Order-6 octagonal tiling
| Order-6 octagonal tiling | |
|---|---|
| Order-6 octagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 86 |
| Schläfli symbol | {8,6} |
| Wythoff symbol | 6 | 8 2 |
| Coxeter diagram | File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png |
| Symmetry group | [8,6], (*862) |
| Dual | Order-8 hexagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
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).
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 File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel n.pngFile:CDel node.png, progressing to infinity.
| Space | Spherical | Compact hyperbolic | Paracompact | |||||
|---|---|---|---|---|---|---|---|---|
| Tiling | File:H2-8-3-dual.svg | File:H2 tiling 248-1.png | File:H2 tiling 258-1.png | File:H2 tiling 268-1.png | File:H2 tiling 278-1.png | File:H2 tiling 288-4.png | File:H2 tiling 28i-4.png | |
| Config. | 8.8 | 83 | 84 | 85 | 86 | 87 | 88 | ...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.
