# 8-cubic honeycomb

8-cubic honeycomb
(no image)
Type Regular 8-dimensional honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,36,4}
{4,35,31,1}
t0,8{4,36,4}
{∞}8
Coxeter-Dynkin diagrams

8-face type {4,36}
7-face type {4,35}
6-face type {4,34}
5-face type {4,33}
4-face type {4,32}
Cell type {4,3}
Face type {4}
Face figure {4,3}
(octahedron)
Edge figure 8 {4,3,3}
(16-cell)
Vertex figure 256 {4,36}
(8-orthoplex)
Coxeter group [4,36,4]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,36,4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,35,31,1}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {∞}8.

## Related honeycombs

The [4,36,4], , Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb.

The 8-cubic honeycomb can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with 8-demicubes, and the alternated gaps are filled by 8-orthoplex facets.

A quadrirectified 8-cubic honeycomb, , containins all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D8* lattice. Facets can be identically colored from a doubled ${\tilde{C}}_8$×2, [[4,36,4]] symmetry, alternately colored from ${\tilde{C}}_8$, [4,36,4] symmetry, three colors from ${\tilde{B}}_8$, [4,35,31,1] symmetry, and 4 colors from ${\tilde{D}}_8$, [31,1,34,31,1] symmetry.