Order-8 tetrahedral honeycomb
| Order-8 tetrahedral honeycomb | |
|---|---|
 Rendered intersection of honeycomb with the ideal plane in Poincare half-space model | |
| Type | Hyperbolic regular honeycomb |
| Schläfli symbols | {3,3,8} {3,(3,4,3)} |
| Coxeter diagrams |      =   
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| Cells | {3,3} 
|
| Faces | {3} |
| Edge figure | {8} |
| Vertex figure | {3,8} 
|
| Dual | {8,3,3} |
| Coxeter group | [3,3,8] [3,((3,4,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ideal vertices with infinitely many tetrahedra existing around each ideal vertex in an order-8 triangular tiling vertex arrangement.
Symmetry constructions
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells.
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with tetrahedral cells.
| {3,3,p} polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | S3 | H3 | |||||||||
| Form | Finite | Paracompact | Noncompact | ||||||||
| Name | {3,3,3}
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{3,3,4} 
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{3,3,5}
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{3,3,6}
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{3,3,7}
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{3,3,8}
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... {3,3,∞} 
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| Image | 
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| Vertex figure |
 {3,3}
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 {3,4}
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 {3,5}
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 {3,6}
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 {3,7}
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 {3,8}
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 {3,∞}
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See also
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) [1]
- Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]