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The truncated square tiling is used in an optical illusion with truncated vertices divides and colored alternately, seeming to twist the grid.. The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}. Uniform color.
In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t 0,1 {4,5}. Related polyhedra and tiling
The truncated order-4 square tiling honeycomb, t 0,1 {4,4,4}, has square tiling and truncated square tiling facets, with a square pyramid vertex figure.
Truncated order-6 square tiling with *443 symmetry mirror lines. 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.
Edges exist between a generator point and its image across a mirror. Up to 3 face types exist centered on the fundamental triangle corners. Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling.
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