Rigidity for sticky disks

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.