Friday, December 14, 2012

Dynamics: The Geometry of Behavior

When I was working at the National Institutes of Health in the 1990s, I ran across a wonderful series of books from the Visual Mathematics Library that had a big impact on the way I thought about math. Dynamics: The Geometry of Behavior, by Ralph Abraham and Christopher Shaw, was published in four volumes: 1 Periodic Behavior, 2 Chaotic Behavior, 3 Global Behavior, and 4 Bifurcation Behavior. The fascinating feature of these books was that they contained almost no equations; everything was explained in pictures. At first glance, they look like comic books, but on closer inspection you realize that the math is presented in a very accurate and rigorous way. There are lots of plots of phase planes, and drawings of experimental apparatus that are being modeled by the math. There is hardly a page without pictures, and 90% of many pages are filled with illustrations. I highly recommend these books for anyone interested in developing an intuitive feeling for nonlinear dynamics (which should be everyone).

Their forward begins
“During the Renaissance, algebra was resumed from Near Eastern sources, and geometry from the Greek. Scholars of the time became familiar with classical mathematics. When calculus was born in 1665, the new ideas spread quickly through the intellectual circles of Europe. Our history shows the importance of the diffusion of these mathematical ideas, and their effects upon the subsequent development of the sciences and technology.

Today, there is a cultural resistance to mathematical ideas. Due to the widespread impression that mathematics is difficult to understand, or to a structural flaw in our educational system, or perhaps to other mechanisms, mathematics has become an esoteric subject. Intellectuals of all sorts now carry on their discourse in nearly total ignorance of mathematical ideas. We cannot help thinking that this is a critical situation, as we hold the view that mathematical ideas are essential for the future evolution of our society.

The absence of visual representations in the curriculum may be part of the problem, contributing to mathematical illiteracy, and to the math-avoidance reflex. This series is based on the idea that mathematical concepts may be communicated easily in a format which combines visual, verbal, and symbolic representations in tight coordination. It aims to attack math ignorance with an abundance of visual representations.

In sum, the purpose of this series is to encourage the diffusion of mathematical ideas, by presenting them visually.”
In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I do not suppress mathematical expressions. In fact, I suspect many of our readers would claim we have too many, rather than too few, equations. Nevertheless, we try to convey our subject in figures as well as math, visually as well as symbolically. We discuss nonlinear dynamics in Chapter 10, and we have some state space figures that are similar to those found in Abraham and Shaw (although they use 4-color figures—green, red, blue, and black—while we use the less attractive black and white). I believe all the illustrations in Abraham and Shaw are hand-drawn, giving them a charm that often is lacking in this age of computer-generated drawings. Unfortunately, Russ and I never cited Abraham and Shaw. One reason I write this blog is to alert our readers to books and articles that don’t appear in the pages of Intermediate Physics for Medicine and Biology.

Dynamics: The Geometry of Behavior is one of those rare gems that you should become familiar with, both for what it can teach and also for its beauty. To learn more about The Visual Mathematics Library, see Ralph Abraham’s webpage.


  1. This is an awesome set I got as a kid. Epiphanous picture books of dynamics. Fun to read in the tub.

  2. One of the best places to read in my experience. Another is while walking. Nice long walk while reading a book, it's sweet!