I was reading MarkCC’s gripe about the misuse of the word dimension and it reminded me about the Hausdorff dimension.  Generally speaking, the Hausdorff dimension matches our normal intuitive definition of dimension, i.e the H-dimension of a smooth curve or line is 1, the H-dimension of a smooth surface is 2, the H-dimension of the interior of a cube is 3. But for fractals, the H-dimension can be a non-integer. For the Cantor set, the H-dimension is 0.63. For more information, check out the Wikipedia article

http://en.wikipedia.org/wiki/Hausdorff_dimension

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