Rhombitriapeirogonal tiling
Rhombitriapeirogonal tiling is a topic that has captured the attention of millions of people around the world. Since its emergence, it has generated great interest and debate in different areas, from politics and economics to culture and entertainment. Its influence has extended to various spheres of life, and its impact continues to be the subject of study and analysis. In this article, we will thoroughly explore Rhombitriapeirogonal tiling and analyze its relevance in today's society. From its origin to its evolution, we will examine its role in the contemporary world and reflect on its meaning for the future.
| Rhombitriapeirogonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.4.∞.4 |
| Schläfli symbol | rr{∞,3} or s2{3,∞} |
| Wythoff symbol | 3 | ∞ 2 |
| Coxeter diagram |     or   
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| Symmetry group | , (*∞32) , (3*∞) |
| Dual | Deltoidal triapeirogonal tiling |
| Properties | Vertex-transitive |
In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
Symmetry
This tiling has , (*∞32) symmetry. There is only one uniform coloring.
Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*∞) orbifold notation. The apeireogons can be considered as truncated, t{∞} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,∞}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an infinite-order triangular tiling results, constructed as a snub triapeirotrigonal tiling, .
Related polyhedra and tiling
| Paracompact uniform tilings in family | ||||||||||
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| Symmetry: , (*∞32) | + (∞32) |
(*∞33) |
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| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h2{∞,3} | s{3,∞} |
| Uniform duals | ||||||||||
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| V∞3 | V3.∞.∞ | V(3.∞)2 | V6.6.∞ | V3∞ | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞)3 | V3.3.3.3.3.∞ | |
Symmetry mutations
This hyperbolic tiling is topologically related as a part of sequence of uniform cantellated polyhedra with vertex configurations (3.4.n.4), and Coxeter group symmetry.
| *n32 symmetry mutation of expanded tilings: 3.4.n.4 | ||||||||||||
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| Symmetry *n32 |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| *232 |
*332 |
*432 |
*532 |
*632 |
*732 |
*832 ... |
*∞32 |
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| Config. | 3.4.2.4 | 3.4.3.4 | 3.4.4.4 | 3.4.5.4 | 3.4.6.4 | 3.4.7.4 | 3.4.8.4 | 3.4.∞.4 | 3.4.12i.4 | 3.4.9i.4 | 3.4.6i.4 | |
See also
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.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
