Cubic-octahedral honeycomb honeycomb is a compact uniform honeycomb, constructed from cube, truncated octahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram . It contains an H2 subgroup tetrahexagonal tiling alternating square and hexagonal faces, with Coxeter diagram or half symmetry : A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,4,3)], , represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram . This lower symmetry can be extended by restoring one mirror as . The truncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated octahedron, truncated cube, rhombicuboctahedron, truncated cuboctahedron cells,
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