Local growth of icosahedral quasicrystalline tilings

TL;DR

This paper introduces a local-rule-based algorithm for growing perfectly ordered icosahedral quasicrystals, demonstrating how local interactions can produce long-range order with minimal defects, relevant for understanding natural and synthetic IQCs.

Contribution

The authors developed a novel local-rule algorithm that produces defect-free IQCs, providing insights into the growth mechanisms of these complex structures.

Findings

Growth is typically infinite when seeded with a defect-containing cluster.

Resultant IQCs exhibit a vanishing density of defects.

The geometric principles can inform experimental and numerical models.

Abstract

Icosahedral quasicrystals (IQCs) with extremely high degrees of translational order have been produced in the laboratory and found in naturally occurring minerals, yet questions remain about how IQCs form. In particular, the fundamental question of how locally determined additions to a growing cluster can lead to the intricate long-range correlations in IQCs remains open. In answer to this question, we have developed an algorithm that is capable of producing a perfectly ordered IQC, yet relies exclusively on local rules for sequential, face-to-face addition of tiles to a cluster. When the algorithm is seeded with a special type of cluster containing a defect, we find that growth is forced to infinity with high probability and that the resultant IQC has a vanishing density of defects. The geometric features underlying this algorithm can inform analyses of experimental systems and numerical models that generate highly ordered quasicrystals.