Snub tetrahexagonal tiling
In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}.
| Snub tetrahexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.3.4.3.6 |
| Schläfli symbol | sr{6,4} or |
| Wythoff symbol | | 6 4 2 |
| Coxeter diagram |    or   |
| Symmetry group | [6,4]+, (642) |
| Dual | Order-6-4 floret pentagonal tiling |
| Properties | Vertex-transitive Chiral |
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetrahexagonal tiling is fifth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
| 4n2 symmetry mutations of snub tilings: 3.3.4.3.n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry 4n2 |
Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| 242 | 342 | 442 | 542 | 642 | 742 | 842 | ∞42 | |
| Snub figures |
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| Config. | 3.3.4.3.2 | 3.3.4.3.3 | 3.3.4.3.4 | 3.3.4.3.5 | 3.3.4.3.6 | 3.3.4.3.7 | 3.3.4.3.8 | 3.3.4.3.∞ |
| Gyro figures |
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| Config. | V3.3.4.3.2 | V3.3.4.3.3 | V3.3.4.3.4 | V3.3.4.3.5 | V3.3.4.3.6 | V3.3.4.3.7 | V3.3.4.3.8 | V3.3.4.3.∞ |
| Uniform tetrahexagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||
  =    =   =  |
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| {6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
| Uniform duals | |||||||||||
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| V64 | V4.12.12 | V(4.6)2 | V6.8.8 | V46 | V4.4.4.6 | V4.8.12 | |||||
| Alternations | |||||||||||
| [1+,6,4] (*443) |
[6+,4] (6*2) |
[6,1+,4] (*3222) |
[6,4+] (4*3) |
[6,4,1+] (*662) |
[(6,4,2+)] (2*32) |
[6,4]+ (642) | |||||
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| h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} | |||||
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 3-3-4-3-6. |
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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