## Table of Contents

47 relations: Complex number, Complex polytope, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, E6 polytope, E8 lattice, E8 polytope, Face (geometry), Fields Medal, Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Kissing number, Maryna Viazovska, Messenger of Mathematics, Modular form, Norman Johnson (mathematician), Regular polytope, Schläfli symbol, Semiregular polytope, Simplex, Sphere packing, Tetrahedron, Thorold Gosset, Triangle, Uniform honeycomb, Uniform k 21 polytope, Vertex arrangement, Vertex figure, Witting polytope, Wythoff construction, 1 52 honeycomb, 2 21 polytope, 2 51 honeycomb, 3 21 polytope, 4 21 polytope, 5-cell, 5-demicube, 5-simplex, 6-simplex, 7-simplex, 8-demicubic honeycomb, 8-orthoplex, 8-simplex, 8-simplex honeycomb.

- 9-polytopes
- E8 (mathematics)

## Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See 5 21 honeycomb and Complex number

## Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

See 5 21 honeycomb and Complex polytope

## 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 5 21 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 5 21 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 5 21 honeycomb and Cross-polytope

## E6 polytope

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

See 5 21 honeycomb and E6 polytope

## E8 lattice

In mathematics, the E lattice is a special lattice in R. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. 5 21 honeycomb and e8 lattice are e8 (mathematics).

See 5 21 honeycomb and E8 lattice

## E8 polytope

In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. 5 21 honeycomb and E8 polytope are e8 (mathematics).

See 5 21 honeycomb and E8 polytope

## 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 5 21 honeycomb and Face (geometry)

## Fields Medal

The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years.

See 5 21 honeycomb and Fields Medal

## Geometry

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

See 5 21 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 5 21 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 5 21 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 5 21 honeycomb and Hyperplane

## 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 5 21 honeycomb and Kissing number

## Maryna Viazovska

Maryna Sergiivna Viazovska (Марина Сергіївна Вязовська,; born 2 December 1984) is a Ukrainian mathematician known for her work in sphere packing.

See 5 21 honeycomb and Maryna Viazovska

## Messenger of Mathematics

The Messenger of Mathematics is a defunct British mathematics journal.

See 5 21 honeycomb and Messenger of Mathematics

## Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane, \,\mathcal\,, that satisfies.

See 5 21 honeycomb and Modular form

## Norman Johnson (mathematician)

Norman Woodason Johnson was a mathematician at Wheaton College, Norton, Massachusetts.

See 5 21 honeycomb and Norman Johnson (mathematician)

## 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 5 21 honeycomb and Regular polytope

## Schläfli symbol

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

See 5 21 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 5 21 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.

See 5 21 honeycomb and Simplex

## Sphere packing

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

See 5 21 honeycomb and Sphere packing

## 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 5 21 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 5 21 honeycomb and Thorold Gosset

## Triangle

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

See 5 21 honeycomb and Triangle

## Uniform honeycomb

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets.

See 5 21 honeycomb and Uniform honeycomb

## 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 5 21 honeycomb and Uniform k 21 polytope

## Vertex arrangement

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

See 5 21 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 5 21 honeycomb and Vertex figure

## Witting polytope

In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3333, and Coxeter diagram.

See 5 21 honeycomb and Witting polytope

## 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 5 21 honeycomb and Wythoff construction

## 1 52 honeycomb

In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. 5 21 honeycomb and 1 52 honeycomb are 9-polytopes and e8 (mathematics).

See 5 21 honeycomb and 1 52 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 5 21 honeycomb and 2 21 polytope

## 2 51 honeycomb

In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. 5 21 honeycomb and 2 51 honeycomb are 9-polytopes and e8 (mathematics).

See 5 21 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 5 21 honeycomb and 3 21 polytope

## 4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. 5 21 honeycomb and 4 21 polytope are e8 (mathematics).

See 5 21 honeycomb and 4 21 polytope

## 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 5 21 honeycomb and 5-demicube

## 5-simplex

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

See 5 21 honeycomb and 5-simplex

## 6-simplex

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

See 5 21 honeycomb and 6-simplex

## 7-simplex

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

See 5 21 honeycomb and 7-simplex

## 8-demicubic honeycomb

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. 5 21 honeycomb and 8-demicubic honeycomb are 9-polytopes.

See 5 21 honeycomb and 8-demicubic honeycomb

## 8-orthoplex

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

See 5 21 honeycomb and 8-orthoplex

## 8-simplex

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

See 5 21 honeycomb and 8-simplex

## 8-simplex honeycomb

In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). 5 21 honeycomb and 8-simplex honeycomb are 9-polytopes.

See 5 21 honeycomb and 8-simplex honeycomb

## See also

### 9-polytopes

- 1 52 honeycomb
- 2 51 honeycomb
- 5 21 honeycomb
- 8-cubic honeycomb
- 8-demicubic honeycomb
- 8-simplex honeycomb
- 9-cube
- 9-demicube
- 9-orthoplex
- 9-simplex
- Cyclotruncated 8-simplex honeycomb
- Omnitruncated 8-simplex honeycomb
- Quarter 8-cubic honeycomb
- Rectified 9-cubes
- Rectified 9-orthoplexes
- Rectified 9-simplexes
- Uniform 9-polytope

### E8 (mathematics)

- 1 42 polytope
- 1 52 honeycomb
- 2 41 polytope
- 2 51 honeycomb
- 4 21 polytope
- 5 21 honeycomb
- E8 (mathematics)
- E8 lattice
- E8 manifold
- E8 polytope
- Heterotic string theory

## References

Also known as E8 honeycomb, Gosset 5 21 honeycomb, Gosset 5 21 lattice.