Order-7 square tiling
In geometry, the order-7 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,7}.
| Order-7 square tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 47 |
| Schläfli symbol | {4,7} |
| Wythoff symbol | 7 | 4 2 |
| Coxeter diagram |     |
| Symmetry group | [7,4], (*742) |
| Dual | Order-4 heptagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
| *n42 symmetry mutation of regular tilings: {4,n} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||||||
 {4,3}  |
 {4,4} |
 {4,5}  |
 {4,6}  |
{4,7} |
 {4,8}...  |
 {4,∞}  | |||||
| Uniform heptagonal/square tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [7,4], (*742) | [7,4]+, (742) | [7+,4], (7*2) | [7,4,1+], (*772) | ||||||||
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| {7,4} | t{7,4} | r{7,4} | 2t{7,4}=t{4,7} | 2r{7,4}={4,7} | rr{7,4} | tr{7,4} | sr{7,4} | s{7,4} | h{4,7} | ||
| Uniform duals | |||||||||||
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| V74 | V4.14.14 | V4.7.4.7 | V7.8.8 | V47 | V4.4.7.4 | V4.8.14 | V3.3.4.3.7 | V3.3.7.3.7 | V77 | ||
This tiling is a part of regular series {n,7}:
| Tiles of the form {n,7} | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Hyperbolic tilings | |||||||
 {2,7}  |
 {3,7} |
{4,7} |
 {5,7} |
 {6,7} |
 {7,7} |
 {8,7} |
... |  {∞,7} |
References
- 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.
See also
Wikimedia Commons has media related to Order-7 square tiling.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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