Tuesday, March 9, 2010

On Fractals, Part 2

Recently I and a friend concurrently stumbled upon this article on how an adaptation of the root-finding algorithm Newton's Method led a mathematician (the co-author of my math text from last year, actually) to discover a fractal pattern in the distribution of which starting points led to which roots of a function. A quick explanation:

Newton's Method is an algorithm invented by physicist, mathematician, alchemist, and all-around genius Isaac Newton for finding roots of a function--places where it equals zero. It's quite simple, and though it has some drawbacks, when it finds roots it finds them incredibly quickly. As my CSci 2031 professor won't let us forget, it's the basis for a lot of numerical answer-finding algorithms today since it's so easy for computers to perform. But it only finds at most one root, so one run of the method isn't sufficient for a function with multiple roots.

The fractal in the first link was generates by testing a variety of points in the complex plane with an adaptation of Newton's Method that apparently works with complex numbers (I'm fuzzy on the details). The function used was f(z) = z^3 - 1, which has only one real root (1) but two more complex ones. Points were colored red, blue, or green depending on which of the three roots Newton's Method found starting from that point, resulting in an infinitely complex, self-similar pattern--a fractal. Not as complicated as the Mandelbrot set since it mostly has one pattern that repeats, but still cool. I don't know why complex functions so often tend to produce mind-blowing patterns like this, but it's one of my favorite math mysteries.

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