6-cubic honeycomb
# 6-cubic honeycomb

6-cubic honeycomb :

| (no image) ||
| - | - |
| Type | Regular 6-honeycomb Uniform 6-honeycomb |
| Family | Hypercube honeycomb |
| Schläfli symbol | ${4,3^4,4}$ ${4,3^3,3^{1,1}}$ |
| Coxeter-Dynkin diagrams | |
| 6-face type | ${4,3^4}$ |
| 5-face type | ${4,3^3}$ |
| 4-face type | {4,3,3} |
| Cell type | {4,3} |
| Face type | {4} |
| Face figure | {4,3} (octahedron) |
| Edge figure | 8 {4,3,3} (16-cell) |
| Vertex figure | 64 ${4,3^4}$ (6-orthoplex) |
| Coxeter group | ${\tilde {C}}_{6}$, $[4,3^4,4]$ ${\tilde {B}}_{6}$, $[4,3^3,3^{1,1}]$ |
| Dual | self-dual |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive |

The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.

## Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol ${4,3^4,4}.$ Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol ${4,3^3,3^{1,1}}.$ The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol ${∞}^{(6)}.$

## Related honeycombs

The $[4,3^4,4],$ , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.

The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.

### Trirectified 6-cubic honeycomb

A trirectified 6-cubic honeycomb, , contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the $D_6^*$ lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{6}$×2, $[[4,3^4,4]]$ symmetry, alternately colored from ${\tilde {C}}_{6}$, $[4,3^4,4]$ symmetry, three colors from ${\tilde {B}}_{6}$, $[4,3^3,3^{1,1}]$ symmetry, and 4 colors from ${\tilde {D}}_{6}$, $[3^{1,1},3,3,3^{1,1}]$ symmetry.

