The Mandlebrot set is a (very important) fractal. A fractal is basically a shape containing smaller copies of itself ad infinitum, usually generated by repeating some simple rule. For the Mandlebrot set the rule is: for every complex number $c$ repeat the operation \[z_{n+1} = z_n^2 + c\] if the size of $z_n$ doesn't tend to infinity then $c$ is in the set, otherwise it isn't. Of course if we ask a computer to do the iterations we can only ask it to do a finite number. We colour the point differently depending on how many iterations it takes to discover if the number is in the set or not up to some number $N$. If we reach $N$ without diverging then we colour the point black and say, to the best of our knowledge, the point is in the set.
There is a lot of fancy maths in this area, but I think that people mainly like making cool pictures, me included. So here is a Mandlebrot set for you to zoom in on. Try and find something nice by clicking on the picture and dragging the red box, look at the edges for interesting swirls. This will go very slowly for a large number of iterations $N$, so be careful you don't put it too high and crash your browser.
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