Pinwheel tiling

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Definition

Let $T$ be the right triangle with side length $1$ , $2$ and $\sqrt{5}$ . Conway noticed that $T$ can be divided in five isometric copies of its image by the dilation of factor $1 / \sqrt{5}$ .

The pinwheel tiling is obtained by repeatedly inflating $T$ by a factor of $\sqrt{5}$ and then subdividing each tile in this manner. Conversely, the tiles of the pinwheel tiling can be grouped into groups of five that form a larger pinwheel tiling. In this tiling, isometric copies of $T$ appear in infinitely many orientations because the small angle of $T$ , $\arctan \frac{1}{2}$ , is not a rational multiple of $π$ . Radin found a collection of five prototiles, each of which is a marking of $T$ , so that the matching rules on these tiles and their reflections enforce the pinwheel tiling. All of the vertices have rational coordinates, and tile orientations are uniformly distributed around the circle.

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Definition

Generalizations

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