Rigidity for sticky discs. (27th February 2019)

• Record Type: Journal Article
• Authors: Connelly, Robert; Gortler, Steven J.; Theran, Louis
• Is Part Of: Proceedings. Volume 475:Number 2222(2019)
• Publication Date: 2019-02-27
• Subjects: rigidity -- circle packings -- jamming; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2018.0773
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : 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 2 n − 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 n − 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.