Pinwheel tiling

In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations. Let be the right triangle with side length , and . Conway noticed that can be divided in five isometric copies of its image by the dilation of factor . By suitably rescaling and translating/rotating, this operation can be iterated to obtain an infinite increasing sequence of growing triangles all made of isometric copies of . The union of all these triangles yields a tiling of the whole plane by isometric copies of . In this tiling, isometric copies of appear in infinitely many orientations (this is due to the angles and of each being algebraically independent to over the reals.). Despite this, all the vertices have rational coordinates. Radin relied on the above construction of Conway to define pinwheel tilings. Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of , in which a tile may intersect another tile only either on a whole side or on half the length side, and such that the following property holds. Given any pinwheel tiling , there is a pinwheel tiling which, once each tile is divided in five following the Conway construction and the result is dilated by a factor , is equal to . In other words, the tiles of any pinwheel tilings can be grouped in sets of five into homothetic tiles, so that these homothetic tiles form (up to rescaling) a new pinwheel tiling. The tiling constructed by Conway is a pinwheel tiling, but there are uncountably many other different pinwheel tiling. They are all locally undistinguishable (i.e., they have the same finite patches). They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates).