Applications of Fractals

Newton’s Method

Besides exotic applications in nature, fractal have important mathematical applications as well. One of them is in the analysis of the Newton’s method, which is used for approximating roots of an equation. For example, it can be used to solve the equation xn = 1, where n is some positive integer. One of the roots of this equation is always 1, and if n is even -1 is also one of the roots. The rest of the roots are complex numbers. If you graph these numbers on a complex plane, you will see that they are equally spaced on the unit circle. The Newton’s method is based on making a guess of the root and then iterating a certain formula to change that number (it is the method used by computers). The number you will end up with is the root that is the closest to the guess... unless you are unlucky enough to pick a number that is equidistant from two different roots. In such case the number will jump around chaotically — and create fractal shapes if you graph it (see chaos and fractals). The fractal will have a different number of "chaos lines" depending on the number of roots (the value of n determines the number of roots). For n=3, for example, the fractal formed is the following:

Notice the circles between every chaos line. Those are the approximate locations of the roots. This figure is indeed fractal, since magnifying it will give us clearly self-similar shapes:

As a different example, let’s draw the fractal for n=10 and use nicer colors: