Mandlebrot Sets

It’s the the set of all complex numbers z for which the sequence defined by the iteration :

z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, ...

remains bounded. This means that there is a number B such that the absolute value of all iterates z(n) never gets larger than B. A bounded sequence may or not have a limit.

For example, if z=0 then z(n) = 0 for all n, so that the limit of the (1) is zero. On the other hand, if z=i ( i being the imaginary unit), then the sequence oscillates between i and i-1, so remains bounded but it does not converge to a limit.

Treating the real and imaginary parts of z as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence crosses an arbitrarily chosen threshold, with a special colour (usually black) used for the values of z for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement).

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

Mandlebrot Set

For a more better Mandlebrot visualization,

Code : https://github.com/Spockuto/Mandelbrot

Live demo : https://vsekar.me/Mandelbrot/