Although chaos would seem to be the antithesis of symmetry, it is possible to find elaborate symmetric patterns in chaotic processes. Shown here is a hexagonal pattern similar to those that sometimes appear on quilts. This image is produced by iterating a map with hexagonal symmetry. Although the iteration possesses all the features of chaos, a surprisingly detailed and intricate symmetric pattern appears when we take a very large number of iterations -- in this case about six billion -- and color each pixel according to the number of times it is visited during iteration. In this sense we are computing an averaged picture. The colors themselves were chosen for their artistic effect. Note that the detail inside each of the hexagons happens to resemble a flower. The fractal quality of the picture is to some extent represented by the reappearance of the hexagonal symmetry on ever-decreasing scales.

In the first picture below [not shown], we show the center of one of the flowers magnified, in area, by a factor of about 400. (The colors used are different from those in the main picture). In the second image, the magnification is about 10,000 and hundreds of hours of cpu time on several high performance workstations were required to compute it.