Order-5 octahedral honeycomb
| Order-5 octahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,5} |
| Coxeter diagrams |     
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| Cells | {3,4} 
|
| Faces | {3} |
| Edge figure | {5} |
| Vertex figure | {4,5} 
|
| Dual | {5,4,3} |
| Coxeter group | [3,4,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.
Images
[edit] Poincaré disk model (cell centered) |
 Ideal surface |
Related polytopes and honeycombs
[edit]It a part of a sequence of regular polychora and honeycombs with octahedral cells: {3,4,p}
| {3,4,p} polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | S3 | H3 | |||||||||
| Form | Finite | Paracompact | Noncompact | ||||||||
| Name | {3,4,3} 
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{3,4,4}    
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{3,4,5}
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{3,4,6} 
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{3,4,7}
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{3,4,8} 
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... {3,4,∞} 
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| Image | 
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| Vertex figure |
 {4,3}  
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 {4,4} 
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{4,5}
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 {4,6}
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 {4,7}
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 {4,8}
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 {4,∞}
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Order-6 octahedral honeycomb
[edit]| Order-6 octahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,6} {3,(3,4,3)} |
| Coxeter diagrams |  =
|
| Cells | {3,4}
|
| Faces | {3} |
| Edge figure | {6} |
| Vertex figure | {4,6} {(4,3,4)} 
|
| Dual | {6,4,3} |
| Coxeter group | [3,4,6] [3,((4,3,4))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,6}. It has six octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-6 square tiling vertex arrangement.
Poincaré disk model (cell centered) |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(4,3,4)}, Coxeter diagram, , with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,6,1+] = [3,((4,3,4))].
Order-7 octahedral honeycomb
[edit]| Order-7 octahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,7} |
| Coxeter diagrams |
|
| Cells | {3,4}
|
| Faces | {3} |
| Edge figure | {7} |
| Vertex figure | {4,7}
|
| Dual | {7,4,3} |
| Coxeter group | [3,4,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,7}. It has seven octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-7 square tiling vertex arrangement.
Poincaré disk model (cell centered) |
 Ideal surface |
Order-8 octahedral honeycomb
[edit]| Order-8 octahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,8} |
| Coxeter diagrams |
|
| Cells | {3,4}
|
| Faces | {3} |
| Edge figure | {8} |
| Vertex figure | {4,8}
|
| Dual | {8,4,3} |
| Coxeter group | [3,4,8] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,8}. It has eight octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-8 square tiling vertex arrangement.
Poincaré disk model (cell centered) |
Infinite-order octahedral honeycomb
[edit]| Infinite-order octahedral honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,∞} {3,(4,∞,4)} |
| Coxeter diagrams | =
|
| Cells | {3,4}
|
| Faces | {3} |
| Edge figure | {∞} |
| Vertex figure | {4,∞} {(4,∞,4)} 
|
| Dual | {∞,4,3} |
| Coxeter group | [∞,4,3] [3,((4,∞,4))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,∞}. It has infinitely many octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an infinite-order square tiling vertex arrangement.
Poincaré disk model (cell centered) |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(4,∞,4)}, Coxeter diagram, = , with alternating types or colors of octahedral cells. In Coxeter notation the half symmetry is [3,4,∞,1+] = [3,((4,∞,4))].
See also
[edit]References
[edit]- 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, LCCN 99-35678, 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)
- 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]
- Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
External links
[edit]- John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
- Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]