Octahedral-hexagonal tiling honeycomb Octahedral-hexagonal tiling honeycomb In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells. A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,4,3)] symmetry, represented by a trigonal trapezohedron fundamental domain, and a Coxeter diagram . The cyclotruncated octahedral-hexagonal tiling honeycomb is a compact uniform honeycomb, constructed from hexagonal tiling, cube, and truncated octahedron cells, in a triangular antiprism vertex figure. It
Order-6 hexagonal tiling honeycomb fundamental domain. It has an octahedral Coxeter diagram with 6 order-3 branches, and 3 infinite-order branches in the shape of a triangular prism, . It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures. There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form. It has a related alternation honeycomb, represented by ↔ , having alternating triangular tiling cells, and a regular form as , called a triangular tiling honeycomb. This honeycomb is a part of a sequence of polychora and