50 relations: Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, En (Lie algebra), Face (geometry), Facet, Facet (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Rectified 8-simplexes, Rectified 9-simplexes, Regular polytope, Schläfli symbol, Semiregular polytope, Simplex, Tetrahedron, Thorold Gosset, Triangle, Uniform 1 k2 polytope, Uniform 2 k1 polytope, Uniform k 21 polytope, Uniform polytope, Vertex figure, Wythoff construction, 1 22 polytope, 1 32 polytope, 1 42 polytope, 1 52 honeycomb, 16-cell, 2 31 polytope, 2 41 polytope, 2 51 honeycomb, 3 21 polytope, 4 21 polytope, 5 21 honeycomb, 5-cell, 5-demicube, 5-orthoplex, 5-simplex, 6-demicube, 6-simplex, 7-demicube, 7-simplex, 8-demicube, 8-simplex, 9-demicube, 9-orthoplex, 9-simplex.

## 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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## Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions.

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## En (Lie algebra)

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k.

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## Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

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## Facet

Facets are flat faces on geometric shapes.

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## Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

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## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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## Gosset–Elte figures

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.

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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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## Hyperplane

In geometry a hyperplane is a subspace of one dimension less than its ambient space.

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## Rectified 8-simplexes

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

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## Rectified 9-simplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

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## Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry.

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## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

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## Semiregular polytope

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes.

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## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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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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## Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.

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## Triangle

A triangle is a polygon with three edges and three vertices.

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## Uniform 1 k2 polytope

In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.

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## Uniform 2 k1 polytope

In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.

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## Uniform k 21 polytope

In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets.

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## Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

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## Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.

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## Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.

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## 1 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.

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## 1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

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## 1 42 polytope

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

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## 1 52 honeycomb

In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.

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## 16-cell

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

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## 2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

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## 2 41 polytope

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

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## 2 51 honeycomb

In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.

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## 3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.

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## 4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.

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## 5 21 honeycomb

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.

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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-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-simplex

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

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## 6-demicube

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices truncated.

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## 6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

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## 7-demicube

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices truncated.

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## 7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.

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## 8-demicube

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices truncated.

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## 8-simplex

In geometry, an 8-simplex is a self-dual regular 8-polytope.

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## 9-demicube

In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices truncated.

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## 9-orthoplex

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

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## 9-simplex

In geometry, a 9-simplex is a self-dual regular 9-polytope.

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