# Truncated pentahexagonal tiling

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Truncated pentahexagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.10.12
Schläfli symbol tr{6,5} or ${\displaystyle t{\begin{Bmatrix}6\\5\end{Bmatrix}}}$
Wythoff symbol 2 6 5 |
Coxeter diagram
Symmetry group [6,5], (*652)
Dual Order 5-6 kisrhombille
Properties Vertex-transitive

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

 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.

Small index subgroups of [6,5], (*652)
Index 1 2 6
Diagram
Coxeter
(orbifold)
[6,5] =
(*652)
[1+,6,5] = =
(*553)
[6,5+] =
(5*3)
[6,5*] =
(*33333)
Direct subgroups
Index 2 4 12
Diagram
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.