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Rigidity for sticky disks

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Author(s)

Robert Connelly, Steven J. Gortler, Louis Simon Theran

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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 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.
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Original languageEnglish
Article number20180773
Number of pages16
JournalProceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Volume475
Issue number2222
DOIs
Publication statusPublished - 27 Feb 2019

    Research areas

  • Rigidity, Circle packings, Jamming

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