The density and minimal gap of visible points in some planar quasicrystals

Abstract

We give formulas for the density of visible points of several families of planar quasicrystals, which include the Ammann-Beenker point set and vertex sets of some rhombic Penrose tilings. These densities are used in order to calculate the limiting minimal normalised gap between the angles of visible points in two families of planar quasicrystals, which include the Ammann-Beenker point set and vertex sets of some rhombic Penrose tilings.