Here's an easier one: can you roll a hoop with r>=1/6 (so that it can cross the largest gap) over the Cantor set? With that set, you necessarily have only single points of contact for the hoop; I assume that the edge of the hoop "sticks" to the point and the diameter traces out a circular arc until the hoop hits the next point. No matter what radius you choose, near the beginning of the trip, the edge of the hoop is very nearly flat compared to the distance d between points, and so the hoop will rotate by an amount very close to 2pi*r/d radians. The missing rotation as compared to rolling along a line (which we can define as the limit as d->0 of "sticky" rolling over a lattice with spacing d) clearly converges. If r is big enough then I claim that rolling the hoop over the Koch curve is identical to rolling it over the Cantor set (once if you go across the bottom, twice if you go across the top), since the hoop can't reach the concave parts of the fractal. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com