Truncated tetrapentagonal tiling
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Short description: A uniform tiling of the hyperbolic plane
In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.
Symmetry
There are four small index subgroup constructed from [5,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A radical subgroup is constructed [5*,4], index 10, as [5+,4], (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]+, index 20, becomes orbifold (22222).
| Small index subgroups of [5,4] | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Index | 1 | 2 | 10 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
[5,4] =     (*542) |
[5,4,1+] =  =   (*552) |
[5+,4] =  (5*2) |
[5*,4] =    (*22222) | |||||||
| Direct subgroups | |||||||||||
| Index | 2 | 4 | 20 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
[5,4]+ = (542) |
[5+,4]+ = =   (552) |
[5*,4]+ = (22222) | ||||||||
Related polyhedra and tiling
See also
- Uniform tilings in hyperbolic plane
- 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)
- Coxeter, H. S. M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 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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