Substitution tilings with statistical circular symmetry

Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all substitution tilings showing tiles in infinitely many orientations.

💡 Research Summary

The paper introduces two novel families of substitution tilings in which the prototiles appear in infinitely many orientations, thereby extending the celebrated pinwheel tiling to a broader class of “infinitely rotating” tilings. After a concise review of substitution tilings and the special role of the pinwheel example, the authors define two concrete substitution rules. The first rule (Rule A) uses a regular pentagon together with a set of interior triangles and quadrilaterals. In each substitution step the pentagon is scaled by a factor λ > 1, the interior pieces are scaled by the reciprocal factor, and the whole patch is rotated by an angle θ drawn from a continuous distribution on