# 2 22 honeycomb

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

222 honeycomb
(no image)
TypeUniform tessellation
Coxeter symbol222
Schläfli symbol{3,3,32,2}
Coxeter diagram
6-face type221
5-face types211
{34}
4-face type{33}
Cell type{3,3}
Face type{3}
Face figure{3}×{3} duoprism
Edge figure{32,2}
Vertex figure122
Coxeter group${\tilde {E}}_{6}$ , [[3,3,3<sup>2,2</sup>]]
Propertiesvertex-transitive, facet-transitive

Its vertex arrangement is the E6 lattice, and the root system of the E6 Lie group so it can also be called the E6 honeycomb.

## Construction

It is created by a Wythoff construction upon a set of 7 hyperplane 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, its only facet type,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, .

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, {3}×{3}, .

## Kissing number

Each vertex of this tessellation is the center of a 5-sphere in the densest known packing in 6 dimensions, with kissing number 72, represented by the vertices of its vertex figure 122.

## E6 lattice

The 222 honeycomb's vertex arrangement is called the E6 lattice.

The E62 lattice, with [[3,3,3<sup>2,2</sup>]] symmetry, can be constructed by the union of two E6 lattices:

The E6* lattice (or E63) with [3[32,2,2]] symmetry. The Voronoi cell of the E6* lattice is the rectified 122 polytope, and the Voronoi tessellation is a bitruncated 222 honeycomb. It is constructed by 3 copies of the E6 lattice vertices, one from each of the three branches of the Coxeter diagram.

= dual to .

## Geometric folding

The ${\tilde {E}}_{6}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.

${\tilde {E}}_{6}$ ${\tilde {F}}_{4}$ {3,3,32,2} {3,3,4,3}

The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with ${\tilde {E}}_{6}$ symmetry. 24 of them have doubled symmetry [[3,3,3<sup>2,2</sup>]] with 2 equally ringed branches, and 7 have sextupled (3!) symmetry [3[32,2,2]] with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 222 and birectified 222 are isotopic, with only one type of facet: 221, and rectified 122 polytopes respectively.

Symmetry Order Honeycombs
[32,2,2] Full

8: , , , , , , , .

[[3,3,3<sup>2,2</sup>]] ×2

24: , , , , , ,

, , , , , ,

, , , , , ,

, , , , , .

[3[32,2,2]] ×6

7: , , , , , , .

### Birectified 222 honeycomb

Birectified 222 honeycomb
(no image)
TypeUniform tessellation
Coxeter symbol0222
Schläfli symbol{32,2,2}
Coxeter diagram
6-face type0221
5-face types022
0211
4-face type021
24-cell 0111
Cell typeTetrahedron 020
Octahedron 011
Face typeTriangle 010
Vertex figureProprism {3}×{3}×{3}
Coxeter group${\tilde {E}}_{6}$ , [3[32,2,2]]
Propertiesvertex-transitive, facet-transitive

The birectified 222 honeycomb , has rectified 1_22 polytope facets, , and a proprism {3}×{3}×{3} vertex figure.

Its facets are centered on the vertex arrangement of E6* lattice, as:

#### Construction

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

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}×{3}×{3}, .

Removing a node on the end of one of the 3-node branches leaves the 122, its only facet type, .

Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 022 and birectified 5-orthoplex, 0211.

Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 021, and 24-cell, 0111.

Removing a fourth end node defines 2 types of cells: octahedron, 011, and tetrahedron, 020.

### k22 polytopes

The 222 honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The final is a paracompact hyperbolic honeycomb, 322. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 ${\tilde {E}}_{6}$ =E6+ ${\bar {T}}_{7}$ =E6++
Coxeter
diagram
Symmetry [[3<sup>2,2,-1</sup>]] [[3<sup>2,2,0</sup>]] [[3<sup>2,2,1</sup>]] [[3<sup>2,2,2</sup>]] [[3<sup>2,2,3</sup>]]
Order 72 1440 103,680
Graph
Name 122 022 122 222 322

The 222 honeycomb is third in another dimensional series 22k.

22k figures of n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 A5 E6 ${\tilde {E}}_{6}$ =E6+ E6++
Coxeter
diagram
Graph
Name 22,-1 220 221 222 223