Rigidity for sticky disks

TL;DR

This paper investigates the rigidity and contact properties of disk packings with generic radii, establishing maximum contact counts and conditions for rigidity, and applies these findings to jamming problems.

Contribution

It provides a new upper bound on the number of contacts in disk packings and characterizes when such packings are rigid, using a novel manifold approach.

Findings

Maximum of 2n-3 contacts in packings with n disks

Rigidity occurs if and only if the contact count is exactly 2n-3

The space of packings with fixed contact graph is a smooth manifold

Abstract

We study the combinatorial and rigidity properties of disk packings with generic radii. We show that a packing of $n$ disks in the plane with generic radii cannot have more than $2n-3$ pairs of disks in contact. The allowed motions of a packing preserve the disjointness of the disk interiors and tangency between pairs already in contact (modeling a collection of sticky disks). 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. on the number of contacts in a jammed packing of disks with generic radii.