Maximally informative dimensions
| Hexagonal tiling honeycomb | |
|---|---|
 Perspective projection view within Poincaré disk model | |
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| Schläfli symbols | {6,3,3} t{3,6,3} 2t{6,3,6} tr{3[3,3]} |
| Coxeter diagrams | Template:CDD Template:CDD Template:CDD Template:CDD |
| Cells | Hexagonal tiling {6,3}
|
| Faces | Hexagon {6} |
| Edge figure | Triangle {3} |
| Vertex figure |  tetrahedron {3,3} |
| Dual | Order-6 tetrahedral honeycomb, {3,3,6} |
| Coxeter groups | , [6,3,3] , [3,6,3] , [6,3,6] , [3[3,3]] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the hexagonal tiling honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol, {6,3,3}, it has three hexagonal tilings around each edge, and four hexagonal tilings around each vertex in an tetrahedral arrangement. [1]
Symmetry constructions
It has a total of four reflectional constructions from four related Coxeter groups all with four mirrors and only the first being regular: Template:CDD [6,3,3], Template:CDD [3,6,3], Template:CDD [6,3,6] and [3[3,3]] Template:CDD, having 1, 4, 6, and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are Template:CDD, Template:CDD, Template:CDD, and Template:CDD, representing different types (colors) of hexagonal tilings in the Wythoff construction.
Related polytopes and 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.
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb, {3,3,6}. Its rectification. t1{6,3,3}, Template:CDD has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure.
Template:CDD Hexagonal tiling honeycomb |
Template:CDD Rectified hexagonal tiling honeycomb |
It is in a sequence with regular polychora: 5-cell {3,3,3}, tesseract {4,3,3}, 120-cell {5,3,3} of Euclidean 4-space, with tetrahedral vertex figures.
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It is a part of sequence of regular honeycombs of the form {6,3,p}, with hexagonal tiling cells:
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See also
References
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- 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 (Chapters 16–17: Geometries on Three-manifolds I,II)
- N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [1] [2]
- N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [3]
- ↑ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III