| Infinite-order truncated triangular tiling | |
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 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | ∞.6.6 |
| Schläfli symbol | t{3,∞} |
| Wythoff symbol | 2 ∞ | 3 |
| Coxeter diagram |       
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| Symmetry group | , (*∞32) |
| Dual | apeirokis apeirogonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.
Symmetry
.The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of , but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.
| Type | Reflectional | Rotational |
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| Index | 1 | 2 |
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| Coxeter (orbifold) |
(*∞33) |
  (∞33) |
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and Coxeter group symmetry.
| *n32 symmetry mutation of truncated tilings: n.6.6 | ||||||||||||
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| Sym. *n42 |
Spherical | Euclid. | Compact | Parac. | Noncompact hyperbolic | |||||||
| *232 |
*332 |
*432 |
*532 |
*632 |
*732 |
*832 ... |
*∞32 |
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| Truncated figures |
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| Config. | 2.6.6 | 3.6.6 | 4.6.6 | 5.6.6 | 6.6.6 | 7.6.6 | 8.6.6 | ∞.6.6 | 12i.6.6 | 9i.6.6 | 6i.6.6 | |
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| Config. | V2.6.6 | V3.6.6 | V4.6.6 | V5.6.6 | V6.6.6 | V7.6.6 | V8.6.6 | V∞.6.6 | V12i.6.6 | V9i.6.6 | V6i.6.6 | |
| Paracompact uniform tilings in family | ||||||||||
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| Symmetry: , (*∞32) | (∞32) |
(*∞33) |
(3*∞) | |||||||
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| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h2{∞,3} | s{3,∞} |
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| V∞ | V3.∞.∞ | V(3.∞) | V6.6.∞ | V3 | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞) | V3.3.3.3.3.∞ | |
| Paracompact hyperbolic uniform tilings in family | |||||||||||
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| Symmetry: , (*∞33) | , (∞33) | ||||||||||
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| (∞,∞,3) | t0,1(∞,3,3) | t1(∞,3,3) | t1,2(∞,3,3) | t2(∞,3,3) | t0,2(∞,3,3) | t0,1,2(∞,3,3) | s(∞,3,3) | ||||
| Dual tilings | |||||||||||
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| V(3.∞) | V3.∞.3.∞ | V(3.∞) | V3.6.∞.6 | V(3.3) | V3.6.∞.6 | V6.6.∞ | V3.3.3.3.3.∞ | ||||
See also
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.
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