Hyperbolic tetrahedral-octahedral honeycomb
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!bgcolor=#e7dcc3 colspan=2|Tetrahedron-octahedron honeycomb | |
| bgcolor=#e7dcc3|Type | Compact uniform honeycomb Semiregular honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {(3,4,3,3)} or {(3,3,4,3)} |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|label4|branch|3ab|branch_10l}} or {{CDD|label4|branch|3ab|branch_01l}} or {{CDD|node_1|split1|nodes|split2-43|node}} |
| bgcolor=#e7dcc3|Cells | {3,3} 40px {3,4} 40px |
| bgcolor=#e7dcc3|Faces | triangular {3} |
| bgcolor=#e7dcc3|Vertex figure | 80px rhombicuboctahedron |
| bgcolor=#e7dcc3|Coxeter group | [(4,3,3,3)] |
| bgcolor=#e7dcc3|Properties | Vertex-transitive, edge-transitive |
In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in a rhombicuboctahedron vertex figure.
{{Honeycomb}}
It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, rectified tetrahedral r{3,3}, becomes the regular octahedron {3,4}.
Images
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See also
- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
- Tetrahedral-octahedral honeycomb - similar Euclidean honeycomb, {{CDD|node_1|split1|nodes|split2|node}}
- Tetrahedral-cubic honeycomb
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, (2015) Chapter 13: Hyperbolic Coxeter groups
{{DEFAULTSORT:Order-4 Dodecahedral Honeycomb}}