7-cubic honeycomb
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Type	Regular 7-honeycomb Uniform 7-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,35,4} {4,34,31,1} {∞}(7)
Coxeter-Dynkin diagrams
7-face type	{4,3,3,3,3,3}
6-face type	{4,3,3,3,3}
5-face type	{4,3,3,3}
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	128 {4,35} (7-orthoplex)
Coxeter group	[4,35,4]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

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 {4,3⁵,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,3⁴,3^1,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁷⁾.

Related honeycombs

The [4,3⁵,4], , 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.

Quadritruncated 7-cubic honeycomb

A quadritruncated 7-cubic honeycomb, , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D₇^* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C))_{7))$×2, [[4,3⁵,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C))_{7))$, [4,3⁵,4] symmetry, three colors from ${\displaystyle {\tilde {B))_{7))$, [4,3⁴,3^1,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D))_{7))$, [3^1,1,3³,3^1,1] symmetry.