A class of fractals known as Mandelbrot sets, named after Benoit Mandelbrot, have pervaded popular culture and are now controlling us. Well, perhaps not quite, but have you ever wondered how they are drawn? Here is an approximation of one:

From now on, *Mandelbrot set* will refer to the following set: for any complex number $ c$, consider the function $ f:\mathbb{C}\to\mathbb{C}$ defined by $ f_c(z) = z^2 + c$. We define the Mandelbrot set to be the set of complex numbers $ c\in\mathbb{C}$ such that the sequence of numbers $ f_c(0), f_c(f_c(0)),f_c(f_c(f_c(0))),\dots$ is bounded.

This is just a special example of a much more general class of functions and fractals. To quickly draw the Mandelbrot set, for each $ c$, compute a fairly large number of iterates of $ f_c$ applied to $ 0$. If the numbers in the sequence are larger in modulus than some fixed large value $ r$, then probably $ c$ is not in the Mandelbrot set. Otherwise, it probably is. We say “probably”, because we are just using numerical evidence to guess whether $ c$ is in the Mandelbrot set.

If we chose $ r^2 = 50$, how many iterations would we need to draw a good picture of the Mandelbrot set? Here is an animation, showing how the drawing improves as the iterations get larger (the number of iterations (iter), as well as the number of points sampled on each axis (prec) is shown at the top):

Doesn’t it look like a cloud of particles are taking a new strange form?