Finite motions from periodic frameworks with added symmetry

TL;DR

This paper develops new Maxwell type counting rules for periodic frameworks with additional symmetry, predicting when such structures are flexible or rigid, which is important for understanding crystal properties.

Contribution

It combines existing counting approaches to derive new criteria for finite flexibility in symmetric, periodic frameworks, extending the theoretical understanding of their rigidity.

Findings

Symmetry can induce additional finite flexibility in periodic frameworks.

New Maxwell type counts for structures with combined periodicity and symmetry.

Results are relevant for understanding physical and chemical properties of crystal structures.

Abstract

Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that for finite frameworks, introducing symmetry modifies the previous general counts, and under some circumstances this symmetrized Maxwell type count can predict added finite flexibility in the structure. In this paper we combine these approaches to present new Maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells. In a number of cases, this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry. Given that many crystal structures have these added symmetries, and that their flexibility may be key to their physical and chemical properties, we present a summary of the results as a way to generate further developments of both a practical and theoretic interest.