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, File:CDel label4.pngFile:CDel branch 10r.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png, and is named by its two regular cells. 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:
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Centered on cube
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Centered on octahedron
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Centered on cuboctahedron
It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, File:CDel nodes 11.pngFile:CDel 3a3b-cross.pngFile:CDel nodes.png, 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 File:CDel node 1.pngFile:CDel splitplit1u.pngFile:CDel branch3u.pngFile:CDel 3a3buc-cross.pngFile:CDel branch3u 11.pngFile:CDel splitplit2u.pngFile:CDel node.png. This lower symmetry can be extended by restoring one mirror as File:CDel branchu 01r.pngFile:CDel 3ab.pngFile:CDel branch 10lru.pngFile:CDel split2-44.pngFile:CDel node.png.
Related honeycombs
There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group File:CDel label4.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png: File:CDel label4.pngFile:CDel branch 10r.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png, File:CDel label4.pngFile:CDel branch 01r.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png, File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png, File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png, File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 11.pngFile:CDel label4.png.
Rectified cubic-octahedral honeycomb
| Rectified cubic-octahedral honeycomb | |
|---|---|
| Type | Compact uniform honeycomb |
| Schläfli symbol | r{(4,3,4,3)} |
| Coxeter diagrams | File:CDel label4.pngFile:CDel branch 01r.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png |
| Cells | r{4,3} File:Uniform polyhedron-43-t1.png rr{3,4} File:Uniform polyhedron-43-t02.png |
| Faces | triangle {3} square {4} |
| Vertex figure | File:Uniform t02 4343 honeycomb verf.png cuboid |
| Coxeter group | [[(4,3)[2]]], File:CDel label4.pngFile:CDel branch c1-2.pngFile:CDel 3ab.pngFile:CDel branch c2-1.pngFile:CDel label4.png |
| 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 File:CDel label4.pngFile:CDel branch 01r.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png. File:H3 4343-1010 center ultrawide.png
- Perspective view from center of rhombicuboctahedron
Cyclotruncated cubic-octahedral honeycomb
| Cyclotruncated cubic-octahedral honeycomb | |
|---|---|
| Type | Compact uniform honeycomb |
| Schläfli symbol | ct{(4,3,4,3)} |
| Coxeter diagrams | File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png |
| Cells | t{4,3} File:Uniform polyhedron-43-t01.png {3,4} File:Uniform polyhedron-43-t2.png |
| Faces | triangle {3} octagon {8} |
| Vertex figure | File:Uniform t01 4343 honeycomb verf.png square antiprism |
| Coxeter group | [[(4,3)[2]]], File:CDel label4.pngFile:CDel branch c1.pngFile:CDel 3ab.pngFile:CDel branch c2.pngFile:CDel label4.png |
| 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 File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png. File:H3 4343-1100 center ultrawide.png
- 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
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 File:CDel label4.pngFile:CDel branch 10r.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png. File:H3 4343-0110 center ultrawide.png
- Perspective view from center of cube
It contains an H2 subgroup tetrahexagonal tiling alternating square and hexagonal faces, with Coxeter diagram File:CDel branch 11.pngFile:CDel split2-44.pngFile:CDel node.png or half symmetry File:CDel nodes 11.pngFile:CDel 3a3b-cross.pngFile:CDel nodes 11.png:
Symmetry
A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,4,3*)], File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel branch.pngFile:CDel labels.png, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram File:CDel node 1.pngFile:CDel splitplit1u.pngFile:CDel branch3u 11.pngFile:CDel 3a3buc-cross.pngFile:CDel branch3u 11.pngFile:CDel splitplit2u.pngFile:CDel node 1.png. This lower symmetry can be extended by restoring one mirror as File:CDel branchu 11.pngFile:CDel 3ab.pngFile:CDel branch 11.pngFile:CDel split2-44.pngFile:CDel node.png.
Truncated cubic-octahedral honeycomb
| Truncated cubic-octahedral honeycomb | |
|---|---|
| Type | Compact uniform honeycomb |
| Schläfli symbol | t{(4,3,4,3)} |
| Coxeter diagrams | File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png |
| Cells | t{3,4} File:Truncated octahedron.png t{4,3} File:Truncated hexahedron.png rr{3,4} File:Small rhombicuboctahedron.png tr{4,3} File:Great rhombicuboctahedron.png |
| Faces | triangle {3} square {4} hexagon {6} octagon {8} |
| Vertex figure | File:Uniform t012 4343 honeycomb verf.png rectangular pyramid |
| Coxeter group | [(4,3)[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 File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile:CDel label4.png. File:H3 4343-1110 center ultrawide.png
- Perspective view from center of rhombicuboctahedron
Omnitruncated cubic-octahedral honeycomb
| Omnitruncated cubic-octahedral honeycomb | |
|---|---|
| Type | Compact uniform honeycomb |
| Schläfli symbol | tr{(4,3,4,3)} |
| Coxeter diagrams | File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 11.pngFile:CDel label4.png |
| Cells | tr{3,4} File:Great rhombicuboctahedron.png |
| Faces | square {4} hexagon {6} octagon {8} |
| Vertex figure | File:Uniform t0123 4343 honeycomb verf.png Rhombic disphenoid |
| Coxeter group | [2[(4,3)[2]]] or [(2,2)+[(4,3)[2]]], File:CDel label4.pngFile:CDel branch c1.pngFile:CDel 3ab.pngFile:CDel branch c1.pngFile:CDel label4.png |
| 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 File:CDel label4.pngFile:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 11.pngFile:CDel label4.png with [2,2]+ (order 4) extended symmetry in its rhombic disphenoid vertex figure. File:H3 4343-1111 center ultrawide.png
- Perspective view from center of truncated cuboctahedron
See also
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
- Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (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