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Unformatted text preview: Some Fractals and Fractal Dimensions • The Cantor set: we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefnitely. • To calculate the fractal / Hausdorff / capacity / boxcounting dimension, we see how many boxes (circles) of diameter 1/r^n we need to cover the set (in this case, we will use r = 3, since it Fts nicely). D = Lim(log(N r )/log(1/r)) = log(2) / log(3) r N r 1 1 1/3 2 1/3^2 2^2 1/3^3 2^3 1/3^n 2^n • The Koch snowfake: We start with an equilateral triangle. We duplicate the middle third oF each side, Forming a smaller equilateral triangle. We repeat the process. • To calculate the fractal / Hausdorff / capacity / boxcounting dimension, we again see how many boxes (circles) of diameter (again)1/3^n we need to cover the set. D = Lim(log(N r )/log(1/r)) = log(4) / log(3) r N r 1 3 1/3 3 * 4 1/3^2 3 * 4^2 1/3^3 3 * 4^3 1/3^n 3 * 4^n • The Sierpinski carpet: We start with a square. We remove the middle square with...
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 Fall '09
 Carter
 Differential Equations, Equations, Cantor, Fractal, Hausdorff, positive Lyapunov exponent

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