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"Jammed packings in the plane" by Robert Connelly
If you have ever driven behind a truck carrying a load of pipes, you might wonder if they if they were secured safely. You should. There are ways to do it that are stable, and ways that are not, and the distinction can be fuzzy. The pipes are essentially packings of circular disks in a planar box. The most dense packing of an infinite number of equal disks in the plane is known. But for pipes and granular materials it is enough that packing be the most dense only for small perturbations of the arrangement. The property of being rigid means that the the packing is jammed, and whether a packing is jammed can be calculated for reasonable containers.
Many interesting containers for circle packings are not boxes but tori, what you get by identifying the left and right sides, and the top and bottom, of a box in the plane so the boundary effects are avoided. There are interesting examples of jammed and not jammed packings of a finite number of circular disks in a torus that have been found by groups of undergraduate students at Cornell and Grand Valley University in Michigan. Their results will be presented.