Abstract

By a sequence of rollings without slipping or twisting along segments of a straight line of the plane a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We provide a new and very short proof that with at most 3 moves we can go from a given initial state to an arbitrary final state. The first an unique proof until now of this result is due to Hammersley 1983. By a sequence of rolling motions without slipping or twisting along arcs of maximal circles outside the surface of a sphere of radius R, a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We prove that with at most 3 moves we can go from a given initial state to an arbitrary final state if the number R is irrational; otherwise and assuming R>1 we also obtain as Frenkel and Garcia 2010 that with at most 4 moves we can go from a given initial state to an arbitrary final state.