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Symmetric Fractals


Table Of Contents

Peter de Jong Map

The symmetric fractals featured on this page were inspired by the book "Symmetry in Chaos" by Michael Field and Martin Golubitsky. All of the fractals here are generated from the same equation:

Symmetric Fractals Equation

where the following are constants (n is an integer and the rest are real):

Symmetric Fractals Parameters

and the iteration begins with:

Symmetric Fractals Initial Conditions

Just as with the Peter de Jong attractors, we look at the probably density function, P(x,y), defined by the probably that some (x_n, y_n) from the map will be in a neighborhood of (x,y). While that sounds complicated, producing a visualization of this PDF is quite simple in practice -- we simply visualize a histogram based on computed values of the map. The first step is to define a rectangular region of the plane over which we wish to visualize the PDF, and then define a set of discrete pixels over that region. Then we compute a few million, or billion, iterations of the map, and count the number of times we hit each pixel. That amounts to just a few lines of code in a sufficiently friendly language like Processing!


Here are a few examples. Most of these were discovered via an automated search program using the technique briefly outlined in the next section. When you draw one of these images with a random set of parameters, the odds are that you are the first human being ever to see that precise image -- with five, 32-bit parameters the odds of someone else picking the same parameters is 1 in 2^160.

sic 0 sic 1 sic 2 sic 3 sic 4 sic 5 sic 6 sic 7 sic 8 sic 9 sic 10 sic 11 sic 12 sic 13 sic 14 sic 15 sic 16 sic 17 sic 18 sic 19 sic 20 sic 21 sic 22 sic 23 sic 24 sic 25 sic 26

Image Generation: Practical Matters

This fractal system presents most of the same practical problems we encountered with the Peter de Jong attractors, but with a few extra wrinkles:

  • The image processing pipeline is both more sophisticated, and varies for each set of parameters.
  • We can't always start with a fixed region of the plane, so this too varies with each set of parameters.
  • More hand work is required when evaluating the results of automated parameter searching -- the highest number of turned on pixels may not be the most interesting image..

Source Code

The source code used to find the parameters and generate the images on this page is distributed with the mraster library as examples. You can find the example source code files on git hub: sic.cpp and sic_search.cpp.


For a very interesting introduction to the history of Chaos I would suggest James Glick's book: "CHAOS: Making a New Science" (ISBN: 0-14-009250-1). This book has almost no mathematics or formal material regarding Fractals or dynamical systems, but is well worth the read for the historical perspective.

If you are looking for a good introduction to dynamical systems, I would suggest Steven Storgatz's book: "Nonlinear Dynamics and Chaos" ISBN 0-7382-0453-6).

For a good, encyclopedic, introduction to the field in general, I strongly suggest "Chaos and Fractals: New Frontiers of Science" by Peitgen, Jurgens, and Saupe. This book is notable because of it's clear presentation and breadth of coverage. It's a great book to have around for the casual reader because it is broken up into semi self-contained sections that one can just pick up and read.

Symmetry in Chaos by Michael Field and Martin Golubitsky is a great book. The authors have some software that can draw the fractals on this page (and many others). Check out the web page!

© 2009 Mitch Richling