Ron Mak @ Apropos Logic

ron@apropos-logic.com
Publications and books by Ronald Mak

Java Number Cruncher

Program 16-3

Newton's Fractal Image

This applet demonstrates how Newton's Method, when applied to the complex plane, can generate a fractal image. The function f(z) = z^3 - 1 has three roots in the complex plane: 1, -0.5+0.87i, and -0.5-0.87i. The applet plots each point in the plane as follows:

  1. Apply Newton's Method using that point as the starting point.
  2. Set the point's color (red, green, or blue) according to which root the algorithm converges to from that point.
  3. Set the color intensity of the point according to the number of iterations required to converge to the root.

The resulting graph is a fractal image.

To run the demo:

  1. The initial fractal image is generated automatically.
  2. Use the mouse to drag a rectangle around an area of the image that you want to zoom into.