Here's another way to clarify my question: Consider a level-zero snowflake curve (a path of length 1) and a level-one snowflake curve (a path consisting of four segments of length 1/3). If we roll a ball along the former, (with its south pole starting facing downward), the final orientation of the ball is NOT THE SAME as if we roll the ball along the latter (also with the south pole starting facing downward). (Actually, I suppose that if one chooses the radius cleverly, the orientation of the ball *could* be the same, but for generic radii, the ball will be in a different orientation.) If we replace the level-one curve by a level-two curve (consisting of sixteen segments of length 1/9), the final orientation of the ball could once again change. And so on. Is there any reason to think that this sequence of orientations of the ball should converge to some particular orientation in the configuration space of the ball (where we mod out by translation --- all we care about is rotation --- so we're really looking at the configuration space of a sphere)? And even if it does converge, is there any reason to think that a different sequence of polygonal approximations to the snowflake curve would give the same limit? Hopefully this is clear. So this I why I don't think the Pearse references are relevant. But thanks anyway to Mike for sending his message and "getting the ball rolling", as it were. :-) Jim Propp