# 2 22 honeycomb

2$_{22}$ honeycomb :
| (no image) ||
| Type | Uniform tessellation |
| Coxeter symbol | 2$_{22}$ |
| Schläfli symbol | {3,3,3$^{2,2}$} |
| Coxeter diagram | |
| 6-face type | 2$_{21}$ |
| 5-face types | 2$_{11}$ {3$^4$} |
| 4-face type | {3$^3$} |
| Cell type | {3,3} |
| Face type | {3} |
| Face figure | {3}×{3} duoprism |
| Edge figure | {3$^{2,2}$} |
| Vertex figure | 1$_{22}$ |
| Coxeter group | ${\tilde {E}}_{6}$, [[3,3,3$^{2,2}$]] |
| Properties | vertex-transitive, facet-transitive |
In geometry, the 2$_{22}$ honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,3$^{2,2}$}. It is constructed from 2$_{21}$ facets and has a 1$_{22}$ vertex figure, with 54 2$_{21}$ polytopes around every vertex.
Its vertex arrangement is the E$_6$ lattice, and the root system of the E$_6$ Lie group so it can also be called the E$_6$ 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 2$_{21}$, its only facet type,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1$_{22}$, .
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t$_2${3$^4$}, .
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 1$_{22}$.

## E$_6$ lattice

The 2$_{22}$ honeycomb's vertex arrangement is called the E$_6$ lattice.
The E$_6$$^2$ lattice, with [[3,3,3$^{2,2}$]] symmetry, can be constructed by the union of two E$_6$ lattices:
The E$_6$$^*$ lattice (or E$_6$$^3$) with [[3,3$^{2,2,2}$]] symmetry. The Voronoi cell of the E$_6$$^*$ lattice is the rectified 1$_{22}$ polytope, and the Voronoi tessellation is a bitruncated 2$_{22}$ honeycomb. It is constructed by 3 copies of the E$_6$ 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.
| ${\tilde {E}}_{6}$ | ${\tilde {F}}_{4}$ |
| {3,3,3$^{2,2}$} | {3,3,4,3} |

## Related honeycombs

The 2$_{22}$ honeycomb is one of 127 uniform honeycombs (39 unique) with ${\tilde {E}}_{6}$ symmetry. 24 of them have doubled symmetry [[3,3,3$^{2,2}$]] with 2 equally ringed branches, and 7 have sextupled (3!) symmetry [[3,3$^{2,2,2}$]] with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 2$_{22}$ and birectified 2$_{22}$ are isotopic, with only one type of facet: 2$_{21}$, and rectified 1$_{22}$ polytopes respectively.
| Symmetry | Order | Honeycombs |
| [3$^{2,2,2}$] | Full | 8: , , , , , , , . |
| [[3,3,3$^{2,2}$]] | ×2 | 24: , , , , , , , , , , , , , , , , , , , , , , , . |
| [[3,3$^{2,2,2}$]] | ×6 | 7: , , , , , , . |

### Birectified 2$_{22}$ honeycomb

Birectified 2$_{22}$ honeycomb :
| (no image) ||
| Type | Uniform tessellation |
| Coxeter symbol | 0$_{222}$ |
| Schläfli symbol | {3$^{2,2,2}$} |
| Coxeter diagram | |
| 6-face type | 0$_{221}$ |
| 5-face types | 0$_{22}$ 0$_{211}$ |
| 4-face type | 0$_{21}$ 24-cell 0$_{111}$ |
| Cell type | Tetrahedron 0$_{20}$ Octahedron 0$_{11}$ |
| Face type | Triangle 0$_{10}$ |
| Vertex figure | Proprism {3}×{3}×{3} |
| Coxeter group | 6×${\tilde {E}}_{6}$, [[3,3$^{2,2,2}$]] |
| Properties | vertex-transitive, facet-transitive |
The birectified 2$_{22}$ honeycomb , has rectified 1 22 polytope facets, , and a proprism {3}×{3}×{3} vertex figure.
Its facets are centered on the vertex arrangement of E$_6$$^*$ 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 rectified 1$_{22}$, its only facet type, .
Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 0$_{22}$ and birectified 5-orthoplex, 0$_{211}$.
Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 0$_{21}$, and 24-cell, 0$_{111}$.
Removing a fourth end node defines 2 types of cells: octahedron, 0$_{11}$, and tetrahedron, 0$_{20}$.

### k$_{22}$ polytopes

The 2$_{22}$ honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k$_{22}$ series. The final is a paracompact hyperbolic honeycomb, 3$_{22}$. Each progressive uniform polytope is constructed from the previous as its vertex figure.
k$_{22}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 |
| Coxeter group | A$_2$A$_2$ | E$_6$ | ${\tilde {E}}_{6}$=E$_6$$^+$ | ${\bar {T}}_{7}$=E$_6$$^{++}$ |
| Coxeter diagram | | | | | |
| Symmetry | [[3$^{2,2,-1}$]] | [[3$^{2,2,0}$]] | [[3$^{2,2,1}$]] | [[3$^{2,2,2}$]] | [[3$^{2,2,3}$]] |
| Order | 72 | 1440 | 103,680 | ∞ | ∞ |
| Graph | | | | ∞ | ∞ |
| Name | −1$_{22}$ | 0$_{22}$ | 1$_{22}$ | 2$_{22}$ | 3$_{22}$ |
The 2$_{22}$ honeycomb is third in another dimensional series 2$_{2k}$.
2$_{2k}$ figures of n dimensions :
| Space | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 |
| Coxeter group | A$_2$A$_2$ | A$_5$ | E$_6$ | ${\tilde {E}}_{6}$=E$_6$$^+$ | E$_6$$^{++}$ |
| Coxeter diagram | | | | | |
| Graph | | | | ∞ | ∞ |
| Name | 2$_{2,-1}$ | 2$_{20}$ | 2$_{21}$ | 2$_{22}$ | 2$_{23}$ |