x. ISSN 1432-0444
• Galloway, Gregory (2000), "Maximum Principles for Null Hypersurfaces and Null Splitting Theorems", Annales de l'Institut Henri Poincaré A, 1 (3): 543–567, arXiv:math/9909158, Bibcode:2000AnHP....1..543G, doi:10.1007/s000230050006, S2CID 9619157.
• Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, Volume 1, Interscience.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 2, Wiley-Interscience, ISBN 978-0-471-15732-8.
• Lohkamp, Joachim (1994), "Metrics of negative Ricci curvature", Annals of Mathematics, Second Series, Annals of Mathematics, 140 (3): 655–683, doi:10.2307/2118620, ISSN 0003-486X, JSTOR 2118620, MR 1307899.
• Moroianu, Andrei (2007), Lectures on Kähler geometry, London Mathematical Society Student Texts, vol. 69, Cambridge University Press, arXiv:math/0402223, doi:10.1017/CBO9780511618666, ISBN 978-0-521-68897-0, MR 2325093
• Nomizu, Katsumi; Sasaki, Takeshi (1994), Affine differential geometry, Cambridge University Press, ISBN 978-0-521-44177-3.
• Ollivier, Yann (2009), "Ricci curvature of Markov chains on metric spaces", Journal of Functional Analysis 256 (3): 810–864. doi:10.1016/j.jfa.2008.11.001. ISSN 0022-1236
• Ricci, G. (1903–1904), "Direzioni e invarianti principali in una varietà qualunque", Atti R. Inst. Veneto, 63 (2): 1233–1239.
• L.A. Sidorov (2001) [1994], "Ricci tensor", Encyclopedia of Mathematics, EMS Press
• L.A. Sidorov (2001) [1994], "Ricci curvature", Encyclopedia of Mathematics, EMS Press
• Najman, Laurent and Romon, Pascal (2017): Modern approaches to discrete curvature, Springer (Cham), Lecture notes in mathematics
External links
• Z. Shen, C. Sormani "The Topology of Open Manifolds with Nonnegative Ricci Curvature" (a survey)
• G. Wei, "Manifolds with A Lower Ricci Curvature Bound" (a survey)
Various notions of curvature defined in differential geometry
Differential geometry
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Riemannian geometry
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Curvature of connections
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Riemannian geometry (Glossary)
Basic concepts
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Types of manifolds
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Mathematicians
• Élie Cartan
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Wikipedia
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Ricci soliton
In differential geometry, a complete Riemannian manifold $(M,g)$ is called a Ricci soliton if, and only if, there exists a smooth vector field $V$ such that
$\operatorname {Ric} (g)=\lambda \,g-{\frac {1}{2}}{\mathcal {L}}_{V}g,$
for some constant $\lambda \in \mathbb {R} $. Here $\operatorname {Ric} $ is the Ricci curvature tensor and ${\mathcal {L}}$ represents the Lie derivative. If there exists a function $f:M\rightarrow \mathbb {R} $ such that $V=\nabla f$ we call $(M,g)$ a gradient Ricci soliton and the soliton equation becomes
$\operatorname {Ric} (g)+\nabla ^{2}f=\lambda \,g.$
Note that when $V=0$ or $f=0$ the above equations reduce to the Einstein equation. For this reason Ricci solitons are a generalization of Einstein manifolds.
Self-similar solutions to Ricci flow
A Ricci soliton $(M,g_{0})$ yields a self-similar solution to the Ricci flow equation
$\partial _{t}g_{t}=-2\operatorname {Ric} (g_{t}).$
In particular, letting
$\sigma (t):=1-2\lambda t$
and integrating the time-dependent vector field $X(t):={\frac {1}{\sigma (t)}}V$ to give a family of diffeomorphisms $\Psi _{t}$, with $\Psi _{0}$ the identity, yields a Ricci flow solution $(M,g_{t})$ by taking
$g_{t}=\sigma (t)\Psi _{t}^{\ast }(g_{0}).$
In this expression $\Psi _{t}^{\ast }(g_{0})$ refers to the pullback of the metric $g_{0}$ by the diffeomorphism $\Psi _{t}$. Therefore, up to diffeomorphism and depending on the sign of $\lambda $, a Ricci soliton homothetically shrinks, remains steady or expands under Ricci flow.
Examples of Ricci solitons
Shrinking ($\lambda >0$)
• Gaussian shrinking soliton $(\mathbb {R} ^{n},g_{eucl},f(x)={\frac {\lambda }{2}}|x|^{2})$
• Shrinking round sphere $S^{n},n\geq 2$
• Shrinking round cylinder $S^{n-1}\times \mathbb {R} ,n\geq 3$
• The four dimensional FIK shrinker [1]
• The four dimensional BCCD shrinker [2]
• Compact gradient Kahler-Ricci shrinkers [3][4][5]
• Einstein manifolds of positive scalar curvature
Steady ($\lambda =0$)
• The 2d cigar soliton (a.k.a. Witten's black hole) $\left(\mathbb {R} ^{2},g={\frac {dx^{2}+dy^{2}}{1+x^{2}+y^{2}}},V=-2(x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}})\right)$
• The 3d rotationally symmetric Bryant soliton and its generalization to higher dimensions [6]
• Ricci flat manifolds
Expanding ($\lambda <0$)
• Expanding Kahler-Ricci solitons on the complex line bundles $O(-k),k>n$ over $\mathbb {C} P^{n},n\geq 1$.[1]
• Einstein manifolds of negative scalar curvature
Singularity models in Ricci flow
Shrinking and steady Ricci solitons are fundamental objects in the study of Ricci flow as they appear as blow-up limits of singularities. In particular, it is known that all Type I singularities are modeled on non-collapsed gradient shrinking Ricci solitons.[7] Type II singularities are expected to be modeled on steady Ricci solitons in general, however to date this has not been proven, even though all known examples are.
Notes
1. Feldman, Mikhail; Ilmanen, Tom; Knopf, Dan (2003), "Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons", Journal of Differential Geometry, 65 (2): 169–209, doi:10.4310/jdg/1090511686
2. Bamler, R.; Cifarelli, C.; Conlon, R.; Deruelle, A. (2022). "A new complete two-dimensional shrinking gradient Kähler-Ricci soliton". arXiv:2206.10785 [math.DG].
3. Koiso, Norihito (1990), "On rotationally symmetric Hamilton's equation for Kahler-Einstein metrics", Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, vol. 18-I, Academic Press, Boston, MA, pp. 327–337, doi:10.2969/aspm/01810327
4. Cao, Huai-Dong (1996), "Existence of gradient Kähler-Ricci solitons", Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, pp. 1–16, arXiv:1203.4794
5. Wang, Xu-Jia; Zhu, Xiaohua (2004), "Kähler-Ricci solitons on toric manifolds with positive first Chern class", Advances in Mathematics, 188 (1): 87–103, doi:10.1016/j.aim.2003.09.009
6. Bryant, Robert L., Ricci flow solitons in dimension three with SO(3)-symmetries (PDF)
7. Enders, Joerg; Müller, Reto; Topping, Peter M. (2011), "On Type I Singularities in Ricci flow", Communications in Analysis and Geometry, 19 (5): 905–922, doi:10.4310/CAG.2011.v19.n5.a4
References
• Cao, Huai-Dong (2010). "Recent Progress on Ricci solitons". arXiv:0908.2006.
• Topping, Peter (2006), Lectures on the Ricci flow, Cambridge University Press, ISBN 978-0521689472
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Wikipedia
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Rice's theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is non-trivial if it is neither true for every partial computable function, nor false for every partial computable function.
Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called trivial if it holds for all partial computable functions or for none, and an effective decision method is called general if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation of 1951 at Syracuse University.
Introduction
Let p be a property of a formal language L that is nontrivial, meaning
1. there exists a recursively enumerable language having the property p,
2. there exists a recursively enumerable language not having the property p,
(that is, p is neither uniformly true nor uniformly false for all recursively enumerable languages).
Then it is undecidable to determine for a given Turing machine M, whether the language recognized by it has the property p.
In practice, this means that there is no machine that can always decide whether the language of a given Turing machine has a particular nontrivial property. Special cases include e.g. the undecidability of whether the language recognized by a Turing machine could be recognized by a nontrivial simpler machine, such as a finite automaton (meaning, it is undecidable whether the language of a Turing machine is regular).
It is important to note that Rice's theorem does not concern the properties of machines or programs; it concerns properties of functions and languages. For example, whether a machine runs for more than 100 steps on a particular input is a decidable property, even though it is non-trivial. Two different machines recognizing exactly the same language might require a different number of steps to recognize the same input string. Similarly, whether a machine has more than five states is a decidable property of the machine, as the number of states can simply be counted. For properties of this kind, which concerns a Turing machine but not the language recognized by it, Rice's theorem does not apply.
Using Rogers' characterization of acceptable programming systems, Rice's theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the behavior of computer programs.
As an example, consider the following variant of the halting problem. Let P be the following property of partial functions F of one argument: P(F) means that F is defined for the argument '1'. It is obviously non-trivial, since there are partial functions that are defined at 1, and others that are undefined at 1. The 1-halting problem is the problem of deciding of any algorithm whether it defines a function with this property, i.e., whether the algorithm halts on input 1. By Rice's theorem, the 1-halting problem is undecidable. Similarly the question of whether a Turing machine T terminates on an initially empty tape (rather than with an initial word w given as second argument in addition to a description of T, as in the full halting problem) is still undecidable.
Formal statement
Let $\mathbb {N} $ denote the natural numbers, and let $\mathbf {P} ^{(1)}$ denote the class of unary (partial) computable functions. Let $\phi \colon \mathbb {N} \to \mathbf {P} ^{(1)}$ be an admissible numbering of the computable functions. Denote by $\phi _{e}:=\phi (e)$ the eth (partial) computable function.
We identify each property that a computable function may have with the subset of $\mathbf {P} ^{(1)}$ consisting of the functions with that property. Thus, given a set $F\subseteq \mathbf {P} ^{(1)}$, a computable function $\phi _{e}$ has property $F$ if and only if $\phi _{e}\in F$. For each property $F\subseteq \mathbf {P} ^{(1)}$ there is an associated membership decision problem $D_{F}$ of determining, given e, whether $\phi _{e}\in F$.
Rice's theorem states that the decision problem $D_{F}$ is decidable (also called recursive or computable) if and only if $F=\varnothing $ or $F=\mathbf {P} ^{(1)}$.
Examples
According to Rice's theorem, if there is at least one partial computable function in a particular class C of partial computable functions and another partial computable function not in C then the problem of deciding whether a particular program computes a function in C is undecidable. For example, Rice's theorem shows that each of the following sets of partial computable functions is undecidable (that is, the set is not recursive, or not computable):
• The class of partial computable functions that return 0 for every input, and its complement.
• The class of partial computable functions that return 0 for at least one input, and its complement.
• The class of partial computable functions that are constant, and its complement.
• The class of partial computable functions that are identical to a given partial computable function, and its complement.
• The class of partial computable functions that diverge (i.e., undefined) for some input, and its complement.
• The class of indices for computable functions that are total.[1]
• The class of indices for recursively enumerable sets that are cofinite.
• The class of indices for recursively enumerable sets that are recursive.
Proof by Kleene's recursion theorem
A corollary to Kleene's recursion theorem states that for every Gödel numbering $\phi \colon \mathbb {N} \to \mathbf {P} ^{(1)}$ of the computable functions and every computable function $Q(x,y)$, there is an index $e$ such that $\phi _{e}(y)$ returns $Q(e,y)$. (In the following, we say that $f(x)$ "returns" $g(x)$ if either $f(x)=g(x)$, or both $f(x)$ and $g(x)$ are undefined.) Intuitively, $\phi _{e}$ is a quine, a function that returns its own source code (Gödel number), except that rather than returning it directly, $\phi _{e}$ passes its Gödel number to $Q$ and returns the result.
Assume for contradiction that $F$ is a set of computable functions such that $\emptyset \neq F\neq \mathbf {P} ^{(1)}$. Then there are computable functions $f\in F$ and $g\notin F$. Suppose that the set of indices $x$ such that $\phi _{x}\in F$ is decidable; then, there exists a function $Q(x,y)$ that returns $g(y)$ if $\phi _{x}\in F$, and $f(y)$ otherwise. By the corollary to the recursion theorem, there is an index $e$ such that $\phi _{e}(y)$ returns $Q(e,y)$. But then, if $\phi _{e}\in F$, then $\phi _{e}$ is the same function as $g$, and therefore $\phi _{e}\notin F$; and if $\phi _{e}\notin F$, then $\phi _{e}$ is $f$, and therefore $\phi _{e}\in F$. In both cases, we have a contradiction.
Proof by reduction from the halting problem
Proof sketch
Suppose, for concreteness, that we have an algorithm for examining a program p and determining infallibly whether p is an implementation of the squaring function, which takes an integer d and returns d2. The proof works just as well if we have an algorithm for deciding any other nontrivial property of program behavior (i.e. a semantic and non-trivial property), and is given in general below.
The claim is that we can convert our algorithm for identifying squaring programs into one that identifies functions that halt. We will describe an algorithm that takes inputs a and i and determines whether program a halts when given input i.
The algorithm for deciding this is conceptually simple: it constructs (the description of) a new program t taking an argument n, which (1) first executes program a on input i (both a and i being hard-coded into the definition of t), and (2) then returns the square of n. If a(i) runs forever, then t never gets to step (2), regardless of n. Then clearly, t is a function for computing squares if and only if step (1) terminates. Since we've assumed that we can infallibly identify programs for computing squares, we can determine whether t, which depends on a and i, is such a program; thus we have obtained a program that decides whether program a halts on input i. Note that our halting-decision algorithm never executes t, but only passes its description to the squaring-identification program, which by assumption always terminates; since the construction of the description of t can also be done in a way that always terminates, the halting-decision cannot fail to halt either.
halts (a,i) {
define t(n) {
a(i)
return n×n
}
return is_a_squaring_function(t)
}
This method doesn't depend specifically on being able to recognize functions that compute squares; as long as some program can do what we're trying to recognize, we can add a call to a to obtain our t. We could have had a method for recognizing programs for computing square roots, or programs for computing the monthly payroll, or programs that halt when given the input "Abraxas"; in each case, we would be able to solve the halting problem similarly.
Formal proof
For the formal proof, algorithms are presumed to define partial functions over strings and are themselves represented by strings. The partial function computed by the algorithm represented by a string a is denoted Fa. This proof proceeds by reductio ad absurdum: we assume that there is a non-trivial property that is decided by an algorithm, and then show that it follows that we can decide the halting problem, which is not possible, and therefore a contradiction.
Let us now assume that P(a) is an algorithm that decides some non-trivial property of Fa. Without loss of generality we may assume that P(no-halt) = "no", with no-halt being the representation of an algorithm that never halts. If this is not true, then this holds for the negation of the property. Since P decides a non-trivial property, it follows that there is a string b that represents an algorithm and P(b) = "yes". We can then define an algorithm H(a, i) as follows:
1. construct a string t that represents an algorithm T(j) such that
• T first simulates the computation of Fa(i),
• then T simulates the computation of Fb(j) and returns its result.
2. return P(t).
We can now show that H decides the halting problem:
• Assume that the algorithm represented by a halts on input i. In this case Ft = Fb and, because P(b) = "yes" and the output of P(x) depends only on Fx, it follows that P(t) = "yes" and, therefore H(a, i) = "yes".
• Assume that the algorithm represented by a does not halt on input i. In this case Ft = Fno-halt, i.e., the partial function that is never defined. Since P(no-halt) = "no" and the output of P(x) depends only on Fx, it follows that P(t) = "no" and, therefore H(a, i) = "no".
Since the halting problem is known to be undecidable, this is a contradiction and the assumption that there is an algorithm P(a) that decides a non-trivial property for the function represented by a must be false.
Rice's theorem and index sets
Rice's theorem can be succinctly stated in terms of index sets:
Let ${\mathcal {C}}$ be a class of partial recursive functions with index set $C$. Then $C$ is recursive if and only if $C=\varnothing $ or $C=\mathbb {N} $.
Here $\mathbb {N} $ is the set of natural numbers, including zero.
An analogue of Rice's theorem for recursive sets
One can regard Rice's theorem as asserting the impossibility of effectively deciding for any recursively enumerable set whether it has a certain nontrivial property.[2] In this section, we give an analogue of Rice's theorem for recursive sets, instead of recursively enumerable sets.[3] Roughly speaking, the analogue says that if one can effectively determine for every recursive set whether it has a certain property, then only finitely many integers determine whether a recursive set has the property. This result is analogous to the original theorem of Rice, because both results assert that a property is "decidable" only if one can determine whether a set has that property by examining for at most finitely many $i$ (for no $i$, for the original theorem), if $i$ belongs to the set.
Let $W$ be a class (called a simple game and thought of as a property) of recursive sets. If $S$ is a recursive set, then for some $e$, computable function $\phi _{e}$ is the characteristic function of $S$. We call $e$ a characteristic index for $S$. (There are infinitely many such $e$.) Let's say the class $W$ is computable if there is an algorithm (computable function) that decides for any nonnegative integer $e$ (not necessarily a characteristic index),
• if $e$ is a characteristic index for a recursive set belonging to $W$, then the algorithm gives "yes";
• if $e$ is a characteristic index for a recursive set not belonging to $W$, then the algorithm gives "no".
A set $S\subseteq \mathbb {N} $ extends a string $\tau $ of 0's and 1's if for every $k<|\tau |$ (the length of $\tau $), the $k$th element of $\tau $ is 1 if $k\in S$; and is 0 otherwise. For example, $S=\{1,3,4,7,\ldots \}$ extends the string $01011001$. A string $\tau $ is winning determining if every recursive set extending $\tau $ belongs to $W$. A string $\tau $ is losing determining if no recursive set extending $\tau $ belongs to $W$.
We can now state the following analogue of Rice's theorem:[4][5]
A class $W$ of recursive sets is computable if and only if there are a recursively enumerable set $T_{0}$ of losing determining strings and a recursively enumerable set $T_{1}$ of winning determining strings such that every recursive set extends a string in $T_{0}\cup T_{1}$.
This result has been applied to foundational problems in computational social choice (more broadly, algorithmic game theory). For instance, Kumabe and Mihara[5][6] apply this result to an investigation of the Nakamura numbers for simple games in cooperative game theory and social choice theory.
See also
• Gödel's incompleteness theorems
• Halting problem
• Recursion theory
• Rice–Shapiro theorem
• Scott–Curry theorem, an analogue to Rice's theorem in lambda calculus
• Turing's proof
• Wittgenstein on Rules and Private Language
Notes
1. Soare, Robert I. (1987). Recursively Enumerable Sets and Degrees. Springer. p. 21. ISBN 9780387152998.
2. A set $S\subseteq \mathbb {N} $ is recursively enumerable if $S=W_{e}:={\textrm {dom}}\,\phi _{e}:=\{x:\phi _{e}(x)\downarrow \}$ for some $e$, where $W_{e}$ is the domain ${\textrm {dom}}\,\phi _{e}$ (the set of inputs $x$ such that $\phi _{e}(x)$ is defined) of $\phi _{e}$. The result for recursively enumerable sets can be obtained from that for (partial) computable functions by considering the class $\{\phi _{e}:{\textrm {dom}}\,\phi _{e}\in C\}$, where $C$ is a class of recursively enumerable sets.
3. A recursively enumerable set $S\subseteq \mathbb {N} $ is recursive if its complement is recursively enumerable. Equivalently, $S$ is recursive if its characteristic function is computable.
4. Kreisel, G.; Lacombe, D.; Shoenfield, J. R. (1959). "Partial recursive functionals and effective operations". In Heyting, A. (ed.). Constructivity in Mathematics. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland. pp. 290–297.
5. Kumabe, M.; Mihara, H. R. (2008). "Computability of simple games: A characterization and application to the core". Journal of Mathematical Economics. 44 (3–4): 348–366. arXiv:0705.3227. doi:10.1016/j.jmateco.2007.05.012. S2CID 8618118.
6. Kumabe, M.; Mihara, H. R. (2008). "The Nakamura numbers for computable simple games". Social Choice and Welfare. 31 (4): 621. arXiv:1107.0439. doi:10.1007/s00355-008-0300-5. S2CID 8106333.
References
• Hopcroft, John E.; Ullman, Jeffrey D. (1979), Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, pp. 185–192
• Rice, H. G. (1953), "Classes of recursively enumerable sets and their decision problems", Transactions of the American Mathematical Society, 74 (2): 358–366, doi:10.1090/s0002-9947-1953-0053041-6, JSTOR 1990888
• Rogers, Hartley Jr. (1987), Theory of Recursive Functions and Effective Computability (2nd ed.), McGraw-Hill, §14.8
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Wikipedia
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Rice's formula
In probability theory, Rice's formula counts the average number of times an ergodic stationary process X(t) per unit time crosses a fixed level u.[1] Adler and Taylor describe the result as "one of the most important results in the applications of smooth stochastic processes."[2] The formula is often used in engineering.[3]
History
The formula was published by Stephen O. Rice in 1944,[4] having previously been discussed in his 1936 note entitled "Singing Transmission Lines."[5][6]
Formula
Write Du for the number of times the ergodic stationary stochastic process x(t) takes the value u in a unit of time (i.e. t ∈ [0,1]). Then Rice's formula states that
$\mathbb {E} (D_{u})=\int _{-\infty }^{\infty }|x'|p(u,x')\,\mathrm {d} x'$
where p(x,x') is the joint probability density of the x(t) and its mean-square derivative x'(t).[7]
If the process x(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give[7][8]
$\mathbb {E} (D_{0})={\frac {1}{\pi }}{\sqrt {-\rho ''(0)}}$
where ρ'' is the second derivative of the normalised autocorrelation of x(t) at 0.
Uses
Rice's formula can be used to approximate an excursion probability[9]
$\mathbb {P} \left\{\sup _{t\in [0,1]}X(t)\geq u\right\}$
as for large values of u the probability that there is a level crossing is approximately the probability of reaching that level.
References
1. Rychlik, I. (2000). "On Some Reliability Applications of Rice's Formula for the Intensity of Level Crossings". Extremes. Kluwer Academic Publishers. 3 (4): 331–348. doi:10.1023/A:1017942408501. S2CID 115235517.
2. Adler, Robert J.; Taylor, Jonathan E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. doi:10.1007/978-0-387-48116-6. ISBN 978-0-387-48112-8.
3. Grigoriu, Mircea (2002). Stochastic Calculus: Applications in Science and Engineering. p. 166. ISBN 978-0-817-64242-6.
4. Rice, S. O. (1944). "Mathematical analysis of random noise" (PDF). Bell System Tech. J. 23 (3): 282–332. doi:10.1002/j.1538-7305.1944.tb00874.x.
5. Rainal, A. J. (1988). "Origin of Rice's formula". IEEE Transactions on Information Theory. 34 (6): 1383–1387. doi:10.1109/18.21276.
6. Borovkov, K.; Last, G. (2012). "On Rice's formula for stationary multivariate piecewise smooth processes". Journal of Applied Probability. 49 (2): 351. arXiv:1009.3885. doi:10.1239/jap/1339878791.
7. Barnett, J. T. (2001). "Zero-Crossings of Random Processes with Application to Estimation Detection". In Marvasti, Farokh A. (ed.). Nonuniform Sampling: Theory and Practice. Springer. ISBN 0306464454.
8. Ylvisaker, N. D. (1965). "The Expected Number of Zeros of a Stationary Gaussian Process". The Annals of Mathematical Statistics. 36 (3): 1043–1046. doi:10.1214/aoms/1177700077.
9. Adler, Robert J.; Taylor, Jonathan E. (2007). "Excursion Probabilities". Random Fields and Geometry. Springer Monographs in Mathematics. pp. 75–76. doi:10.1007/978-0-387-48116-6_4. ISBN 978-0-387-48112-8.
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Wikipedia
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Rice–Shapiro theorem
In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro.[1]
Formal statement
Let A be a set of partial-recursive unary functions on the domain of natural numbers such that the set $Ix(A):=\{n\mid \varphi _{n}\in A\}$ is recursively enumerable, where $\varphi _{n}$ denotes the $n$-th partial-recursive function in a Gödel numbering.
Then for any unary partial-recursive function $\psi $, we have:
$\psi \in A\Leftrightarrow \exists $ a finite function $\theta \subseteq \psi $ such that $\theta \in A.$
In the given statement, a finite function is a function with a finite domain $x_{1},x_{2},...,x_{m}$ and $\theta \subseteq \psi $ means that for every $x\in \{x_{1},x_{2},...,x_{m}\}$ it holds that $\psi (x)$ is defined and equal to $\theta (x)$.
Perspective from effective topology
For any finite unary function $\theta $ on integers, let $C(\theta )$ denote the 'frustum' of all partial-recursive functions that are defined, and agree with $\theta $, on $\theta $'s domain.
Equip the set of all partial-recursive functions with the topology generated by these frusta as base. Note that for every frustum $C$, $Ix(C)$ is recursively enumerable. More generally it holds for every set $A$ of partial-recursive functions:
$Ix(A)$ is recursively enumerable iff $A$ is a recursively enumerable union of frusta.
Notes
1. Rogers Jr., Hartley (1987). Theory of Recursive Functions and Effective Computability. MIT Press. ISBN 0-262-68052-1.
References
• Cutland, Nigel (1980). Computability: an introduction to recursive function theory. Cambridge University Press.; Theorem 7-2.16.
• Rogers Jr., Hartley (1987). Theory of Recursive Functions and Effective Computability. MIT Press. p. 482. ISBN 0-262-68052-1.
• Odifreddi, Piergiorgio (1989). Classical Recursion Theory. North Holland.
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Ricerche di Matematica
Ricerche di Matematica is a peer-reviewed mathematics journal on applied mathematics and pure mathematics. It was established in 1952 by Carlo Miranda with the collaboration of Renato Caccioppoli and other members of the Istituto di Matematica of the University of Naples Federico II. From 1952 to 2005 the journal was published in 54 volumes in Naples with articles in Italian, English, or French. From 2006 "Ricerche di Matematica" (with articles only in English) is published by Springer-Verlag under the auspices of the Dipartimento di Matematica e Applicazioni "Renato Caccioppoli"; a board of professors in this department at the University of Naples Federico II appoints and supports the journal's editors.[2] The journal is indexed by Mathematical Reviews and Zentralblatt MATH. Its 2018 h-index was 14, and its 2018 impact factor was 1.16.[3]
Ricerche di Matematica
DisciplinePure and applied mathematics
LanguageEnglish
Edited bySalvatore Rionero
Publication details
History1952–present
Publisher
Springer-Verlag Italia S.r.l., Milano
FrequencyBiannual[1]
Standard abbreviations
ISO 4 (alt) · Bluebook (alt1 · alt2)
NLM (alt) · MathSciNet (alt )
ISO 4Ric. Mat.
Indexing
CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt)
MIAR · NLM (alt) · Scopus
ISSN0035-5038 (print)
1827-3491 (web)
LCCN2019204734
OCLC no.1640734
Links
• Journal homepage
References
1. "Ricerche di Matematica, All Volumes and Issues". link.springer.com.
2. Official website
3. "Ricerche di Matematica". resurchify.com.
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Richard Baldus
Richard Baldus (11 May 1885, in Salonika – 28 January 1945, in Munich) was a German mathematician, specializing in geometry.[1]
Richard Baldus was the son of a station chief of the Anatolian Railway. After his graduation (Abitur) in 1904 at Wilhelmsgymnasium München,[2] he studied in Munich and at the University of Erlangen, where he received his Ph.D. (Promotierung) in 1910 under Max Noether with thesis Über Strahlensysteme, welche unendlich viele Regelflächen 2. Grades enthalten[3] and where he received his Habilitierung in 1911. He became in 1919 Professor für Geometrie at the Technische Hochschule Karlsruhe and served there as rector in 1923–1924. In 1932 he became Professor für Geometrie (as successor to Sebastian Finsterwalder) at TU München, where in 1934 he also became the successor to the professorial chair of Walther von Dyck, upon the latter's retirement.
In 1933 Baldus was the president of the Deutsche Mathematiker-Vereinigung. He was an invited speaker at the International Congress of Mathematicians in 1928 at Bologna. He was elected in 1929 a member of the Heidelberger Akademie der Wissenschaften and in 1935 a member of the Bayerische Akademie der Wissenschaften.
Selected publications
• Über Strahlensysteme, welche unendlich viele Regelflächen 2. Grades enthalten. Erlangen: Junge & Sohn. 1910.
• "Mathematik und räumliche Anschauung". Jahresbericht der Deutschen Mathematiker-Vereinigung. 30: 1–15. 1921.
• Baldus, Richard (1923). "Zur Steinerschen Definition der Projektivität". Mathematische Annalen. 90 (1): 86–102. doi:10.1007/BF01456243. S2CID 121669489.
• Formalismus und Intuitionismus in der Mathematik. Karlsruhe: G. Braun. 1924.
• Nichteuklidische Geometrie - hyperbolische Geometrie der Ebene. Sammlung Göschen, de Gruyter. 1927.[4]
• Zur Klassifikation der ebenen und räumlichen Kollineationen. Verlag d. Bayer. Akademie d. Wissenschaften. 1928.
• Baldus, Richard (1928). "Zur Axiomatik der Geometrie. I". Mathematische Annalen. 100: 321–333. doi:10.1007/BF01448848. S2CID 124779504.
• "Zur Axiomatik der Geometrie II. Vereinfachungen des archimedischen und cantorschen Axioms" (PDF). Atti del Congresso Internazionale die Mathematici. 1928.
• Zur Axiomatik der Geometrie. III. Über das Archimedische und das Cantorsche Axiom. Verlag der Bayer. Akad. der Wiss. 1930.
References
1. Georg Faber (1953), "Baldus, Richard", Neue Deutsche Biographie (in German), vol. 1, Berlin: Duncker & Humblot, pp. 558–558; (full text online)
2. Jahresbericht vom K. Wilhelms-Gymnasium zu München. ZDB-ID 12448436, 1903/04
3. Richard Baldus at the Mathematics Genealogy Project
4. Allen, Edward Switzer (1929). "Three books on non-euclidean geometry". Bull. Amer. Math. Soc. 35: 271–276. doi:10.1090/S0002-9904-1929-04726-8. (See p. 274.)
External links
• Literature by and about Richard Baldus in the German National Library catalogue
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Richard Bruce Paris
Richard Bruce Paris (23 January 1946 – 8 July 2022[1]) was a British mathematician and reader at the Abertay University in Dundee, who specialized in calculus. He also had a honorary readership of the University of St. Andrews, Scotland. The research activity of Paris particularly concerned the asymptotics of integrals and properties of special functions. He is the author of Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descent as well as the co-author of Asymptotics and Mellin-Barnes Integrals and of Asymptotics of High Order Differential Equations. In addition, he contributed to the NIST Handbook of Mathematical Functions and also released numerous papers for Proceedings of the Royal Society A, Methods and Applications of Analysis and the Journal of Computational and Applied Mathematics.
Richard Bruce Paris
Richard Bruce Paris in 2003
Born(1946-01-23)23 January 1946
Bradford, UK
Died8 July 2022(2022-07-08) (aged 76)
NationalityBritish
Alma materUniversity of Manchester
Known forspecial functions, Mellin-Barnes integrals, hyperasymptotics, Hadamard expansions
Scientific career
FieldsMathematics
InstitutionsAbertay University, Dundee
ThesisThe Role of the Magnetic Field in Cosmogony (1971)
Doctoral advisorLeon Mestel
Personal life
Born in 1946, Richard Bruce Paris was the son of an engineer. He spent his early childhood in the Yorkshire area until his family moved to the Wirral Peninsula, Cheshire, in the mid-1950s, due to the work of his father. There, Paris visited the Calday Grange Grammar School in West Kirkby to eventually discover his interest in mathematics. [2]
Paris was married to Jocelyne Marie-Louise Neidinger with whom he has a son Simon and a daughter Gaëlle. [2]
Career
In 1967, Paris acquired a first class honours degree in Mechanical Engineering from the Victoria University of Manchester. He continued his study at the university's department of mathematics, which he graduated as a Doctor of Philosophy in 1971. [2] Paris was a doctoral student of the British-Australian astronomer Leon Mestel. His PhD thesis was finished under the title The Role of the Magnetic Field in Cosmogony. [3]
After Paris finished his doctoral thesis, in 1974 he moved to France to work for Euratom at the Department of Plasma Physics and Controlled Fusion in Fontenay-aux-Roses. In addition, from the mid-1970s to the mid-1980s, Paris did several research visits in Los Alamos, USA. Finally, in 1984 he had to move to Southern France, due to a job transfer to Cadarache. In 1987, Paris quit his job at Euratom and returned to Scotland to work as a senior lecturer at the Abertay University in Dundee. A year leater, in 1988, he received the honorary readership of the University of St. Andrews, Scotland. In 1999, he also achieved the degree of a Doctor of Science at the University of Manchester. Paris stayed at the University of Abertay, where he eventually obtained the status of a reader, until his retirement in 2010. Yet, this was not the end of his mathematical work but he kept contributing until his unexpected death in July 2022. In fact, one month earlier he shared his final article on ResearchGate. [2]
In 1986, Paris became an elected fellow of the British Institute of Mathematics and its Applications.[4]
Work
The work of Paris deals with the asymptotic behaviour of a wide scope of special functions, in many case with a connection to physical problems.
In collaboration with David Kaminski, associate professor of mathematics at the University of Lethbridge, Paris published the monograph Asymptotics and Mellin-Barnes integrals. It is one of the few textbooks that extensively treats the application of Mellin transforms particularly to different asymptotic problems. Mellin-Barnes integrals constitute a special class of contour integrals that feature special functions in the integrand, most frequently products of gamma functions. Their evaluation relies on the residue theorem and requires appropriate manipulations of the integration path. The name is due to the mathematicians R. H. Mellin and E. W. Barnes. Many integrals can be transformed to a Mellin-Barnes representation, by writing their integrands in terms of inverse Mellin transforms. In the context of Laplace-type integrals, this technique provides a powerful alternative to Laplace's method. In general, however, it admits a broader applicability and much space for modifications. This versatility is shown by means of several examples from number theory and integrals of higher dimension.[5]
In his monograph Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descent, by means of theoretical and numerical examples, Paris illustrates the application of Laplace's method and possibilities to achieve a higher accuracy. The term Hadamard expansions describes a special kind of asymptotic expansions whose coefficients are again series. It refers to the French mathematician Jacques Hadamard who introduced the first series of this kind in 1908 in his paper Sur l'expression asymptotique de la fonction de Bessel.[6]
Paris also organized the chapters 8 and 11, respectively about the incomplete Gamma and about the Struve functions and related functions, of the NIST Digital Library of Mathematical Functions and of the NIST Handbook of Mathematical Functions. He validated the original release in 2010 and was the Associate Editor for his chapters from 2015 until his death.[4]
Publications
• with A. D. Wood: Asymptotics of Higher Order Differential Equations, Longman Scientific and Technical, 1986, ISBN 0-470-20375-7
• with D. Kaminski: Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001, ISBN 978-0-521-79001-7 (vol. 85 of the Encyclopedia of Mathematics and its Applications)
• Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descent, Cambridge University Press, 2011, ISBN 978-1-107-00258-6 (vol. 141 of the Encyclopedia of Mathematics and its Applications)
• with F. W. J. Olver, R. Askey et al.: NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, Hardback ISBN 978-0-521-19225-5, Paperback ISBN 978-0-521-14063-8
References
1. "Hommage à Richard Bruce Paris, mathématicien - Le Blog de Sylvie Neidinger". blogdesylvieneidinger.blogspirit.com (in French). Retrieved 2023-01-20.
2. OP-SF NET - Volume 30, No. 1 - Jan. 15, 2023 - The Electronic News Net of the SIAM Activity Group on Orthogonal Polynomials and Special Functions - http://math.nist.gov/opsf
3. "U376701 | University of Manchester Library Universal Access". uomlibrary.access.preservica.com. Retrieved 2023-01-10.
4. "DLMF: Profile Richard B. Paris ‣ About the Project". dlmf.nist.gov. Retrieved 2023-01-02.
5. Paris, R. B. (2001). Asymptotics and Mellin-Barnes integrals. D. Kaminski. Cambridge: Cambridge University Press. ISBN 0-521-79001-8. OCLC 70756548.
6. Paris, R. B. (2011). Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents. Cambridge: Cambridge University Press. ISBN 978-1-107-08985-3. OCLC 847526828.
External links
• Richard Bruce Paris at ResearchGate
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Richard C. DiPrima Prize
The Richard C. DiPrima Prize is awarded every two years by the Society for Industrial and Applied Mathematics to an early career researcher who has done outstanding research in applied mathematics. First awarded in 1988, it honors the memory of Richard C. DiPrima, a former president of SIAM who also served for many years as a member of its council and board of trustees, as vice president for programs, and as a dedicated and committed member.[1]
Recipients
The recipients of the Richard C. DiPrima Prize are:[2]
• 1988: Mary E. Brewster
• 1990: No award
• 1992: Anne Bourlioux
• 1992: Robin Carl Young
• 1994: Stephen Jonathan Chapman
• 1996: David Paul Williamson
• 1998: Bart De Schutter
• 2000: Keith Lindsay
• 2002: Gang Hu
• 2004: Diego Dominici
• 2006: Xinwei Yu
• 2008: Daan Huybrechs
• 2010: Colin B. Macdonald
• 2012: Thomas Goldstein
• 2014: Thomas D. Trogdon
• 2016: Blake H. Barker
• 2018: Peter Gangl
• 2020: Anna Seigal
See also
• List of mathematics awards
References
1. Recognizing excellence in the mathematical sciences : an international compilation of awards, prizes, and recipients. Jaguszewski, Janice M. Greenwich, Conn.: JAI Press. 1997. ISBN 0762302356. OCLC 37513025.{{cite book}}: CS1 maint: others (link)
2. "SIAM: The Richard C. DiPrima Prize". www.siam.org. Retrieved 2018-12-09.
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Richard Cleve
Richard Erwin Cleve is a Canadian professor of computer science at the David R. Cheriton School of Computer Science at the University of Waterloo, where he holds the Institute for Quantum Computing Chair in quantum computing, and an associate member of the Perimeter Institute for Theoretical Physics.[1]
Richard Erwin Cleve
Alma materUniversity of Waterloo
University of Toronto
AwardsCAP-CRM Prize in Theoretical and Mathematical Physics
Scientific career
FieldsComputer science
InstitutionsUniversity of Calgary
University of Waterloo
Institute for Quantum Computing
Perimeter Institute for Theoretical Physics
Doctoral advisorCharles Rackoff
Education
He obtained his BMath and MMath from the University of Waterloo,[2] and his Ph.D. in 1989 at the University of Toronto under the supervision of Charles Rackoff.[3]
Research
He was the recipient of the 2008 CAP-CRM Prize in Theoretical and Mathematical Physics, awarded for "fundamental results in quantum information theory, including the structure of quantum algorithms and the foundations of quantum communication complexity."[4] He has authored several highly cited papers in quantum information,[5][6][7] and is one of the creators of the field of quantum communication complexity.[4][8] He is also one of the founding managing editors of the journal Quantum Information & Computation,[9] a founding fellow of the Quantum Information Processing program at the Canadian Institute for Advanced Research, and a Team Leader at QuantumWorks.[4]
References
1. Richard Cleve at the IQC directory.
2. Richard Cleve at the University of Waterloo website.
3. Richard Cleve at the Mathematics Genealogy Project.
4. 2008 CAP/CRM Prize in Theoretical and Mathematical Physics
5. Barenco, Adriano; Charles H. Bennett; Richard Cleve; David P. DiVincenzo; Norman Margolus; Peter Shor; Tycho Sleator; John A. Smolin; Harald Weinfurter (1995-11-01). "Elementary gates for quantum computation". Physical Review A. 52 (5): 3457–3467. arXiv:quant-ph/9503016. Bibcode:1995PhRvA..52.3457B. doi:10.1103/PhysRevA.52.3457. PMID 9912645. S2CID 8764584. Retrieved 2009-08-18.
6. Childs, Andrew M.; Richard Cleve; Enrico Deotto; Edward Farhi; Sam Gutmann; Daniel A. Spielman (2003). "Exponential algorithmic speedup by a quantum walk". Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. San Diego, CA, USA: ACM. pp. 59–68. arXiv:quant-ph/0209131. doi:10.1145/780542.780552. ISBN 1-58113-674-9. Retrieved 2009-08-18.
7. Beals, Robert; Harry Buhrman; Richard Cleve; Michele Mosca; Ronald de Wolf (2001). "Quantum lower bounds by polynomials". J. ACM. 48 (4): 778–797. arXiv:quant-ph/9802049. doi:10.1145/502090.502097. Retrieved 2009-08-18.
8. Buhrman, Harry; Richard Cleve; Avi Wigderson (1998). "Quantum vs. classical communication and computation". Proceedings of the thirtieth annual ACM symposium on Theory of computing. Dallas, Texas, United States: ACM. pp. 63–68. arXiv:quant-ph/9802040. doi:10.1145/276698.276713. ISBN 0-89791-962-9. Retrieved 2009-08-18.
9. List of editors of Quantum Information & Computation
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Richard D. Gill
Richard David Gill (born 11 September 1951) is a mathematician born in the United Kingdom who has lived in the Netherlands since 1974. As a probability theorist and statistician, Gill is most well known for his research on counting processes and survival analysis, some of which has appeared in an advanced textbook. Now retired, he was the chair of mathematical statistics at Leiden University. Gill is also known for his pro bono consulting and advocacy on behalf of victims of incompetent statistical testimony, including a Dutch nurse who was wrongfully convicted and jailed for six years.
Richard David Gill
Richard D. Gill
Born (1951-09-11) 11 September 1951
Redhill, Surrey
CitizenshipBritish
Alma materUniversity of Cambridge
Free University of Amsterdam (PhD)
Scientific career
InstitutionsUtrecht University
Leiden University
Doctoral studentsSara van de Geer, Mark van der Laan
Biography
He studied mathematics at the University of Cambridge (1970–1973), and subsequently followed the Diploma of Statistics course there (1973–1974).
Marrying a Dutch woman, he moved to the Netherlands where he worked from 1974 to 1988 at the Mathematical Centre (later renamed Centrum Wiskunde & Informatica, or CWI) of Amsterdam. In 1979, Gill obtained his PhD with the thesis Censoring and Stochastic Integrals, which was supervised by Jacobus Oosterhoff of the Vrije Universiteit, which awarded the doctorate.[1] Gill spent Autumn 1980 at the Statistical Research Unit at the University of Copenhagen. Gill continued to collaborate with Danish (and Norwegian) statisticians for ten years, helping to write the book Statistical models based on counting processes, which is often referred to as "ABGK" (for the authors Andersen, Borgan, Gill, and Keiding).[2] In 1983 he became the head of the Department of Mathematical Statistics at CWI.
In 1988 he moved to the Department of Mathematics of Utrecht University. Gill became the chair in mathematical stochastics—this chair represented the three mathematical sciences of mathematical statistics, probability theory, and operations research. His PhD students include Sara van de Geer and Mark van der Laan.[1]
In 2006, he moved to the Department of Mathematics at Leiden University, where he became the chair of mathematical statistics. Since then, he has conducted statistical research in the theory of quantum information, forensic statistics, scientific integrity and in biostatistics. He has also worked on survival analysis, semiparametric models, causality, missing data, machine learning, and statistics in image analysis. Gill also publishes on the foundations of several mathematical sciences: the foundations of statistics, of probability, of mathematics, and of quantum physics. He reached the mandatory retirement age in 2017, and continues with research and consultancy.
Statistical advocacy against wrongful convictions
In recent years he has lobbied for retrials for Lucia de Berk, Kevin Sweeney and Benjamin Geen. The nurse Lucia de Berk was sentenced to life imprisonment, after a legal psychologist gave testimony that there was great likelihood that de Berk committed a string of murders.
The court was told by Dr Henk Elffers of the Netherlands Institute for the Study of Crime and Law Enforcement that more children had died on her shifts than appeared possible by chance. He put the odds of her presence being a mere coincidence at one in 342 million, a figure that seemed to have blinded the court to any alternative explanation of the deaths.[3]
This statistical testimony was shown to be fallacious by professional statisticians, notably Gill. Continued scrutiny showed that the data had also been collected to support the prosecutor's conviction of Berk, which further invalidated the pseudo-statistical testimony.[4][5][6]
The conduct of the case, in Professor Gill's account, was extraordinary. Convinced she was guilty, the police and the managers of the Juliana Children's Hospital assembled a dossier in which it seemed every death became unnatural when it had occurred during, or after, a shift in which she had worked. For one of the alleged murders, it was established on appeal she had not even been in the hospital for three days around the time it occurred. Using more appropriate statistical methods reduced the odds from one in 342 million to one in 48. A further analysis by Professor Gill further reduces the odds to one in nine.[3]
Professor Gill helped in the campaign to have a new trial. Consequently, a retrial was ordered, and de Berk was found not guilty, and received a public apology from the Dutch government, along with financial compensation (amount unknown) for her six years of incarceration.[7][8]
Honors
Richard Gill is a member of the Royal Netherlands Academy of Arts and Sciences.[9] He is a past president of The Netherlands Society for Statistics and Operations Research, which publishes the journal Statistica Neerlandica.[10] Gill was selected as the 2010–2011 Distinguished Lorentz Fellow by the Netherlands Institute for Advanced Study in Humanities and Social Sciences.[11]
References
1. Richard David Gill at the Mathematics Genealogy Project
2. Andersen, Per Kragh; Borgan, Ørnulf; Gill, Richard D.; Keiding, Niels (1993). Statistical models based on counting processes. Springer series in statistics. New York: Springer-Verlag. pp. xii+767. ISBN 978-0-387-97872-7. MR 1198884.
3. Hawkes, Nigel (10 April 2010). "Did statistics damn Lucia de Berk?: Behind the numbers". The Independent.
4. Mark Buchanan (18 January 2007). "Statistics: conviction by numbers" (PDF). Nature. 445 (7125): 254–255. Bibcode:2007Natur.445..254B. doi:10.1038/445254a. PMID 17230166. S2CID 4419275.
5. Persbericht CWI Archived 8 April 2008 at the Wayback Machine Petitie 2 November 2007
6. "Expert on the most important proof in the Lucia de B. case: 'This baby has not been poisoned'". Archived 4 October 2008 at the Wayback Machine NOVA. 29 September 2007.
7. "Nurse Lucia de Berk finally found not guilty of murdering seven patients". 14 April 2010. Archived from the original on 24 July 2011.
8. "Apology for nurse jailed for murdering seven patients", AP, The Independent 14 April 2010.
9. "Richard Gill" (in Dutch). Royal Netherlands Academy of Arts and Sciences. Archived from the original on 15 July 2015. Retrieved 15 July 2015.
10. Board Archived 13 November 2009 at the Wayback Machine, Netherlands Society for Statistics and Operations Research. Accessed 23 January 2010
11. Richard Gill Distinguished Lorentz Fellow 2010–2011, News release, Leiden University. Accessed 23 January 2010.
External links
• Richard Gill's homepage at Leiden University.
• Mathematical Reviews. "Richard D. Gill". Retrieved 22 February 2011.
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Richard E. Bellman
Richard Ernest Bellman[2] (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founded the leading biomathematical journal Mathematical Biosciences.
Richard Ernest Bellman
Born
Richard Ernest Bellman
(1920-08-26)August 26, 1920
New York City, New York, U.S.
DiedMarch 19, 1984(1984-03-19) (aged 63)
Los Angeles, California, U.S.
Alma mater
• Brooklyn College (BS)
• University of Wisconsin (MA)
• Princeton University (PhD)
Known forDynamic programming
Stochastic dynamic programming
Curse of dimensionality
Linear search problem
Bellman equation
Bellman–Ford algorithm
Bellman's lost in a forest problem
Bellman–Held–Karp algorithm
Grönwall–Bellman inequality
Hamilton–Jacobi–Bellman equation
AwardsJohn von Neumann Theory Prize (1976)
IEEE Medal of Honor (1979)
Richard E. Bellman Control Heritage Award (1984)
Scientific career
FieldsMathematics and Control theory
InstitutionsUniversity of Southern California
Rand Corporation
Stanford University
ThesisOn the Boundedness of Solutions of Non-Linear Differential and Difference Equations[1]
Doctoral advisorSolomon Lefschetz[1]
Doctoral studentsChristine Shoemaker[1]
Biography
Bellman was born in 1920 in New York City to non-practising[3] Jewish parents of Polish and Russian descent, Pearl (née Saffian) and John James Bellman,[4] who ran a small grocery store on Bergen Street near Prospect Park, Brooklyn.[5] On his religious views, he was an atheist.[6] He attended Abraham Lincoln High School, Brooklyn in 1937,[4] and studied mathematics at Brooklyn College where he earned a BA in 1941. He later earned an MA from the University of Wisconsin. During World War II he worked for a Theoretical Physics Division group in Los Alamos. In 1946 he received his Ph.D. at Princeton University under the supervision of Solomon Lefschetz.[7] Beginning 1949 Bellman worked for many years at RAND corporation and it was during this time that he developed dynamic programming.[8]
Later in life, Richard Bellman's interests began to emphasize biology and medicine, which he identified as "the frontiers of contemporary science". In 1967, he became founding editor of the journal Mathematical Biosciences, which rapidly became (and remains) one of the most important journals in the field of Mathematical Biology. In 1985, the Bellman Prize in Mathematical Biosciences was created in his honor, being awarded biannually to the journal's best research paper.
Bellman was diagnosed with a brain tumor in 1973, which was removed but resulted in complications that left him severely disabled. He was a professor at the University of Southern California, a Fellow in the American Academy of Arts and Sciences (1975),[9] a member of the National Academy of Engineering (1977),[10] and a member of the National Academy of Sciences (1983).
He was awarded the IEEE Medal of Honor in 1979, "for contributions to decision processes and control system theory, particularly the creation and application of dynamic programming".[11] His key work is the Bellman equation.
Work
Bellman equation
A Bellman equation, also known as a dynamic programming equation, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. Almost any problem which can be solved using optimal control theory can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory.[12]
Hamilton–Jacobi–Bellman equation
The Hamilton–Jacobi–Bellman equation (HJB) is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given dynamical system with an associated cost function. Classical variational problems, for example, the brachistochrone problem can be solved using this method as well. The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton–Jacobi equation by William Rowan Hamilton and Carl Gustav Jacob Jacobi.[13]
Curse of dimensionality
Main article: Curse of dimensionality
The curse of dimensionality is an expression coined by Bellman to describe the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space. One implication of the curse of dimensionality is that some methods for numerical solution of the Bellman equation require vastly more computer time when there are more state variables in the value function. For example, 100 evenly spaced sample points suffice to sample a unit interval with no more than 0.01 distance between points; an equivalent sampling of a 10-dimensional unit hypercube with a lattice with a spacing of 0.01 between adjacent points would require 1020 sample points: thus, in some sense, the 10-dimensional hypercube can be said to be a factor of 1018 "larger" than the unit interval. (Adapted from an example by R. E. Bellman, see below.) [14]
Bellman–Ford algorithm
Though discovering the algorithm after Ford he is referred to in the Bellman–Ford algorithm, also sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a weighted digraph where some of the edge weights may be negative. Dijkstra's algorithm accomplishes the same problem with a lower running time, but requires edge weights to be non-negative.
Publications
Over the course of his career he published 619 papers and 39 books. During the last 11 years of his life he published over 100 papers despite suffering from crippling complications of brain surgery (Dreyfus, 2003). A selection:[4]
• 1957. Dynamic Programming
• 1959. Asymptotic Behavior of Solutions of Differential Equations
• 1961. An Introduction to Inequalities
• 1961. Adaptive Control Processes: A Guided Tour
• 1962. Applied Dynamic Programming
• 1967. Introduction to the Mathematical Theory of Control Processes
• 1970. Algorithms, Graphs and Computers
• 1972. Dynamic Programming and Partial Differential Equations
• 1982. Mathematical Aspects of Scheduling and Applications
• 1983. Mathematical Methods in Medicine
• 1984. Partial Differential Equations
• 1984. Eye of the Hurricane: An Autobiography, World Scientific Publishing.
• 1985. Artificial Intelligence
• 1995. Modern Elementary Differential Equations
• 1997. Introduction to Matrix Analysis
• 2003. Dynamic Programming
• 2003. Perturbation Techniques in Mathematics, Engineering and Physics
• 2003. Stability Theory of Differential Equations (originally publ. 1953)[15]
References
1. Richard E. Bellman at the Mathematics Genealogy Project
2. Richard Bellman's Biography
3. Robert S. Roth, ed. (1986). The Bellman Continuum: A Collection of the Works of Richard E. Bellman. World Scientific. p. 4. ISBN 9789971500900. He was raised by his father to be a religious skeptic. He was taken to a different church every week to observe different ceremonies. He was struck by the contrast between the ideals of various religions and the history of cruelty and hypocrisy done in God's name. He was well aware of the intellectual giants who believed in God, but if asked, he would say that each person had to make their own choice. Statements such as "By the State of New York and God ..." struck him as ludicrous. From his childhood he recalled a particularly unpleasant scene between his parents just before they sent him to the store. He ran down the street saying over and over again, "I wish there was a God, I wish there was a God."
4. Salvador Sanabria. Richard Bellman profile at http://www-math.cudenver.edu; retrieved October 3, 2008.
5. Bellman biodata at history.mcs.st-andrews.ac.uk; retrieved August 10, 2013.
6. Richard Bellman (June 1984). "Growing Up in New York City". Eye Of The Hurricane. World Scientific Publishing Company. p. 7. ISBN 9789814635707. Retrieved 5 July 2021. Naturally, I was raised as an atheist. This was quite easy since the only one in the family that had any religion was my grandmother, and she was of German stock. Although she believed in God, and went to the synagogue on the high holy days, there was no nonsense about ritual. I well remember when I went off to the army, she said, "God will protect you." I smiled politely. She added, "I know you don't believe in God, but he will protect you anyway." I know many sophisticated and highly intelligent people who are practicing Catholics, Protestants, Jews, Mormons, Hindus, Buddhists, etc., feel strongly that religion, or lack of it, is a highly personal matter. My own attitude is like Lagrange's. One day, he was asked by Napoleon whether he believed in God. "Sire," he said, "I have no need of that hypothesis."
7. Mathematics Genealogy Project
8. Bellman R: An introduction to the theory of dynamic programming RAND Corp. Report 1953 (Based on unpublished researches from 1949. It contained the first statement of the principle of optimality)
9. "Book of Members, 1780–2010: Chapter B" (PDF). American Academy of Arts and Sciences. Retrieved April 6, 2011.
10. "NAE Members Directory – Dr. Richard Bellman profile". NAE. Retrieved April 6, 2011.
11. "IEEE Medal of Honor Recipients" (PDF). IEEE. Retrieved April 6, 2011.
12. Ljungqvist, Lars; Sargent, Thomas J. (2012). Recursive Macroeconomic Theory (3rd ed.). MIT Press. ISBN 978-0-262-31202-8.
13. Kamien, Morton I.; Schwartz, Nancy L. (1991). Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management (2nd ed.). Amsterdam: Elsevier. pp. 259–263. ISBN 9780486488561.
14. Richard Bellman (1961). Adaptive control processes: a guided tour. Princeton University Press.
15. Haas, F. (1954). "Review: Stability theory of differential equations, by R. Bellman". Bull. Amer. Math. Soc. 60 (4): 400–401. doi:10.1090/s0002-9904-1954-09830-0.
Further reading
• Bellman, Richard (1984). Eye of the Hurricane: An Autobiography, World Scientific.
• Stuart Dreyfus (2002). "Richard Bellman on the Birth of Dynamic Programming". In: Operations Research. Vol. 50, No. 1, Jan–Feb 2002, pp. 48–51.
• J.J. O'Connor and E.F. Robertson (2005). Biography of Richard Bellman from the MacTutor History of Mathematics.
• Stuart Dreyfus (2003) "Richard Ernest Bellman". In: International Transactions in Operational Research. Vol 10, no. 5, pp. 543–545.
Articles
• Bellman, R.E, Kalaba, R.E, Dynamic Programming and Feedback Control, RAND Corporation, P-1778, 1959.
External links
• "IEEE Global History Network – Richard Bellman". IEEE. 14 August 2017. Retrieved April 6, 2011.
• Harold J. Kushner's speech on Richard Bellman, when accepting the Richard E. Bellman Control Heritage Award (click on "2004: Harold J. Kushner")
• IEEE biography
• Richard E. Bellman at the Mathematics Genealogy Project
• Author profile in the database zbMATH
• Biography of Richard Bellman from the Institute for Operations Research and the Management Sciences (INFORMS)
IEEE Medal of Honor
1976–2000
• H. Earle Vaughan (1977)
• Robert Noyce (1978)
• Richard Bellman (1979)
• William Shockley (1980)
• Sidney Darlington (1981)
• John Tukey (1982)
• Nicolaas Bloembergen (1983)
• Norman Ramsey (1984)
• John Roy Whinnery (1985)
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AACC Richard E. Bellman Control Heritage Award
1979–2000
• Hendrik Wade Bode (1979)
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• Lotfi A. Zadeh (1998)
• Yu-Chi Ho (1999)
• W. Harmon Ray (2000)
2001–present
• A. V. Balakrishnan (2001)
• Petar V. Kokotovic (2002)
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• Tamer Başar (2006)
• Sanjoy K. Mitter (2007)
• Pravin Varaiya (2008)
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• Manfred Morari (2011)
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• Dimitri Bertsekas (2014)
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John von Neumann Theory Prize
1975–1999
• George Dantzig (1975)
• Richard Bellman (1976)
• Felix Pollaczek (1977)
• John F. Nash / Carlton E. Lemke (1978)
• David Blackwell (1979)
• David Gale / Harold W. Kuhn / Albert W. Tucker (1980)
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• Dimitri Bertsekas / John Tsitsiklis (2018)
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Richard Ehrenborg
Richard Ehrenborg is a Swedish mathematician working in algebraic combinatorics.[1] He is known for developing the quasisymmetric function of a poset.[2] He currently holds the Ralph E. and Norma L. Edwards Research Professorship at the University of Kentucky [3] and is the first recipient of the Royster Research Professor at University of Kentucky.[4]
Richard Ehrenborg
NationalitySwedish
Alma materMassachusetts Institute of Technology
Scientific career
FieldsMathematics
InstitutionsUniversity of Kentucky
ThesisCombinatorial methods in multilinear algebra (1993)
Doctoral advisorGian-Carlo Rota
Websitehttps://www.ms.uky.edu/~jrge/
Ehrenborg earned his Ph.D. from MIT in 1993[5] under the supervision of Gian-Carlo Rota. He is a descendant of another Richard Ehrenborg,[3] (born 1655) who was a professor and Rektor of Lund University. He is also a juggler and magician.[6]
Selected publications
• Ehrenborg, Richard (1996). "On posets and Hopf algebras". Advances in Mathematics. 119 (1): 1–25. doi:10.1006/aima.1996.0026. ISSN 0001-8708. MR 1383883.
• Ehrenborg, Richard; Skinner, Chris M (1995). "The blind bartender's problem". Journal of Combinatorial Theory, Series A. 70 (2): 249–266. doi:10.1016/0097-3165(95)90092-6. ISSN 0097-3165. MR 1329391.
See also
• Four glasses puzzle
External links
• "Richard Ehrenborg's Homepage". Retrieved September 27, 2019.
References
1. "Richard Ehrenborg profile page at IAS". Retrieved September 27, 2019.
2. Ehrenborg, Richard (1996). "On posets and Hopf algebras". Advances in Mathematics. 119 (1): 1–25. doi:10.1006/aima.1996.0026. ISSN 0001-8708. MR 1383883.
3. Allen, Jennifer T. "Mathematics Names New Royster, Edwards Chairs - Mathematics". University of Kentucky Mathematics Department. Retrieved September 27, 2019.
4. "Faculty News" (PDF). University of Kentucky Mathematics Department newsletter. Fall 2006. Retrieved September 27, 2019.
5. Richard Ehrenborg at the Mathematics Genealogy Project
6. "Department of Mathematics to present distinguished researcher Richard Ehrenborg". Marshall University newsletter. Retrieved September 27, 2019.
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Richard Eliot Chamberlin
Richard Eliot Chamberlin (20 March 1923, Cambridge, Massachusetts – 14 March 1994)[1] was an American mathematician, specializing in geometric topology.
R. Eliot Chamberlin's father was Ralph Vary Chamberlin. Eliot Chamberlin attended East High School in Salt Lake City.[1] He received his bachelor's degree from the University of Utah. In the early 1940s he was a teaching fellow in physics at the University of Utah and then the Massachusetts Institute of Technology. After serving as an instructor of physics at Northeastern University, he served two years in the United States Navy during World War II. After discharge from the Navy, he entered graduate school in mathematics at Harvard University, and received his Ph.D. in 1950 with thesis supervisor Hassler Whitney.[2]
Chamberlin joined the faculty of the mathematics department at the University of Utah in 1949 and retired there as professor emeritus on 1 July 1988.[3] Chamberlin gave an invited address at the International Congress of Mathematicians in 1950 in Cambridge, Massachusetts.
Selected publications
• Chamberlin, Richard Eliot; Wolfe, Jr., James Harold (1953). "Multiplicative homomorphisms of matrices". Proceedings of the American Mathematical Society. 4 (1): 37–42. doi:10.2307/2032198. JSTOR 2032198. MR 0052382.
• Chamberlin, Richard Eliot; Wolfe, Jr., James Harold (1954). "Note on a converse of Lucas's theorem". Proceedings of the American Mathematical Society. 5 (2): 203–205. doi:10.2307/2032224. JSTOR 2032224. MR 0061207.
• Chamberlin, Richard Eliot (1959). "A class of unknotted curves in 3-space". Proceedings of the American Mathematical Society. 10 (1): 149–157. doi:10.2307/2032904. JSTOR 2032904. MR 0100270.
• Chamberlin, Richard Eliot; Case, James Hughson (1960). "Characterizations of tree-like continua". Pacific Journal of Mathematics. 10 (1): 73–84. doi:10.2140/pjm.1960.10.73. MR 0111000.
References
1. Death: Dr. R. Eliot Chamberlin, Deseret News, 16 March 1994
2. Richard Eliot Chamberlin at the Mathematics Genealogy Project
3. Mathematics Department Newsletter, (1987–1988), math.utah.edu
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Richard Harington
The Rev. Richard Harington (5 May 1800 – 13 December 1853) was an Oxford college head in the 19th century.[1]
Barker was born in Hanover Square, Westminster and educated at Brasenose College, Oxford.[2] A mathematician, he was Principal of Brasenose[3] from 1842[4] until his death.[5]
Notes
1. Victorian Web
2. Foster, Joseph (1888–1892). "Harington, Richard (2)" . Alumni Oxonienses: the Members of the University of Oxford, 1715–1886. Oxford: Parker and Co – via Wikisource.
3. BNC web-site
4. 'University And Clerical Intelligence' The Times Monday, 13 June 1842 Issue 18008 p.6
5. 'Deaths' The Times Friday, 16 December 1853 Issue 21613 p.9
Principals of Brasenose College, Oxford
• Matthew Smyth
• John Hawarden
• Thomas Blanchard
• Richard Harris
• Alexander Nowell
• Thomas Singleton
• Samuel Radcliffe
• Daniel Greenwood
• Thomas Yate
• John Meare
• Robert Shippen
• Francis Yarborough
• William Gwyn
• Ralph Cawley
• Thomas Barker
• William Cleaver
• Frodsham Hodson
• Ashurst Gilbert
• Richard Harington
• Edward Hartopp Cradock
• Albert Watson
• Charles Buller Heberden
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• William Stallybrass
• Hugh Last
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R. H. Bruck
Richard Hubert Bruck (December 26, 1914 – December 18, 1991) was an American mathematician best known for his work in the field of algebra, especially in its relation to projective geometry and combinatorics.
R. H. Bruck
Bruck (right) with Karl W. Gruenberg (center) and Kurt Hirsch
Born(1914-12-26)December 26, 1914
DiedDecember 18, 1991(1991-12-18) (aged 76)
Alma materUniversity of Toronto
Known forLoops, Bruck–Ryser Theorem, Finite Nets, Bruck–Bose Construction
SpouseHelen
AwardsGuggenheim Fellowship
Chauvenet Prize (1956)
Scientific career
FieldsMathematics
InstitutionsUniversity of Wisconsin–Madison
ThesisThe General Linear Group in a Field of Characteristic p (1940)
Doctoral advisorRichard Brauer
Doctoral studentsGeorge I. Glauberman
Michael G. Aschbacher
Sue Whitesides
Bruck studied at the University of Toronto, where he received his doctorate in 1940 under the supervision of Richard Brauer.[1] He spent most his career as a professor at University of Wisconsin–Madison, advising at least 31 doctoral students.
He is best known for his 1949 paper coauthored with H. J. Ryser, the results of which became known as the Bruck–Ryser theorem (now known in a generalized form as the Bruck-Ryser-Chowla theorem), concerning the possible orders of finite projective planes.
In 1946, he was awarded a Guggenheim Fellowship. In 1956, he was awarded the Chauvenet Prize for his article Recent Advances in the Foundations of Euclidean Plane Geometry.[2] In 1962, he was an invited speaker at the International Congress of Mathematicians in Stockholm. In 1963, he was a Fulbright Lecturer at the University of Canberra. In 1965 a Groups and Geometry conference was held at the University of Wisconsin in honor of Bruck's retirement.
Dick Bruck and his wife Helen were supporters of the fine arts. They were patrons of the regional American Players Theatre in Wisconsin.[3]
Selected publications
• Bruck, R.H. (1946), "Contributions to the theory of loops", Transactions of the American Mathematical Society, 60 (2): 245–354, doi:10.2307/1990147, JSTOR 1990147
• Bruck, R. H.; Ryser, H. J. (1949). "The nonexistence of certain finite projective planes". Canadian Journal of Mathematics. 1: 88–92. doi:10.4153/CJM-1949-009-2. S2CID 123440808.
• Bruck, R.H.; Kleinfeld, Erwin (1951), "The structure of alternative division rings", Proceedings of the American Mathematical Society, 2 (6): 878–890, doi:10.1090/s0002-9939-1951-0045099-9, PMC 1063309, PMID 16578361
• Bruck, R.H. (1951), "Finite Nets.I.Numerical invariants", Canadian Journal of Mathematics, 3 (1): 94–106, doi:10.4153/cjm-1951-012-7, hdl:10338.dmlcz/101384, S2CID 124575806
• Bruck, R.H. (1963), "Finite Nets.II.Uniqueness and imbedding", Pacific Journal of Mathematics, 13 (2): 421–457, doi:10.2140/pjm.1963.13.421
• Bruck, R. H. (1955). "Recent Advances in the Foundations of Euclidean Geometry". The American Mathematical Monthly. Mathematical Association of America. 62 (7): 2–17. doi:10.2307/2308175. JSTOR 2308175.
• Bruck, R.H. (1955), "Difference sets in a finite group", Transactions of the American Mathematical Society, 78 (2): 464–481, doi:10.1090/s0002-9947-1955-0069791-3
• Bruck, R. H. (1958), A Survey of Binary Systems, Berlin: Springer-Verlag (3rd ed. in 1971, ISBN 978-0-387-03497-3)
• Bruck, R. H. (1960), "Some theorems on Moufang loops", Math. Z., 73 (1): 59–78, doi:10.1007/bf01163269, S2CID 121239766
• Bruck, R.H.; Bose, R.C. (1964), "The construction of translation planes from projective spaces", Journal of Algebra, 1: 85–102, doi:10.1016/0021-8693(64)90010-9
Notes
1. R. H. Bruck at the Mathematics Genealogy Project
2. "The Mathematical Association of America's Chauvenet Prize". Mathematical Association of America. Retrieved August 10, 2012.
3. Listed in the summer 1985 playbill of the APT as patron contributors in 1984.
External links
• Biography at the University of Texas
• Bruck–Ryser–Chowla Theorem at Mathworld
Chauvenet Prize recipients
• 1925 G. A. Bliss
• 1929 T. H. Hildebrandt
• 1932 G. H. Hardy
• 1935 Dunham Jackson
• 1938 G. T. Whyburn
• 1941 Saunders Mac Lane
• 1944 R. H. Cameron
• 1947 Paul Halmos
• 1950 Mark Kac
• 1953 E. J. McShane
• 1956 Richard H. Bruck
• 1960 Cornelius Lanczos
• 1963 Philip J. Davis
• 1964 Leon Henkin
• 1965 Jack K. Hale and Joseph P. LaSalle
• 1967 Guido Weiss
• 1968 Mark Kac
• 1970 Shiing-Shen Chern
• 1971 Norman Levinson
• 1972 François Trèves
• 1973 Carl D. Olds
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• 1976 Lawrence Zalcman
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• 1978 Shreeram S. Abhyankar
• 1979 Neil J. A. Sloane
• 1980 Heinz Bauer
• 1981 Kenneth I. Gross
• 1982 No award given.
• 1983 No award given.
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Richard Jack (mathematician)
Richard Jack (died 1759) was a Scottish mathematician, astronomer, and engineer active in the mid-18th century. He provided the only testimony against Lt. Gen. John Cope at the court martial following the Battle of Prestonpans during the 1745 Jacobite uprising, but Having exaggerated his own accomplishments and lacking corroboration, Jack had his testimony discounted by the judges, who found Cope blameless. Jack was later involved in the development of achromatic lenses and his work on conic sections was a source for the mathematical sections of the first edition of the Encyclopaedia Britannica.
Life
Richard Jack was born in Scotland, in the Great Britain, probably between 1710 and 1715.[1] He married Elizabeth Brown on 14 March 1737,[2] around the time that he was teaching mathematics in Newcastle-upon-Tyne in 1737.[1] The couple later had a son, also named Richard.[3] Jack lectured on mathematics in Edinburgh, probably from 1739 to 1743 when he ran advertisements in the Caledonian Mercury.[1] He enjoyed the patronage of Hugh Hume-Campbell, the 3rd earl of Marchmont,[4] whom he was helping to observe sunspots.[5]
During Charles Edward Stuart's Jacobite uprising in 1745, Jack remained loyal to George II. Volunteering, he was entrusted with arranging some of the placement of the cannon at Edinburgh[6] as a fortifications engineer[4] under the guidance of Prof. Colin MacLaurin.[7] He fled with most of the other Hanoverian forces ahead of Charles's unopposed entrance into Edinburgh on 17 September.[4] He then performed reconnaissance on the Stuart forces, counting and evaluating the men in the main force encamped on Arthur's Seat on 19 September.[6] He claimed that on the next day he had assisted with the planning of the artillery placement[8] and then personally fired two cannons, dislodging Stuart men from the church at Tranent. Other witnesses later averred Jack had claimed knowledge of the theory of gunnery but had not been involved in any of the army's strategy;[9] he had scouted some areas and helped direct artillery fire against the men in Tranent after almost being killed by them, but he had proven so completely inept at the cannons' operation that he never fired them himself.[9] The following day was the 21 September Battle of Prestonpans, during which Jack said he and four sailors worked the same two cannon while Lt. Col. Alan Whiteford and five sailors worked the other four; Lt. Col. Whiteford and other witnesses, however, said the nine sailors had fled before the battle, Jack had been sent away as useless and wasn't seen in the fighting, and Whiteford had been forced to operate all six guns on his own.[10] A year later, on 24 September 1746,[6] Jack was the only eyewitness to testify under oath at the court martial of the Whig commander Lt. Gen. John Cope.[1] He testified that he had seen three officers—probably but not certainly including Cope—flee the battlefield ahead of the general defeat after a Highland charge.[11] Given his inflated claims in other matters and lack of corroboration, however, the court discounted his testimony and exonerated Cope,[4][12] although the general never again held high position.[13]
On 25 May 1750, Jack received patent #656 for a "quadrant for taking the altitude of the sun or moon by refraction" and also "a refracting telescope with four spherical lenses"[14] jointly with the successful London instrument maker George Adams.[5][15] They claimed that their design offered 30 levels of magnification[16] and eliminated color aberrations. Although those claims were vigorously disputed by rival instrument makers,[17] Jack and Adams seem to have been vindicated by some practical and authoritative test in early 1752 and made a sizeable profit on the design.[18] None of the devices are known to still exist[19] but, on the basis of surviving records, Millburn considers it likely that representatives of the Admiralty or Board of Ordnance praised the patent telescope's high level of magnification—the most essential attribute for long-range fire—despite the accuracy of other complaints about its faults.[20] Jack advertised his lectures on math in London in 1751 and 1754.[1] He also lectured on experimental philosophy, fortification, and gunnery.[21]
Jack died in 1759.[1] The advertisement for his probate auction stated that he had been "assistant engineer in the late expedition against Guadaloupe",[3] a French colony captured by British forces under Maj. Gen. Peregrine Hopson after a six-month siege during the Seven Years' War. His effects included an air pump, microscope, telescopes, and other scientific instruments.[19]
Works
Jack wrote three major works:[1]
• Elements of Conic Sections in Three Books, in Which Are Demonstrated the Principal Properties of the Parabola, Ellipse, and Hyperbola, Edinburgh: Walter and Thomas Ruddiman, 1742.
• The Mathematical Principles of Theology, or, The Existence of God Geometrically Demonstrated in Three Books..., London: George Hawkins, 1747.
• Euclid's Data, Restored to Their True and Genuine Order, Agreeable to Pappus Alexandrinus's Account of Them, in His Preface to the Seventh Book of His Mathematical Collections, London: Andrew Millar, 1756.
Jack also composed most of a fourth book, The Doctrine of Proportion Geometrically Demonstrated, prior to 1745 but lost its manuscript when Stuart forces ransacked his home during their occupation of Edinburgh.[22]
Jack's work on conic sections was a major reference for the 1771 first edition of the Encyclopaedia Britannica.[1] His work on a geometrical proof of the existence of God, however, was generally held in low repute and considered by MacFarlane to be "one of the most absurd" attempts to apply mathematical reasoning to theological questions.[23] Gillespie described it as "a specimen of impure Mathematics, gone deplorably out of their road".[24] Writing under the pen name Antitheos, George Simpson offered that Jack's arguments "may afford grounds for curious speculation respecting that bias toward absurdity which is too frequently found to beset the human mind".[25]
References
Wikisource has original text related to this article:
Author:Richard Jack
Citations
1. Johnson (1993), p. 225.
2. Grant (1915), p. 63.
3. Johnson (1993), p. 230.
4. Johnson (1993), p. 227.
5. Johnson (1993), p. 229.
6. Robins (1749), p. 83.
7. Robins (1749), p. 84.
8. Robins (1749), p. 85.
9. Robins (1749), p. 86.
10. Robins (1749), p. 88–89.
11. Robins (1749), p. 90.
12. Robins (1749), p. 100.
13. Dalton (1904), p. 269.
14. Biagioli (2006), p. i.
15. Gee (2014), p. 55.
16. Gee (2014), p. 60.
17. Gee (2014), p. 56.
18. Millburn (2017), p. 62–63.
19. Millburn (2017), p. 61.
20. Millburn (2017), p. 63–65.
21. Millburn (2017), p. 60–61.
22. Gillespie (1863), p. 72.
23. MacFarlane (1836), pp. 203.
24. Gillespie (1863), p. 63.
25. Gillespie (1863), p. 70.
Bibliography
• Biagioli, Mario (2006), "From Prints to Patents: Living on Instruments in Early Modern Europe", Appendix: "Early Modern Instruments Patents Database, 1500–1800" (PDF), History of Science, pp. 139–186.
• Dalton, Charles (1904), English Army Lists and Commission Registers, 1661–1714, vol. V, London: Eyre and Spottiswood
• Gee, Brian (2014), Francis Watkins and the Dolland Telescope Patent Controversy, Abingdon: Routledge, ISBN 9781317133308.
• Gillespie, William Honyman (1863), "A Review of Mr Richard Jack's Work, Entitled 'Mathematical Principles of Theology, or The Existence of God Geometrically Demonstrated'", The Necessary Existence of God, London: Houlston & Wright, pp. 63–73.
• Grant, Francis James, ed. (1915), Parish of Holyroodhouse or Canongate: Register of Marriages, 1564–1800, Edinburgh: Scottish Record Society.
• Johnson, William (July 1993), "Richard Jack and Henry Baker, F.R.S., in the Late Summer of 1746", Notes and Records of the Royal Society of London, vol. 47, London: Royal Society, pp. 225–231.
• MacFarlane, Charles, ed. (1836), "XXVIII. Miscellaneous Points in the History of the Sciences", The Book of Table-Talk, vol. I, London: Charles Knight, pp. 188-167.
• Millburn, John R. (2017), Adams of Fleet Street: Instrument Makers to King George III, Abingdon: Routledge, ISBN 9781351960830.
• Robins, Benjamin (1749), A Report of the Proceedings and Opinions of the Board of General Officers, on Their Examination into the Conduct, Behaviour, and Proceedings of Lieutenant-General Sir John Cope, Knight of the Bath, Colonel Peregrine Lascelles, and Brigadier-General Thomas Fowke from the Time of the Breaking Out of the Rebellion in North-Britain in the Year 1845, till the Action at Preston-Pans Inclusive..., Dublin: George Faulkner.
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Wikipedia
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Richard K. Guy
Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary.[1] He is known for his work in number theory, geometry, recreational mathematics, combinatorics, and graph theory.[2][3] He is best known for co-authorship (with John Conway and Elwyn Berlekamp) of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory.[4] He published more than 300 scholarly articles.[5] Guy proposed the partially tongue-in-cheek "strong law of small numbers", which says there are not enough small integers available for the many tasks assigned to them – thus explaining many coincidences and patterns found among numerous cultures.[6] For this paper he received the MAA Lester R. Ford Award.[7]
Richard K. Guy
Guy in 2005
Born
Richard Kenneth Guy
(1916-09-30)30 September 1916
Nuneaton, England
Died9 March 2020(2020-03-09) (aged 103)
Calgary, Alberta, Canada
NationalityBritish/Canadian
Alma materGonville and Caius College, Cambridge
(B.A. in 1938, M.A. in 1941)
Known forRecreational mathematics
Strong law of small numbers
Unistable polyhedron
AwardsLester R. Ford Award (1989)
Scientific career
FieldsMathematics
InstitutionsUniversity of Calgary
Websitescience.ucalgary.ca/mathematics-statistics/about/richard-guy
Biography
Early life
Guy was born 30 September 1916 in Nuneaton, Warwickshire, England, to Adeline Augusta Tanner and William Alexander Charles Guy. Both of his parents were teachers, rising to the rank of headmistress and headmaster, respectively. He attended Warwick School for Boys, the third oldest school in Britain, but was not enthusiastic about most of the curriculum. He was good at sports and excelled in mathematics. At the age of 17 he read Dickson's History of the Theory of Numbers. He said it was better than "the whole works of Shakespeare", solidifying his lifelong interest in mathematics.[8]
In 1935 Guy entered Gonville and Caius College, Cambridge, as a result of winning several scholarships. To win the most important of these he had to travel to Cambridge and write exams for two days. His interest in games began while at Cambridge where he became an avid composer of chess problems.[9] In 1938, he was graduated with a second-class honours degree; he would later state that his failure to get a first may have been related to his obsession with chess.[10] Although his parents strongly advised against it, Guy decided to become a teacher and got a teaching diploma at the University of Birmingham. He met his future wife, Nancy Louise Thirian, through her brother Michael, who was a fellow scholarship winner at Gonville and Caius. He and Louise shared loves of mountain climbing and dancing. They married in December 1940.
War years
In November 1942, Guy received an emergency commission in the Meteorological Branch of the Royal Air Force, with the rank of flight lieutenant.[11] He was posted to Reykjavik, and later to Bermuda, as a meteorologist. He tried to get permission for Louise to join him but was refused. While in Iceland, he did some glacier travel, skiing, and mountain climbing, marking the beginning of another long love affair, this one with snow and ice.[12] When Guy returned to England after the war, he went back to teaching, this time at Stockport Grammar School, but stayed only two years. In 1947 the family moved to London, where he got a job teaching mathematics at Goldsmiths' College.[13]
Later life and death
Wikinews has related news:
• British mathematician Richard K. Guy dies at 103
In 1951 he moved to Singapore, where he taught at the University of Malaya until 1962. He then spent a few years at the Indian Institute of Technology in Delhi, India. While they were in India, he and Louise went mountaineering in the foothills of the Himalayas.[14] Guy moved to Canada in 1965, settling down at the University of Calgary in Alberta, where he obtained a professorship.[15][16] Although he officially retired in 1982, he still went to the office five days a week to work, even as he passed the age of 100.[17] Along with George Thomas and John Selfridge, Guy taught at Canada/USA Mathcamp during its early years.[18]
In 1991 the University of Calgary awarded him an honorary doctorate. Guy said that they gave him the degree out of embarrassment, although the university stated that "his extensive research efforts and prolific writings in the field of number theory and combinatorics have added much to the underpinnings of game theory and its extensive application to many forms of human activity."[19] Guy and his wife Louise (who died in 2010) remained very committed to mountain hiking and environmentalism even in their later years. In 2014, he donated $100,000 to the Alpine Club of Canada for the training of amateur leaders.[20] In turn, the Alpine Club has honoured them by building the Louise and Richard Guy Hut near the base of Mont des Poilus.[21] They had three children, among them computer scientist and mathematician Michael J. T. Guy.
Guy died on 9 March 2020 at the age of 103.[22][23]
Mathematics
I love mathematics so much, and I love anybody who can do it well, so I just like to hang on and try to copy them as best I can, even though I'm not really in their league.[24]
– R. K. Guy
While teaching in Singapore in 1960 Guy met the Hungarian mathematician Paul Erdős. Erdős was noted for posing and solving difficult mathematical problems and shared several of them with Guy.[25] Guy later recalled "I made some progress in each of them. This gave me encouragement, and I began to think of myself as possibly being something of a research mathematician, which I hadn't done before."[26] Eventually he wrote four papers with Erdős, giving him an Erdős number of 1,[27] and solved one of Erdős' problems.[28] Guy was intrigued by unsolved problems and wrote two books devoted to them.[29][30] Many number theorists got their start trying to solve problems from Guy's book Unsolved problems in number theory.[31]
Guy described himself as an amateur mathematician,[32] although his work was widely respected by professionals.[33] In a career that spans eight decades he wrote or co-authored more than a dozen books and collaborated with some of the most important mathematicians of the twentieth century.[34] Paul Erdős, John H. Conway, Donald Knuth, and Martin Gardner were among his collaborators, as were Elwyn Berlekamp, John L. Selfridge, Kenneth Falconer, Frank Harary, Lee Sallows, Gerhard Ringel, Béla Bollobás, C. B. Lacampagne, Bruce Sagan, and Neil Sloane.[35]
Over the course of his career Guy published more than 100 research papers in mathematics, including four with Erdős.[36][37][38][39][40]
Guy was influential in the field of recreational mathematics. He collaborated with Berlekamp and Conway on two volumes of Winning Ways, which Martin Gardner described in 1998 as "the greatest contribution to recreational mathematics in this century".[41][42] Guy was considered briefly as a replacement for Gardner when the latter retired from the Mathematical Games column at Scientific American.[43] Guy conducted extensive research on Conway's Game of Life, and in 1970, discovered the game's glider.[44][45] Around 1968, Guy discovered a unistable polyhedron with 19 faces; no such construct with fewer faces was found until 2012. As of 2016 Guy still was active in conducting mathematical work.[46] To mark his 100th birthday friends and colleagues organised a celebration of his life and a tribute song and video was released by Gathering 4 Gardner.[47]
Guy was one of the original directors of the Number Theory Foundation and played an active role in supporting their efforts to "foster a spirit of cooperation and goodwill among the family of number theorists" for more than twenty years.[48][49]
Chess problems
From 1947 to 1951 Guy was the endings editor for British Chess Magazine.[50] He is known for almost 200 endgame studies. Along with Hugh Blandford and John Roycroft, he is one of the inventors of the GBR code (Guy–Blandford–Roycroft code), a system of representing the position of chess pieces on a chessboard. Publications including EG use it to classify endgame types and to index endgame studies.[51]
Richard Guy endgame composition: 1938
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution:
1. Kd1 Ka3
2. Kc1 a5
3. h4 a4
4. h5 Ka2
5. h6 a3
6. h7 Ka1
7. h8=N a2
8. Ng6 fxg6
9. f7 g5
10. f8=N g4
11. Ne6 dxe6
12. d7 e5
13. d8=N e4
14. Nc6 bxc6
15. b7 c5
16. Kd1 Kb2
17. b8=Q+ 1-0
Selected publications
Books
• 1975 (with John L. Selfridge) Optimal coverings of the square, North-Holland, Amsterdam, OCLC Number: 897757276.
• 1976 Packing [1, n] with solutions of ax + by = cz — the unity of combinatorics Atti dei Conv. Lincei, 17, Tomo II, 173–179
• 1981 Unsolved problems in number theory, Springer-Verlag in New York, ISBN 0-387-90593-6
• 1982 Sets of integers whose subsets have distinct sums, North-Holland, OCLC Number: 897757256.
• 1982 (with Elwyn Berlekamp and John H. Conway) Winning Ways for your Mathematical Plays, Academic Press, ISBN 0120911507.
• 1987 Six phases for the eight-lambdas and eight-deltas configurations, North-Holland, OCLC Number: 897693235.
• 1989 Fair game how to play impartial combinatorial games, COMAP in Arlington, MA, ISBN 0912843160.
• 1991 Graphs and the strong law of small numbers in 'Graph Theory, Combinatorics, and Applications, Wiley, OCLC Number: 897682607. ISBN 9780471532194
• 1994 (with Hallard T. Croft and Kenneth Falconer) Unsolved problems in geometry, Springer-Verlag, ISBN 0387975063.
• 1996 (with John H. Conway) The book of numbers, Copernicus, ISBN 9780387979939.
• 2002 (with Paul Vaderlind and Loren C. Larson) The inquisitive problem solver, Mathematical Association of America, ISBN 0883858061.
• 2020 (with Ezra A. Brown) The Unity of Combinatorics, Mathematical Association of America, ISBN 978-1-4704-5279-7
Papers
• Guy, R. K.; Smith, Cedric A. B. (1956). "The G-values of various games". Math. Proc. Camb. Philos. Soc. 52 (3): 514–526. Bibcode:1956PCPS...52..514G. doi:10.1017/S0305004100031509. S2CID 120605511.
• Guy, R. K. (1958). "Two theorems on partitions". Math. Gaz. 42 (340): 84–86. doi:10.2307/3609388. JSTOR 3609388. S2CID 125687055.
• Guy, R. K.; Harary, Frank (1967). "On the Mobius ladders". Can. Math. Bull. 10 (4): 493–496. doi:10.4153/CMB-1967-046-4. S2CID 124320546.
• Bremner, Andrew; Goggins, Joseph R.; Guy, Michael J. T.; Guy, R. K. (2000). "On rational Morley triangles". Acta Arith. 93 (2): 177–187. doi:10.4064/aa-93-2-177-187.
• Sallows, Lee; Guy, R. K.; Gardner, Martin; Knuth, Donald (1992). "New pathways in serial isogons". Math. Intell. 14 (2): 55–67. doi:10.1007/BF03025216. S2CID 121493484.
• Guy, R. K. (1967). "A coarseness conjecture of Erdös". J. Comb. Theory. 3: 38–42. doi:10.1016/S0021-9800(67)80014-0.
• Guy, R. K.; Kelly, Patrick A. (1968). "The no-three-in-line problem". Can. Math. Bull. 11 (4): 527–531. doi:10.4153/CMB-1968-062-3. S2CID 120649715.
• Guy, R. K.; Jenkyns, Tom; Schaer, Jonathan (1968). "The toroidal crossing number of the complete graph". J. Comb. Theory. 4 (4): 376–390. doi:10.1016/S0021-9800(68)80063-8.
• Guy, R. K. (1969). "A many-facetted problem of zarankiewicz". The Many Facets of Graph theory. Lecture Notes in Mathematics. Vol. 110. pp. 129–148. doi:10.1007/BFb0060112. ISBN 978-3-540-04629-5.
• Guy, R. K.; Jenkyns, Tom (1969). "The toroidal crossing number of K(m,n)". J. Comb. Theory. 6 (3): 236–250. doi:10.1016/S0021-9800(69)80084-0.
• Guy, R. K. (1970). "Latest results on crossing numbers". Recent Trends in Graph Theory. Lecture Notes in Mathematics. Vol. 186. pp. 143–156. doi:10.1007/BFb0059432. ISBN 978-3-540-05386-6.
• Guy, R. K. (1972). "The slimming number and genus of graphs". Can. Math. Bull. 15 (2): 195–200. doi:10.4153/CMB-1972-035-8. S2CID 123893633.
• Guy, R. K. (1972). "Crossing numbers of graphs". Graph Theory and applications. Lecture Notes in Mathematics. Vol. 303. pp. 111–124. doi:10.1007/BFb0067363. ISBN 978-3-540-06096-3.
• Guy, R. K.; Selfridge, J. L. (1975). "What drives an aliquot sequence?". Math. Comput. 29 (129): 101–107. doi:10.1090/S0025-5718-1975-0384669-X.
• Guy, R. K.; Ringel, Gerhard (1976). "Triangular embedding of Kn – K6". J. Comb. Theory B. 21 (2): 140–145. doi:10.1016/0095-8956(76)90054-X.
• Béla Bollobás, R. K. Guy (1983). "Equitable and proportional coloring of trees". J. Comb. Theory B. 34 (2): 177–186. doi:10.1016/0095-8956(83)90017-5.
• Guy, R. K.; Selfridge, J. L. (1980). "Corrigendum to 'What drives an aliquot sequence?'". Math. Comput. 34 (149): 319–321. doi:10.1090/S0025-5718-1980-0551309-8.
• Guy, R. K. (1983). "Conway's prime producing machine". Math. Mag. 56 (1): 26–33. doi:10.2307/2690263. JSTOR 2690263.
• Guy, R. K.; Lacampagne, C. B.; Selfridge, J. L. (1987). "Primes at a glance". Math. Comput. 48 (177): 183–202. doi:10.1090/S0025-5718-1987-0866108-3.
• Guy, R. K. (1988). "The strong law of small numbers". Am. Math. Mon. 95 (8): 697–712. doi:10.2307/2322249. JSTOR 2322249.
• Bremner, Andrew; Guy, R. K. (1988). "A dozen difficult diophantine dilemmas". Am. Math. Mon. 95 (1): 31–36. doi:10.2307/2323442. JSTOR 2323442.
• Guy, R. K. (1990). "The second strong law of small numbers". Am. Math. Mon. 63 (1): 3–20. doi:10.2307/2691503. JSTOR 2691503.
• Bremner, Andrew; Guy, R. K. (1992). "Nu-configurations in tiling the square". Math. Comput. 59 (199): 195–202. Bibcode:1992MaCom..59..195B. doi:10.1090/S0025-5718-1992-1134716-2.
• Guy, R. K.; Krattenthaler, C.; Sagan, Bruce E. (1992). "Lattice paths, reflections, and dimension-changing bijections". Ars Combinatoria. 34: 15. CiteSeerX 10.1.1.32.294.
• Bremner, Andrew; Guy, R. K.; Nowakowski, Richard J. (1993). "Which integers are representable as the product of the sum of three integers with the sum of their reciprocals?". Math. Comput. 61 (203): 117–130. Bibcode:1993MaCom..61..117B. doi:10.1090/S0025-5718-1993-1189516-5.
• Guy, R. K. (1994). "Every number is expressible as the sum of how many polygonal numbers?". Am. Math. Mon. 101 (2): 169–72. doi:10.2307/2324367. JSTOR 2324367.
• Guy, R. K.; Nowakowski, Richard (1995). "Coin-Weighing Problems". Am. Math. Mon. 102 (2): 164–167. doi:10.2307/2975353. JSTOR 2975353.
• Guy, R. K. (2000). "Catwalks, sandsteps and pascal pyramids". J. Integer Seq. 3: 00.1.6. Bibcode:2000JIntS...3...16G.
• Conway, John H.; Guy, R. K.; Schneeberger, W. A.; Sloane, N. J. A. (1996–1997). "The primary pretenders". Acta Arith. 78 (4): 307–313. doi:10.4064/aa-78-4-307-313.
References
1. Albers & Alexanderson (2011) p. 320
2. MMA (2016)
3. Author biography from Winning Ways for your Mathematical Plays, Vol. I, 2nd ed., AK Peters, 2001.
4. Roberts (2016)
5. Scott (2012) p. 29
6. Guy, Richard K. (October 1988). "The Strong Law of Small Numbers" (PDF). Am. Math. Mon. 95 (8): 697–712. doi:10.2307/2322249. ISSN 0002-9890. JSTOR 2322249.
7. MMA (2016)
8. Scott (2012) p. 6
9. Roberts (2016)
10. Albers & Alexanderson (2011) p. 169
11. "No. 35894". The London Gazette (Supplement). 5 February 1943. p. 707.
12. Scott (2012) p. 29: Richard has often told me that he has had three loves in his life: Louise and mountains of course are two of them, but his first love was mathematics.
13. Scott (2012) p. 11
14. Guiltenane (2016)
15. University of Calgary (2016)
16. Roberts (2016)
17. Guiltenane (2016): Guy has said, "I didn't retire, they just stopped paying me."
18. Siobahn Roberts (2010), "Profile of Scott Aaronson", Finding Nirvana in Numbers, Simons Foundation, retrieved 13 March 2020
19. Scott (2012) p. 31
20. Scott (2012) p. 39
21. Alpine Club of Canada (30 October 2014). "Introducing the Louise & Richard Guy Hut". Archived from the original on 11 October 2016.
22. "Remembering Richard Guy: 1916-2020". University of Calgary. 10 March 2020. Retrieved 10 March 2020.
23. "Canadian Climbing Legend Richard Guy Dies at 103". Gripped. 10 March 2020.
24. Roberts (2016) p.30
25. Roberts (2016)
26. Albers & Alexanderson (2011) p. 176
27. Coauthors of Paul Erdos
28. Brent Wittmeier, "Math genius left unclaimed sum," Edmonton Journal, 28 September 2010.
29. Unsolved problems in number theory and Unsolved problems in combinatorial games
30. Albers (2011): p. 165
31. Scott (2016) p. 30: It is no exaggeration to say that Unsolved Problems in Number Theory has inspired generations of aspiring Number Theorists!
32. Scot (2012) p. 29
33. Roberts (2016): "He pushes the boundaries of that definition."
34. Scott (2016)
35. Albers (2011)
36. "Richard K. Guy". Mathematical Reviews. American Mathematical Society. Retrieved 13 March 2020.
37. P. Erdős; R. K. Guy; J. L. Selfridge (1982). "Another property of 239 and some related questions". Congr. Numer. 34: 243–257. MR 0681710.
38. P. Erdős; R. K. Guy; J. W. Moon (1974). "On refining partitions". J. London Math. Soc. 9: 565–570. MR 0360302.
39. P. Erdős; R. K. Guy (1973). "Crossing number problems". Amer. Math. Monthly. 80: 52–58. doi:10.1080/00029890.1973.11993230. MR 0382006.
40. P. Erdős; R. K. Guy (1970). "Distinct distances between lattice points". Elem. Math. 25: 121–123. MR 0281691.
41. A Quarter-Century of Recreational Mathematics by Martin Gardner, Scientific American, August 1998
42. Scott (2016) p. 30: Mathematician Michael Bennett calls Winning Ways for your Mathematical Plays the bible of Combinatorial Game Theory.
43. Mulcahy (2016): Richard also reveals a little known fact about the end of Gardner's quarter-century column run for that publication, "There was serious consideration given to my taking over the column from him. I'm glad that it didn't happen, because you can't follow Martin Gardner!".
44. Mulcahy (2016)
45. Gardner, Martin (1970). The fantastic combinations of John Conway's new solitaire game "life" Scientific American: Mathematical Games. October 1970.
46. Kenneth Falconer (3 October 2016). "Richard Guy at 100". London Mathematical Society Newsletter. Archived from the original on 29 December 2017.
47. Richard Guy 100th Birthday Tribute Song video
48. William Blair. "Chair's Corner" (PDF). NIU Department of Mathematical Sciences Newsletter. University of Northern Illinois. Retrieved 13 March 2020.
49. "In Memoriam". The Number Theory Foundation. Number Theory Foundation. Retrieved 10 March 2020.
50. The Chess Endgame Study: A Comprehensive Introduction By A. J. Roycroft, New York : Dover Publications, 1981, p. 58, ISBN 0486241866
51. Hooper, David; Whyld, Kenneth (1992) The Oxford Companion to Chess, "GBR code", p. 353, Oxford University Press, ISBN 0-19-280049-3
Sources
• Albers, Donald J.; Alexanderson, Gerald L. (1985). Mathematical People: Profiles and Interviews, John Horton Conway by Richard K. Guy: pp. 36–46, Princeton University Press, ISBN 0817631917
• Albers, Donald J.; Alexanderson, Gerald L. (2011). Fascinating Mathematical People : interviews and memoirs, Interview with Richard K. Guy: pp. 165–192, Princeton University Press, ISBN 0691148295
• Berlekamp, Elwyn R. (2014). The Mathematical Legacy of Martin Gardner Society for Industrial and Applied Mathematics (SIAM), 2 September 2014
• Fortney, Valerie (2015). "Richard Guy to visit his namesake alpine hut" The Calgary Herald, 10 September 2015
• Guiltenane, Erin (2016). Emeritus professor marks a century of life and learning University of Calgary: Faculty of Science, 29 September 2016
• MMA (2016). Happy Birthday, Richard Guy! Mathematical Association of America, 30 September 2016
• Mulcahy, Colm (2016). Richard K. Guy turns 100 MMA: CardColm, 30 September 2016
• Roberts, Siobhan (2016). An “Infinitely Rich” Mathematician Turns 100, 30 September 2016
• Scott, Chic (2012). Young at Heart: The Inspirational Lives of Richard and Louise Guy, Pub by The Alpine Club of Canada, Canmore, Alberta, ISBN 978-0-920330-24-1
External links
Wikinews has related news:
• British mathematician Richard K. Guy dies at 103
• Richard K. Guy author profile on MathSciNet
• Personal web page
• Richard K. Guy at the Mathematics Genealogy Project
• Granville, Andrew; Pomerance, Carl (April 2022). "The Man Who Loved Problems: Richard K. Guy" (PDF). Notices of the American Mathematical Society. 69 (4): 574–585. doi:10.1090/noti2456.
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Richard Kadison
Richard Vincent Kadison (July 25, 1925 – August 22, 2018)[2] was an American mathematician known for his contributions to the study of operator algebras.
Richard Kadison
Born(1925-07-25)July 25, 1925
New York City, New York, U.S.
DiedAugust 22, 2018(2018-08-22) (aged 93)
Narberth, Pennsylvania, U.S.
NationalityAmerican
Alma materUniversity of Chicago
Known forKadison–Kaplansky conjecture
Kadison's inequality
Kadison–Singer problem[1]
Kadison transitivity theorem
Kadison–Sakai theorem
Kadison–Kastler metric
AwardsSteele Prize (1999)
Scientific career
FieldsMathematics
InstitutionsUniversity of Pennsylvania
Doctoral advisorMarshall Harvey Stone
Doctoral studentsJames Glimm
Richard Lashof
Marc Rieffel
Mikael Rørdam
Erling Størmer
Work
Born in New York City in 1925,[3][4] Kadison was a Gustave C. Kuemmerle Professor in the Department of Mathematics of the University of Pennsylvania.[5]
Kadison was a member of the U.S. National Academy of Sciences (elected in 1996),[6][7] and a foreign member of the Royal Danish Academy of Sciences and Letters[2] and of the Norwegian Academy of Science and Letters.[8] He was a 1969 Guggenheim Fellow.[9]
Kadison was awarded the 1999 Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society.[5][10] In 2012 he became a fellow of the American Mathematical Society.[11]
Personal
Kadison was a skilled gymnast with a specialty in rings, making the 1952 US Olympic Team but later withdrawing due to an injury.[12] He married Karen M. Holm on June 5, 1956, and they had one son, Lars.[12]
Kadison died after a short illness on August 22, 2018.[3]
Selected publications
Books
• with John Ringrose: Fundamentals of the theory of operator algebras. 2 vols., Academic Press 1983; new edition, Fundamentals of the theory of operator algebras: Elementary theory, Vol. 1, 1997 Fundamentals of the theory of operator algebras: Advanced theory, Vol. 2, 1997 AMS 1997[13][14]
• with John Ringrose: Fundamentals of the theory of operator algebras, III-IV. An exercise approach, Birkhäuser, Basel, III: 1991, xiv+273 pp., ISBN 0-8176-3497-5; IV: 1992, xiv+586 pp., ISBN 0-8176-3498-3[15][16][17]
PNAS articles
• Kadison, R. V. (1998). "On representations of finite type". Proc Natl Acad Sci U S A. 95 (23): 13392–6. Bibcode:1998PNAS...9513392K. doi:10.1073/pnas.95.23.13392. PMC 24829. PMID 9811810.
• with I. M. Singer: Kadison, R. V.; Singer, I. M. (1952). "Some Remarks on Representations of Connected Groups". Proc Natl Acad Sci U S A. 38 (5): 419–23. Bibcode:1952PNAS...38..419K. doi:10.1073/pnas.38.5.419. PMC 1063576. PMID 16589115.
• with Bent Fuglede: Fuglede, B.; Kadison, R. V. (1951). "On a Conjecture of Murray and von Neumann". Proc Natl Acad Sci U S A. 37 (7): 420–5. Bibcode:1951PNAS...37..420F. doi:10.1073/pnas.37.7.420. PMC 1063392. PMID 16578376.
• with Zhe Liu: Kadison, Richard V.; Liu, Zhe (2014). "A note on derivations of Murray–von Neumann algebras". Proc Natl Acad Sci U S A. 111 (6): 2087–93. Bibcode:2014PNAS..111.2087K. doi:10.1073/pnas.1321358111. PMC 3926033. PMID 24469831.
• Kadison, R. V. (2002). "The Pythagorean Theorem: II. The infinite discrete case". Proc Natl Acad Sci U S A. 99 (8): 5217–22. Bibcode:2002PNAS...99.5217K. doi:10.1073/pnas.032677299. PMC 122749. PMID 16578869.
• Kadison, R. V. (2002). "The Pythagorean Theorem: I. The finite case". Proc Natl Acad Sci U S A. 99 (7): 4178–84. Bibcode:2002PNAS...99.4178K. doi:10.1073/pnas.032677199. PMC 123622. PMID 11929992.
• Kadison, R. V. (1957). "Irreducible Operator Algebras". Proc Natl Acad Sci U S A. 43 (3): 273–6. Bibcode:1957PNAS...43..273K. doi:10.1073/pnas.43.3.273. PMC 528430. PMID 16590013.
• Kadison, R. V. (1955). "On the Additivity of the Trace in Finite Factors". Proc Natl Acad Sci U S A. 41 (6): 385–7. Bibcode:1955PNAS...41..385K. doi:10.1073/pnas.41.6.385. PMC 528101. PMID 16589685.
• Kadison, R. V. (1955). "Multiplicity Theory for Operator Algebras". Proc Natl Acad Sci U S A. 41 (3): 169–73. Bibcode:1955PNAS...41..169K. doi:10.1073/pnas.41.3.169. PMC 528046. PMID 16589638.
• with Bent Fuglede: Fuglede, B.; Kadison, R. V. (1951). "On Determinants and a Property of the Trace in Finite Factors". Proc Natl Acad Sci U S A. 37 (7): 425–31. Bibcode:1951PNAS...37..425F. doi:10.1073/pnas.37.7.425. PMC 1063393. PMID 16578377.
References
1. Kadison–Singer Conjecture Succumbs to Proof | Mathematical Association of America
2. Foreign Members list. Royal Danish Academy of Sciences and Letters. Accessed January 12, 2010
3. "In Memoriam, University of Pennsylvania, Department of Mathematics". University of Pennsylvania.
4. "Library of Congress Name Authority File". Library of Congress.
5. Richard Kadison wins 1999 AMS Steele Prize. Archived 2010-06-19 at the Wayback Machine Department of Mathematics, University of Pennsylvania. Accessed January 12, 2010.
6. Kadison, Richard V., U.S. National Academy of Sciences. Accessed January 12, 2010. Election citation: "Kadison has been the principal figure in the American school of study of operator algebras in Hilbert space since the Second World War and one of the central leaders of the world development leading to applications in quantum field theory, statistical mechanics, noncommutative geometry, and knot theory."
7. "National Academy of Sciences Elects New Members". Science. 272 (5263): 808–0. 1996. doi:10.1126/science.272.5263.808. S2CID 220101739.
8. Academy members list, Mathematical Sciences, Norwegian Academy of Science and Letters. Accessed January 12, 2010.
9. Guggenheim Fellow list, John Simon Guggenheim Memorial Foundation. Accessed January 12, 2010.
10. "1999 Steele Prizes" (PDF). Notices of the American Mathematical Society. 46 (4): 457–462. 1999.
11. List of Fellows of the American Mathematical Society, retrieved 2013-01-27.
12. Ge, Liming; Jaffe, Arthur; Rieffel, Marc; Rørdam, Mikael (October 2019). "In Memoriam: Richard Kadison (1925–2018)" (PDF). Notices of the American Mathematical Society. 66 (9): 1453–1463. doi:10.1090/noti1949.
13. Fundamentals of the Theory of Operator Algebras. Volume I, AMS website
14. Fundamentals of the Theory of Operator Algebras. Volume II, AMS website
15. Pedersen, Gert K. (1994). "Review of Fundamentals of the theory of operator algebras, III-IV. An exercise approach by Richard Kadison and John Ringrose" (PDF). Bull. Amer. Math. Soc. (N.S.). 31 (2): 275–277. doi:10.1090/s0273-0979-1994-00531-2.
16. Fundamentals of the Theory of Operator Algebras. Volume III, AMS website
17. Fundamentals of the Theory of Operator Algebras. Volume IV, AMS website
External links
• Richard Kadison at the Mathematics Genealogy Project
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