Cubic-octahedral honeycomb explained

bgcolor=#e7dcc3 colspan=2	Cube-octahedron honeycomb
Type	Compact uniform honeycomb
Schläfli symbol	or
Coxeter diagrams	↔ ↔
Cells
Faces
Vertex figure	rhombicuboctahedron
Coxeter group	[(4,3)<sup>[2]]
Properties	Vertex-transitive, edge-transitive

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.

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

bgcolor=#e7dcc3 colspan=2	Rectified cubic-octahedral honeycomb
Type	Compact uniform honeycomb
Schläfli symbol	r
Coxeter diagrams
Cells
Faces
Vertex figure	cuboid
Coxeter group	[[(4,3)[2]]],
Properties	Vertex-transitive, edge-transitive

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

bgcolor=#e7dcc3 colspan=2	Cyclotruncated cubic-octahedral honeycomb
Type	Compact uniform honeycomb
Schläfli symbol	ct
Coxeter diagrams
Cells
Faces
Vertex figure	square antiprism
Coxeter group	[[(4,3)[2]]],
Properties	Vertex-transitive, edge-transitive

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:

Cyclotruncated octahedral-cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Cyclotruncated octahedral-cubic honeycomb
Type	Compact uniform honeycomb
Schläfli symbol	ct
Coxeter diagrams	↔ ↔
Cells
Faces
Vertex figure	triangular antiprism
Coxeter group	[[(4,3)[2]]],
Properties	Vertex-transitive, edge-transitive

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 :

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

bgcolor=#e7dcc3 colspan=2	Truncated cubic-octahedral honeycomb
Type	Compact uniform honeycomb
Schläfli symbol	t
Coxeter diagrams
Cells
Faces
Vertex figure	rectangular pyramid
Coxeter group	[(4,3)<sup>[2]]
Properties	Vertex-transitive

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

bgcolor=#e7dcc3 colspan=2	Omnitruncated cubic-octahedral honeycomb
Type	Compact uniform honeycomb
Schläfli symbol	tr
Coxeter diagrams
Cells
Faces
Vertex figure	Rhombic disphenoid
Coxeter group	[2[(4,3)<sup>[2]]] or [(2,2)<sup>+</sup>[(4,3)<sup>[2]]],
Properties	Vertex-transitive, edge-transitive, cell-transitive

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

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
• Jeffrey R. Weeks The Shape of Space, 2nd edition (Chapter 16-17: Geometries on Three-manifolds I, II)
• Norman Johnson Uniform Polytopes, Manuscript
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups