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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 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
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.
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.
An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling Regular tilings and their duals drawn by Max Brückner in Vielecke und Vielflache (1900) This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane , and their dual tilings.
Examples The hexagonal prism is constructed from both the (6 2 2) and (3 2 2) families.: The truncated square tiling is constructed by two different symmetry positions in the (4 4 2) family.
The truncated square tiling honeycomb, t{4,4,3}, has cube and truncated square tiling facets, with a triangular pyramid vertex figure. It is the same as the cantitruncated order-4 square tiling honeycomb , tr{4,4,4}, .
The runcitruncated order-4 square tiling honeycomb, t 0,1,3 {4,4,4}, has square tiling, truncated square tiling, cube, and octagonal prism facets, with a square pyramid vertex figure. The runcicantellated order-4 square tiling honeycomb is equivalent to the runcitruncated order-4 square tiling honeycomb.