Snub hexahexagonal tiling
In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}.
| 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 |
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
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
| 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
- 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
Wikimedia Commons has media related to Uniform tiling 3-3-6-3-6.
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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.