Re: [math-fun] rolling a ball along a fractal
Mike Stay wrote:
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
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
Can someone who knows more geometry than I do explain the connection between my question and Pearse's work? (I've taken a look at the links Mike provided, but I don't know how things like tube formulas and complex dimensions are pertinent to questions about shadowing a path in the plane with a compatible path in the configuration space of a sphere.) Jim Propp
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. On Mon, Aug 4, 2008 at 1:50 PM, James Propp <jpropp@cs.uml.edu> wrote:
Mike Stay wrote:
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
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
Can someone who knows more geometry than I do explain the connection between my question and Pearse's work? (I've taken a look at the links Mike provided, but I don't know how things like tube formulas and complex dimensions are pertinent to questions about shadowing a path in the plane with a compatible path in the configuration space of a sphere.)
Jim Propp
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Mike Stay