Fractals are something I find really interesting, but their not exactly solvable, they tend to be infinite and a computer is needed to solve any decent amount of them. So, show me you fractal solving bash scripts! (Actually I’m looking for a good free software fractal renderer. XaoS is all I can find.)

#### The Mandelbrot set

lets define w as some complex number.

lets define the set z as:

z_0 = w, z_1 = z_0^2 + w, z_n = z_(n-1)^2 + w

For what values of w does z converge? An number w where its set z converges to |z| <= 2 is part of the Mandelbrot set.

These points can be rendered black on a plot of the Mandelbrot set, any number w where its set z leaves the area of radius 2 is colour coded by the number of iterations it takes to leave that area. This is what leads to the fractal pattern.