Tiling Vertices and the Spacing Distribution of Their Radial Projection

Abstract

The Fourier-based di raction approach is an established method to extract order and symmetry properties from a given point set. We want to investigate a di erent method for planar sets which works in direct space and relies on reduction of the point set information to its angular component relative to a chosen reference frame. The object of interest is the distribution of the spacings of these angular components, which can for instance be encoded as a density function on R+. In fact, this radial projection method is not entirely new, and the most natural choice of a point set, the integer lattice Z, is already well understood. We focus on the radial projection of aperiodic point sets and study the relation between the resulting distribution and properties of the underlying tiling, like symmetry, order and the algebraic type of the in ation multiplier.