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What is the dimension of the Mandelbrot Set?

April 26, 2017dimension Mandelbrot

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What is the dimension of the Mandelbrot Set?

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A6j: The Mandelbrot Set has a dimension of 2. The Mandelbrot Set contains and is contained in a disk. A disk has a dimension of 2, thus so does the Mandelbrot Set.The Koch snowflake (Hausdorff dimension 1.2619…) does not satisfy this condition because it is a thin boundary curve, thus containing no disk. If you add the region inside the curve then it does have dimension of 2.The boundary of the Mandelbrot set and the Julia set of a generic c in M have Hausdorff dimension 2 and have topological dimension 1. The proof is based on the study of the bifurcation of parabolic periodic points. (Since the boundary has empty interior, the topological dimension is less than 2, and thus is 1.