Order-8 hexagonal tiling
Order-8 hexagonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 68 |
Schläfli symbol | {6,8} |
Wythoff symbol | 8 | 6 2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [8,6], (*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 [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [6,8,1+], gives [(6,6,4)], (*664). Removing the mirror between the order 8 and 6 points, [6,1+,8], gives (*4232). Removing two mirrors as [6,8*], leaves remaining mirrors (*33333333).
Uniform Coloring | ![]() | ![]() | ||
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Symmetry | [6,8] (*862) ![]() ![]() ![]() ![]() ![]() | [6,8,1+] = [(6,6,4)] (*664) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | [6,1+,8] (*4232) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | [6,8*] (*33333333) |
Symbol | {6,8} | {6,8}1⁄2 | r(8,6,8) | {6,8}1⁄8 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 [8,6*], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.
Related polyhedra and tiling
Uniform octagonal/hexagonal tilings
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Symmetry: [8,6], (*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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V86 | V6.16.16 | V(6.8)2 | V8.12.12 | V68 | V4.6.4.8 | V4.12.16 |
Alternations | ||||||
[1+,8,6] (*466) | [8+,6] (8*3) | [8,1+,6] (*4232) | [8,6+] (6*4) | [8,6,1+] (*883) | [(8,6,2+)] (2*43) | [8,6]+ (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} |
Alternation duals | ||||||
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V(4.6)6 | V3.3.8.3.8.3 | V(3.4.4.4)2 | V3.4.3.4.3.6 | V(3.8)8 | V3.45 | V3.3.6.3.8 |
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