Order-4 heptagonal tiling
In this article, we are going to delve into the fascinating world of Order-4 heptagonal tiling. Whether we are talking about Order-4 heptagonal tiling's life, a relevant event related to Order-4 heptagonal tiling, or Order-4 heptagonal tiling's influence on today's society, this topic deserves to be explored in depth. Throughout the next few lines, we will analyze various aspects that will allow us to better understand the importance of Order-4 heptagonal tiling and its impact in different areas. Without a doubt, it is an exciting topic that arouses the interest of a wide range of people, so we should not underestimate its relevance today.
| Order-4 heptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 74 |
| Schläfli symbol | {7,4} r{7,7} |
| Wythoff symbol | 4 | 7 2 2 | 7 7 |
| Coxeter diagram |    
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| Symmetry group | , (*742) , (*772) |
| Dual | Order-7 square tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 7 mirrors meeting as edges of a regular heptagon. This symmetry by orbifold notation is called *2222222 with 7 order-2 mirror intersections. In Coxeter notation can be represented as , removing two of three mirrors (passing through the heptagon center) in the symmetry.
The kaleidoscopic domains can be seen as bicolored heptagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling.
Related polyhedra and tiling
| 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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| 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 | ||
| 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} | ||||
| Uniform duals | |||||||||||
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| V77 | V7.14.14 | V7.7.7.7 | V7.14.14 | V77 | V4.7.4.7 | V4.14.14 | V3.3.7.3.7 | ||||
This tiling is topologically related as a part of sequence of regular tilings with heptagonal faces, starting with the heptagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram 
, progressing to infinity.
 {7,3} 
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{7,4}
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 {7,5} 
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 {7,6} 
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{7,7}
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This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
| *n42 symmetry mutation of regular tilings: {n,4} | |||||||
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| Spherical | Euclidean | Hyperbolic tilings | |||||
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| 24 | 34 | 44 | 54 | 64 | 74 | 84 | ...∞4 |
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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