Computational Diversions: Julia Variations

Abstract

One of the favorite display shelves in every recreational programmer’s cabinet of curiosities belongs to the Julia sets. Julia sets, named after the French mathematician Gaston Julia, have a variety of equivalent mathematical definitions; they are also remarkably easy graphical entities to generate with a minimal degree of programming. Here’s the basic idea: suppose you have a complex polynomial function f(z). (A good example-to-think-with is the quadratic function f(z) = z 2 ? c, where c is some constant complex number.) If you iterate f(z) a very large number of times, starting from a particular initial value z0, and follow the values of the successive points under iteration, you will find that the sequence of complex number values either becomes arbitrarily large (goes to infinity), or it doesn’t. So some starting z0 points ‘‘go to infinity’’ and some don’t. Plot the values of z0 whose sequences go to infinity in one color (say, white); and those whose sequences don’t in another (say, black). The black points constitute the filled Julia set of the function f, and the boundary between black and white is the Julia set. Given this definition, you might imagine that these sets are of interest only to a dedicated subculture among mathematicians; but the graphical plots of the Julia sets are in fact remarkably appealing, and form an easily-programmed class of fractals for computer programming students. Here’s the general algorithm (written in pseudocode) through which you can program a Julia set for a quadratic polynomial: