Is there a sensible way to say what it means to roll a ball across a plane, without slipping, so that the point of contact traces out a snowflake curve? (Equivalently, the center of the ball should trace out a snowflake curve on a parallel plane.) It's clear how to roll a ball so that the point of contact traces out a polygonal path, and from this one can define what it means to roll a ball so that the point of contact traces out a specified rectifiable path (undoubtedly there's a classic theorem that says that the behavior of the ball converges to a limit as one looks at ever-finer polygonal approximations to the path), but it's unclear whether this limit theorem would apply to a non-rectifiable curve like the snowflake curve. Maybe there's a dichotomy here, and it makes sense to roll a ball along a fractal curve as long as its fractal dimension is sufficiently low, but not if the fractal dimension exceeds some cutoff? (This question has "Dan Asimov" written all over it, but others may reply as well. :-) ) Jim Propp