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On the Complex Plane

Brian Hayes

Indra's Pearls: The Vision of Felix Klein. David Mumford, Caroline Series and David Wright. xx + 396 pp. Cambridge University Press, 2002. $50.

Swirly polychrome pictures of the Mandelbrot set and other intricate fractal objects were much in vogue 25 years ago, when computer graphics was still a novelty. Many of those images now seem quaint and dated, like paisley neckties, and one can't help wondering if the mathematics behind them was also merely a passing fad. Indra's Pearls makes a strong claim to the contrary; the mathematics here is unquestionably genuine. The swirly fractal images are also pretty impressive. (Even the Mandelbrot set makes a cameo appearance, albeit in black and white.)

In this computer-generated rendering . . .Click to Enlarge Image

The particular fractal patterns explored by David Mumford, Caroline Series and David Wright have their roots in a simple geometric operation: inverting a circle. In this context "inverting" does not mean turning the circle upside down but rather turning it inside out, mapping every point of the interior onto a point outside the circle, and vice versa. Suppose you have several circles, like coins lying on a table, and you apply the same inverting transformation repeatedly to all of them. Then each circle appears reflected in all the others, and the reflections also get reflected, forming a series of nested disks. In Buddhist tradition this is the vision of Indra's net, which is studded with infinitely many shiny pearls. As the authors put it,

The pearls in the net reflect each other, the reflections themselves containing not merely the other pearls but also the reflections of the other pearls. In fact the entire universe is to be found not only in each pearl, but also in each reflection in each pearl, and so ad infinitum.

The mathematical counterparts of Indra's pearls live on the plane of complex numbers, and the process of reflection is modeled by a transformation applied to this plane. The transformation, or mapping, is defined by a 2 x 2 matrix; multiplying the coordinates of each point by the matrix yields a new point. Depending on the nature of the matrix, the transformation might merely change the position or orientation of figures drawn in the plane, or it might also alter shapes and sizes. Of particular interest are a family of transformations called Möbius maps, after the German mathematician August Ferdinand Möbius (who also lends his name to the famous one-sided loop). A Möbius mapping distorts some shapes, but it has the special property that it always maps circles into circles—which is just what's needed to reproduce the reflections of Indra's pearls.

Mumford, Series and Wright are interested in the "limit set" of this family of transformations. When the mapping is applied repeatedly, the reflected circles become more numerous, but they also get steadily smaller; in the limiting case of infinitely many iterations, there are infinitely many circles, but they shrink away to dimensionless points, usually forming a disconnected "dust."

Indra's Pearls records a 20-year collaborative effort to describe and explore the limit set of this process. Among books of mathematics, it is unusual in two respects. First, it focuses on the journey rather than the destination. The reader is invited to tag along and watch the passing scenery—and maybe even help paddle the boat from time to time—but the guides can't say at the outset where the voyage will end. Second, theorems, pictures and computer programs are all equally important in this story. I don't mean to suggest that the authors are practicing some sort of postmodern mathematics in which pictures or programs take the place of proofs. In their account, however, the process of discovery usually begins with "Let's write the program and see what happens"; the proof comes later.

The mathematics presented is not difficult, but there is a lot of it: Group theory, topology, non-Euclidean geometry, linear algebra, limits. All of it is patiently explained, but the reader too must be patient. By the time you finish, you'll know your way around the complex plane.—Brian Hayes

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