How do you find the equation of the main cardioid in the Mandelbrot set?

[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)?

 

[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

 

[pierce15]

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

How would one show/prove that?

 

[kreil]

http://cosinekitty.com/mandel_orbits_analysis.html