 | ||
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.
Contents
- Images
- Symmetry
- Related honeycombs
- Rectified cubic octahedral honeycomb
- Cyclotruncated cubic octahedral honeycomb
- Cyclotruncated octahedral cubic honeycomb
- Truncated cubic octahedral honeycomb
- Omnitruncated cubic octahedral honeycomb
- References
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Images
Wide-angle perspective views:
It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, , with vertex figure (3.4)4.
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 rhombicuboctahedronCyclotruncated cubic-octahedral honeycomb
The cyclotruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cube, octahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram .
Perspective view from center of octahedronIt can be seen as somewhat analogous to the trioctagonal tiling with truncated square and triangle facets:
Cyclotruncated octahedral-cubic honeycomb
The cyclotruncated octahedral-cubic honeycomb is a compact uniform honeycomb, constructed from cube, truncated octahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .
Perspective view from center of cubeIt contains an H2 subgroup tetrahexagonal tiling alternating square and hexagonal faces, with Coxeter diagram or half symmetry :
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, truncated cuboctahedron cells, in a rectangular pyramid vertex figure. It has a Coxeter diagram .
Perspective view from center of rhombicuboctahedronOmnitruncated 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