Order-8 octagonal tiling
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| Order-8 octagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 88 |
| Schläfli symbol | {8,8} |
| Wythoff symbol | 8 | 8 2 |
| Coxeter diagram |    |
| Symmetry group | [8,8], (*882) |
| Dual | self dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} (eight octagons around each vertex) and is self-dual.
Symmetry
[edit]This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *44444444 with 8 order-4 mirror intersections. In Coxeter notation can be represented as [8,8*], removing two of three mirrors (passing through the octagon center) in the [8,8] symmetry.
Related polyhedra and tiling
[edit]This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram 
, progressing to infinity.
| Space | Spherical | Compact hyperbolic | Paracompact | |||||
|---|---|---|---|---|---|---|---|---|
| Tiling |  |  |  |  |  |  |  | |
| Config. | 8.8 | 83 | 84 | 85 | 86 | 87 | 88 | ...8∞ |
| Regular tilings: {n,8} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Spherical | Hyperbolic tilings | ||||||||||
 {2,8}  |  {3,8}  |  {4,8}  |  {5,8}  |  {6,8}  | {7,8}  | {8,8} | ... |  {∞,8}  | |||
| Uniform octaoctagonal tilings | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry: [8,8], (*882) | |||||||||||
=   =  | = = | =  =  | =  = | = = | =  = | = = | |||||
| |  |  |  | |  |  | |||||
| {8,8} | t{8,8} | r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
| Uniform duals | |||||||||||
 | | | | | | | |||||
 |  |  |  |  |  |  | |||||
| V88 | V8.16.16 | V8.8.8.8 | V8.16.16 | V88 | V4.8.4.8 | V4.16.16 | |||||
| Alternations | |||||||||||
| [1+,8,8] (*884) | [8+,8] (8*4) | [8,1+,8] (*4242) | [8,8+] (8*4) | [8,8,1+] (*884) | [(8,8,2+)] (2*44) | [8,8]+ (882) | |||||
=   |  | =  | | =   | =  = | = = | |||||
 |  | |  |  | |||||||
| h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
| Alternation duals | |||||||||||
 | | | | | | | |||||
 |  | ||||||||||
| V(4.8)8 | V3.4.3.8.3.8 | V(4.4)4 | V3.4.3.8.3.8 | V(4.8)8 | V46 | V3.3.8.3.8 | |||||
See also
[edit]
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.
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
