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 +
0*i*). 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 ...

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