Mike Stay writes:

A tube formula talks about the area within a certain distance of the fractal; essentially you're rolling a circle or a ball over the surface of the fractal. By defining the zeta function of a fractal and looking at its zeros, you get the Hausdorff dimension as one of the zeros and a lattice of other zeros. The tube formula is a sum over terms involving those complex zeros of the fractal's zeta.

So a tube formula is about the area of the Minkowski sum of the fractal and a disk (or, one dimension up, the volume of the Minkowski sum of the fractal and a ball). But that's very different from what I'm asking about, which is how a ball rotates if it rolls "without slipping" so that the point of contact with the plane follows the fractal. Now that I think about it, I'm not sure how to phrase the "without slipping" condition in a way that depends solely on rectifiability of the curve. It seems that the derivative is involved. So for a nowhere-differentiable curve like the Koch snowflake, I don't even know how to abstractly characterize the property of the path in the configuration space of a sphere that I'm looking for. But it still makes sense to approximate the snowflake curve by polygonal paths, and we can ask whether the path in sphere-configuration-space converges as the mesh of our polygonal approximation goes to zero, so we can duck the issue of what "no-slip" really means for smooth curves. A related issue is parallel transport of tangent planes. If you have a particle moving smoothly on the surface of the sphere, there's a natural way to carry a coordinate frame along with the particle. So you get a way to identify points in one tangent plane with points in another tangent plane. (But watch out: if you come back to where you started, your coordinate frame will have rotated, according to how much area on the sphere your path has enclosed.) Piecewise smooth motions are okay too. But what about a fractal curve on the sphere? It's not at all clear to me when and how parallel transport would work. Note that, as in my problem, we're trying to shadow one path with another, but this time we're going in the other direction (from the sphere to the plane instead of the reverse). I hope I've made it clear how my question is different from Pearse's. If not, I trust someone will ask me to clarify (or tell me what I'm missing). Jim Propp