Truncated tetraheptagonal tiling
In today's article we are going to delve into the topic of Truncated tetraheptagonal tiling, exploring its implications, characteristics and possible applications. Truncated tetraheptagonal tiling is a topic that has been the subject of interest and debate in various areas, generating conflicting opinions and challenging established concepts. Throughout this article, we will delve into the history of Truncated tetraheptagonal tiling, analyze its relevance today, and examine its impact in different contexts. In addition, we will stop at the different perspectives that exist around Truncated tetraheptagonal tiling, offering a panoramic view that allows us to understand the complexity of this topic. With a critical and enriching look, we will address the multiple facets of Truncated tetraheptagonal tiling, with the aim of enriching knowledge and encouraging deep reflection on its meaning and significance.
| Truncated tetraheptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.8.14 |
| Schläfli symbol | tr{7,4} or |
| Wythoff symbol | 2 7 4 | |
| Coxeter diagram |   
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| Symmetry group | , (*742) |
| Dual | Order-4-7 kisrhombille tiling |
| Properties | Vertex-transitive |
In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.
Images
Poincaré disk projection, centered on 14-gon:
Symmetry
The dual to this tiling represents the fundamental domains of (*742) symmetry. There are three small index subgroups constructed from by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
| Small index subgroups of (*742) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Index | 1 | 2 | 14 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
= (*742) |
=  =   (*772) |
=  (7*2) |
=   (*2222222) | |||||||
| Index | 2 | 4 | 28 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
+ = (742) |
+ = =   (772) |
+ = (2222222) | ||||||||
Related polyhedra and tiling
| Uniform heptagonal/square tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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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| 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 | ||
| *n42 symmetry mutation of omnitruncated tilings: 4.8.2n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry *n42 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| *242 |
*342 |
*442 |
*542 |
*642 |
*742 |
*842 ... |
*∞42 | |
| Omnitruncated figure |
 4.8.4 |
 4.8.6 |
 4.8.8 |
 4.8.10 |
 4.8.12 |
4.8.14 |
 4.8.16 |
 4.8.∞ |
| Omnitruncated duals |
 V4.8.4 |
 V4.8.6 |
 V4.8.8 |
 V4.8.10 |
 V4.8.12 |
V4.8.14 |
 V4.8.16 |
 V4.8.∞ |
| *nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *nn2 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||||||||
| *222 |
*332 |
*442 |
*552 |
*662 |
*772 |
*882 ... |
*∞∞2 | |||||||
| Figure | 
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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.∞.∞ | ||||||
| Dual | 
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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.∞.∞ | ||||||
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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