Rigidity for sticky discs

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 475; no. 2222; pp. 1 - 16
• Main Authors: Connelly, Robert, Gortler, Steven J., Theran, Louis
• Format: Journal Article
• Language: English
• Published: Royal Society 06-02-2019
• ISSN: 1364-5021; 1471-2946

We study the combinatorial and rigidity properties of disc packings with generic radii. We show that a packing of n discs in the plane with generic radii cannot have more than 2n − 3 pairs of discs in contact. The allowed motions of a packing preserve the disjointness of the disc interiors and tangency between pairs already in contact (modelling a collection of sticky discs). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly 2n − 3 contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy–Alexandrov stress lemma. Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly et al. (Connelly et al. 2018 (http://arxiv.org/abs/ 1702.08442)) on the number of contacts in a jammed packing of discs with generic radii.