Truncated order-4 hexagonal tiling


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

Constructions

There are two uniform constructions of this tiling, first from kaleidoscope, and a lower symmetry by removing the last mirror, , gives ,.
NameTetrahexagonalTruncated hexahexagonal
Image
Symmetry

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Symbolttr
Coxeter diagram

Dual tiling

Related polyhedra and tiling

Symmetry

The dual of the tiling represents the fundamental domains of orbifold symmetry. From symmetry, there are 15 small index subgroup 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, is the commutator subgroup of .
Larger subgroup constructed as , removing the gyration points of, index 12 becomes.
The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.