There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. 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 [4,33,4], , 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.
It is also related to the regular 6-cube which exists in 6-space with three 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.
The Penrose tilings are 2-dimensional aperiodic tilings that can be obtained as a projection of the 5-cubic honeycomb along a 5-fold rotational axis of symmetry. The vertices correspond to points in the 5-dimensional cubic lattice, and the tiles are formed by connecting points in a predefined manner.[1]
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 , [4,33,4] symmetry, three colors from , [4,3,3,31,1] symmetry, and 4 colors from , [31,1,3,31,1] symmetry.