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I was curios about Hausdorff dimensions. They seem to neatly describe rough surfaces. So I was wondering if there are common applications of Hausdorff dimensions in things like complicated friction research? I have a little bit of a desire to learn specifically about formation of cracks in surfaces, which seem as though friction and perhaps a little chaos might have an intuitive factor in their formation.

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Fractal dimensions are being used to describe the structure of exhaust gases, like bunches of soot. –  NikolajK May 3 '13 at 7:43
    
Fractal dimensions are also used to describe internittency in turbulence theory sometimes. –  Dilaton May 11 '13 at 8:06

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Fifteen years ago, there was much research interest in the applicability of fractals to the description of fractures in rocks. In particular, as a mathematical technique for modeling fracture populations, fracture networks, or fracture surfaces in systems exhibiting brittle-failure, fluid flow through porous rock, and sliding friction. I wouldn't say any of the applications became 'common' however, you probably still find most references to this in academic papers. Rather than searching for the term Hausdorff dimensions, I suggest searching for some combination of "fractal" and "fracture." At the time, much of this work was motivated (paid for) by interest in geothermal energy extraction and development.

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