Order-5 tesseractic honeycomb
In the geometry of hyperbolic 4-space, the order-5 tesseractic honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {4,3,3,5}, it has five 8-cells (also known as tesseracts) around each face. Its dual is the order-4 120-cell honeycomb, {5,3,3,4}.
| Order-5 tesseractic honeycomb | |
|---|---|
| (No image) | |
| Type | Hyperbolic regular honeycomb |
| Schläfli symbol | {4,3,3,5} |
| Coxeter diagram |      |
| 4-faces |  {4,3,3} |
| Cells |  {4,3} |
| Faces |  {4} |
| Face figure |  {5} |
| Edge figure |  {3,5} |
| Vertex figure |  {3,3,5} |
| Dual | Order-4 120-cell honeycomb |
| Coxeter group | BH4, [5,3,3,4] |
| Properties | Regular |
Related polytopes and honeycombs
It is related to the Euclidean 4-space (order-4) tesseractic honeycomb, {4,3,3,4}, and the 5-cube, {4,3,3,3} in Euclidean 5-space. The 5-cube can also be seen as an order-3 tesseractic honeycomb on the surface of a 4-sphere.
It is analogous to the order-5 cubic honeycomb {4,3,5} and order-5 square tiling {4,5}.
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)
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