Truncated order-4 hexagonal tiling

Truncated order-4 hexagonal tiling Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.12.12
Schläfli symbol t{6,4}
tr{6,6} or $t{\begin{Bmatrix}6\\6\end{Bmatrix}}$ Wythoff symbol 2 4 | 6
2 6 6 |
Coxeter diagram          or   Symmetry group [6,4], (*642)
[6,6], (*662)
Dual Order-6 tetrakis square tiling
Properties Vertex-transitive

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

Constructions

There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662).

Two uniform constructions of 4.6.4.6
Name Tetrahexagonal Truncated hexahexagonal
Image
Symmetry [6,4]
(*642)

[6,6] = [6,4,1+]
(*662)
=
Symbol t{6,4} tr{6,6}
Coxeter diagram

Dual tiling  The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group.

Related polyhedra and tiling

Symmetry

The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,6,1+,6,1+] (3333) is the commutator subgroup of [6,6].

Larger subgroup constructed as [6,6*], removing the gyration points of (6*3), index 12 becomes (*333333).

The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.