Snub tetrahexagonal tiling
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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [6,4]+, (642) |
Dual | Order-6-4 floret pentagonal tiling |
Properties | Vertex-transitive Chiral |
In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}.
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 | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
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 | ![]() | ![]() | ![]() | ![]() | ||||
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) | |||||||||||
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() = ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() = ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() = ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | |||||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||||
{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
Uniform duals | |||||||||||
![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | |||||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||||
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) | |||||
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | |||||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||||
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-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
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
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