## Table of Contents

32 relations: Coxeter group, Coxeter–Dynkin diagram, Facet (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hosohedron, Hyperplane, Isogonal figure, Kissing number, Rectified 5-simplexes, Schläfli symbol, Sphere packing, Tetrahedral prism, Tetrahedron, Triangle, Uniform 8-polytope, Vertex arrangement, Vertex figure, Voronoi diagram, Wythoff construction, 1 32 polytope, 1 33 honeycomb, 2 21 polytope, 2 31 polytope, 3 21 polytope, 5-cell, 5-orthoplex, 5-simplex, 6-demicube, 6-simplex, 7-simplex.

- 8-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 3 31 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 3 31 honeycomb and Coxeter–Dynkin diagram

## 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 3 31 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.

See 3 31 honeycomb and Geometry

## 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 3 31 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 3 31 honeycomb and Harold Scott MacDonald Coxeter

## Hosohedron

In spherical geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

See 3 31 honeycomb and Hosohedron

## Hyperplane

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

See 3 31 honeycomb and Hyperplane

## Isogonal figure

In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

See 3 31 honeycomb and Isogonal figure

## Kissing number

In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere.

See 3 31 honeycomb and Kissing number

## Rectified 5-simplexes

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

See 3 31 honeycomb and Rectified 5-simplexes

## Schläfli symbol

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

See 3 31 honeycomb and Schläfli symbol

## Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.

See 3 31 honeycomb and Sphere packing

## Tetrahedral prism

In geometry, a tetrahedral prism is a convex uniform 4-polytope.

See 3 31 honeycomb and Tetrahedral prism

## 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 3 31 honeycomb and Tetrahedron

## Triangle

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

See 3 31 honeycomb and Triangle

## Uniform 8-polytope

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. 3 31 honeycomb and Uniform 8-polytope are 8-polytopes.

See 3 31 honeycomb and Uniform 8-polytope

## Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions.

See 3 31 honeycomb and Vertex arrangement

## 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 3 31 honeycomb and Vertex figure

## Voronoi diagram

In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects.

See 3 31 honeycomb and Voronoi diagram

## 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 3 31 honeycomb and Wythoff construction

## 1 32 polytope

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

See 3 31 honeycomb and 1 32 polytope

## 1 33 honeycomb

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol, and is composed of 132 facets. 3 31 honeycomb and 1 33 honeycomb are 8-polytopes.

See 3 31 honeycomb and 1 33 honeycomb

## 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.

See 3 31 honeycomb and 2 21 polytope

## 2 31 polytope

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

See 3 31 honeycomb and 2 31 polytope

## 3 21 polytope

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

See 3 31 honeycomb and 3 21 polytope

## 5-cell

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

## 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 3 31 honeycomb and 5-orthoplex

## 5-simplex

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

See 3 31 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 3 31 honeycomb and 6-demicube

## 6-simplex

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

See 3 31 honeycomb and 6-simplex

## 7-simplex

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

See 3 31 honeycomb and 7-simplex

## See also

### 8-polytopes

- 1 33 honeycomb
- 1 42 polytope
- 2 41 polytope
- 3 31 honeycomb
- 4 21 polytope
- 7-cubic honeycomb
- 7-demicubic honeycomb
- 7-simplex honeycomb
- 8-cube
- 8-demicube
- 8-orthoplex
- 8-simplex
- A8 polytope
- B8 polytope
- Cantellated 8-simplexes
- Cantic 8-cube
- Cyclotruncated 7-simplex honeycomb
- D8 polytope
- E8 polytope
- Heptellated 8-simplexes
- Hexicated 8-simplexes
- Omnitruncated 7-simplex honeycomb
- Pentellated 8-simplexes
- Quarter 7-cubic honeycomb
- Rectified 8-cubes
- Rectified 8-orthoplexes
- Rectified 8-simplexes
- Runcinated 8-simplexes
- Stericated 8-simplexes
- Truncated 8-cubes
- Truncated 8-orthoplexes
- Truncated 8-simplexes
- Uniform 8-polytope

## References

Also known as E7 honeycomb, E7 lattice, Gosset 3 31 honeycomb.