Order-5-4 square honeycomb
 | This article may be too technical for most readers to understand. (June 2020) |
| Order-4-5 square honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {4,5,4} |
| Coxeter diagrams |    
|
| Cells | {4,5} 
|
| Faces | {4} |
| Edge figure | {4} |
| Vertex figure | {5,4} |
| Dual | self-dual |
| Coxeter group | [4,5,4] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.
Geometry[edit]
All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 pentagonal tiling vertex figure.
 Poincaré disk model |
 Ideal surface |
Related polytopes and honeycombs[edit]
It a part of a sequence of regular polychora and honeycombs {p,5,p}:
| {p,5,p} regular honeycombs | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | H3 | ||||||||||
| Form | Compact | Noncompact | |||||||||
| Name | {3,5,3} | {4,5,4} | {5,5,5} | {6,5,6} | {7,5,7} | {8,5,8} | ...{∞,5,∞} | ||||
| Image | 
|
|
|
|
| ||||||
| Cells {p,5} |
 {3,5} |
{4,5} |
 {5,5} |
 {6,5} |
 {7,5} |
 {8,5} |
 {∞,5} | ||||
| Vertex figure {5,p} |
 {5,3} |
 {5,4} |
 {5,5} |
 {5,6} |
 {5,7} |
 {5,8} |
 {5,∞} | ||||
Order-5-5 pentagonal honeycomb[edit]
| Order-5-5 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,5,5} |
| Coxeter diagrams |
|
| Cells | {5,5}
|
| Faces | {5} |
| Edge figure | {5} |
| Vertex figure | {5,5} |
| Dual | self-dual |
| Coxeter group | [5,5,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-5 pentagonal honeycomb (or 5,5,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,5,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-5 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
Poincaré disk model |
 Ideal surface |
Order-5-6 hexagonal honeycomb[edit]
| Order-5-6 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {6,5,6} {6,(5,3,5)} |
| Coxeter diagrams |   =  
|
| Cells | {6,5}
|
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {5,6} {(5,3,5)} 
|
| Dual | self-dual |
| Coxeter group | [6,5,6] [6,((5,3,5))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.
Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(5,3,5)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1+] = [6,((5,3,5))].
Order-5-7 heptagonal honeycomb[edit]
| Order-5-7 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {7,5,7} |
| Coxeter diagrams | 
|
| Cells | {7,5}
|
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {5,7}
|
| Dual | self-dual |
| Coxeter group | [7,5,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-7 heptagonal honeycomb (or 7,5,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,5,7}. It has seven order-5 heptagonal tilings, {7,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
 Ideal surface |
Order-5-infinite apeirogonal honeycomb[edit]
| Order-5-infinite apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbols | {∞,5,∞} {∞,(5,∞,5)} |
| Coxeter diagrams |  ↔ 
|
| Cells | {∞,5}
|
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure | {5,∞} {(5,∞,5)}
|
| Dual | self-dual |
| Coxeter group | [∞,5,∞] [∞,((5,∞,5))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-infinite apeirogonal honeycomb (or ∞,5,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,5,∞}. It has infinitely many order-5 apeirogonal tilings {∞,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement.
Poincaré disk model |
 Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, , with alternating types or colors of cells.
See also[edit]
- Convex uniform honeycombs in hyperbolic space
- List of regular polytopes
- Infinite-order dodecahedral honeycomb
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]