Order-8 hexagonal tiling
From Wikipedia, the free encyclopedia
| Order-8 hexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex figure | 68 |
| Schläfli symbol | {6,8} |
| Wythoff symbol | 8 | 6 2 |
| Coxeter diagram |     |
| Symmetry group | [8,6], (*862) |
| Dual | Order-6 octagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*444444) with 6 order-4 mirror intersections. In Coxeter notation can be represented as [8,1+,6,1+], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.
Related polyhedra and tiling
| 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
 |
Wikimedia Commons has media related to Order-8 hexagonal tiling. |
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-6 (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.
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 available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.