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| 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 | , (*∞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,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.
Uniform colorings
There is a half symmetry form, , seen with alternating colors:
Symmetry
This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of (*2) orbifold symmetry.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4).
| *n42 symmetry mutation of regular tilings: {4,n} | |||||||||||
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| Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||||||
 {4,3} 
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 {4,4}
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 {4,5} 
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 {4,6} 
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 {4,7} 
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 {4,8}... 
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{4,∞}
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| Paracompact uniform tilings in 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∞ | V4.∞.∞ | V(4.∞) | V8.8.∞ | V4 | V4.∞ | V4.8.∞ | |
| Alternations | |||||||
(*44∞) |
(∞*2) |
(*2∞2∞) |
(4*∞) |
(*∞∞2) |
(2*2∞) |
(∞42) | |
 =  
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=  
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| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} | |
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| V(∞.4) | V3.(3.∞) | V(4.∞.4) | V3.∞.(3.4) | V∞ | V∞.4 | V3.3.4.3.∞ | |
See also
References
- John H. Conway; Heidi Burgiel; Chaim Goodman-Strauss (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN 978-1-56881-220-5.
- H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery Archived 2013-03-24 at the Wayback Machine
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