This might be interesting on a surface of constant negative curvature (which each surface of higher genus can be made to have). But it reminds me of another question I've wondered about but never answered: Question: --------- Suppose a unit 2-sphere, initially with the North Pole at top, is rolled on a horizontal plane without slipping or twisting. What is the shortest simple closed curve in the plane that the sphere can be rolled around so the final point of tangency is the North Pole? ----- Also of interest would be simple closed curve with the same property that bounds the least area (are they the same?). —Dan Bill Gosper wrote: ----- Choose a point on a 2-sphere and traverse 90º of a great circle. Hang a 90º left turn. Traverse another 90º arc. Turn. Iterate. When, if ever, do you return to your starting point? -----