Reuleaux Pairs and Surfaces That Cannot Be Gripped

It is well known that there exist surfaces whose motion cannot be completely constrained by non-frictional contact forces. We give a new proof of the classification of these surfaces based on group theory. Having derived a simple character ization of these "surfaces that cannot be gripped, " we show that they are equivalent to the Reuleaux lower pairs. The proof emphasizes the symmetry of the surfaces rather than their analytic form. We also show that the screw system of such a surface is isomorphic to the Lie algebra of the surface's symmetry group.