The mathematical ideas of this semi-expository paper arose first as physical arguments one of which then gave rise to new mathematical proof.
1. A new proof of the Gauss-Bonnet theorem based on its reduction to a theorem about dual cones is given. This reduction was motivated by a simple observation on the motion of rigid bodies.
2. Two different proofs of the dual cones theorem are given, one based on "moving fronts" and another on a heuristic mechanical analogy.
3. As a corollary of this approach we obtain the observation that the total geodesic curvature of a curve on a surface is an invariant of the Gauss map, and that dual curves have reciprocal geodesic curvatures.
4. The ``bicycle wheel'' idea is applied (a) to construct a ``spherimeter'', i.e. a mechanical device which calculates areas of spherical regions; (b) to describe an aspect of wave propagation in a waveguide, such as an optical fiber, and (c) to describe twisting of an elastic object such as a beam or a rope.
5. Finally, the relationship with the writhing number and the Berry's phase is described.