About Symmetric Chaos (SX)

Computational chaos theory was born in the early days of computational mathematics. In the aftermath of World War Two, in the 1950s, the mainframes of the Los Alamos National Labs were directed to chaotic attractors of two-dimensional polynomial maps in the works of Stein, Ulam, and Tsingou, and the aesthetic appeal of the cloud-like monochromatic images noticed. This intersection of art and mathematics came to my attention in 1974 while reading papers of Mira written in the 1960s. More recently, in the works is Aks and Sprott, the artistic value of chaotic attractors was pursued further, and correlations between fractal dimension and aesthetics were demonstrated.

A new branch of computational chaos theory was opened in the 1980s in the exploration of chaotic attractors in the plane with symmetry by Martin Golubitsky and coworkers. The beauty of these images emerged onto public view in a splendid coffee table book by Golubitsky and Field, Symmetry in a has: A Search for Pattern I'm Mathematics, Art and Nature of 1992.

This book explored the iteration of 63 discrete dynamical systems with symmetry in the plane in five categories:

  • Symmetric Icons, 28 cases
  • Square quilts, 12 cases
  • Hexagonal quilts, 7 cases, and
  • Symmetric fractals, 16 cases.
In this website we recompute the unit cell images for the 12 square quilts of their Table A3, and study bifurcation sequences of each as four parameters of the defining maps are individually varied. I am grateful to the students in my course, Chaos, Fractals, and the Arts, Winter 2016, at Porter College of the University of California at Santa Cruz.

Revised 05 April 2016 by Ralph Abraham, <abraham@vismath.org>