Pushing things around

Unsolved for over twenty-five years, a surprisingly difficult conjecture stated that a non-crossing polygonal chain of fixed-length edges in the plane can be continuously opened without crossing. Gunter Rote, Erik Demaine and I proved this Carpenter's Rule conjecture in 2000.

Unsolved for almost fifty years, M. Kneser and E. M. Poulsen conjectured that if a finite collection of circular disks in the plane are picked up and re-positioned in the plane so that no pair of centers gets closer together, then the area of the union never gets smaller. Karoly Bezdek and I proved this Kneser-Poulsen conjecture in 2002.

These two seemingly disparate results, the proofs of the Carpenter's Rule conjecture and the Kneser-Poulsen conjecture, can be combined to extend the Carpenter's Rule result to chains of appropriately slender sets, which can open without self-intersection. One of the critical ingredients in the Kneser-Poulsen result is a formula by Balazs Csikos for the change in the area of unions and intersections of disks as the centers are moved continuously. This brings up a large class of problems simultaneously generalizing the notions of packings and coverings of disks. For example, it shows how to find the critical area configurations of the union of one set of disks, with one larger radius, intersected with the union of another set of disks, with another smaller radius.