6-simplex honeycomb
# 6-simplex honeycomb

6-simplex honeycomb :

| (No image) ||
| - | - |
| Type | Uniform 6-honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | ${3^{[7]}}$ |
| Coxeter diagram | |
| 6-face types | ${3^5}$ , $t_1{3^5}$ $t_2{3^5}$ |
| 5-face types | ${3^4}$ , $t_1{3^4}$ $t_2{3^4}$ |
| 4-face types | ${3^3}$ , $t_1{3^3}$ |
| Cell types | {3,3} , $t_1{3,3}$ |
| Face types | {3} |
| Vertex figure | $t_{0,5}{3^5}$ |
| Symmetry | ${\tilde {A}}_{6}$×2, $[[3^{[7]}]]$ |
| Properties | vertex-transitive |

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

## A6 lattice

This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the ${\tilde {A}}_{6}$ Coxeter group. It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.

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 | $_3$ |

## Projection by folding

The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

* ${\tilde {A}}_{6}$
* ${\tilde {C}}_{3}$
