| Snub hexaoctagonal tiling | |
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 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic uniform tiling |
| Vertex configuration | 3.3.6.3.8 |
| Schläfli symbol | sr{8,6} or |
| Wythoff symbol | | 8 6 2 |
| Coxeter diagram |    or  
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| Symmetry group | , (862) |
| Dual | Order-8-6 floret pentagonal tiling |
| Properties | Vertex-transitive Chiral |
In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are three triangles, one hexagon, and one octagon on each vertex. It has Schläfli symbol of sr{8,6}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full symmetry, and 7 with subsymmetry.
| Uniform octagonal/hexagonal tilings | ||||||
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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} |
| Alternation duals | ||||||
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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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