Infinite-order square tiling
| Infinite-order square tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 4∞ |
| Schläfli symbol | {4,∞} |
| Wythoff symbol | ∞ | 4 2 |
| Coxeter diagram |       
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| Symmetry group | [∞,4], (*∞42) |
| Dual | Order-4 apeirogonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
| Uniform tilings based on square tiling symmetry | |||||||||||
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| Symmetry: [4,4], (*442) | [4,4]+, (442) | [4,4+], (4*2) | |||||||||
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| {4,4} | t{4,4} | r{4,4} | t{4,4} | {4,4} | rr{4,4} | tr{4,4} | sr{4,4} | s{4,4} | |||
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| V4.4.4.4 | V4.8.8 | V4.4.4.4 | V4.8.8 | V4.4.4.4 | V4.4.4.4 | V4.8.8 | V3.3.4.3.4 | ||||
| Paracompact uniform tilings in [∞,4] family | |||||||
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| {∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4} | |
| Dual figures | |||||||
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| V∞4 | V4.∞.∞ | V(4.∞)2 | V8.8.∞ | V4∞ | V43.∞ | V4.8.∞ | |
| Alternations | |||||||
| [1+,∞,4] (*44∞) |
[∞+,4] (∞*2) |
[∞,1+,4] (*2∞2∞) |
[∞,4+] (4*∞) |
[∞,4,1+] (*∞∞2) |
[(∞,4,2+)] (2*2∞) |
[∞,4]+ (∞42) | |
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| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} | |
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| V(∞.4)4 | V3.(3.∞)2 | V(4.∞.4)2 | V3.∞.(3.4)2 | V∞∞ | V∞.44 | V3.3.4.3.∞ | |
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-6 (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.