Autumn Books | Published:

Mathematics: A fractal life

Nature volume 490, pages 476477 (25 October 2012) | Download Citation

Mark Buchanan enjoys the quirky memoir of a mathematical rebel — the late Benoît Mandelbrot.

The Fractalist: Memoir of a Scientific Maverick

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Pantheon: 2012. 352 pp. $30 9780307377357

At the start of The Fractalist is a photograph dated June 1930. In a Jewish family's apartment in Warsaw, four Polish mathematicians are hosting a meal for an honoured guest — French mathematician Jacques Hadamard. Somewhere in the same house, although not pictured, would have been a six-year-old named Benoît. The family's surname was Mandelbrot.

A hugely productive theorist of geometry and roughness, and the “father of fractals”, Benoît Mandelbrot died aged 85 on 14 October 2010. The Fractalist is his posthumously published autobiography, polished with help from his former assistant and colleagues.

A collection of poignant vignettes, the memoir paints Mandelbrot as an intellectual individualist who created his own field. He was reputedly a prickly character, self-important and quick to take offence. Traces of his strong opinions come through, but they are overwhelmed by warm and delightful reminiscences of family and colleagues.

Mandelbrot explains that in that 1930 photo are several of the most important people in his life. Key among them is his mathematician uncle, Szolem. The first in the family to attend university, Szolem Mandelbrot had travelled to Paris to study mathematics under Hadamard, and went on to hold the same chair at the Collège de France as Henri Poincaré. He was Mandelbrot's informal mentor, and linked the family to international science.

In 1936, as European civilization began to dissolve, Mandelbrot's “lucid and decisive” parents took the family to Paris to join Szolem. Benoît recalls learning about the history and architecture of the city during long walks. When the war started, the family fled occupied Paris for rural Tulle in the unoccupied Limousin region.

Danger was never far away, and in 1943, the family went into hiding and the parents and children split up. Posing as apprentice tool-makers, Benoît and his cousin Leon aroused suspicion and were arrested by the police, who were seeking the culprits of a bombing. Days of anguish followed before a helpful official intervened on their behalf.

Through such kindnesses, and the determination of his parents, Mandelbrot survived the war. Back in Paris, his mathematical skill gained him entrance to the prestigious École Polytechnique. This would have guaranteed him a comfortable career in France, but Mandelbrot never settled for the safe option.

In one vignette, Mandelbrot recalls a “life-altering verbal lashing from Szolem”. It was 1952 and Mandelbrot, aged 28, was back in Paris after studying fluid dynamics and aircraft design at the California Institute of Technology in Pasadena. He hadn't chosen a topic for his PhD, and Szolem berated him for excessive book learning and preparation to be a “well-trained monkey”.

Szolem suggested pursuing quadratic dynamics, the sometimes complex behaviour of mathematical functions when iterated — plug in a number, take the result and plug that back in again, endlessly. Mandelbrot looked into the topic, then dropped it. But the lashing had an effect. Soon, Szolem pointed Mandelbrot to the linguist George Zipf, who had found a scaling law in the relative frequency of word use that seemed to be universal across languages. “Silly stuff only you can like” was Szolem's view. Benoît, to the alarm of his professors, chose it as his thesis topic — and so set a course to revolutionize science.

Mandelbrot's insight that the apparently random might harbour hidden order broadened over his postdoc positions at the Massachusetts Institute of Technology (MIT) in Cambridge; Princeton University in New Jersey; and the University of Geneva in Switzerland. In 1958, he began a 35-year career at the IBM Research Center in Yorktown Heights, New York.

His work on market fluctuations in the 1960s could have revolutionized financial economics if the field's equilibrium-focused ideology hadn't pushed it aside for 30 years. Perceiving parallels in everything from fractured surfaces to stock-market movements, Mandelbrot coined the term fractals and did more than anyone to make it possible to talk about natural roughness and disorder in a precise, scientific way.

In 1979, he returned to Szolem's suggestion of quadratic dynamics. Within a year, his explorations of functional iterations led him to discover the Mandelbrot set — an astonishing pattern of infinite richness produced by simple rules. This launched a whole branch of mathematics.

Much of what makes The Fractalist fun to read is Mandelbrot's scattered recollection of encounters with luminaries. At MIT he discussed linguistics with the young Noam Chomsky and argued with anthropologist Margaret Mead. Robert Oppenheimer liked his ideas, and Mandelbrot worked with the famed psychologist Jean Piaget in Geneva and computing pioneer John von Neumann at Princeton.

Mandelbrot recalls how after one of his lectures, a famous mathematician objected, saying he had made “absolutely no sense at all”. Oppenheimer and von Neumann sprang into action, explaining points that even Mandelbrot hadn't noticed. The meeting, he remembers, “went from abysmally low to unforgettably high”. His appreciation of friends, music and quirky mathematics colours every page of The Fractalist. Mandelbrot's odd habits explain why he was so original: he avoided work involving direct competition with others, and naturally worked in the gaps between fields, in blind spots.

“A youthful decision set me on a maverick's lonely ride,” he writes. “Its consequences took a long time to develop.”

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  1. Mark Buchanan is a physicist and writer based in France. His latest book is The Social Atom.

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Correspondence to Mark Buchanan.

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