Modified strip projection method

TL;DR

This paper introduces a modified strip projection method for quasicrystals that improves cluster occupation by prioritizing points based on neighbor counts and occupancy thresholds, leading to more complete and less overlapping quasiperiodic packings.

Contribution

The paper presents a novel modification to the strip projection method that enhances cluster occupation and reduces overlaps in quasicrystal pattern generation.

Findings

Increased cluster occupation in generated patterns.

Reduced superposition of fully occupied clusters.

More uniform and complete quasiperiodic packings.

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.