| Truncated heptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.14.14 |
| Schläfli symbol | t{7,4} |
| Wythoff symbol | 2 4 | 7 2 7 7 | |
| Coxeter diagram |     or  
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| Symmetry group | , (*742) , (*772) |
| Dual | Order-7 tetrakis square tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}.
Constructions
There are two uniform constructions of this tiling, first by the kaleidoscope, and second by removing the last mirror, , gives , (*772).
| Name | Tetraheptagonal | Truncated heptaheptagonal |
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| Image | 
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| Symmetry | (*742)   
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| Symbol | t{7,4} | tr{7,7} |
| Coxeter diagram |
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Symmetry
There is only one simple subgroup , index 2, removing all the mirrors. This symmetry can be doubled to 742 symmetry by adding a bisecting mirror.
| Type | Reflectional | Rotational |
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| Index | 1 | 2 |
| Diagram | 
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= (*772) |
=  (772) |
Related polyhedra and tiling
| *n42 symmetry mutation of truncated tilings: 4.2n.2n | |||||||||||
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| Symmetry *n42 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | |||||||
| *242 |
*342 |
*442 |
*542 |
*642 |
*742 |
*842 ... |
*∞42 | ||||
| Truncated figures |
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| Config. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | |||
| n-kis figures |
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| Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ | |||
| Uniform heptagonal/square tilings | |||||||||||
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| Symmetry: , (*742) | , (742) | , (7*2) | , (*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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| V7 | V4.14.14 | V4.7.4.7 | V7.8.8 | V4 | V4.4.7.4 | V4.8.14 | V3.3.4.3.7 | V3.3.7.3.7 | V7 | ||
| Uniform heptaheptagonal tilings | |||||||||||
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| Symmetry: , (*772) | , (772) | ||||||||||
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| {7,7} | t{7,7} |
r{7,7} | 2t{7,7}=t{7,7} | 2r{7,7}={7,7} | rr{7,7} | tr{7,7} | sr{7,7} | ||||
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| V7 | V7.14.14 | V7.7.7.7 | V7.14.14 | V7 | V4.7.4.7 | V4.14.14 | V3.3.7.3.7 | ||||
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
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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