| Order-8 hexagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 6 |
| Schläfli symbol | {6,8} |
| Wythoff symbol | 8 | 6 2 |
| Coxeter diagram |    
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| Symmetry group | , (*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}.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the kaleidoscope. Removing the mirror between the order 2 and 6 points, , gives , (*664). Removing the mirror between the order 8 and 6 points, , gives (*4232). Removing two mirrors as , leaves remaining mirrors (*33333333).
| Uniform Coloring |
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| Symmetry | (*862)   
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= (*664)  =  
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(*4232) =   
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(*33333333) |
| Symbol | {6,8} | {6,8}1⁄2 | r(8,6,8) | {6,8}1⁄8 |
| Coxeter diagram |
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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 , removing two of three mirrors (passing through the square center) in the symmetry.
Related polyhedra and tiling
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| Symmetry: , (*862) | ||||||
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| {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 | ||||||
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| V8 | V6.16.16 | V(6.8) | V8.12.12 | V6 | V4.6.4.8 | V4.12.16 |
| Alternations | ||||||
(*466) |
(8*3) |
(*4232) |
(6*4) |
(*883) |
(2*43) |
(862) |
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| h{8,6} | s{8,6} | hr{8,6} | s{6,8} | h{6,8} | hrr{8,6} | sr{8,6} |
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| V(4.6) | V3.3.8.3.8.3 | V(3.4.4.4) | V3.4.3.4.3.6 | V(3.8) | V3.4 | V3.3.6.3.8 |
See also
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
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