The Nonholonomy of the Rolling Sphere

An old problem in the field of holonomy asks: Given a pair of orientations for a sphere resting on a plane, is there a closed path along which one can roll the sphere (without slipping or twisting), starting with the first orientation, and return to the origin with the sphere in the second orientation? (See Figure 1.) The answer is yes, and the goal of this article is to provide an elementary proof of this fact.