Uniquely realizable crystalline structures

TL;DR

This paper develops conditions for the global rigidity of infinite periodic tensegrity frameworks in Euclidean space, extending finite framework results to periodic structures with various lattice constraints.

Contribution

It introduces infinite periodic stress matrices and provides necessary and sufficient conditions for global rigidity in periodic frameworks with different lattice flexibility options.

Findings

Established sufficient conditions for global rigidity in periodic frameworks.

Derived necessary and sufficient conditions for generic frameworks with fixed and flexible lattices.

Extended finite framework rigidity results to infinite periodic structures.

Abstract

We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in $\mathbb{R}^d$ in the cases when the lattice is either fixed, fully flexible, or flexible with a volume constraint for the fundamental domain. For the fixed and fully flexible lattice variants, we also establish necessary and sufficient conditions for generic infinite periodic bar-joint frameworks to be globally rigid in $\mathbb{R}^d$. These results provide periodic versions of the fundamental results of Connelly, as well as Gortler, Healy and Thurston on the global rigidity of generic finite bar-joint frameworks.