16-3 Newton’s Fractal Image

This program 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 program 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.

The original Java Applets do not run in modern browsers, but you can download the Java code and run it locally on your desktop. Check back for JavaScript and Python versions - coming soon!


Download Demos in Java