Rhombitetraapeirogonal tiling
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| Rhombitetraapeirogonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex figure | 4.4.∞.4 |
| Schläfli symbol | rr{∞,4} |
| Wythoff symbol | 4 | ∞ 2 |
| Coxeter diagram |     |
| Symmetry group | [∞,4], (*∞42) |
| Dual | Deltoidal tetraapeirogonal tiling |
| Properties | Vertex-transitive |
In geometry, the rhombitetrapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.
Constructions
There are two uniform constructions of this tiling, one from [∞,4] or (*∞42) symmetry, and secondly removing the miror middle, [∞,1+,4], gives a rectangular fundamental domain [∞,∞,∞], (*∞222).
| Name | Rhombitetrahexagonal tiling | |
|---|---|---|
| Image | |
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| Symmetry | [∞,4] (*∞42)    |
[∞,∞,∞] = [∞,1+,4] (*∞222)   |
| Schläfli symbol | rr{∞,4} | t0,1,2,3{∞,∞,∞} |
| Coxeter diagram | |
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Symmetry
The dual of this tiling, called a deltoidal tetraapeirogonal tiling represents the fundamental domains of (*∞222) orbifold symmetry. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.
Related polyhedra and tiling
| Symmetry [n,4], (*n42) |
Spherical | Euclidean | Hyperbolic tiling | ||||
|---|---|---|---|---|---|---|---|
| *342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4] |
*∞42 [∞,4] | |
| Quasiregular figures |
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| Coxeter Schläfli |
 rr{3,4} |
rr{4,4} |
 rr{5,4} |
 rr{6,4} |
 rr{7,4} |
 rr{8,4} |
rr{∞,4} |
| Dual (rhombic) figures configuration |
 V3.4.4.4 |
 V4.4.4.4 |
 5.4.4.4 |
 V6.4.4.4 |
 V7.4.4.4 |
 V8.4.4.4 |
 V∞.4.4.4 |
| Coxeter |  |
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| Symmetry: [∞,4], (*∞42) | |||||||
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| {∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4} | |
| Dual figures | |||||||
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| V∞4 | V4.∞.∞ | V(4.∞)2 | V8.8.∞ | V4∞ | V43.∞ | V4.8.∞ | |
| Alternations | |||||||
| [1+,∞,4] (*44∞) |
[∞+,4] (∞*2) |
[∞,1+,4] (*2∞2∞) |
[∞,4+] (4*∞) |
[∞,4,1+] (*∞∞2) |
[(∞,4,2+)] (2*2∞) |
[∞,4]+ (∞42) | |
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| h{∞,4} | s{∞,4} | hr{∞,4} | s{4,∞} | h{4,∞} | hrr{∞,4} | s{∞,4} | |
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| Alternation duals | |||||||
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| V(∞.4)4 | V3.(3.∞)2 | V(4.∞.4)2 | V3.∞.(3.4)2 | V∞∞ | V∞.44 | V3.3.4.3.∞ | |
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.
See also
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Wikimedia Commons has media related to Uniform tiling 4-4-4-i. |
- Square tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
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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