Truncated order-6 pentagonal tiling
In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.
| Truncated order-6 pentagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 6.10.10 |
| Schläfli symbol | t{5,6} t(5,5,3) |
| Wythoff symbol | 5 3 5 5 | |
| Coxeter diagram |       |
| Symmetry group | [6,5], (*652) [(5,5,3)], (*553) |
| Dual | Order-5 hexakis hexagonal tiling |
| Properties | Vertex-transitive |
Uniform colorings
 t012(5,5,3) |
 With mirrors |
| An alternate construction exists from the [(5,5,3)] family, as the omnitruncation t012(5,5,3). It is shown with two (colors) of decagons. | |
Symmetry
The dual of this tiling represents the fundamental domains of the *553 symmetry. There are no mirror removal subgroups of [(5,5,3)], but this symmetry group can be doubled to 652 symmetry by adding a bisecting mirror to the fundamental domains.
| Type | Reflective domains | Rotational symmetry |
|---|---|---|
| Index | 1 | 2 |
| Diagram |  |
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| Coxeter (orbifold) |
[(5,5,3)] =    (*553) |
[(5,5,3)]+ =   (553) |
Related polyhedra and tiling
| Uniform hexagonal/pentagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [6,5], (*652) | [6,5]+, (652) | [6,5+], (5*3) | [1+,6,5], (*553) | ||||||||
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| {6,5} | t{6,5} | r{6,5} | 2t{6,5}=t{5,6} | 2r{6,5}={5,6} | rr{6,5} | tr{6,5} | sr{6,5} | s{5,6} | h{6,5} | ||
| Uniform duals | |||||||||||
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| V65 | V5.12.12 | V5.6.5.6 | V6.10.10 | V56 | V4.5.4.6 | V4.10.12 | V3.3.5.3.6 | V3.3.3.5.3.5 | V(3.5)5 | ||
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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.
See also
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Wikimedia Commons has media related to Uniform tiling 6-10-10. |
- Square tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
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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