## Table of Contents

51 relations: Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, E6 polytope, E8 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, 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, ... Expand index (1 more) »

- 10-polytopes

## 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).

See E9 honeycomb 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.

See E9 honeycomb and Coxeter–Dynkin diagram

## Cross-polytope

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.

See E9 honeycomb and Cross-polytope

## E6 polytope

In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry.

See E9 honeycomb and E6 polytope

## E8 polytope

In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry.

See E9 honeycomb and E8 polytope

## 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.

See E9 honeycomb and En (Lie algebra)

## Face (geometry)

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

See E9 honeycomb and Face (geometry)

## Facet

Facets are flat faces on geometric shapes.

## 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.

See E9 honeycomb and Facet (geometry)

## Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

## 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.

See E9 honeycomb and Gosset–Elte figures

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.

See E9 honeycomb and Harold Scott MacDonald Coxeter

## Hyperplane

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.

See E9 honeycomb and Hyperplane

## 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.

See E9 honeycomb and Rectified 8-simplexes

## 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.

See E9 honeycomb and Rectified 9-simplexes

## Regular polytope

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

See E9 honeycomb and Regular polytope

## Schläfli symbol

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

See E9 honeycomb and Schläfli symbol

## Semiregular polytope

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

See E9 honeycomb and Semiregular polytope

## Simplex

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

## 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 E9 honeycomb and Tetrahedron

## Thorold Gosset

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

See E9 honeycomb and Thorold Gosset

## Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

## Uniform 1 k2 polytope

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

See E9 honeycomb and Uniform 1 k2 polytope

## Uniform 2 k1 polytope

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

See E9 honeycomb and Uniform 2 k1 polytope

## 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.

See E9 honeycomb and Uniform k 21 polytope

## Uniform polytope

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

See E9 honeycomb and Uniform polytope

## Vertex figure

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

See E9 honeycomb and Vertex figure

## Wythoff construction

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

See E9 honeycomb and Wythoff construction

## 1 22 polytope

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

See E9 honeycomb and 1 22 polytope

## 1 32 polytope

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

See E9 honeycomb and 1 32 polytope

## 1 42 polytope

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

See E9 honeycomb and 1 42 polytope

## 1 52 honeycomb

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

See E9 honeycomb and 1 52 honeycomb

## 16-cell

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

## 2 31 polytope

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

See E9 honeycomb and 2 31 polytope

## 2 41 polytope

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

See E9 honeycomb and 2 41 polytope

## 2 51 honeycomb

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

See E9 honeycomb and 2 51 honeycomb

## 3 21 polytope

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

See E9 honeycomb and 3 21 polytope

## 5 21 honeycomb

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

See E9 honeycomb and 5 21 honeycomb

## 5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.

## 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 E9 honeycomb 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 E9 honeycomb and 5-orthoplex

## 5-simplex

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

See E9 honeycomb and 5-simplex

## 6-demicube

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

See E9 honeycomb and 6-demicube

## 6-simplex

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

See E9 honeycomb and 6-simplex

## 7-demicube

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

See E9 honeycomb and 7-demicube

## 7-simplex

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

See E9 honeycomb and 7-simplex

## 8-demicube

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

See E9 honeycomb and 8-demicube

## 8-simplex

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

See E9 honeycomb and 8-simplex

## 9-demicube

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

See E9 honeycomb and 9-demicube

## 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.

See E9 honeycomb and 9-orthoplex

## 9-simplex

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

See E9 honeycomb and 9-simplex

## See also

### 10-polytopes

- 10-cube
- 10-demicube
- 10-orthoplex
- 10-simplex
- E9 honeycomb
- Rectified 10-cubes
- Rectified 10-orthoplexes
- Rectified 10-simplexes
- Uniform 10-polytope

## References

Also known as 1 62 honeycomb, 2 61 honeycomb, 6 21 honeycomb, 7 21 honeycomb, E10 honeycomb.