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What exactly is the importance of the Mandelbrot set in general?

From what I've read, it seems more of a mathematical play thing than anything else.. there must be more to it than the disturbing pictures, no?

Also, is there an easily understandable proof anywhere showing that the boundary of a Mandelbrot set is a fractal?

I know Shishikura proved it in the 90s, but I haven't been able to find his proof, nor do I believe I would have the ability to decipher it without an intermediary source breaking it down a bit. Maybe even just an explanation of his process?

Thanks for any assistance!

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# (any) Applications of Mandelbrot sets? Proof of fractal?

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