Abstract
The model sets (also called cut and project sets), first defined by Yves Meyer in harmonic analysis, play a central role in quasicrystal modelling. Each of them is defined by using a cut and project scheme containing two projectors and a lattice. We present a method which can be used to study the self-similarities of a model set based on the matrices of these projectors in a basis of the lattice. This method also allows one to study the self-similarities of the diffraction spectrum of a model set because, generally, the Bragg peaks with intensity above a given threshold also form a model set. The diffraction pattern corresponding to a quasicrystal is invariant under a finite group G, and the local structure of the quasicrystal can be described by using a finite union of orbits of G, called a G cluster. The neighbours of each atom belong to some orbits of G, and the quasicrystal can be regarded as a union of interpenetrating partially occupied translations of the corresponding G cluster. We present a method to obtain a model set (called the G-model set) by starting from a G cluster. The experimental diffraction patterns allow one to determine the symmetry group G, and high-resolution electron microscopy images enable one to choose a G cluster describing the local structure. The existing computer programs for the cut and project method allow one to pass directly from the local structure of the quasicrystal to a mathematical model, to compute the theoretical diffraction spectrum and to compare it with the experimental data.