Truncated infinite-order triangular tiling
Appearance
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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Truncated_infinite-order_triangular_tiling_with_mirrors.png/220px-Truncated_infinite-order_triangular_tiling_with_mirrors.png)
![](http://upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png)
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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 |
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Index | 1 | 2 |
Diagram | ![]() |
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Coxeter (orbifold) |
[(∞,3,3)]![]() ![]() ![]() ![]() (*∞33) |
[(∞,3,3)]+![]() ![]() ![]() ![]() (∞33) |
Related polyhedra and tiling
[edit]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.
*n32 symmetry mutation of truncated tilings: n.6.6 | ||||||||||||
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Sym. *n42 [n,3] |
Spherical | Euclid. | Compact | Parac. | Noncompact hyperbolic | |||||||
*232 [2,3] |
*332 [3,3] |
*432 [4,3] |
*532 [5,3] |
*632 [6,3] |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] |
[12i,3] | [9i,3] | [6i,3] | ||
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 | |
n-kis figures |
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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 [∞,3] family | ||||||||||
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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,∞} |
Uniform duals | ||||||||||
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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.∞ |
Paracompact hyperbolic uniform tilings in [(∞,3,3)] family | |||||||||||
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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
[edit]![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Uniform tiling 6-6-i.
References
[edit]- 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.