In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an E₆ honeycomb (or 2₂₂ honeycomb) is a tessellation of uniform polytopes in 6-dimensional Euclidean space.

is a affine Coxeter group. 127 uniform honeycombs can be generated from this family by all ring permutations of its Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but there are one simplest ones, with a single ring at the end of one of its 3 branches: 2₂₂.

The 2₂₂ honeycomb's vertex arrangement

Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....

 is called the E6 lattice.

2₂₂ honeycomb

222 honeycomb
(no image)
Type	Uniform tessellation
Coxeter symbol	222
Schläfli symbol	{3,3,32,2}
Coxeter–Dynkin diagram
6-face type	221 Gosset 2 21 polytope In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure....
5-face types	211 Pentacross 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 hypercells.... {34}
4-face type	{33} Pentachoron In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...
Cell type	{3,3} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
Face type	{3} Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
Face figure	{3}×{3} duoprism Duoprism In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
Edge figure	t2{34}
Vertex figure	122 Gosset 1 22 polytope In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 ....
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	, [32,2,2]
Properties	vertex-transitive Vertex-transitive In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same... , facet-transitive

The 2₂₂ honeycomb is a uniform tessellation

Uniform tessellation

In geometry, a uniform tessellation is a vertex-transitive tessellations made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex....

. It can also be represented by the Schlafli symbol {3,3,3^2,2}. It is constructed from 2₂₁

Gosset 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure....

 facets and has a 1₂₂

Gosset 1 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 ....

 vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

, with 54 2₂₁ polytopes around every vertex.

Construction

It is created by a Wythoff construction

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...

 upon a set of 7 hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

 mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter–Dynkin diagram, .

Removing a node on the end of one of the 2-node branches leaves the 2₂₁

Gosset 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure....

, its only facet

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

 type,

The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 is determined by removing the ringed node and ringing the neighboring node. This makes 1₂₂

Gosset 1 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 ....

, .

The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t₂{3⁴}, .

The face figure is the vertex figure of the edge figure, here being a triangular duoprism

Duoprism

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...

, {3}×{3}, .

Kissing number

Each vertex of this tessellation is the center of a 5-sphere in the densest known packing

Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space...

 in 6 dimensions, with kissing number 72.

Geometric folding

The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

{3,3,32,2}	{3,3,4,3}