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
![]() | Original source: https://en.wikipedia.org/wiki/Truncated pentahexagonal tiling.
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