Infinite-order triangular tiling
| Infinite-order triangular tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 3∞ |
| Schläfli symbol | {3,∞} |
| Wythoff symbol | ∞ | 3 2 |
| Coxeter diagram |       
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| Symmetry group | [∞,3], (*∞32) |
| Dual | Order-3 apeirogonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the Infinite-order trianglar tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}.
| *n32 symmetry mutation of regular tilings: {3,n} | |||||||||||
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| Spherical | Euclid. | Compact hyper. | Paraco. | Noncompact hyperbolic | |||||||
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| 3.3 | 33 | 34 | 35 | 36 | 37 | 38 | 3∞ | 312i | 39i | 36i | 33i |
| 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.∞ | |
= Other infinite-order trianglar tilings
A nonregular infinite-order trianglar tiling can be generated by a cursive process from a central triangle shown here:
See also
- List of regular polytopes
- List of uniform planar tilings
- Tilings of regular polygons
- Triangular tiling
- 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.