Rhombihexaoctagonal tiling
From HandWiki
| Rhombihexaoctagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 6.4.8.4 |
| Schläfli symbol | rr{8,6} or |
| Wythoff symbol | 6 | 8 2 |
| Coxeter diagram |    
|
| Symmetry group | [8,6], (*862) |
| Dual | Deltoidal hexaoctagonal tiling |
| Properties | Vertex-transitive |
In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}.
Symmetry
The dual tiling, called a deltoidal hexaoctagonal tiling represent the fundamental domains of *4232 symmetry, a half symmetry of [8,6], (*862) as [8,1+,6].
- 160px
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
| Uniform octagonal/hexagonal tilings | ||||||
|---|---|---|---|---|---|---|
| Symmetry: [8,6], (*862) | ||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| {8,6} | t{8,6} |
r{8,6} | 2t{8,6}=t{6,8} | 2r{8,6}={6,8} | rr{8,6} | tr{8,6} |
| Uniform duals | ||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| V86 | V6.16.16 | V(6.8)2 | V8.12.12 | V68 | V4.6.4.8 | V4.12.16 |
| Alternations | ||||||
| [1+,8,6] (*466) |
[8+,6] (8*3) |
[8,1+,6] (*4232) |
[8,6+] (6*4) |
[8,6,1+] (*883) |
[(8,6,2+)] (2*43) |
[8,6]+ (862) |
|
|
|
|
|
|
|
|
|
| ||||
| h{8,6} | s{8,6} | hr{8,6} | s{6,8} | h{6,8} | hrr{8,6} | sr{8,6} |
| Alternation duals | ||||||
|
|
|
|
|
|
|
|
||||||
| V(4.6)6 | V3.3.8.3.8.3 | V(3.4.4.4)2 | V3.4.3.4.3.6 | V(3.8)8 | V3.45 | V3.3.6.3.8 |
See also
- Tilings of regular polygons
- List of uniform planar tilings
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.
External links
- Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html.
- Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
 |  |
