Tetraoctagonal tiling
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In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
| Name | Tetra-octagonal tiling | Rhombi-octaoctagonal tiling | ||
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| Symmetry | [8,4] (*842)     
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[8,8] = [8,4,1+] (*882)  =  
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[(4,4,4)] = [1+,8,4] (*444) =   
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[(∞,4,∞,4)] = [1+,8,4,1+] (*4242) =   or 
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| Schläfli | r{8,4} | rr{8,8} =r{8,4}1/2 |
r(4,4,4) =r{4,8}1/2 |
t0,1,2,3(∞,4,∞,4) =r{8,4}1/4 |
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= or
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Symmetry
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.
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Related polyhedra and tiling
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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