Trigonal trapezohedral honeycomb into squares. It is dual to the quarter cubic honeycomb with tetrahedral and truncated tetrahedral cells: Trigonal trapezohedral honeycomb The trigonal trapezohedral honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. Cells are identical trigonal trapezohedron or rhombohedra. John Horton Conway calls it an oblate cubille. This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 4 trigonal trapezohedra or rhombohedra. It is analogous to the regular hexagonal being dissectable into 3 rhombi and tiling the plane as a rhombille. The rhombille tiling is actually an orthogonal projection of
Icosahedral honeycomb part of a sequence of regular polychora and honeycombs {3,"p",3}: It a part of a sequence of regular polychora and honeycombs {"p",5,"p"}: There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t{3,5,3}, , also called "truncated dodecahedral honeycomb", each of whose cells are truncated dodecahedra. The rectified icosahedral honeycomb, t{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure: There are four rectified compact regular honeycombs: The truncated icosahedral honeycomb, t{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex