Tuesday, 26 August 2014

The Deep Connection Between Julia and Mandelbrot Fractals

The Julia and Mandelbrot fractals are intimately connected.

They are both generated by iterating the simple function z2 + c.

For the Mandelbrot set, z starts as the value of the point being tested on the complex plane, and c is zero. In effect the function becomes z2 because c is zero.

For the Julia sets, c is set to a contant value throughout all the calculations. In thisway, c uniquely defined that particular Julia fractal. That same c always generates that same pattern.

If you've explored the Julia and Mandelbrot sets, you may suspect that there is a connection between the two. In fact there is:

  1. Julia sets whose unqiue c value falls inside the Mandelbrot set are connected - that is, they are all one piece.
  2. Julia sets whose unique c value fall outside the Mandelbrot set are not connected - that is, they consist of many disconnected pieces. 
  3. Julia sets whose c lies further away from the Mandelbrot set have greater fragmentation, until they become almost dust like.

The following digram summarises this deep connection: (click to enlarge)