Jim Propp originally asked:

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?

Oh, you mean rolling a *ball* *across* a plane!! Like rolling it on a table. Not rolling a circle around the outside of the curve. (Nor rolling a ball over the surface of a sponge.) We are talking about curves rather than dust, right? All the fractal curves I have heard about have details that get smaller as you approximate them more closely. If not, I don't know what you have.

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?

So the Q is whether rolling the ball along the curve approaches (for small details) stamping it like silly putty on newsprint. Think of rolling the contact point of the ball around a tiny circle over and over. The ball probably slowly precesses. Maybe the precession could "unwind" the snakings of a non-rectifiable curve so that although it converged on the plane it would diverge on the sphere? Or maybe just the rotation of the sphere could diverge even if the distance it rolled would converge. Or maybe they both have to converge no matter what (finite) dimension of the curve. Example: Start with a regular clockwise polygon and replace the sides with clockwise epicycles nearly as big as the original. But, to continue talking about the question Jim *didn't* ask,

Mike Stay wrote:

...

Well, Erin Pearse wrote all over *about* the question: http://arxiv.org/abs/math-ph/0412029 http://arxiv.org/abs/math.DS/0605527 http://www.math.cornell.edu/~erin/dissertation/dissertation.pdf

Okay, beginner questions. Looking at the dissertation, I was confused about Pearse's meaning of "convex hull", which I thought was an outline. The outlines of all his examples were simple polygons. For instance, the Koch curve's outline is a triangle, the Koch snoflake's is a hexagon, etc. To me, those are the hulls and there's nothing fractal about those hulls. There are definitely fractals whose outlines are also fractals, he just doesn't use any of them as examples. --Steve