Fascinating fractals and other mysteries of math

THE MATHEMATICAL TOURIST by Ivars Peterson

New York: W.H. Freeman. 240 pp. $17.95

THIS book rates attention. No fun-with-numbers matchsticks or bits-of-string book, ``The Mathematical Tourist'' takes readers into the key concerns of modern mathematics and shows how relevant the mathematician's preoccupations can be to the real world.

Subtitled ``Snapshots of Modern Mathematics,'' Ivars Peterson's book is written in a lucid style using well-chosen examples and analogies to give insight into abstruse ideas. Clear explanations of difficult concepts abound. Not only does Peterson give understandable examples of the meaning of the fourth dimension and beyond, but he also enters the fantastic landscape of fractals. These forms can have dimensionalities somewhere between the one dimension of a line and the two dimensions of figures on a flat surface, or between two and the three dimensions of the world of objects.

Fractals were at first regarded as ``skeletons in the closet of otherwise orderly mathematics.'' These infinitely complex figures are derived by changing a basic figure in a way that replicates it on a smaller scale, then repeating the operation ad infinitum, so that the details of the figure appear the same at any magnification. Fractals are creations of the computer age. They can be evolved and displayed effectively only with computer graphic techniques.

With fractals, computers can create images of the physical world. When random skewness is injected into the operation, the process can make figures like fern fronds and mountain scenes. These landscape generators can be used in applications (including video games) that require a moving or changing background. The advantage in producing the landscape by a computational process is that there is no need to store the images as millions of pieces of data. Another use is simulating radio transmission noise. Finding and transmitting the rules for constructing complex images with fractals may be more efficient than transmitting the image data for every point.

``The Mathematical Tourist'' covers some of the same ground included in ``Chaos,'' by James Gleick, another science writer. Both books open areas of recent scientific understanding to the view of a more general audience.

Topology, minimal surfaces, Mandelbrot sets, linear programming, are only a few of the other topics Peterson illuminates under grabby chapter titles like ``Ants in Labyrinths'' and ``The Dragons of Chaos.'' Sixteen color plates and numerous black and white line drawings and computer graphics reinforce the text.

From this book the reader finds that modern mathematics is both theoretical and empirical. It uses computers to clarify where mathematicians should focus their analyses but depends more on insight than on computers to reach conclusions. It searches for proofs rather than relying on the predominance of instances and looks for simple criteria that point to the solvability - or unsolvability - of problems.

The tourist of this book is an armchair traveler. The book is about ideas, not things to do. It can be read beginning to end, or sampled at any point for pleasure and profit. Only a few mathematical expressions are used, and the reader is expected only to behold, not manipulate them. Peterson provides no step-by-step directions for hands-on exploration of the concepts, but he gives ample leads to further reading.