Omnitruncated 6-simplex honeycomb
# Omnitruncated 6-simplex honeycomb

Omnitruncated 6-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform honeycomb |
| Family | Omnitruncated simplectic honeycomb |
| Schläfli symbol | ${3^{[8]}}$ |
| Coxeter–Dynkin diagrams | |
| Facets | $t_{0,1,2,3,4,5}{3,3,3,3,3}$ |
| Vertex figure | Irr. 6-simplex |
| Symmetry | ${\tilde {A}}_{7}$×14, $[7[3^{[7]}]]$ |
| Properties | vertex-transitive |

In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-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).

## $A^*$ $_6$ lattice

The $A^*$ $_6$ lattice (also called $A^7$ $_6)$ is the union of seven $A_6$ lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.

∪ ∪ ∪ ∪ ∪ ∪ = dual of

## Related polytopes and honeycombs

This honeycomb is one of 17 unique uniform honeycombs constructed by the ${\tilde {A}}_{6}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs :

| Heptagon symmetry | Extended symmetry | Extended group | Honeycombs |
| - | - | - | - |
| a1 | $[3^{[7]}]$ | ${\tilde {A}}_{6}$ | |
| i2 | $[[3^{[7]}]]$ | ${\tilde {A}}_{6}$×2 | $_1$ $_2$ |
| r14 | $[7[3^{[7]}]]$ | ${\tilde {A}}_{6}$×14 | $_3$ |
