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.
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.
Small index subgroups of [6,5], (*652)
| Index
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1
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2
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6
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| Diagram
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Coxeter
(orbifold)
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[6,5] =    
(*652)
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[1+,6,5] =  =  
(*553)
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[6,5+] = 
(5*3)
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[6,5*] =  
(*33333)
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| Direct subgroups
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| Index
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2
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4
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12
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| Diagram
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Coxeter
(orbifold)
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[6,5]+ =
(652)
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[6,5+]+ = = 
(553)
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[6,5*]+ =
(33333)
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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.
| Uniform hexagonal/pentagonal tilings
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| Symmetry: [6,5], (*652)
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[6,5]+, (652)
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[6,5+], (5*3)
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[1+,6,5], (*553)
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| {6,5}
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t{6,5}
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r{6,5}
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2t{6,5}=t{5,6}
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2r{6,5}={5,6}
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rr{6,5}
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tr{6,5}
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sr{6,5}
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s{5,6}
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h{6,5}
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| Uniform duals
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| V65
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V5.12.12
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V5.6.5.6
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V6.10.10
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V56
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V4.5.4.6
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V4.10.12
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V3.3.5.3.6
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V3.3.3.5.3.5
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V(3.5)5
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