Cubic-octahedral honeycomb Cubic-octahedral honeycomb In the geometry of hyperbolic 3-space, the cube-octahedron honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells. Wide-angle perspective views: It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, , with vertex figure (3.4). A lower symmetry form, index 6, of this honeycomb can be constructed with [(4,3,4,3)] symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram . This lower symmetry can be extended by restoring one mirror as .
Cubic honeycomb It is composed of cubes and triangular prisms in a ratio of 1:2. It is constructed from a snub square tiling extruded into prisms. It is one of 28 convex uniform honeycombs. Cubic honeycomb The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille. The Cartesian coordinates of the vertices are: