Truncated order-6 square tiling
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.
Uniform colorings
 The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram    .
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Symmetry
The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.
A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).
The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.
| Small index subgroups of [(4,4,3)] (*443) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Index | 1 | 2 | 6 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
[(4,4,3)] =    (*443) |
[(4,1+,4,3)] =   =  (*3232) |
[(4,4,3+)] =  (3*22) |
[(4,4,3*)] =   (*222222) | |||||||
| Direct subgroups | |||||||||||
| Index | 2 | 4 | 12 | ||||||||
| Diagram | 
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| Coxeter (orbifold) |
[(4,4,3)]+ =  (443) |
[(4,4,3+)]+ = =  (3232) |
[(4,4,3*)]+ = (222222) | ||||||||
Related polyhedra and tilings
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 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 8 forms.
It can also be generated from the (4 4 3) hyperbolic tilings:
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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