Foias constant
| Triangular tiling honeycomb | |
|---|---|
| |
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| Schläfli symbol | {3,6,3} h{6,3[3]} = {3[3,3]} |
| Coxeter-Dynkin diagrams | Template:CDD Template:CDD =Template:CDD |
| Cells | Triangular tiling {3,6}
|
| Faces | triangle {3} |
| Edge figure | triangle {3} |
| Vertex figure | Hexagonal tiling, {6,3} 
|
| Dual | Self-dual |
| Coxeter groups | Template:Overline3, [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. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.
Symmetry
It has one lower reflective symmetry construction, as Template:CDD, which alternates 3 types (colors) of triangular tilings around every edge. In Coxeter notation, the removal of the 3rd and 4th mirrors, [3,6,3*] creates a new Coxeter group [3[3,3]], Template:CDD, subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: Template:CDD = Template:CDD.
Related honeycombs
It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.
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}, Template:CDD.
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)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)