Newton's Fractal

Newton never knew that he invented a fractal. Recently, though, a fractal was discovered in Newton's method. Newton's method is a way to find the root(zero) of a function given the function, its derivative, and an initial guess. There may be more than one root to a function. If that is the case, the initial guess will determine which root the method will find. The fractal is found in the relationship between the initial guess and the final answer.

This applet shows the fractal created when trying to find the cube root of 1. (By using the pull-down menu, you can set it to find the 4th, 5th, 6th, or 7th root of one, or one of four polynomials, creating a different fractal.) If you use complex numbers, there are three correct answers to this problem. For each point on the complex plane, the color is set according to which answer is found by using that point as the initial guess. The stripes are determined by how many iterations of newton's method are required to get an answer. The black areas are areas where the calculations have not yet been finished.

I came up with four polynomials to create more interesting, often asymmetrical, images. They are listed below.

  1. (x+1)(x-1)(x+1+i)(x+1-i)
  2. x(x-1)(x-i)(x-i+1)
  3. (x+1)(x-1)(x+1+i)(x+1+i)(x+1-i) This one is like #1, except that it has a double root. This makes it more asymmetrical.
  4. x^5+4x This one has the shape of a cross.
Click to zoom in, Shift-click to zoom out. Be patient with this applet; it is the most demanding applet I have on this site in terms of calculations and memory usage.
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