Infinite-order dodecahedral honeycomb
| Infinite-order dodecahedral honeycomb | |
|---|---|
 Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model | |
| Type | Hyperbolic regular honeycomb |
| Schläfli symbols | {5,3,∞} {5,(3,∞,3)} |
| Coxeter diagrams |       =    |
| Cells | {5,3}  |
| Faces | {5} |
| Edge figure | {∞} |
| Vertex figure | {3,∞}, {(3,∞,3)}  |
| Dual | {∞,3,5} |
| Coxeter group | [5,3,∞] [5,((3,∞,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
Symmetry constructions
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.
Related polytopes and honeycombs
It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells.
| {5,3,p} polytopes | |||||||
|---|---|---|---|---|---|---|---|
| Space | S3 | H3 | |||||
| Form | Finite | Compact | Paracompact | Noncompact | |||
| Name | {5,3,3} | {5,3,4} | {5,3,5} | {5,3,6} | {5,3,7} | {5,3,8} | ... {5,3,∞} |
| Image |  |
 |
 |
 |
| ||
| Vertex figure |
 {3,3} |
 {3,4} |
 {3,5} |
 {3,6} |
 {3,7} |
 {3,8} |
{3,∞} |
See also
- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
- Infinite-order hexagonal tiling honeycomb
References
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
- George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
- Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
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