Robust matching rules for real quasicrystals

TL;DR

This paper introduces a geometric and homological framework for extracting minimal-range matching rules from quasicrystal diffraction data, ensuring robustness against defects and capturing long-range order.

Contribution

It presents a novel algorithmic approach combining geometric and homological methods to derive minimal-range matching rules directly from diffraction data.

Findings

Successfully extracts shortest possible matching rules from diffraction data.

Framework tolerates defects, maintaining robustness.

Ensures long-range quasiperiodic order in derived rules.

Abstract

We consider the problem of extraction and validation of matching rules, directly from the phased diffraction data of a quasicrystal, and propose an algorithmic procedure to produce the rules of the shortest possible range. We have developed a geometric framework to express such rules together with a homological mechanism enforcing the long-range quasiperiodic order. This mechanism tolerates the presence of defects in a robust way.