Snub hexahexagonal tiling
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| Snub hexahexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.3.6.3.6 |
| Schläfli symbol | s{6,4} sr{6,6} |
| Wythoff symbol | | 6 6 2 |
| Coxeter diagram |     |
| Symmetry group | [6,6]+, (662) [6+,4], (6*2) |
| Dual | Order-6-6 floret hexagonal tiling |
| Properties | Vertex-transitive |
In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}.
Images
[edit]Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
[edit]A higher symmetry coloring can be constructed from [6,4] symmetry as s{6,4}, . In this construction there is only one color of hexagon.
Related polyhedra and tiling
[edit]| Uniform hexahexagonal tilings | ||||||
|---|---|---|---|---|---|---|
| Symmetry: [6,6], (*662) | ||||||
 =   =  | = = | =  =  | =  = | = = | =  = | = = |
 |  |  |  |  |  |  |
| {6,6} = h{4,6} | t{6,6} = h2{4,6} | r{6,6} {6,4} | t{6,6} = h2{4,6} | {6,6} = h{4,6} | rr{6,6} r{6,4} | tr{6,6} t{6,4} |
| Uniform duals | ||||||
 | | | | | | |
 |  |  |  |  |  |  |
| V66 | V6.12.12 | V6.6.6.6 | V6.12.12 | V66 | V4.6.4.6 | V4.12.12 |
| Alternations | ||||||
| [1+,6,6] (*663) | [6+,6] (6*3) | [6,1+,6] (*3232) | [6,6+] (6*3) | [6,6,1+] (*663) | [(6,6,2+)] (2*33) | [6,6]+ (662) |
=  | | =  | | =   | | |
| | | | | | | |
 |  | |  | | ||
| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
| 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} | |||||
| 4n2 symmetry mutations of snub tilings: 3.3.n.3.n | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry 4n2 | Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||
| 222 | 322 | 442 | 552 | 662 | 772 | 882 | ∞∞2 | ||||
| Snub figures |  |  |  |  | |  |  |  | |||
| Config. | 3.3.2.3.2 | 3.3.3.3.3 | 3.3.4.3.4 | 3.3.5.3.5 | 3.3.6.3.6 | 3.3.7.3.7 | 3.3.8.3.8 | 3.3.∞.3.∞ | |||
| Gyro figures |  |  |  |  | |||||||
| Config. | V3.3.2.3.2 | V3.3.3.3.3 | V3.3.4.3.4 | V3.3.5.3.5 | V3.3.6.3.6 | V3.3.7.3.7 | V3.3.8.3.8 | V3.3.∞.3.∞ | |||
References
[edit]- 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
[edit]
Wikimedia Commons has media related to Uniform tiling 3-3-6-3-6.
External links
[edit]- 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
