Triangular tiling honeycomb
| Triangular tiling honeycomb | |
|---|---|
| (No image yet) | |
| Type | regular hyperbolic honeycomb |
| Schläfli symbol | {3,6,3} |
| Coxeter-Dynkin diagram |       
|
| Cells | Triangular tiling {3,6} |
| Faces | triangle {3} |
| Edge figure | triangle {3} |
| Vertex figure | Hexagonal tiling, {6,3} |
| Cells/edge | {3,6}3 |
| Dual | Self-dual |
| Coxeter group | Y3, [3,6,3] , [3[3,3]] |
| Properties | Regular |
The triangular tiling honeycomb is one of 15 regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,6,3}, there are three icosahedra surround each edge, and infinitely many triangular tiling surround each vertex, in a regular hexagonal tiling vertex figure.
Symmetry
It has two symmetry constructions, the lower one being , from the complete graph Coxeter group 
, which alternates 3 types (colors) of triangular tilings around every edge.
Related honeycombs
There are nine uniform honeycombs in the [3,6,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,6,3}, .
See also
- Seifert–Weber space
- List of regular polytopes
- Convex uniform honeycombs in hyperbolic space
- 11-cell - An abstract regular polychoron which shares the {3,5,3} Schläfli symbol.
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)