Trigonal trapezohedral honeycomb 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 the "trigonal trapezohedral honeycomb". A different orthogonal projection produces the quadrille where the rhombi are distorted
Honeycomb (geometry) The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers. A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a "space-filling polyhedron". A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of