Truncated pentahexagonal tiling
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.
Dual tiling
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| The dual tiling is called an order-5-6 kisrhombille tiling, made as a complete bisection of the order-5 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,5] (*652) symmetry. | |
Symmetry
There are four small index subgroup from [6,5] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
| Index | 1 | 2 | 6 | |
|---|---|---|---|---|
| Diagram | 
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| Coxeter (orbifold) |
[6,5] =     (*652) |
[1+,6,5] =  =   (*553) |
[6,5+] =  (5*3) |
[6,5*] =   (*33333) |
| Direct subgroups | ||||
| Index | 2 | 4 | 12 | |
| Diagram | 
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| Coxeter (orbifold) |
[6,5]+ = (652) |
[6,5+]+ = =  (553) |
[6,5*]+ = (33333) | |
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal 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 [6,5] symmetry, and 3 with subsymmetry.
See also
- Tilings of regular polygons
- List of uniform planar tilings
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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