Cubic-square tiling honeycomb

In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram,, and is named by its two regular cells.

It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified square tiling r{4,4}, becomes the regular square tiling {4,4}.

Symmetry
A lower symmetry form, index 6, of this honeycomb can be constructed with [(4,4,4,3*] symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram . Another lower symmetry constructions exists with symmetry [(4,4,(4,3)*)], index 48 and an ideal regular octahedral fundamental domain.