Snub tetraoctagonal tiling
| Snub tetraoctagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.3.4.3.8 |
| Schläfli symbol | sr{8,4} or |
| Wythoff symbol | | 8 4 2 |
| Coxeter diagram |    |
| Symmetry group | [8,4]+, (842) |
| Dual | Order-8-4 floret pentagonal tiling |
| Properties | Vertex-transitive Chiral |
In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetraoctagonal tiling is seventh 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 octagonal/square tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| [8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) | |||||||||||
  =    =    =  | =  | = =  = | = | =  = | = | | |||||
 |  |  |  |  |  |  | |||||
| {8,4} | t{8,4} | r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} | |||||
| Uniform duals | |||||||||||
 | | | | | | | |||||
 |  |  |  |  |  |  | |||||
| V84 | V4.16.16 | V(4.8)2 | V8.8.8 | V48 | V4.4.4.8 | V4.8.16 | |||||
| Alternations | |||||||||||
| [1+,8,4] (*444) | [8+,4] (8*2) | [8,1+,4] (*4222) | [8,4+] (4*4) | [8,4,1+] (*882) | [(8,4,2+)] (2*42) | [8,4]+ (842) | |||||
 =  | =  | =   | =  | =  | = | | |||||
 |  |  |  |  |  | | |||||
| h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||
| Alternation duals | |||||||||||
 | | | | | | | |||||
 |  |  |  |  | |||||||
| V(4.4)4 | V3.(3.8)2 | V(4.4.4)2 | V(3.4)3 | V88 | V4.44 | V3.3.4.3.8 | |||||
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
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
