Uniquely realizable crystalline structures

TL;DR

This work addresses the global rigidity of infinite periodic bar-joint frameworks by introducing periodic stress matrices suitable for three lattice regimes: fixed, fully flexible, and volume-bounded. It develops a rigorous algebraic and geometric framework based on $\mathbb{Z}^d$-gain graphs and associated incidence structures to analyze periodic rigidity. The authors derive sufficient conditions for global rigidity of periodic tensegrities and establish generic equivalences to infinitesimal rigidity with equilibrium stresses under fully flexible and fixed lattices, extending classical finite-framework results to the periodic setting. A notable contribution is the volume-constrained case, where a non-convex optimization arises but every local minimizer is global, providing a complete global rigidity criterion under that constraint; overall, the paper yields periodic analogs of Connelly's and Gortler–Healy–Thurston's results for generic finite frameworks and advances the understanding of crystalline rigidity across lattice types.

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.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4