Event Details

Rigidity for sticky disks

The Köbe-Andreev-Thurston (KAT) theorem says that any planar graph has a realisation as the contract graph of a disk packing in the Euclidean plane. Since any disk packing gives rise to a non-crossing drawing of its contact graph, the KAT theorem exactly characterises the combinatorics of disk packings in the plane. I’ll talk about what happens when we consider packings where the disks have generic radii, a setup motivated by applications to colloids, “polydisperse" jammed systems, and geometric constraint solving. The generic case turns out to be closely related to the rigidity theory of frameworks in the Euclidean plane.

A packing $P$ of $n$ disks with generic radii in the Euclidean plane can have at most $2n - 3$ pairs of disks in contact. Moreover, if $P$ has $2n -3$ contacts, all the deformations of $P$ that preserve contacts, the radii of the disks, and the non-overlapping property of a packing arise from rigid body motions. The latter result is an analogue of Laman’s Theorem from framework rigidity for disk packings. The proof techniques also yield a simple proof for a finite version of the recent “Isostatic Theorem” of Connelly, et. al.

This is joint work with Shlomo Gortler and Bob Connelly.