Truncated infinite-order square tiling
| Infinite-order truncated square tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | ∞.8.8 |
| Schläfli symbol | t{4,∞} |
| Wythoff symbol | 2 ∞ | 4 |
| Coxeter diagram |    
|
| Symmetry group | [∞,4], (*∞42) |
| Dual | apeirokis apeirogonal tiling |
| Properties | Vertex-transitive |
In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,∞}.
Symmetry
The dual to this tiling represents (*∞44) orbifold symmetry. In this lower symmetry this tiling has 3 colors. Bisecting the isosceles triangle domains can double the symmetry to *∞42 symmetry
Related polyhedra and tiling
| *n42 symmetry mutation of truncated tilings: n.8.8 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||
| *242 [2,4] |
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | ||||
| Truncated figures |
|
|
|
|
|
|
|
| |||
| Config. | 2.8.8 | 3.8.8 | 4.8.8 | 5.8.8 | 6.8.8 | 7.8.8 | 8.8.8 | ∞.8.8 | |||
| n-kis figures |
|
|
|
|
|
|
|
| |||
| Config. | V2.8.8 | V3.8.8 | V4.8.8 | V5.8.8 | V6.8.8 | V7.8.8 | V8.8.8 | V∞.8.8 | |||
| Paracompact uniform tilings in [∞,4] family | |||||||
|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
| |
| {∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4} | |
| Dual figures | |||||||
|
|
|
|
|
|
| |
|
|
|
|
|
|
| |
| 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) | |
 =  
|
|
|
|
=  
|
|
| |
| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} | |
|
|
|
| ||||
| Alternation duals | |||||||
|
|
|
|
|
|
| |
|
|
||||||
| 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 978-1-56881-220-5 (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.