Most of us have discovered by now how useful root-finding algorithms can be in rendering surfaces to create images.  Newton’s root-finding method, however, is also often used to create beautiful and varied images by generating fractals.

            Fractals, made famous by Benoit Mandelbrot, are mathematically generated images with patterns of self similarity that cannot be described by normal geometric shapes.  Since they are of infinitely fine shape, the closer you look at a fractal, the more detail you see of the repeating edge pattern.  The mathematical relationship of self-similarity can be particularly well-demonstrated on a computer, and in the case of fractals, this often done by generating an image with different colors representing different characteristics of the function.  As such, fractals are a more public accessible aspect to the intricacies of mathematics and computing.

There are many sets of fractals such as the Mandelbrot set and the Phoenix set, but Newton fractals are particularly interesting to us since, through our study of numerical analysis, we thoroughly understand the algorithm that generates them.  They are actually dependent on the fact that the Newton technique of root finding can be very sensitive to the starting point before iterating through the method.  For a given function in the complex plane, often a polynomial, the Newton fractal is created by finding which initial points converge to each specific root using Newton’s method. The points are colored differently based on which root they go to so that the graph is split into multiple basins of attraction, or regions where the point converge to the same route.  For instance, the polynomial z^4-1 has 4 roots: 1, -1 , i, -i, so there would be 4 different colorings designating which point the polynomial converges to in the complex plane.  This is the rather ordinary and symmetric looking type of example that is shown in the applets below, but an interesting fractal can created by just about any polynomial with a couple of complex roots.  From looking at these fractals, one can see another important property of Newton’s method in regards to numerical analysis: there are always some regions, usually colored black, that don’t ever converge to a root.  The boundaries of the basins of attraction are an example of one of these regions. This is discussed briefly in the text (Numerical and Mathematical Computing by Ward Cheney, David R. Kincaid, 99-100).

Overall, Newton fractals are a very interesting application of Newton’s method. They demonstrate some of the method’s important characteristics: initials point dependency, and lack of convergence for some cases, and they provide a glimpse into the further mathematical intricacies that Newton’s method has to offer.

For a general explanation of fractals:

http://en.wikipedia.org/wiki/Fractal

More specific information on Newton fractals:

http://www2.vo.lu/homepages/phahn/fractals/newton.htm

http://en.wikipedia.org/wiki/Newton_fractal

Applets that helpfully demonstrate qualities of Newton fractals:

http://www.geocities.com/capecanaveral/hangar/7959/newtonapplet.html

http://www.apropos-logic.com/nc/NewtonsFractal.html

http://www.math.umn.edu/~jhall/Newton5.htm