String-Wrapped Rotating Disks

Abstract

Let the centers of a finite number of disjoint, closed disks be pinned to the plane, but with each free to rotate about its center. Given an arrangement of such disks with each labeled + or −, we investigate the question of whether they can be all wrapped by a single loop of string so that, when the string is taut and circulates, it rotates by friction all the ⊕-disks counterclockwise and all the ⊝-disks clockwise, without any string-rubbing conflicts. We show that although this is not always possible, natural disk-separation conditions guarantee a solution. We also characterize the hexagonal “penny-packing” arrangements that are wrappable.