Topological densities of quasiperiodic structures

Abstract We study the topological densities of quasiperiodic graphs, whose set of vertices is a cut-and-project set. We use two approaches: a calculation based on the asymptotics of the coordination sequence and a calculation based on the growth forms of the graphs. The main result is an explicit formula expressing the topological density of quasiperiodic graphs in terms of the growth form and cut-and-project construction of the graph. Both approaches are illustrated with three examples: the graphs of vertices of the Penrose and Ammann-Beenker tilings and the Rauzy tiling.