Order-5 pentagonal tiling
| 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 | 5 2 |
| Coxeter diagram |    |
| Symmetry group | [5,5], (*552) |
| Dual | self dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
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.
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) | ||||||||||
=  | = | =  | = | = | = | = |  = | ||||
 |  |  |  |  |  |  |  | ||||
| Order-5 pentagonal tiling {5,5} | Truncated order-5 pentagonal tiling t{5,5} | Order-4 pentagonal tiling r{5,5} | Truncated order-5 pentagonal tiling 2t{5,5} = t{5,5} | Order-5 pentagonal tiling 2r{5,5} = {5,5} | Tetrapentagonal tiling rr{5,5} | Truncated order-4 pentagonal tiling tr{5,5} | Snub pentapentagonal tiling sr{5,5} | ||||
| Uniform duals | |||||||||||
 | | | | | | |  | ||||
| |  |  | | |  |  | |||||
| Order-5 pentagonal tiling V5.5.5.5.5 | V5.10.10 | Order-5 square tiling V5.5.5.5 | V5.10.10 | Order-5 pentagonal tiling V5.5.5.5.5 | V4.5.4.5 | V4.10.10 | V3.3.5.3.5 | ||||
See also
Wikimedia Commons has media related to Order-5 pentagonal tiling.
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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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