Order-8 octagonal tiling
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
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
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 | |||||
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Tiling | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |
Config. | 8.8 | 83 | 84 | 85 | 86 | 87 | 88 | ...8∞ |
Regular tilings: {n,8}
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Spherical | Hyperbolic tilings | ||||||||||
![]() {2,8} ![]() ![]() ![]() ![]() ![]() | ![]() {3,8} ![]() ![]() ![]() ![]() ![]() | ![]() {4,8} ![]() ![]() ![]() ![]() ![]() | ![]() {5,8} ![]() ![]() ![]() ![]() ![]() | ![]() {6,8} ![]() ![]() ![]() ![]() ![]() | ![]() {7,8} ![]() ![]() ![]() ![]() ![]() | ![]() {8,8} ![]() ![]() ![]() ![]() ![]() | ... | ![]() {∞,8} ![]() ![]() ![]() ![]() ![]() |
Uniform octaoctagonal tilings
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Symmetry: [8,8], (*882) | |||||||||||
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||||
{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 | |||||||||||
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||||
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 | |||||||||||
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![]() | ![]() | ||||||||||
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
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
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. 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