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 .

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)Index126DiagramCoxeter(orbifold)[(4,4,3)] = (*443)[(4,1+,4,3)] = = (*3232)[(4,4,3+)] = (3*22)[(4,4,3*)] = (*222222)Direct subgroupsIndex2412DiagramCoxeter(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:

Truncated order-6 square tiling

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