Order-5-3 square honeycomb
| Order-5-3 square honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {4,5,3} |
| Coxeter diagram |     
|
| Cells | {4,5} 
|
| Faces | {4} |
| Vertex figure | {5,3} |
| Dual | {3,5,4} |
| Coxeter group | [4,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Geometry
The Schläfli symbol of the order-5-3 square honeycomb is {4,5,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |
 Ideal surface |
Related polytopes and honeycombs
It is a part of a series of regular polytopes and honeycombs with {p,5,3} Schläfli symbol, and dodecahedral vertex figures:
Order-5-3 pentagonal honeycomb
| Order-5-3 pentagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {5,5,3} |
| Coxeter diagram |
|
| Cells | {5,5} 
|
| Faces | {5} |
| Vertex figure | {5,3} |
| Dual | {3,5,5} |
| Coxeter group | [5,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 pentagonal honeycomb or 5,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-5-3 pentagonal honeycomb is {5,5,3}, with three order-5 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |
 Ideal surface |
Order-5-3 hexagonal honeycomb
| Order-5-3 hexagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {6,5,3} |
| Coxeter diagram | 
|
| Cells | {6,5} 
|
| Faces | {6} |
| Vertex figure | {5,3} |
| Dual | {3,5,6} |
| Coxeter group | [6,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 hexagonal honeycomb or 6,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-5-3 hexagonal honeycomb is {6,5,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |
 Ideal surface |
Order-5-3 heptagonal honeycomb
| Order-5-3 heptagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {7,5,3} |
| Coxeter diagram | 
|
| Cells | {7,5} 
|
| Faces | {7} |
| Vertex figure | {5,3} |
| Dual | {3,5,7} |
| Coxeter group | [7,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 heptagonal honeycomb or 7,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-5-3 heptagonal honeycomb is {7,5,3}, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |
 Ideal surface |
Order-5-3 octagonal honeycomb
| Order-5-3 octagonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {8,5,3} |
| Coxeter diagram | 
|
| Cells | {8,5} 
|
| Faces | {8} |
| Vertex figure | {5,3} |
| Dual | {3,5,8} |
| Coxeter group | [8,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 octagonal honeycomb or 8,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-5-3 octagonal honeycomb is {8,5,3}, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
 Poincaré disk model (Vertex centered) |
Order-5-3 apeirogonal honeycomb
| Order-5-3 apeirogonal honeycomb | |
|---|---|
| Type | Regular honeycomb |
| Schläfli symbol | {∞,5,3} |
| Coxeter diagram | 
|
| Cells | {∞,5} 
|
| Faces | Apeirogon {∞} |
| Vertex figure | {5,3} |
| Dual | {3,5,∞} |
| Coxeter group | [∞,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 apeirogonal honeycomb or ∞,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,5,3}, with three order-5 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
 Poincaré disk model (Vertex centered) |
 Ideal surface |
See also
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: {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]