Cubic honeycomb honeycomb
| Cubic honeycomb honeycomb | |
|---|---|
| (No image) | |
| Type | Hyperbolic regular honeycomb |
| Schläfli symbol | {4,3,4,3} {4,31,1,1} |
| Coxeter diagram |       ↔         ↔
|
| 4-faces | 50px {4,3,4} |
| Cells | 30px {4,3} |
| Faces | 30px {4} |
| Face figure | 30px {3} |
| Edge figure | 30px {4,3} |
| Vertex figure | 50px {3,4,3} |
| Dual | Order-4 24-cell honeycomb |
| Coxeter group | R4, [4,3,4,3] |
| 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. Acronym: chont[1]
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
Notes
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, pp. 212–213)
- Klitzing, Richard. "4D Tetracombs". https://bendwavy.org/klitzing/dimensions/hyperbolic.htm#4D-compact. o3o4o3o4x - chont
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