Modified strip projection method

Abstract

The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quasicrystal can be regarded as a quasiperiodic packing of copies of a G-cluster C, joined by glue atoms. The physical space E containing C can be embedded into a higher-dimensional space R^k such that, up to an inflation factor, C is the orthogonal projection of the set {(\pm 1,0,...,0), (0,\pm 1,0,...,0), ... (0,...,0,\pm 1)}. The projections of the points of Z^k lying in the strip S=E+[-1/2,1/2]^k+t obtained by shifting a hypercube [-1/2,1/2]^k+t along E is a quasiperiodic packing of partially occupied copies of C, but unfortunately, the occupation of clusters is very low. In our modified strip projection method we firstly determine for each point x\in Z^k\cap S the number n(x) of all the arithmetic neighbours of x lying in the strip S, and project the points of Z^k\cap S on E in the decreasing order of the occupation number n(x). In the case when n(x) represents more than p% of all the points of the cluster C we project all the arithmetic neighbours of x (lying inside or outside S). We choose p such that to avoid the superposition of the fully occupied clusters. The projection of a point x with n(x) less than p% of all the points of the cluster C is added to the pattern only if it is not too close to the already obtained points.