Triapeirogonal tiling
| Triapeirogonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex figure | 3.∞.3.∞ |
| Schläfli symbol | r{∞,3} |
| Wythoff symbol | 2 | ∞ 3 |
| Coxeter diagram |     |
| Symmetry group | [∞,3], (*∞32) |
| Dual | Order-3-infinite rhombille tiling |
| Properties | Vertex-transitive edge-transitive |
In geometry, the triapeirogonal tiling (or trigonal-horocyclic tiling) is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,3}.
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform quasiregular polyhedra with vertex configurations (3.n.3.n), and [n,3] Coxeter group symmetry.
| Sym. *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
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| *332 [3,3] Td |
*432 [4,3] Oh |
*532 [5,3] Ih |
*632 [6,3] p6m |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] |
[12i,3] | [9i,3] | [6i,3] | [3i,3] | ||
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| Config. | r{3,3} | r{4,3} | r{5,3} | r{6,3} | r{7,3} | r{8,3} | r{∞,3} | r{12i,3} | r{9i,3} | r{6i,3} | r{3i,3} | |
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| Dual uniform figures | ||||||||||||
| Dual conf. |
 V(3.3)2 |
 V(3.4)2 |
 V(3.5)2 |
 V(3.6)2 |
 V(3.7)2 |
 V(3.8)2 |
 V(3.∞)2 |
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| Symmetry: [∞,3], (*∞32) | [∞,3]+ (∞32) |
[1+,∞,3] (*∞33) |
[∞,3+] (3*∞) | |||||||
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| {∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h2{∞,3} | s{3,∞} |
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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: [(∞,3,3)], (*∞33) | [(∞,3,3)]+, (∞33) | ||||||
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| {(∞,∞,3)} | t0,1{(∞,3,3)} | t1(∞,3,3) | t1,2(∞,3,3) | t2{(∞,3,3)} | t0,2(∞,3,3) | t0,1,2{(∞,3,3)} | s(∞,3,3) |
| Dual tilings | |||||||
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| V(3.∞)3 | V3.∞.3.∞ | V(3.∞)3 | V3.6.∞.6 | V(3.3)∞ | V3.6.∞.6 | V6.6.∞ | V3.3.3.3.3.∞ |
See also
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Wikimedia Commons has media related to Uniform tiling 3-i-3-i. |
- List of uniform planar tilings
- Tilings of regular polygons
- Uniform tilings in hyperbolic plane
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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.
- http://bendwavy.org/klitzing/incmats/o3xinfino.htm
- Richard Klitzing, 2D, Non-Compact Tilings o3x∞o
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