Cyclotruncated 6-simplex honeycomb
# Cyclotruncated 6-simplex honeycomb

Cyclotruncated 6-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform honeycomb |
| Family | Cyclotruncated simplectic honeycomb |
| Schläfli symbol | $t_{0,1}{3^{[7]}}$ |
| Coxeter diagram | |
| 6-face types | ${3^5}$ $t{3^5}$ $2t{3^5}$ $3t{3^5}$ |
| Vertex figure | Elongated 5-simplex antiprism |
| Symmetry | ${\tilde {A}}_{6}$×2, $[[3^{[7]}]]$ |
| Properties | vertex-transitive |

In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

## Structure

It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.

## 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 | $_3$ |
