Pythagorean tiling

4 years ago by in Featured, Geometry, Geometry

5Pythagorean tiling or two squares tessellation is a tessellation of an Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Because of numerous proofs of the Pythagorean theorem based on such a tiling, it is called a Pythagorean tiling. It also is commonly used as a pattern for floor tiles; in this context it is also known as a hopscotch pattern.

Topology and Symmetry

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).

3Topologically, 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: 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.

A uniform tiling is a tiling in which each tile is a regular polygon and in which there is a symmetry, mapping every vertex to every other vertex. Usually, uniform tilings additionally are required to have tiles that meet edge-to-edge, but if this requirement is relaxed then there are eight additional uniform tilings: four formed from infinite strips of squares or equilateral triangles, three formed from equilateral triangles and regular hexagons, and one more, the Pythagorean tiling.


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