The Mandelbrot set, generated by iterating the map f(z)=z2+c, looks something like this, when colored by the number of iterations to blowup:

Corresponding to each point in and around the Mandelbrot set is a Julia set, created by a similar iteration. In fact, a point is in the Mandelbrot set if and only if its Julia set is connected. If we start with a point near the origin, and move it in a circle around the main bulb of the Mandelbrot set, the series of corresponding Julia sets morph beautifully into one another, slowly completing a counter-clockwise rotation.