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Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including ''The Fifty-Nine Icosahedra'' (1938) and ''Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson (mathematician), Norman Johnson. Reflectional groups For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors. The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the ''A''''n'' group is represented by [3''n''−1], to imply ''n'' nodes connected by ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter–Dynkin Diagram
In geometry, a Harold Scott MacDonald Coxeter, Coxeter–Eugene Dynkin, Dynkin diagram (or Coxeter diagram, Coxeter graph) is a Graph (discrete mathematics), graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group. A class of closely related objects is the Dynkin diagrams, which differ from Coxeter diagrams in two respects: firstly, branches labeled "" or greater are Directed graph, directed, while Coxeter diagrams are Undirected graph, undirected; secondly, Dynkin diagrams must satisfy an additional (Crystallographic restriction theorem, crystallographic) restriction, namely that the only allowed branch labels are and Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras. Description A Coxeter group is a group that admits a presentation: \langle r_0,r_1,\dots,r_n \mid (r_i r_j)^ = 1 \rangle where the are integers that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Element
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number of an irreducible root system. *The Coxeter number is the order of any Coxeter element;. *The Coxeter number is where is the rank, and is the number of reflections. In the crystallographic case, is half the number of roots; and is the dimension of the corresponding semisimple Lie algebra. *If the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Graph
In the mathematics, mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic graph, cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter. Properties The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth (graph theory), girth 7. It is also a 3-k-vertex-connected graph, vertex-connected graph and a 3-k-edge-connected graph, edge-connected graph. It has book thickness 3 and queue number 2. The Coxeter graph is hypohamiltonian graph, hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has Crossing number (graph theory), rectilinear crossing number 11, and is the smallest cubic graph with that crossing number . Construction The simplest construction of a Coxeter graph is from a Fano plane. Take the Combination, 7C3 = 35 possible 3-combinations on 7 obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boerdijk–Coxeter Helix
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and , is a linear stacking of regular tetrahedron, tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helix, helices. There are two Chirality (mathematics), chiral forms, with either right-handed or left-handed windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex 4-polytope, polychora. Buckminster Fuller named it a ''tetrahelix'' and considered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Skew Apeirohedron
In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron. They have either Skew polygon, skew regular Face (geometry), faces or skew regular vertex figures. History In 1926 John Flinders Petrie took the concept of a regular skew polygons, polygons whose vertices are not all in the same plane, and extended it to polyhedra. While regular tiling, apeirohedra are typically required to tile the 2-dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon vertex figure. Petrie discovered two regular skew apeirohedra, the mucube and the muoctahedron. H.S.M. Coxeter, Harold Scott MacDonald Coxeter derived a third, the mutetrahedron, and proved that these three were complete. Under Coxeter and Petrie's definition, requiring convex faces and allowing a skew vertex figure, the three were not only the only skew apeirohedra in 3-dimensional Euclidean space, but they were the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte–Coxeter Graph
In the mathematics, mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth (graph theory), girth 8, it is a cage (graph theory), cage and a Moore graph. It is bipartite graph, bipartite, and can be constructed as the Levi graph of the generalized quadrangle ''W''2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a). All the cubic graph, cubic distance-regular graphs are known. The Tutte–Coxeter is one of the 13 such graphs. It has Crossing number (graph theory), crossing number 13, book thickness 3 and queue number 2.Wolz, Jessica; ''Engineering Linear Layouts with SAT.'' Master Thesis, Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldberg–Coxeter Construction
The Goldberg–Coxeter construction or Goldberg–Coxeter operation (GC construction or GC operation) is a graph operation defined on regular graph, regular polyhedral graphs with Degree (graph theory), degree 3 or 4. It also applies to the dual graph of these graphs, i.e. graphs with triangular or quadrilateral "faces". The GC construction can be thought of as subdividing the faces of a polyhedron with a lattice of triangular, square, or hexagonal polygons, possibly skewed with regards to the original face: it is an extension of concepts introduced by the Goldberg polyhedra and geodesic polyhedra. The GC construction is primarily studied in organic chemistry for its application to fullerenes, but it has been applied to nanoparticles, computer-aided design, basket weaving, and the general study of graph theory and polyhedra. The Goldberg–Coxeter construction may be denoted as GC_(G_0), where G_0 is the graph being operated on, k and l are integers, k >0, and \ell \ge 0. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter–Todd Lattice
In mathematics, the Coxeter–Todd lattice K12, discovered by , is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is analogous to the Barnes–Wall lattice. The automorphism group of the Coxeter–Todd lattice has order 210·37·5·7=78382080, and there are 756 vectors in this lattice of norm 4 (the shortest nonzero vectors in this lattice). Properties The Coxeter–Todd lattice can be made into a 6-dimensional lattice self dual over the Eisenstein integers. The automorphism group of this complex lattice has index 2 in the full automorphism group of the Coxeter–Todd lattice and is a complex reflection group (number 34 on the list) with structure 6.PSU4(F3).2, called the Mitchell group. The genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant tax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter's Loxodromic Sequence Of Tangent Circles
In geometry, Coxeter's loxodromic sequence of tangent circles is an infinite sequence of circles arranged so that any four consecutive circles in the sequence are pairwise mutually tangent. This means that each circle in the sequence is tangent to the three circles that precede it and also to the three circles that follow it. Properties The radii of the circles in the sequence form a geometric progression with ratio k=\varphi + \sqrt \approx 2.89005 \ , where \varphi is the golden ratio. This ratio k and its reciprocal satisfy the equation (1+x+x^2+x^3)^2=2(1+x^2+x^4+x^6)\ , and so any four consecutive circles in the sequence meet the conditions of Descartes' theorem. The centres of the circles in the sequence lie on a logarithmic spiral. Viewed from the centre of the spiral, the angle between the centres of successive circles is \cos^ \left( \frac \right) \approx 128.173 ^ \circ \ . The angle between consecutive triples of centers is \theta=\cos^\frac\approx 51.8273^\circ, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Todd–Coxeter Algorithm
In group theory, the Todd–Coxeter algorithm, created by J. A. Todd and H. S. M. Coxeter in 1936, is an algorithm for solving the coset enumeration problem. Given a presentation of a group ''G'' by generators and relations and a subgroup ''H'' of ''G'', the algorithm enumerates the cosets of ''H'' on ''G'' and describes the permutation representation of ''G'' on the space of the cosets (given by the left multiplication action). If the order of a group ''G'' is relatively small and the subgroup ''H'' is known to be uncomplicated (for example, a cyclic group), then the algorithm can be carried out by hand and gives a reasonable description of the group ''G''. Using their algorithm, Coxeter and Todd showed that certain systems of relations between generators of known groups are complete, i.e. constitute systems of defining relations. The Todd–Coxeter algorithm can be applied to infinite groups and is known to terminate in a finite number of steps, provided that the index of ''H'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
