The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3<sup>5</sup>,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,3<sup>4</sup>,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {âÂÂ}<sup>(7)</sup>.
The [4,3<sup>5</sup>,4], , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.
The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.
A quadritruncated 7-cubic honeycomb, , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D<sub>7</sub><sup>*</sup> lattice. Facets can be identically colored from a doubled ÃÂ2, <nowiki>[[</nowiki>4,3<sup>5</sup>,4] symmetry, alternately colored from , [4,3<sup>5</sup>,4] symmetry, three colors from , [4,3<sup>4</sup>,3<sup>1,1</sup>] symmetry, and 4 colors from , [3<sup>1,1</sup>,3<sup>3</sup>,3<sup>1,1</sup>] symmetry.