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Mandelbrot set question

  1. Jan 9, 2013 #1
    Assuming that we could interpret the imaginary axis in the complex plane as the output of a relation, how would we find the equation of the curve that bounds the main cardioid of the M-set? Is there a way to find the equation of the main cardioid on a "minibrot" (e.g. if I zoom in on the fractal very deeply and find another quasi-similar M-set)?
     
  2. jcsd
  3. Jan 10, 2013 #2

    kreil

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    Gold Member

    The Mandelbrot set is obtained from the recursion relation,

    [tex]
    z_{n+1} = z_n^2 +C
    [/tex]

    The kidney bean-shaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations1

    [tex]
    4x = 2 \cos(t) - \cos(2t)
    [/tex]
    [tex]
    4x = 2 \sin(t) - \sin(2t)
    [/tex]


    1 http://mathworld.wolfram.com/MandelbrotSet.html
     
  4. Jan 10, 2013 #3
    How would one show/prove that?
     
  5. Jan 11, 2013 #4

    kreil

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