Sunday, February 2, 2014

The Mandelbrot Set (Chaos Part II)

In a previous post, we discussed chaos from a mathematical perspective. In everyday life we use the term "chaos" when events are unfolding in an unpredictable manner. In mathematics, functions and algorithms can exhibit chaotic behavior when it is hard to predict the output given the input. This is the same as saying two input values can have dramatically different output values, even when the input values are nearly the same.

A famous algorithm that exhibits chaotic behavior is the following. Given a complex number \(c\), let \(z_0=c\), \(z_1 = z_0^2+c\), \(z_2 = z_1^2+c\), etc. If you start with \(c=0\), the resulting sequence is rather boring: all \(z_i=0.\) If you start with \(c=1\), you get \(z_1=2\), \(z_2=5\), \(z_3=26\), etc. This sequence will contain very large numbers very quickly.

Other sequences show interesting patterns. With \(c=i\), we have \(z_1=-1+i\), \(z_2 = -i\), \(z_3=-1+i\), \(z_4=-i,\ldots\) This pattern repeats forever.

The Mandelbrot Set is the set of all complex numbers \(c\) for which the above algorithm does not result in numbers approaching \(\infty\). So \(0\) is in the set, as is \(i\). From what we saw above, the number \(1\) is not.

We can visualize the Mandelbrot Set by coloring all the points in the complex plane that are in the set white, leaving the rest black. That is what we see in the animation above. The animation also illustrates one of the amazing - and chaotic - properties of the Mandelbrot Set. Some values are very, very close together where one approaches \(\infty\) and the other does not: one value is not in the Mandelbrot Set, and the other is.

Another amazing thing about the Mandelbrot Set is its self similarity. The animation zooms in on a region that looks "black." That is, on a region that seems to contain only points not in the set. However, as we zoom in, we begin to see that some of the points are in the set. And they seem to be grouped in a way that looks awfully similar to the original image.

So two nearby points can act very differently. And points in the set cluster in groups that look just like each other. That's chaos. And that's awesome.

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