Cyclotruncated 7-simplex honeycomb
# Cyclotruncated 7-simplex honeycomb

Cyclotruncated 7-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform honeycomb |
| Family | Cyclotruncated simplectic honeycomb |
| Schläfli symbol | $t_{0,1}{3^{[8]}}$ |
| Coxeter diagram | |
| 7-face types | ${3^6}$ $t_{0,1}{3^6}$ $t_{1,2}{3^6}$ $t_{2,3}{3^6}$ |
| Vertex figure | Elongated 6-simplex antiprism |
| Symmetry | ${\tilde {A}}_{7}$$×2_2,$ $[[3^{[8]}]]$ |
| Properties | vertex-transitive |

In seven-dimensional Euclidean geometry, the cyclotruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, truncated 7-simplex, bitruncated 7-simplex, and tritruncated 7-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.

## Structure

It can be constructed by eight sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 6-simplex honeycomb divisions on each hyperplane.

## 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 | $_3$ |
