The Mandelbrot Set: Ordering the Julia Sets

Abstract

The Mandelbrot set is certainly the most popular fractal, probably the most popular object of contemporary mathematics of all. Some people claim that it is not only the most beautiful but also the most complex object which has been seen, i.e., made visible. Since Mandelbrot made his extraordinary experiment in 1979, it has been duplicated by tens of thousands of amateur scientists around the world.²

In the Mandelbrot set, nature (or is it mathematics?) provides us with a powerful visual counterpart of the musical idea of ‘theme and variation’: the same shapes are repeated everywhere, yet each repetition is somewhat different. It would have been impossible to discover this property of iteration if we had been reduced to hand calculation, and I think that no one would have been sufficiently bright or ingenious to ‘invent’ this rich and complicated theme and variations. It leaves us no way to become bored, because new things appear all the time, and no way to become lost, because familiar things come back time and time again. Because this constant novelty, this set is not truly fractal by most definitions; we may call it a borderline fractal, a limit fractal that contains many fractals. Compared to actual fractals, its structures are more numerous, its harmonies are richer, and its unexpectedness is more unexpected.

Benoit Mandelbrot¹