On the notions of symmetry and aperiodicity for Delone sets

TL;DR

This paper reviews and clarifies the concepts of symmetry and aperiodicity in Delone sets, emphasizing the importance of the hull in understanding non-periodic systems in crystallography.

Contribution

It provides a systematic and consistent framework for understanding symmetry and aperiodicity in Delone sets, focusing on the hull concept.

Findings

Clarified the role of the hull in Delone set analysis

Unified various notions of symmetry and aperiodicity

Enhanced understanding of non-periodic systems in crystallography

Abstract

Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.