Truncated order-4 pentagonal tiling
In today's world, Truncated order-4 pentagonal tiling has become a topic of great relevance and interest to a wide spectrum of people. From professionals to amateurs, Truncated order-4 pentagonal tiling arouses curiosity and debate in different areas. With a rich and varied history, Truncated order-4 pentagonal tiling has significantly impacted society and the way we approach different aspects of life. In this article, we will explore various perspectives and aspects related to Truncated order-4 pentagonal tiling, with the aim of providing a comprehensive and enriching vision on this topic.
| Truncated pentagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 4.10.10 |
| Schläfli symbol | t{5,4} |
| Wythoff symbol | 2 4 | 5 2 5 5 | |
| Coxeter diagram |     or  
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| Symmetry group | , (*542) , (*552) |
| Dual | Order-5 tetrakis square tiling |
| Properties | Vertex-transitive |
In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}.
Uniform colorings
A half symmetry = coloring can be constructed with two colors of decagons. This coloring is called a truncated pentapentagonal tiling.
Symmetry
There is only one subgroup of , +, removing all the mirrors. This symmetry can be doubled to 542 symmetry by adding a bisecting mirror.
| Type | Reflective domains | Rotational symmetry |
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| Index | 1 | 2 |
| Diagram | 
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| Coxeter (orbifold) |
=  =   (*552) |
+ =  =  (552) |
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.∞.∞ | |||
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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 pentagonal/square tilings | |||||||||||
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| Symmetry: , (*542) | +, (542) | , (5*2) | , (*552) | ||||||||
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| {5,4} | t{5,4} | r{5,4} | 2t{5,4}=t{4,5} | 2r{5,4}={4,5} | rr{5,4} | tr{5,4} | sr{5,4} | s{5,4} | h{4,5} | ||
| Uniform duals | |||||||||||
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| V54 | V4.10.10 | V4.5.4.5 | V5.8.8 | V45 | V4.4.5.4 | V4.8.10 | V3.3.4.3.5 | V3.3.5.3.5 | V55 | ||
| Uniform pentapentagonal tilings | |||||||||||
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| Symmetry: , (*552) | +, (552) | ||||||||||
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| Order-5 pentagonal tiling {5,5} |
Truncated order-5 pentagonal tiling t{5,5} |
Order-4 pentagonal tiling r{5,5} |
Truncated order-5 pentagonal tiling 2t{5,5} = t{5,5} |
Order-5 pentagonal tiling 2r{5,5} = {5,5} |
Tetrapentagonal tiling rr{5,5} |
Truncated order-4 pentagonal tiling tr{5,5} |
Snub pentapentagonal tiling sr{5,5} | ||||
| Uniform duals | |||||||||||
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| Order-5 pentagonal tiling V5.5.5.5.5 |
V5.10.10 | Order-5 square tiling V5.5.5.5 |
V5.10.10 | Order-5 pentagonal tiling V5.5.5.5.5 |
V4.5.4.5 | V4.10.10 | V3.3.5.3.5 | ||||
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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