Order-5 apeirogonal tiling
In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.
| Order-5 apeirogonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | ∞5 |
| Schläfli symbol | {∞,5} |
| Wythoff symbol | 5 | ∞ 2 |
| Coxeter diagram |     |
| Symmetry group | [∞,5], (*∞52) |
| Dual | Infinite-order pentagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive edge-transitive |
Symmetry
The dual to this tiling represents the fundamental domains of [∞,5*] symmetry, orbifold notation *∞∞∞∞∞ symmetry, a pentagonal domain with five ideal vertices.
The order-5 apeirogonal tiling can be uniformly colored with 5 colored apeirogons around each vertex, and coxeter diagram: 
, except ultraparallel branches on the diagonals.
Related polyhedra and tiling
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with five faces per vertex, starting with the icosahedron, with Schläfli symbol {n,5}, and Coxeter diagram 
, with n progressing to infinity.
| Spherical | Hyperbolic tilings | |||||||
|---|---|---|---|---|---|---|---|---|
 {2,5}  |
 {3,5}  |
 {4,5}  |
 {5,5} |
 {6,5}  |
 {7,5}  |
 {8,5}  |
... | {∞,5} |
| Paracompact uniform apeirogonal/pentagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [∞,5], (*∞52) | [∞,5]+ (∞52) |
[1+,∞,5] (*∞55) |
[∞,5+] (5*∞) | ||||||||
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| {∞,5} | t{∞,5} | r{∞,5} | 2t{∞,5}=t{5,∞} | 2r{∞,5}={5,∞} | rr{∞,5} | tr{∞,5} | sr{∞,5} | h{∞,5} | h2{∞,5} | s{5,∞} | |
| Uniform duals | |||||||||||
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| V∞5 | V5.∞.∞ | V5.∞.5.∞ | V∞.10.10 | V5∞ | V4.5.4.∞ | V4.10.∞ | V3.3.5.3.∞ | V(∞.5)5 | V3.5.3.5.3.∞ | ||
See also
Wikimedia Commons has media related to Order-5 apeirogonal tiling.
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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