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Coxeter element

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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. [1]

48 relations: Cambridge University Press, Conjugacy class, Conjugate element (field theory), Coxeter group, Coxeter–Dynkin diagram, Cube, Dihedral group, Dodecahedron, Dynkin diagram, Harold Scott MacDonald Coxeter, Icosahedral symmetry, Icosahedron, Improper rotation, John Horton Conway, 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), Regular 4-polytope, Regular polyhedron, Regular Polytopes (book), Root of unity, Rotations in 4-dimensional Euclidean space, Séminaire Lotharingien de Combinatoire, Symmetric group, Tesseract, Tetrahedral symmetry, Tetrahedron, Transactions of the American Mathematical Society, 120-cell, 16-cell, 24-cell, 5-cell, 5-cube, 5-demicube, 5-orthoplex, 5-polytope, 5-simplex, 600-cell.

Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

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Conjugate element (field theory)

In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial pK,α(x) of α over K. Conjugate elements are also called Galois conjugates, or simply conjugates.

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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).

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Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

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Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

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Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

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Dodecahedron

In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.

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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).

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Icosahedral symmetry

A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.

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Icosahedron

In geometry, an icosahedron is a polyhedron with 20 faces.

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Improper rotation

In geometry, an improper rotation,.

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John Horton Conway

John Horton Conway FRS (born 26 December 1937) is a British mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

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Lie algebra

In mathematics, a Lie algebra (not) is a vector space together with a non-associative multiplication called "Lie bracket".

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List of spherical symmetry groups

Spherical symmetry groups are also called point groups in three dimensions; however, this article is limited to the finite symmetries.

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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.

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Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

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Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.

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Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces.

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Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

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Oxford

Oxford is a city in the South East region of England and the county town of Oxfordshire.

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Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second-oldest, after Cambridge University Press.

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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.

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Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive sides (but no n) belong to one of the facets.

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Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

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Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

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Regular polyhedron

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.

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Regular Polytopes (book)

Regular Polytopes is a mathematical geometry book written by Canadian mathematician H.S.M. Coxeter.

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Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

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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).

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Séminaire Lotharingien de Combinatoire

The Séminaire Lotharingien de Combinatoire (Lotharingian Seminar of Combinatorics) is a peer-reviewed academic journal specialising in combinatorial mathematics, named after Lotharingia.

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Symmetric group

In abstract algebra, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutation operations that can be performed on n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself.

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Tesseract

In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square.

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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.

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Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, three of which meet at each corner or vertex.

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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.

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120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.

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16-cell

In four-dimensional geometry, a 16-cell, is a regular convex 4-polytope.

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24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

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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.

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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 truncated.

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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.

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5-polytope

In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets.

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5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

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600-cell

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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Coxeter element of a Coxeter group, Coxeter number, Coxeter plane, Dual Coxeter number.

References

[1] https://en.wikipedia.org/wiki/Coxeter_element

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