Octahedral-hexagonal tiling honeycomb
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!bgcolor=#e7dcc3 colspan=2|Octahedron-hexagonal tiling honeycomb | |
| bgcolor=#e7dcc3|Type | Paracompact uniform honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {(3,4,3,6)} or {(6,3,4,3)} |
| bgcolor=#e7dcc3|Coxeter diagrams | {{CDD|label6|branch_10r|3ab|branch|label4}} or {{CDD|label6|branch_01r|3ab|branch|label4}} File:CDel_K6_634_11.png |
| bgcolor=#e7dcc3|Cells | {3,4} 40px {6,3} 40px r{6,3} 40px |
| bgcolor=#e7dcc3|Faces | triangular {3} square {4} hexagon {6} |
| bgcolor=#e7dcc3|Vertex figure | 80px rhombicuboctahedron |
| bgcolor=#e7dcc3|Coxeter group | [(6,3,4,3)] |
| bgcolor=#e7dcc3|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, {{CDD|label6|branch_10r|3ab|branch|label4}}, and is named by its two regular cells.
{{Honeycomb}}
Symmetry
A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,4,3*)] symmetry, represented by a trigonal trapezohedron fundamental domain, and a Coxeter diagram File:CDel_K6_634_10.png.
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Related honeycombs
= Cyclotruncated octahedral-hexagonal tiling honeycomb =
| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Cyclotruncated octahedral-hexagonal tiling honeycomb | |
| bgcolor=#e7dcc3|Type | Paracompact uniform honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | ct{(3,4,3,6)} or ct{(3,6,3,4)} |
| bgcolor=#e7dcc3|Coxeter diagrams | {{CDD|label6|branch_10r|3ab|branch_10l|label4}} or {{CDD|label6|branch_01r|3ab|branch_01l|label4}} File:CDel_K6_634_11.png |
| bgcolor=#e7dcc3|Cells | {6,3} 40px 40px {4,3} 40px t{3,4} 40px |
| bgcolor=#e7dcc3|Faces | triangular {3} square {4} hexagon {6} |
| bgcolor=#e7dcc3|Vertex figure | 80px triangular antiprism |
| bgcolor=#e7dcc3|Coxeter group | [(6,3,4,3)] |
| bgcolor=#e7dcc3|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 {{CDD|label6|branch_10r|3ab|branch_10l|label4}}.
== Symmetry==
A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,6,3*)], represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram File:CDel_K6_634_11.png.
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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
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