Pythagorean tiling

[8] It is a chiral pattern, meaning that it is impossible to superpose it on top of its mirror image using only translations and rotations.

[9] This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth-century Islamic mathematicians Al-Nayrizi and Thābit ibn Qurra, and by the 19th-century British amateur mathematician Henry Perigal.

[10] Although the Pythagorean tiling is itself periodic (it has a square lattice of translational symmetries) its cross sections can be used to generate one-dimensional aperiodic sequences.

[14] If x is chosen as the golden ratio, the sequence of 0s and 1s generated in this way has the same recursive structure as the Fibonacci word: it can be split into substrings of the form "01" and "0" (that is, there are no two consecutive ones) and if these two substrings are consistently replaced by the shorter strings "0" and "1" then another string with the same structure results.

He conjectures that, in any dimension greater than three, there is again a unique unilateral and equitransitive way of tiling space by hypercubes of two different sizes.

[17] An earlier paper by Danzer, Grünbaum, and Shephard provides another example, a convex pentagon that tiles the plane only when combined in two sizes.

An early structural application of the Pythagorean tiling appears in the works of Leonardo da Vinci, who considered it among several other potential patterns for floor joists.

[1] It has been suggested that seeing a similar tiling in the palace of Polycrates may have provided Pythagoras with the original inspiration for his theorem.