Order-6 octagonal tiling
In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.
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 *33333333 with 8 order-3 mirror intersections. In Coxeter notation can be represented as [8*,6], removing two of three mirrors (passing through the octagon center) in the [8,6] symmetry.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [8,6,1+], gives [(8,8,3)], (*883). Removing two mirrors as [8,6*], leaves remaining mirrors (*444444).
| Uniform Coloring |
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| Symmetry | [8,6] (*862)     
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[8,6,1+] = [(8,8,3)] (*883)  =  
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[8,1+,6] (*4232) =   
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[8,6*] (*444444)  
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| Symbol | {8,6} | {8,6}1⁄2 | r(8,6,8) | |
| Coxeter diagram |
 
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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 | 
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| Config. | 8.8 | 83 | 84 | 85 | 86 | 87 | 88 | ...8∞ |
See also
- Square tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
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
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