# 3 31 honeycomb

3$_{31}$ honeycomb :
| (no image) ||
| Type | Uniform tessellation |
| Schläfli symbol | {3,3,3,3$^{3,1}$} |
| Coxeter symbol | 3$_{31}$ |
| Coxeter-Dynkin diagram | |
| 7-face types | 3$_{21}$ {3$^6$} |
| 6-face types | 2$_{21}$ {3$^5$} |
| 5-face types | 2$_{11}$ {3$^4$} |
| 4-face type | {3$^3$} |
| Cell type | {3$^2$} |
| Face type | {3} |
| Face figure | 0$_{31}$ |
| Edge figure | 1$_{31}$ |
| Vertex figure | 2$_{31}$ |
| Coxeter group | ${\tilde {E}}_{7}$, [3$^{3,3,1}$] |
| Properties | vertex-transitive |
In 7-dimensional geometry, the 3$_{31}$ honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3$^{3,1}$} and is composed of 3$_{21}$ and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

## Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the 6-simplex facet:
Removing the node on the end of the 3-length branch leaves the 3$_{21}$ facet:
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2$_{31}$ polytope.
The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (1$_{31}$).
The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (0$_{31}$).
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism ×{3,3}.

## Kissing number

Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 2$_{31}$.

## E7 lattice

The 3$_{31}$ honeycomb's vertex arrangement is called the E$_7$ lattice.
${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144. Both ${\tilde {E}}_{7}$ and ${\tilde {A}}_{7}$ can be seen as affine extension from $A_{7}$ from different nodes:
The E$_7$ lattice can also be expressed as a union of the vertices of two A$_7$ lattices, also called A$_7$$^2$:
The E$_7$$^*$ lattice (also called E$_7$$^2$) has double the symmetry, represented by [[3,3$^{3,3}$]]. The Voronoi cell of the E$_7$$^*$ lattice is the 1$_{32}$ polytope, and voronoi tessellation the 1$_{33}$ honeycomb. The E$_7$$^*$ lattice is constructed by 2 copies of the E$_7$ lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A$_7$$^*$ lattices, also called A$_7$$^4$:
    ∪ = ∪ ∪ ∪ = dual of .

## Related honeycombs

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3$_{k1}$ series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.
3$_{k1}$ dimensional figures :
| Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$=E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| Coxeter diagram | | | | | | |
| Symmetry | [3$^{−1,3,1}$] | [3$^{0,3,1}$] | [[3$^{1,3,1}$]] = [4,3,3,3,3] | [3$^{2,3,1}$] | [3$^{3,3,1}$] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | 3$_{1,-1}$ | 3$_{10}$ | 3$_{11}$ | 3$_{21}$ | 3$_{31}$ | 3$_{41}$ |