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# 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 ...

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