Pentahexagonal tiling
In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}.
| Pentahexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | (5.62 |
| Schläfli symbol | r{6,5} or |
| Wythoff symbol | 2 | 6 5 |
| Coxeter diagram |     |
| Symmetry group | [6,5], (*652) |
| Dual | Order-6-5 rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
Uniform colorings
Related polyhedra and tiling
| Uniform hexagonal/pentagonal tilings | |||||||||||
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| 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 | ||
| *5n2 symmetry mutations of quasiregular tilings: (5.n)2 | ||||||||
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| Symmetry *5n2 [n,5] |
Spherical | Hyperbolic | Paracompact | Noncompact | ||||
| *352 [3,5] |
*452 [4,5] |
*552 [5,5] |
*652 [6,5] |
*752 [7,5] |
*852 [8,5]... |
*∞52 [∞,5] |
[ni,5] | |
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| Config. | (5.3)2 | (5.4)2 | (5.5)2 | (5.6)2 | (5.7)2 | (5.8)2 | (5.∞)2 | (5.ni)2 |
| Rhombic figures |
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| Config. | V(5.3)2 | V(5.4)2 | V(5.5)2 | V(5.6)2 | V(5.7)2 | V(5.8)2 | V(5.∞)2 | V(5.∞)2 |
| Symmetry mutation of quasiregular tilings: 6.n.6.n | |||||||||||
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| Symmetry *6n2 [n,6] |
Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||||||
| *632 [3,6] |
*642 [4,6] |
*652 [5,6] |
*662 [6,6] |
*762 [7,6] |
*862 [8,6]... |
*∞62 [∞,6] |
[iπ/λ,6] | ||||
| Quasiregular figures configuration |
 6.3.6.3 |
 6.4.6.4 |
6.5.6.5 |
 6.6.6.6 |
 6.7.6.7 |
 6.8.6.8 |
 6.∞.6.∞ |
6.∞.6.∞ | |||
| Dual figures | |||||||||||
| Rhombic figures configuration |
 V6.3.6.3 |
 V6.4.6.4 |
V6.5.6.5 |
 V6.6.6.6 |
V6.7.6.7 |
 V6.8.6.8 |
 V6.∞.6.∞ |
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| [(5,5,3)] reflective symmetry uniform tilings | ||||||
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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 5-6-5-6. |
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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