Omnitruncated 7-simplex honeycomb
# Omnitruncated 7-simplex honeycomb

Omnitruncated 7-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform honeycomb |
| Family | Omnitruncated simplectic honeycomb |
| Schläfli symbol | ${3^{[8]}}$ |
| Coxeter–Dynkin diagrams | |
| 6-face types | $t_{0123456}{3,3,3,3,3,3}$ |
| Vertex figure | Irr. 7-simplex |
| Symmetry | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {{\tilde {A}}}_{8} ×16, $[8[3^{[8]}]]$ |
| Properties | vertex-transitive |

In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-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).

## $A_7^*$ lattice

The $A^*$ $_7$ lattice (also called $A^8$ $_7)$ is the union of eight $A_7$ lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.

∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

## Related polytopes and honeycombs

This honeycomb is one of 29 unique uniform honeycombs constructed by the ${\tilde {A}}_{7}$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

A7 honeycombs :

| Octagon symmetry | Extended symmetry | Extended group | Honeycombs |
| - | - | - | - |
| a1 | $[3^{[8]}]$ | ${\tilde {A}}_{7}$ | |
| d2 | $<[3^{[8]}]>$ | ${\tilde {A}}_{7}$$×2_1$ | $_1$ |
| p2 | $[[3^{[8]}]]$ | ${\tilde {A}}_{7}$$×2_2$ | $_2$ |
| d4 | $<2[3^{[8]}]>$ | ${\tilde {A}}_{7}$$×4_1$ | |
| p4 | $[2[3^{[8]}]]$ | ${\tilde {A}}_{7}$$×4_2$ | |
| d8 | $[4[3^{[8]}]]$ | ${\tilde {A}}_{7}$×8 | |
| r16 | $[8[3^{[8]}]]$ | ${\tilde {A}}_{7}$×16 | $_3$ |
