Fractal Theory

 

Newton Sets

The Newton-Raphson iteration method provides a way for finding the root(s) of a polynomial function by using the first derivative of the function to make an approximation. A root is a number which when substituted as a variable in an function makes the value of the function equal to '0'. (i.e. numbers x where f(x)=0) This method makes use of the Taylor series and is discussed in more detail by following the link below to Wolfram Mathworld. This method was first proposed by Sir Isaac Newton and further developed by Joseph Raphson in the late 17th century.

The basic idea is that you can determine a solution for the root of a polynomial function by repeated applications of the formula x(n+1) = x(n) - f(x) / f'(x). After a number of iterations, the value x quickly converges to the approximate root value. This can be generalized to complex numbers by substituting the complex number z for the real number x which changes the equation to z(n+1) = z(n) - f(z) / f'(z). Z is a complex number, f(z) and the first derivative f'(z) are complex functions. It looks similar to the iterative concept used to create the Mandlebrot and Julia sets, doesn't it?

Click to enlargeThe Newton set is defined by grouping the numbers Z in the complex plane which are found by repeated applications of the equation. The fractal to the left is a fifth order Newton set using the function f(z) = z^5-1. The derivative of this function is f'(z) = 5z^4. Therefore substituting in the formula we get z = z - (z^5-1) / (5z^4). The iteration of this formula generates a set which when plotted creates the fractal to the left. For those of you who wish to try to recreate this fractal, enter this equation into Fractal Explorer and play with the parameters.

The important issue for us here is that the Newton-Raphson equation is iterative, similar to the Mandlebrot and Julia set equations, and lends itself to generating fractal images. As with Mandlebrot and Julia set fractals, these fractals allow you to zoom into the image and see detailed pieces of the image repeated indefinitely.

Click to enlargeClick to enlarge

There are different Newton sets that you may create by changing the exponent in the equation. For example, suppose we define a seventh order Newton function, f(z) = z^7-1. Here the first derivative of f(z) is f'(z) = 7z^6. When substituted in the Newton formula we get z = z - (z^7-1) / 7z^6. Upon iteration this equation generates a set of points that when plotted create the fractal to the left. By adjusting the filters and properties using Fractal Explorer, I created the image to the right from the same function.

This discussion only scrapes the surface of fractal theory, but I hope that it gives you some insight into the mathematical basis for the images formed. For more detailed information on fractal theory, see the links below.

Fractal Geometry - Michael Frame, Benoit Mandelbrot, and Nial Neger (Yale University)

Fractals and Fractal Geometry - Thinkquest.org

Introduction to Fractals - Written by Paul Bourke

Newton's Method - Wolfram Mathworld

 

Page 2 <-Back