Mathematical puzzles to please the eye

Fractals and Chaos
August 5, 2005

Do mathematicians discover mathematics or do they create it? One has only to look at the intricate beauty of the Mandelbrot set to marvel at what was waiting to be found. But to explain its structure is an unfinished task that has taken some of the most creative mathematics of our time.

Benoit Mandelbrot was 80 last year, and it is 25 years since he first saw the ubiquitous set that bears his name. This is the collection of all complex numbers, "c", such that the repeated operation of squaring followed by adding c, beginning with zero, gives a sequence that is bounded. In 1979-80, Mandelbrot started investigating the values of c for which this sequence tends to a periodic cycle, and was astonished at the extraordinary structure that emerged in his computer plots. His pictures revealed a central "mainland" surrounded by a network of "islands", each a scaled-down copy of the mainland (these islands were removed from the first published pictures by a zealous printer who believed them to be "dust"!).

Adrien Douady named this structure the "Mandelbrot set". In 1982, Douady and John H. Hubbard proved that it is connected and formulated the conjecture ("MLC") that it is locally connected - a conjecture that if ever proved would finally resolve its remaining mysteries.

This book is a selection of articles from the 1980s and early 1990s, together with previously unpublished material from the same period, linked by additional chapters containing commentaries and reminiscences. The papers are grouped under five main headings, from quadratic Julia and Mandelbrot sets, to the limit sets of Kleinian groups and invariant multifractal measures. They present Mandelbrot's personal account of his discoveries and his mathematical philosophy rather than a comprehensive guide to the subject.

Mandelbrot's work is driven by an insatiable curiosity in what he sees, guided by a mathematical eye strongly influenced by the heritage of Henri Poincare, who initiated the study of Kleinian groups and whose ideas about dynamics inspired the theory of iterated complex maps developed by Pierre Fatou and Gaston Julia. Fatou-Julia theory, a triumph of complex analysis, was neglected in the mid-20th century - even by Mandelbrot as a student, despite his having Julia as a teacher at the Ecole Polytechnique. The arrival of the microcomputer brought a huge new flowering of theory, as well as experiments, in the 1980s, interacting with and complementing remarkable advances in nonlinear dynamics and hyperbolic geometry.

One does not need to be a professional mathematician to read these papers, though it helps to have a scientific background. They include very readable historical and polemical commentaries - Mandelbrot certainly lets you know his views on the 20th-century trend towards general theory and abstraction.

The numerous computer plots are pleasing to the eye, and the primitive early ones are of particular interest. Fractals were brought to a wider public by the superb colour graphics of Heinz-Otto Peitgen and Peter Richter ( The Beauty of Fractals , 1986). The recent book by David Mumford, Caroline Series and David Wright ( Indra's Pearls: The Vision of Felix Klein ), containing stunning pictures of limit sets of Kleinian groups, is also highly recommended by Mandelbrot.

The informal mix of mathematics and commentary in Mandelbrot's book provides a fascinating insight into his motivation and method. Often at odds with the mathematical mainstream, he reminds us of the value of images and of basing our mathematics on what we see. Mandelbrot looked at the world and saw fractals (a term he coined in 1975), helping to change how we view the mathematical and physical universe.

Shaun Bullett is professor of mathematics, Queen Mary, University of London.

Fractals and Chaos: The Mandelbrot Set and Beyond

Author - Benoit B. Mandelbrot
Publisher - Springer
Pages - 308
Price - £38.50
ISBN - 0 387 20158 0

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