Pythagorean tiling

A Pythagorean tiling
Street Musicians at the Door, Jacob Ochtervelt, 1665. As observed by Nelsen[1] the floor tiles in this painting are set in the 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,[2] explaining its name.[1] It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern[3] or pinwheel pattern,[4] but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.[5]

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.

  1. ^ a b Cite error: The named reference rbn was invoked but never defined (see the help page).
  2. ^ Wells, David (1991), "two squares tessellation", The Penguin Dictionary of Curious and Interesting Geometry, New York: Penguin Books, pp. 260–261, ISBN 0-14-011813-6.
  3. ^ "How to Install Hopscotch Pattern Tiles", Home Guides, San Francisco Chronicle, retrieved 2016-12-12.
  4. ^ Editors of Fine Homebuilding (2013), Bathroom Remodeling, Taunton Press, p. 45, ISBN 978-1-62710-078-6. A schematic diagram illustrating this floor tile pattern appears earlier, on p. 42.
  5. ^ Radin, C. (1994), "The Pinwheel Tilings of the Plane", Annals of Mathematics, 139 (3): 661–702, doi:10.2307/2118575, JSTOR 2118575