Truncated infinite-order triangular tiling
| Infinite-order truncated triangular tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex figure | ∞.6.6 |
| Schläfli symbol | t{3,∞} |
| Wythoff symbol | 2 ∞ | 3 |
| Coxeter diagram |     |
| Symmetry group | [∞,3], (*∞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
Truncated infinite-order triangular tiling with mirror lines
The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.
| Type | Reflectional | Rotational |
|---|---|---|
| Index | 1 | 2 |
| Diagram |  |
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| Coxeter (orbifold) |
[(∞,3,3)]    (*∞33) |
[(∞,3,3)]+  (∞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 [n,3] Coxeter group symmetry.
| Sym. *n42 [n,3] |
Spherical | Euclid. | Compact hyperb. | Parac. | Noncompact hyperbolic | |||||||
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| *232 [2,3] D3h |
*332 [3,3] Td |
*432 [4,3] Oh |
*532 [5,3] Ih |
*632 [6,3] P6m |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] |
[12i,3] | [9i,3] | [6i,3] | [3i,3] | |
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| Schläfli | t{3,2} | t{3,3} | t{3,4} | t{3,5} | t{3,6} | t{3,7} | t{3,8} | t{3,∞} | t{3,12i} | t{3,9i} | t{3,6i} | t{3,3i} |
| Coxeter |  |
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| Uniform dual figures | ||||||||||||
| n-kis figures 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 | V3i.6.6 |
| Coxeter |  |
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| Symmetry: [∞,3], (*∞32) | [∞,3]+ (∞32) |
[1+,∞,3] (*∞33) |
[∞,3+] (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∞3 | V3.∞.∞ | V(3.∞)2 | V6.6.∞ | V3∞ | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞)3 | V3.3.3.3.3.∞ | |
| Symmetry: [(∞,3,3)], (*∞33) | [(∞,3,3)]+, (∞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.∞)3 | V3.∞.3.∞ | V(3.∞)3 | V3.6.∞.6 | V(3.3)∞ | V3.6.∞.6 | V6.6.∞ | V3.3.3.3.3.∞ |
See also
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Wikimedia Commons has media related to Uniform tiling 6-6-i. |
- List of uniform planar tilings
- Tilings of regular polygons
- Uniform tilings in hyperbolic plane
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.
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