Tesseractic honeycomb honeycomb
{{Short description|Geometrical concept}}
| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Tesseractic honeycomb honeycomb | ||
| bgcolor=#ffffff align=center colspan=2|(No image) | ||
| bgcolor=#e7dcc3|Type | Hyperbolic regular honeycomb | |
| bgcolor=#e7dcc3|Schläfli symbol | {4,3,3,4,3} {4,3,31,1,1} | |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|node_1|4|node|3|node|3|node|4|node|3|node}} {{CDD|node_1|4|node|3|node|splitsplit1|branch3|node}} ↔ {{CDD|node_1|4|node|3|node | 3|node|4|node_g|3sg|node_g}} {{CDD|nodes_10r|4a4b|nodes|split2|node|3|node}} |
| bgcolor=#e7dcc3|5-faces | 50px {4,3,3,4} | |
| bgcolor=#e7dcc3|4-faces | 50px {4,3,3} | |
| bgcolor=#e7dcc3|Cells | 50px {4,3} | |
| bgcolor=#e7dcc3|Faces | 50px {4} | |
| bgcolor=#e7dcc3|Cell figure | 50px {3} | |
| bgcolor=#e7dcc3|Face figure | 50px {4,3} | |
| bgcolor=#e7dcc3|Edge figure | 50px {3,4,3} | |
| bgcolor=#e7dcc3|Vertex figure | 50px {3,3,4,3} | |
| bgcolor=#e7dcc3|Dual | Order-4 24-cell honeycomb honeycomb | |
| bgcolor=#e7dcc3|Coxeter group | {{overline|R}}5, [3,4,3,3,4] | |
| bgcolor=#e7dcc3|Properties | Regular |
In the geometry of hyperbolic 5-space, the tesseractic honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,3,4,3}, it has three tesseractic honeycombs around each cell. It is dual to the order-4 24-cell honeycomb honeycomb.
Related honeycombs
It is related to the regular Euclidean 4-space tesseractic honeycomb, {4,3,3,4}.
It is analogous to the paracompact cubic honeycomb honeycomb, {4,3,4,3}, in 4-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)