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 (3.4)4.
 * Uniform tiling verf 34343434.png

Symmetry
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.

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.


 * Perspective view from center of rhombicuboctahedron

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.


 * Perspective view from center of octahedron

It can be seen as somewhat analogous to the trioctagonal tiling, which has truncated square and triangle facets:
 * Uniform tiling 433-t01.png

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.


 * Perspective view from center of cube

It contains an H2 subgroup tetrahexagonal tiling alternating square and hexagonal faces, with Coxeter diagram or half symmetry :
 * H2 tiling 344-5.png 3222-uniform_tiling-verf4646.png

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.

Truncated cubic-octahedral honeycomb
The truncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated octahedron, truncated cube, rhombicuboctahedron, and truncated cuboctahedron cells, in a rectangular pyramid vertex figure. It has a Coxeter diagram.


 * Perspective view from center of rhombicuboctahedron

Omnitruncated cubic-octahedral honeycomb
The omnitruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cuboctahedron cells, in a rhombic disphenoid vertex figure. It has a Coxeter diagram with [2,2]+ (order 4) extended symmetry in its rhombic disphenoid vertex figure.


 * Perspective view from center of truncated cuboctahedron