Pythagorean tiling

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.

This tiling has four-way rotational symmetry around each of its squares. When the ratio of the side lengths of the two squares is an irrational number such as the golden ratio, its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions have also been studied.

The Pythagorean tiling is the unique tiling by squares of two different sizes that is both unilateral (no two squares have a common side) and equitransitive (each two squares of the same size can be mapped into each other by a symmetry of the tiling).

Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagons. The smaller squares in the Pythagorean tiling are adjacent to four larger tiles, as are the squares in the truncated square tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate between large and small, just as the octagons in the truncated square tiling. However, the two tilings have different sets of symmetries, because the truncated square tiling is symmetric under mirror reflections whereas the Pythagorean tiling isn't. Mathematically, this can be explained by saying that the truncated square tiling has dihedral symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic set of symmetries around the corresponding points, giving it p4 symmetry. It is a chiral pattern, meaning that it is impossible to superpose it on top of its mirror image using only translations and rotations.

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