What is the exact shape of the main component of the Mandelbrot set? I’m referring to the heart-shaped area centered at at the origin. Is there a simple way to express this shape in Cartesian or polar coordinates?
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See https://sites.google.com/site/fabstellar84/fractals/cardioid and https://math.stackexchange.com/questions/1341171/why-does-the-boundary-of-the-mandelbrot-set-contain-a-cardioid – lhf Jul 02 '20 at 14:39
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It's a cardioid. The Wikipedia article on the Mandelbrot set explains this in some detail, and even has a section titled “main cardioid”:
Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid is the region of parameters $c$ for which $P_c$ has an attracting fixed point. It consists of all parameters of the form $$c = \frac\mu2(1-\frac\mu 2)$$ for some $\mu$ in the open unit disc.
The answers at Why does the boundary of the Mandelbrot set contain a cardioid? have a presentation of the cardioid's boundary that may be easier to deal with.
MJD
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