Rigidity for sticky discs

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 475; no. 2222; p. 20180773
• Main Authors: Connelly, Robert, Gortler, Steven J, Theran, Louis
• Format: Journal Article
• Language: English
• Published: England The Royal Society Publishing 01.02.2019
• Subjects: rigidity; jamming; circle packings
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2018.0773

We study the combinatorial and rigidity properties of disc packings with generic radii. We show that a packing of discs in the plane with generic radii cannot have more than 2  - 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 2  - 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 (Connelly . 2018 (http://arxiv.org/abs/1702.08442)) on the number of contacts in a jammed packing of discs with generic radii.