It’s hard to find a fractal more famous than the Mandelbrot set. It’s one of a class called Iterated Function Systems, which unsurprisingly all revolve around repeatedly applying functions to points and analyzing the output. Here, those points are in the complex plane, and the function is \(f(z) = z^2 + c\), where \(c\) is the initial value of the point. For example, the point \(i\) gives \(f(i) = i^2 + i = -1 + i\) when run through the function, and then \(-1 + i\) gives \(f(-1 + i) = (-1 + i)^2 + i = -i\). If the numbers in this sequence stay small forever, then the point is in the Mandelbrot set, and we color it black. If not, then we color the point according to how slowly its sequence grew in size — bright for slow and dark for fast.
Of course, the Mandelbrot set is most well-known for the tiny features only noticible by zooming in. Hover over the Mandelbrot set to view that area up close, and click to generate a higher-resolution image.