Omnitruncated 8-simplex honeycomb
# Omnitruncated 8-simplex honeycomb

Omnitruncated 8-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform honeycomb |
| Family | Omnitruncated simplectic honeycomb |
| Schläfli symbol | ${3^{[9]}}$ |
| Coxeter–Dynkin diagrams | |
| 7-face types | $t_{01234567}{3,3,3,3,3,3,3}$ |
| Vertex figure | Irr. 8-simplex |
| Symmetry | ${\tilde {A}}_{9}$×18, $[9[3^{[9]}]]$ |
| Properties | vertex-transitive |

In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-simplex facets.

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

## $A^*$ $_8$ lattice

The $A^*$ $_8$ lattice (also called $A^9$ $_8)$ is the union of nine $A_8$ lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

## Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honeycombs constructed by the ${\tilde {A}}_{8}$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

A8 honeycombs :

| Enneagon symmetry | Symmetry | Extended group | Honeycombs |
| - | - | - | - |
| a1 | $[3^{[9]}]$ | ${\tilde {A}}_{8}$ | |
| i2 | $[[3^{[9]}]]$ | ${\tilde {A}}_{8}$×2 | $_1$ $_2$ |
| i6 | $[3[3^{[9]}]]$ | ${\tilde {A}}_{8}$×6 | |
| r18 | $[9[3^{[9]}]]$ | ${\tilde {A}}_{8}$×18 | $_3$ |
