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.

Let me make sure I understand Mike's idea for the level-one approximation to the Cantor set: you roll the hoop so that the point of contact goes from (0,0) to (1/3,0), then swing the hoop down pivoting on the point (1/3,0) until the hoop touches the Cantor set at both (1/3,0) and (2/3,0) (so that the bottom point of the hoop is at (1/2,s) for some s < 0), then swing the hoop up pivoting on the point (2/3,0) until the hoop's lowest point is (2/3,0), and then roll the hoop so that the point of contact goes from (2/3,0) to (1,0). Have I got that right? It's a cute idea, but I don't see how it applies to the Koch curve problem.

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.

Can you elaborate? Jim Propp