Cyclotruncated 8-simplex honeycomb
# Cyclotruncated 8-simplex honeycomb

Cyclotruncated 8-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform honeycomb |
| Family | Cyclotruncated simplectic honeycomb |
| Schläfli symbol | $t_{0,1}{3^{[9]}}$ |
| Coxeter diagram | |
| 8-face types | ${3^7}$ , $t_{0,1}{3^7}$ $t_{1,2}{3^7}$ , $t_{2,3}{3^7}$ $t_{3,4}{3^7}$ |
| Vertex figure | Elongated 7-simplex antiprism |
| Symmetry | ${\tilde {A}}_{8}$×2, $[[3^{[9]}]]$ |
| Properties | vertex-transitive |

In eight-dimensional Euclidean geometry, the cyclotruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, truncated 8-simplex, bitruncated 8-simplex, tritruncated 8-simplex, and quadritruncated 8-simplex facets. These facet types occur in proportions of 2:2:2:2:1 respectively in the whole honeycomb.

## Structure

It can be constructed by nine sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 7-simplex honeycomb divisions on each hyperplane.

## 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 | $_3$ |
