# 1 33 honeycomb

1$_{33}$ honeycomb :
| (no image) ||
| Type | Uniform tessellation |
| Schläfli symbol | {3,3$^{3,3}$} |
| Coxeter symbol | 1$_{33}$ |
| Coxeter-Dynkin diagram | or |
| 7-face type | 1$_{32}$ |
| 6-face types | 1$_{22}$ 1$_{31}$ |
| 5-face types | 1$_{21}$ {3$^4$} |
| 4-face type | 1$_{11}$ {3$^3$} |
| Cell type | 1$_{01}$ |
| Face type | {3} |
| Cell figure | Square |
| Face figure | Triangular duoprism |
| Edge figure | Tetrahedral duoprism |
| Vertex figure | Trirectified 7-simplex |
| Coxeter group | ${\tilde {E}}_{7}$, [[3,3$^{3,3}$]] |
| Properties | vertex-transitive, facet-transitive |
In 7-dimensional geometry, 1$_{33}$ is a uniform honeycomb, also given by Schläfli symbol {3,3$^{3,3}$}, and is composed of 1$_{32}$ facets.

## 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 a node on the end of one of the 3-length branch leaves the 1$_{32}$, its only facet type.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 0$_{33}$.
The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.

## Kissing number

Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.

## Geometric folding

The ${\tilde {E}}_{7}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
| ${\tilde {E}}_{7}$ | ${\tilde {F}}_{4}$ |
| {3,3$^{3,3}$} | {3,3,4,3} |

## 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$$^*$ lattice (also called E$_7$$^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 polytopes and honeycombs

The 1$_{33}$ is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1$_{3k}$ series. The final is a noncompact hyperbolic honeycomb, 1$_{34}$.
1$_{3k}$ 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$^{1,3,1}$] | [3$^{2,3,1}$] | [[3$^{3,3,1}$]] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | 1$_{3,-1}$ | 1$_{30}$ | 1$_{31}$ | 1$_{32}$ | 1$_{33}$ | 1$_{34}$ |

#### Rectified 1$_{33}$ honeycomb

Rectified 1$_{33}$ honeycomb :
| (no image) ||
| Type | Uniform tessellation |
| Schläfli symbol | {3$^{3,3,1}$} |
| Coxeter symbol | 0$_{331}$ |
| Coxeter-Dynkin diagram | or |
| 7-face type | Trirectified 7-simplex Rectified 1_32 |
| 6-face types | Birectified 6-simplex Birectified 6-cube Rectified 1_22 |
| 5-face types | Rectified 5-simplex Birectified 5-simplex Birectified 5-orthoplex |
| 4-face type | 5-cell Rectified 5-cell 24-cell |
| Cell type | {3,3} {3,4} |
| Face type | {3} |
| Vertex figure |×{3,3}×{3,3} |
| Coxeter group | ${\tilde {E}}_{7}$, [[3,3$^{3,3}$]] |
| Properties | vertex-transitive, facet-transitive |
The rectified 1$_{33}$ or 0$_{331}$, Coxeter diagram has facets and , and vertex figure .