MandelbrotMengen

The MandelbrotMengen, commonly called the Mandelbrot set, is a set of complex numbers c for which the quadratic polynomial f_c(z) = z^2 + c, iterated from z_0 = 0, yields a bounded orbit. In practical terms, the sequence z_{n+1} = z_n^2 + c remains bounded. The set is compact and connected in the complex plane, and its boundary is a famous fractal.

For a given c, the corresponding Julia set J_c has the property that J_c is connected if and only if c lies in the MandelbrotMengen; if c lies outside, J_c is a Cantor set. The MandelbrotMengen exhibits a rich internal structure: a large main cardioid, with infinitely many circular bulbs attached at various locations, corresponding to periodic dynamics. Points in the interior of the set belong to hyperbolic components where the orbit tends to an attracting cycle.

The boundary is highly intricate and is studied as a paradigmatic example of a fractal. The exact Hausdorff dimension of the boundary is unknown, though it is believed to be very close to 2. The set is connected, but the question of local connectivity (the Mandelbrot set being locally connected) remains an open problem known as the MLC conjecture.

Historically, the concept was developed in the 1980s through work by Benoit Mandelbrot and by Adrien Douady and John Hubbard. In visualization, the escape-time algorithm colors points by how quickly their orbit escapes, revealing the characteristic cardioid form and self-similar features upon zooming.