Table of Contents
57 relations: Cambridge University Press, Conjugacy class, Coxeter group, Coxeter–Dynkin diagram, Cube, Dihedral group, Dodecahedron, Dynkin diagram, E6 (mathematics), E7 (mathematics), E8 (mathematics), Harold Scott MacDonald Coxeter, Icosahedral symmetry, Icosahedron, Improper rotation, John Horton Conway, John Stembridge, Lie algebra, List of spherical symmetry groups, Longest element of a Coxeter group, Mathematics, Octahedral symmetry, Octahedron, Order (group theory), Oxford, Oxford University Press, Patrick du Val, Petrie polygon, Platonic solid, Projection (linear algebra), Quiver (mathematics), Reflection group, Regular 4-polytope, Regular polyhedron, Regular Polytopes (book), Root of unity, Root system, Rotations in 4-dimensional Euclidean space, Séminaire Lotharingien de Combinatoire, Symmetric group, Tesseract, Tetrahedral symmetry, Tetrahedron, Transactions of the American Mathematical Society, 1 22 polytope, 120-cell, 16-cell, 2 31 polytope, 24-cell, 4 21 polytope, ... Expand index (7 more) »
- Coxeter groups
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge.
See Coxeter element and Cambridge University Press
Conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b.
See Coxeter element and Conjugacy class
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). Coxeter element and Coxeter group are Coxeter groups.
See Coxeter element and Coxeter group
Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group. Coxeter element and Coxeter–Dynkin diagram are Coxeter groups.
See Coxeter element and Coxeter–Dynkin diagram
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces.
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.
See Coxeter element and Dihedral group
Dodecahedron
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces.
See Coxeter element and Dodecahedron
Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).
See Coxeter element and Dynkin diagram
E6 (mathematics)
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. Coxeter element and E6 (mathematics) are lie groups.
See Coxeter element and E6 (mathematics)
E7 (mathematics)
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. Coxeter element and e7 (mathematics) are lie groups.
See Coxeter element and E7 (mathematics)
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. Coxeter element and e8 (mathematics) are lie groups.
See Coxeter element and E8 (mathematics)
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.
See Coxeter element and Harold Scott MacDonald Coxeter
Icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron.
See Coxeter element and Icosahedral symmetry
Icosahedron
In geometry, an icosahedron is a polyhedron with 20 faces.
See Coxeter element and Icosahedron
Improper rotation
In geometry, an improper rotation. Coxeter element and improper rotation are lie groups.
See Coxeter element and Improper rotation
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
See Coxeter element and John Horton Conway
John Stembridge
John Stembridge is a Professor of Mathematics at the University of Michigan.
See Coxeter element and John Stembridge
Lie algebra
In mathematics, a Lie algebra (pronounced) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. Coxeter element and Lie algebra are lie groups.
See Coxeter element and Lie algebra
List of spherical symmetry groups
Finite spherical symmetry groups are also called point groups in three dimensions.
See Coxeter element and List of spherical symmetry groups
Longest element of a Coxeter group
In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. Coxeter element and longest element of a Coxeter group are Coxeter groups.
See Coxeter element and Longest element of a Coxeter group
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Coxeter element and Mathematics
Octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether.
See Coxeter element and Octahedral symmetry
Octahedron
In geometry, an octahedron (octahedra or octahedrons) is a polyhedron with eight faces.
See Coxeter element and Octahedron
Order (group theory)
In mathematics, the order of a finite group is the number of its elements.
See Coxeter element and Order (group theory)
Oxford
Oxford is a city and non-metropolitan district in Oxfordshire, England, of which it is the county town.
See Coxeter element and Oxford
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford.
See Coxeter element and Oxford University Press
Patrick du Val
Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity.
See Coxeter element and Patrick du Val
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no) belongs to one of the facets.
See Coxeter element and Petrie polygon
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space.
See Coxeter element and Platonic solid
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P.
See Coxeter element and Projection (linear algebra)
Quiver (mathematics)
In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows between two vertices are allowed.
See Coxeter element and Quiver (mathematics)
Reflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. Coxeter element and reflection group are Coxeter groups.
See Coxeter element and Reflection group
Regular 4-polytope
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope.
See Coxeter element and Regular 4-polytope
Regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.
See Coxeter element and Regular polyhedron
Regular Polytopes (book)
Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter.
See Coxeter element and Regular Polytopes (book)
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power.
See Coxeter element and Root of unity
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. Coxeter element and root system are lie groups.
See Coxeter element and Root system
Rotations in 4-dimensional Euclidean space
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4).
See Coxeter element and Rotations in 4-dimensional Euclidean space
Séminaire Lotharingien de Combinatoire
The Séminaire Lotharingien de Combinatoire (English: Lotharingian Seminar of Combinatorics) is a peer-reviewed academic journal specialising in combinatorial mathematics, named after Lotharingia.
See Coxeter element and Séminaire Lotharingien de Combinatoire
Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
See Coxeter element and Symmetric group
Tesseract
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube.
See Coxeter element and Tesseract
Tetrahedral symmetry
A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
See Coxeter element and Tetrahedral symmetry
Tetrahedron
In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.
See Coxeter element and Tetrahedron
Transactions of the American Mathematical Society
The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.
See Coxeter element and Transactions of the American Mathematical Society
1 22 polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
See Coxeter element and 1 22 polytope
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Coxeter element and 120-cell
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Coxeter element and 16-cell
2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
See Coxeter element and 2 31 polytope
24-cell
In four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Coxeter element and 24-cell
4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
See Coxeter element and 4 21 polytope
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.
See Coxeter element and 5-cell
5-cube
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
See Coxeter element and 5-cube
5-demicube
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
See Coxeter element and 5-demicube
5-orthoplex
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
See Coxeter element and 5-orthoplex
5-polytope
In geometry, a five-dimensional polytope (or 5-polytope or polyteron) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
See Coxeter element and 5-polytope
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.
See Coxeter element and 5-simplex
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Coxeter element and 600-cell
See also
Coxeter groups
- Affine symmetric group
- Bruhat order
- Coxeter complex
- Coxeter element
- Coxeter group
- Coxeter matroid
- Coxeter–Dynkin diagram
- Longest element of a Coxeter group
- Parabolic subgroup of a reflection group
- Reflection group
- Triangle group
- Uzi Vishne
References
Also known as Coxeter element of a Coxeter group, Coxeter number, Coxeter plane, Dual Coxeter number.