Order-5 pentagonal tiling
In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,5}, constructed from five pentagons around every vertex. As such, it is self-dual.
| Order-5 pentagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 55 |
| Schläfli symbol | {5,5} |
| Wythoff symbol | 5 2 |
| Coxeter diagram |    |
| Symmetry group | [5,5], (*552) |
| Dual | self dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
Related tilings
| Spherical | Hyperbolic tilings | |||||||
|---|---|---|---|---|---|---|---|---|
 {2,5}  |
 {3,5}  |
 {4,5}  |
{5,5} |
 {6,5}  |
 {7,5}  |
 {8,5}  |
... |  {∞,5}  |
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).
| Finite | Compact hyperbolic | Paracompact | ||||
|---|---|---|---|---|---|---|
 {5,3} |
 {5,4} |
 {5,5} |
 {5,6} |
 {5,7} |
 {5,8}... |
 {5,∞} |
| Uniform pentapentagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [5,5], (*552) | [5,5]+, (552) | ||||||||||
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| {5,5} | t{5,5} |
r{5,5} | 2t{5,5}=t{5,5} | 2r{5,5}={5,5} | rr{5,5} | tr{5,5} | sr{5,5} | ||||
| Uniform duals | |||||||||||
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| V5.5.5.5.5 | V5.10.10 | V5.5.5.5 | V5.10.10 | V5.5.5.5.5 | V4.5.4.5 | V4.10.10 | V3.3.5.3.5 | ||||
See also
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Wikimedia Commons has media related to Order-5 pentagonal 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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