Cubic-octahedral honeycomb
In the geometry of hyperbolic 3-space, the cubic-octahedral 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.
Images
Wide-angle perspective views:It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling,, with vertex figure 4.
Symmetry
A lower symmetry form, index 6, of this honeycomb can be constructed with symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram. This lower symmetry can be extended by restoring one mirror as.| ↔ = | ↔ = | ↔ = |
Related honeycombs
There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group :,,,,.Rectified cubic-octahedral honeycomb
The rectified cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cuboctahedron and rhombicuboctahedron cells, in a cuboid vertex figure. It has a Coxeter diagram.Cyclotruncated cubic-octahedral honeycomb
The cyclotruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cube and octahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram.It can be seen as somewhat analogous to the trioctagonal tiling, which has truncated square and triangle facets:
Cyclotruncated octahedral-cubic honeycomb
The cyclotruncated octahedral-cubic honeycomb is a compact uniform honeycomb, constructed from cube and 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 :