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
Tetrahedral-octahedral honeycomb Conway calls this honeycomb a 3-RCO-trille, and its dual quarter cubille. The dual of a "runcic cubic honeycomb" is called a quarter cubille, with Coxeter diagram , with faces in 2 of 4 hyerplanes of the formula_3, [4,3] symmetry fundamental domain. Cells can be seen as 1/4 of dissected cube, using 4 vertices and the center. Four cells exist around 6 edges, and 3 cells around 3 edges. It is related to the runcinated cubic honeycomb, with quarter of the cubes alternated into tetrahedra, and half expanded into rhombicuboctahedra. This honeycomb can be divided on truncated square tiling planes, using