Books and papers on chaos and fractals have proliferated enormously in recent years. This book brings something new to the subject. It explains a simple new technique for generating a class of fractals called “strange attractors.” It shows the reader how to produce heretofore unknown patterns, featuring an infinite variety of displays and musical sounds, using a single program. The large collection of such objects offers many interesting possibilities to artists and scientists alike.
The book consists of eight chapters and six appendices. The first chapter, “Order and Chaos,” is an introduction to the essential ideas, mathematical techniques, and programming tools necessary for a good understanding of the rest of the book. The goal of the second chapter, “Wiggly Lines,” is to show the reader how to search for chaotic solutions to simple equations with a single variable using the computer. If the graphical results are not appealing, a musical interpretation is attempted. In some cases, patterns and structure in data can be more readily heard than seen. The repetitive sound of a simple limit cycle contrasts sharply with the nonrepetitive waverings of a chaotic time series.
The third chapter, “Pieces of Planes,” deals with two-dimensional maps whose graphs are pieces of planes, and that thus produce more interesting displays. This chapter provides the minimum tools for creating attractors that qualify as art. A two-dimensional world is a mere shadow of reality. Therefore, the fourth chapter, “Attractors of Depth,” emphasizes new visualization techniques and provides many new examples of strange attractors that have depth as well as width and height. The effects obtained are delightful. For example, the point source of illumination is moved slightly off to one side, which produces the appearance of shadows.
Although we normally think of space as three-dimensional, mathematics is not so constrained. Strange attractors can be embedded in space of four or more dimensions. The fifth chapter exploits a number of appropriate visualization techniques after explaining why dimensions beyond the third are useful for describing the world in which we live.
The attractors presented in chapter 6, “Fields and Flows,” are produced by differential equations. They consist of continuous lines whose weavings and waverings describe the trajectory, and yield objects of considerable beauty. The seventh chapter, “Further Fascinating Functions,” proposes the use of functions other than polynomials. The effects are fascinating. A pair of 3-D glasses accompanying the book provides a better perception of the images created.
The epilogue reveals the real goal of this work. The main objective of the ideas presented is not to produce pretty pictures but to suggest scientific applications of the method used to generate strange attractors. The epilogue also suggests some additional explorations one might undertake as an extension of both the scientific and the artistic aspects of the work described.
Six appendices complete the information: a rich annotated bibliography; the complete BASIC programs readers should have developed if they followed the exercises in the book; considerations for using the programs with non–IBM-compatible computers and with different dialects of BASIC; the C version of one of the more interesting programs; a list of all the equations solved by the program to produce the attractors; alphabetical listings of the codes of those attractors whose figures have incomplete coding; a selection of additional interesting cases; and a list of special cases of historical or mathematical significance. An accompanying floppy disk may be used to produce a collection of strange attractors. A complete index is provided, and the typography is good.
Chaotic processes are all around us. Their mathematical solutions usually produce chaotic strange attractors, whose diversity and beauty we are invited to explore.
