Octahedral-hexagonal tiling honeycomb explained
| bgcolor=#e7dcc3 colspan=2 | Octahedron-hexagonal tiling honeycomb |
|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | or |
| Coxeter diagrams | or
|
| Cells | |
| Faces | |
| Vertex figure |
rhombicuboctahedron |
| Coxeter group | [(6,3,4,3)] |
| Properties | Vertex-transitive, edge-transitive | |
In the
geometry of
hyperbolic 3-space, the
octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from
octahedron,
hexagonal tiling, and
trihexagonal tiling cells, in a
rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram,, and is named by its two regular cells.
Symmetry
A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,4,3<sup>*</sup>)] symmetry, represented by a trigonal trapezohedron fundamental domain, and a Coxeter diagram .
Related honeycombs
Cyclotruncated octahedral-hexagonal tiling honeycomb
| bgcolor=#e7dcc3 colspan=2 | Cyclotruncated octahedral-hexagonal tiling honeycomb |
|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | ct or ct |
| Coxeter diagrams | or
|
| Cells | |
| Faces | |
| Vertex figure |
triangular antiprism |
| Coxeter group | [(6,3,4,3)] |
| Properties | Vertex-transitive | |
The
cyclotruncated octahedral-hexagonal tiling honeycomb is a compact uniform
honeycomb, constructed from
hexagonal tiling,
cube, and
truncated octahedron cells, in a triangular antiprism
vertex figure. It has a Coxeter diagram .
Symmetry
A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,6,3<sup>*</sup>)], represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram .
See also
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
