Advanced Perl Programming

A Detour into Fractals

This chapter would be incomplete and dry without a small foray into Mandelbrot sets and the implementation of draw_mandel.

For starters, I highly recommend Ivars Peterson’s book The Mathematical Tourist [Section 18.7] for its engaging style and treatment of a surprisingly wide set of mathematical topics. We’ll begin by assuming that you already know about complex numbers.

We know that a complex number a + b i is composed of two parts, the real part a, and the imaginary part b, that taken together constitute a point on a graph. Now consider the expression z = z ² - 1, where z is a complex number. We start with a complex number (z ⁰) and plot it. We then substitute it in the above expression to produce a new complex number and plot this number. This exercise is repeated, say, 20 or 30 times. We find that different starting values of z ⁰ result either in this series trailing off to infinity, or remaining confined within a boundary. All z ⁰’s that result in a bounded series belong to a Julia set, named after the mathematician Gaston Julia. In other words, if we plot all the z ⁰’s that result in a bounded series, we will see a nice fractal picture (no, not the one we saw earlier).

Now, let us make the equation a bit more general: z ← z ² + c, where c is a complex number (the discussion above was for c = -1 + 0i). Now, if we plot the Julia sets for different values of c, we find that some plots show beautiful connected shapes while other disperse into a cloud of ...