Triangular tiling honeycomb infinite cells or vertex figures. There are nine uniform honeycombs in the [3,6,3] Coxeter group family, including this regular form as well as the bitruncated form, t{3,6,3}, with all truncated hexagonal tiling facets. The rectified triangular tiling honeycomb, , has alternating trihexagonal tiling and hexagonal tiling cells, with a triangular prism vertex figure. A lower symmetry of this honeycomb can be constructed as a "cantic order-6 hexagonal tiling honeycomb", ↔ . A second lower construction, index is ↔ . The truncated triangular tiling honeycomb, is the same as the hexagonal tiling honeycomb, . The bitruncated triangular tiling honeycomb, , has
Bitruncated cubic honeycomb Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction. The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb. The [4,3], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the formula_1 Coxeter group. The symmetry can be multiplied by the symmetry of rings in