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?
2026-03-28 20:51:04.1774731064
Mandelbrot Set Main Shape
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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”:
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.