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Exploring Symmetry in Chaos

By Michael Field and Martin Golubitsky

The much-anticipated second edition of Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature, by Michael Field and Martin Golubitsky, was published by SIAM in 2009. It features numerous new illustrations, addresses recent progress in the mathematics that underlies symmetric chaos, and serves as a follow-up to the first edition, which was released by Oxford University Press in 1992. 

The book is written for a general audience and illustrates the ways in which classical symmetry and modern chaotic dynamical systems can interact to produce a set of striking images. It explains the relevant mathematical background, provides a detailed description of how the images are produced, and describes several implications of the mixture of symmetry and chaos. 

The following excerpt begins a discussion of the production process for the many images of symmetric chaos. This text comes from chapter 1, “Introduction to Symmetry and Chaos,” and is modified slightly for clarity.

Pixel Rules 

Figure 1. Pixels on a \(10 \times 10\) grid.
If we intentionally confuse pixels on the screen with points (in the plane), then the rules that make our pictures are similar to the arithmetic rules we use. However, unlike the doubling rule, we do not want our rules on repeated application to grow without bound (otherwise, points would soon leave the computer screen). We think of a pixel rule as a rule that has pixels as input and pixels as output. The pixel rule may depend on some complicated mathematical formula, but for the moment we wish to keep the arithmetic hidden. To make a black and white picture, we assume that the screen is black. We choose one pixel and turn it on — the corresponding point on the screen will then be white. Then we invoke our pixel rule, beginning with the first pixel as input, and obtain a new pixel that we turn on. Finally, we repeat this rule over and over again until we decide to stop. The whole process is called iteration. In this scheme, there is no reason why one pixel cannot be visited more than once.

As an example of a very simple pixel rule, choose one pixel from the screen, say the top left. We define a pixel rule by requiring that we always select the top left pixel as output, regardless of the pixel we choose from the screen as input. However many times we apply the rule, we never see more than two pixels lit on the screen: the initial pixel and the top left pixel.

Next we look at a slightly more complicated pixel rule. Following Figure 1, suppose that the monitor screen has 100 pixels arranged in a \(10 \times 10\) grid. Choose a pixel \(\mathcal{P}\) from the screen and the direction left. The pixel rule has two parts: if you can, move one pixel in the direction you are going; if you cannot, turn right one quarter of a turn. The picture that will result from this pixel rule is easy to describe. There is an initial segment moving left from the initial point \(\mathcal{P}\) to the boundary of the grid, followed by a never-ending circumnavigation of the boundary in the clockwise direction (see Figure 2).

Figure 2. Dynamics on the pixel grid.
Even though the rules we describe here are rather simple, there are one or two interesting features that we want to single out for special mention.

First of all, note that the first part of the pixel sequence is different from its long-term behavior. In particular, the pixels on the initial line segment—labelled \(L\) in Figure 2—are never revisited. We say that this part of the pixel sequence represents the initial behavior and often use the term transient to describe the initial behavior. The transient behavior is seen at the beginning but not in the long term. The part of the pixel sequence that begins at the boundary represents the long-term behavior.

A second important observation about this example is that the long-term behavior repeats ad infinitum. We refer to this characteristic as periodicity. Since there are 36 pixels on the perimeter, this pixel rule repeats itself every 36 iterates (ignoring the initial transient).

Indeed, if we apply any pixel rule enough times, at least one pixel will eventually be revisited. To see why this is so, suppose that there are 100 pixels on the screen (any large number will do equally well). After 100 iterates we have “lit” 101 pixels, so at least one pixel must have been “lit” twice (this argument is an example of the pigeonhole principle: if 101 letters are to be put in 100 pigeonholes, at least one pigeonhole must contain at least two letters). It follows that if the rule we used to create our pictures was actually a pixel rule, then after an initial transient we would have to find periodic behavior. In general, our picture rules do not lead to this simple kind of periodic behavior, and color can be used to understand this point.

Coloring by Number

Figure 3. Emperor’s cloak: pentagonal symmetry and a color bar.
We now say more about how we color our figures. The basic idea is quite simple. Start with a mathematical formula that generates a picture, such as Figure 3. Choose an initial point and apply the rule a large number of times, typically between 20,000,000 and 100,000,000. Ignore the transient part of the pixel sequence that is produced (in practice, we only count pixel hits after the first 1,000 applications of the rule). Record the number of times each pixel is hit and color the pixel according to the value of that number. This process is no more than coloring by number. The actual colors are chosen according to which colors best bring out the underlying structure. Figure 3 shows the result of coloring a figure with five-fold symmetry after 667,000,000 iterations on a \(3,000 \times 3,000\) pixel grid. Since there are 9,000,000 pixels, it follows by the pigeonhole principle that some pixels must have been hit more than once. In practice, many pixels are hit more than once; the color band in Figure 3 shows the colors assigned to pixels based on the number of times they have been hit. As we usually do, we leave the pixel black if it has not been hit. We color white shading to yellow if the pixel has been hit between one and 10 times, yellow if the pixel has been hit between 11 and 30 times, yellow shading to red if the pixel has been hit between 31 and 270 times, and so on, ultimately ending up with navy blue if the pixel has been hit at least 2,370 times (the maximum number of hits on an individual pixel was 42,534).

Thus far, we have confused pixels and points on the screen and regarded our mathematical formula as a pixel rule. However, when making a large number of applications of our rule, we really must distinguish the underlying arithmetical rule from a pixel rule. To see why this is so, recall that a pixel rule begins with a transient and then behaves periodically. A consequence is that the only sensible choice of coloring for pixels that are chosen using a pixel rule would be one color for the transient pixels (those visited only once) and another color for the pixels that are visited periodically. If we look at the colorings of Figure 3, we see that the picture represents a process that is far from periodic.

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  Michael Field is a professor in the Department of Mechanical Engineering at the University of California, Santa Barbara. He is currently working on theoretical problems in non-convex optimization and machine learning using ideas that originate in dynamics and symmetry.  
  Martin Golubitsky is a distinguished professor of mathematics at the Ohio State University. He is the founding editor-in-chief of the SIAM Journal on Applied Dynamical Systems and a past president of SIAM.