5-cubic honeycomb


The 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
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.

Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol. Another form has two alternating 5-cube facets with Schläfli symbol. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol 5.

Related polytopes and honeycombs

The ,, Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.
It is also related to the regular 6-cube which exists in 6-space with 3 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb,.

Tritruncated 5-cubic honeycomb

A tritruncated 5-cubic honeycomb,, contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled ×2, 4,33,4 symmetry, alternately colored from, symmetry, three colors from, symmetry, and 4 colors from, symmetry.