In geometry, 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 {4,3<sup>6</sup>,4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,3<sup>5</sup>,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {âÂÂ}<sup>(8)</sup>.
The [4,3<sup>6</sup>,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, , contains all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D<sub>8</sub><sup>*</sup> lattice. Facets can be identically colored from a doubled ÃÂ2, <nowiki></nowiki>4,3<sup>6</sup>,4<nowiki></nowiki> symmetry, alternately colored from , [4,3<sup>6</sup>,4] symmetry, three colors from , [4,3<sup>5</sup>,3<sup>1,1</sup>] symmetry, and 4 colors from , [3<sup>1,1</sup>,3<sup>4</sup>,3<sup>1,1</sup>] symmetry.