Mandelbrot set question

1. Jan 9, 2013

pierce15

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. Jan 10, 2013

kreil

The Mandelbrot set is obtained from the recursion relation,

$$z_{n+1} = z_n^2 +C$$

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

$$4x = 2 \cos(t) - \cos(2t)$$
$$4x = 2 \sin(t) - \sin(2t)$$

1 http://mathworld.wolfram.com/MandelbrotSet.html

3. Jan 10, 2013

pierce15

How would one show/prove that?

4. Jan 11, 2013