This chapter would be incomplete and dry
without a small foray into Mandelbrot sets and the implementation of
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
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 is a complex number.
We start with a complex number
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
0 result either in
this series trailing off to infinity, or remaining confined within a
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
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:
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 ...