Purely local growth of a quasicrystal

TL;DR

This paper proves that aperiodic quasicrystals, modeled by Golden-Octagonal tilings, can grow purely locally and deterministically, challenging the common belief that such growth requires non-local information.

Contribution

It establishes a theorem demonstrating the possibility of purely local and deterministic growth of aperiodic tilings, specifically Golden-Octagonal tilings, as a model for quasicrystals.

Findings

Purely local growth of aperiodic tilings is possible.

Contradicts the common belief that non-local information is needed.

Provides a mathematical model for quasicrystal growth.

Abstract

Self-assembly is the process in which the components of a system, whether molecules, polymers, or macroscopic particles, are organized into ordered structures as a result of local interactions between the components themselves, without exterior guidance. In this paper, we speak about the self-assembly of aperiodic tilings. Aperiodic tilings serve as a mathematical model for quasicrystals - crystals that do not have any translational symmetry. Because of the specific atomic arrangement of these crystals, the question of how they grow remains open. In this paper, we state the theorem regarding purely local and deterministic growth of Golden-Octagonal tilings. Showing, contrary to the popular belief, that local growth of aperiodic tilings is possible.