# 6-cubic honeycomb

6-cubic honeycomb
(no image)
Type Regular 6-space honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,34,4}
{4,33,31,1}
Coxeter-Dynkin diagrams

6-face type {4,34}
5-face type {4,33}
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 {4,34}
(6-orthoplex)
Coxeter group ${\displaystyle {\tilde {C}}_{6}}$, [4,34,4]
${\displaystyle {\tilde {B}}_{6}}$, [4,33,31,1]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 6-cube 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 {4,34,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,33,31,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}6.

## Related honeycombs

The [4,34,4], , 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, , containins all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C}}_{6}}$×2, [[4,34,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C}}_{6}}$, [4,34,4] symmetry, three colors from ${\displaystyle {\tilde {B}}_{6}}$, [4,33,31,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{6}}$, [31,1,3,3,31,1] symmetry.

Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$