Order-4-5 pentagonal honeycomb
| Order-4-5 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,4,5} |
| Coxeter diagrams |    
|
| Cells | {5,4} 
|
| Faces | {5} |
| Edge figure | {5} |
| Vertex figure | {4,5} |
| Dual | self-dual |
| Coxeter group | [5,4,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,4,5}.
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-4 pentagonal tilings existing around each edge and with an order-5 square tiling vertex figure.
 Poincaré disk model |
 Ideal surface |
Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs {p,4,p}:
Order-4-6 hexagonal honeycomb
| Order-4-6 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {6,4,6} {6,(4,3,4)} |
| Coxeter diagrams |   =   
|
| Cells | {6,4} 
|
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {4,6}  {(4,3,4)} 
|
| Dual | self-dual |
| Coxeter group | [6,4,6] [6,((4,3,4))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings, {6,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 square tiling vertex arrangement.
 Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(4,3,4)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,4,6,1+] = [6,((4,3,4))].
Order-4-infinite apeirogonal honeycomb
| Order-4-infinite apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {∞,4,∞} {∞,(4,∞,4)} |
| Coxeter diagrams |  ↔ 
|
| Cells | {∞,4} 
|
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure |  {4,∞} {(4,∞,4)}
|
| Dual | self-dual |
| Coxeter group | [∞,4,∞] [∞,((4,∞,4))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement.
 Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(4,∞,4)}, Coxeter diagram, , with alternating types or colors of cells.
See also
- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
- Infinite-order dodecahedral honeycomb
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, 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
- John Baez, Visual insights: {5,4,3} Honeycomb (2014/08/01) {5,4,3} 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]
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