Cubic honeycomb honeycomb
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!bgcolor=#e7dcc3 colspan=2|Cubic honeycomb honeycomb | |
| bgcolor=#ffffff align=center colspan=2|(No image) | |
| bgcolor=#e7dcc3|Type | Hyperbolic regular honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | {4,3,4,3} {4,31,1,1} |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|node_1|4|node|3|node|4|node|3|node}} {{CDD|node_1|4|node|splitsplit1|branch3|node}} ↔ {{CDD|node_1|4|node|3|node|4|node_g|3sg|node_g}} {{CDD|node_1|4|node|3|node|split1-43|nodes}} {{CDD|nodes_11|2a2b-cross|nodes|split2|node|3|node|ultra|node_1}} ↔ {{CDD|node_1|4|node_g|3sg|node_g|4|node|3|node}} |
| bgcolor=#e7dcc3|4-faces | 50px {4,3,4} |
| bgcolor=#e7dcc3|Cells | 30px {4,3} |
| bgcolor=#e7dcc3|Faces | 30px {4} |
| bgcolor=#e7dcc3|Face figure | 30px {3} |
| bgcolor=#e7dcc3|Edge figure | 30px {4,3} |
| bgcolor=#e7dcc3|Vertex figure | 50px {3,4,3} |
| bgcolor=#e7dcc3|Dual | Order-4 24-cell honeycomb |
| bgcolor=#e7dcc3|Coxeter group | {{overline|R}}4, [4,3,4,3] |
| bgcolor=#e7dcc3|Properties | Regular |
In the geometry of hyperbolic 4-space, the cubic honeycomb honeycomb is one of two paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite facets, whose vertices exist on 3-horospheres and converge to a single ideal point at infinity. With Schläfli symbol {4,3,4,3}, it has three cubic honeycombs around each face, and with a {3,4,3} vertex figure. It is dual to the order-4 24-cell honeycomb.
Related honeycombs
It is related to the Euclidean 4-space 16-cell honeycomb, {3,3,4,3}, which also has a 24-cell vertex figure.
It is analogous to the paracompact tesseractic honeycomb honeycomb, {4,3,3,4,3}, in 5-dimensional hyperbolic space, square tiling honeycomb, {4,4,3}, in 3-dimensional hyperbolic space, and the order-3 apeirogonal tiling, {∞,3} of 2-dimensional hyperbolic space, each with hypercube honeycomb facets.
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)