Tiling Vertices and the Spacing Distribution of their Radial Projection

Abstract

The Fourier-based diffraction approach is an established method to extract order and symmetry propertiesfrom a given point set. We want to investigate a different method for planar sets which works in direct spaceand relies on reduction of the point set information to its angular component relative to a chosen referenceframe. The object of interest is the distribution of the spacings of these angular components, which can for instance beencoded as a density function on $\Real_{+}$. In fact, this \emph{radial projection} method is not entirely new, andthe most natural choice of a point set, the integer lattice $\Integer^2$, is already well understood. We focus on the radial projection of aperiodic point sets and study the relation between the resultingdistribution and properties of the underlying tiling, like symmetry, order and the algebraic type of theinflation multiplier.