- The Fractalist: Memoir of a Scientific Maverick by Benoit Mandelbrot
Pantheon, 324 pp, £22.50, October 2012, ISBN 978 0 307 37735 7
Benoit Mandelbrot, who died in 2010, was a Polish-born, French-educated mathematician who flourished and became famous in America. His special genius was his ability to disregard disciplinary boundaries and find a common pattern underlying disparate phenomena. From adolescence on he was possessed by an urgent desire to invent a mathematical object that would transform the way we look at the world. By most accounts, certainly his own, he was successful: ‘What shape,’ he asks,
is a mountain, a coastline, a river, or a dividing line between two river watersheds? What shape is a cloud, a flame, or a welding? How dense is the distribution of galaxies in the universe? How can one describe – to be able to act upon – the volatility of prices quoted in financial markets? How to compare and measure the vocabularies of different writers? … These questions, as well as a host of others, are scattered across a multitude of sciences and have been faced only recently … by me.
His achievement was to give mathematical form to the feature all these shapes have in common: namely that each exhibits a similarity with itself at different scales. For example, the chart of daily price variation of a commodity over a year will be statistically similar to the chart of its minute by minute variation over the course of a day; the nature of this similarity is a measure of the chart’s intrinsic ‘roughness’.
Mandelbrot’s hero and scientific model, and the dedicatee of his memoir, is Johannes Kepler, ‘who brought ancient data and ancient toys together and founded science’. Kepler’s toys were the ellipses that replaced circles as the true paths of the planets. The wish to emulate Kepler fuelled the search for the mathematical structures Mandelbrot was to call fractals, the founding concept of his science of roughness. What are fractals? The short answer is mathematical models of self-similarity and self-resemblance: objects whose parts mimic the whole. Natural examples abound: a cauliflower is made up of florets, or miniature cauliflowers, each of which in turn is composed of smaller florets, and so on. A tree – a trunk and two branches – repeats itself: each branch behaves like the trunk and forms two branchlets, and so on. The peaks on a mountain mimic its overall jaggedness, the rocks on each peak in turn mimic them, and so on. One gets the idea.
The memoir opens with a family photograph, a group portrait taken at a dinner party in June 1930 in the house in the Warsaw ghetto where Mandelbrot was born, every person at which ‘deeply affected either my blood or my spirit’. In the place of honour sits the mathematician Jacques Hadamard, ‘arguably the greatest mathematician in France at that time’. Far left is Szolem Mandelbrot, Benoit’s mathematician uncle, who was to play an important role in his scientific life. The dinner party was held to celebrate Szolem being chosen as one of four professors to represent France at the First Congress of Mathematics of the Soviet Union. Two other French mathematicians are present, Arnaud Denjoy and Paul Montel, who had supervised Szolem’s PhD thesis. The others at the dinner table are Benoit’s father (second from left, referred to throughout as ‘Father’), his grandfather Szlomo, the white-bearded patriarch of the family who spoke only Yiddish, and his cousin Leon, the editor of a Polish-language Jewish newspaper. At the back stand the only woman, his aunt Helena Loterman, who looked after her father, Szlomo, and her husband, both of whom ‘vanished in the Holocaust’. Not in the photo is the six-year-old Benoit, his younger brother, Léon, or their mother (always ‘Mother’).
Shortly after the photograph was taken the Depression hit and Father left for Paris to save his wholesale clothing business. In 1936 he decided the whole family must move to Paris, to a two-room flat in Belleville, then a slum quarter. Within a year, after intense drilling from Mother, Mandelbrot passed the dreaded certificat d’études élémentaires and entered a lycée, where he was soon ‘way ahead, reading and dreaming on my own’: dreaming over the strange shapes in an out-of-date mathematical textbook and devouring an obsolete multi-volume Larousse Encyclopedia Father had lugged home. Father and son would go on educational walks across Paris; a sign reading Ecole Polytechnique in faded gold letters prompted the fatherly dream that Benoit might one day attend that illustrious institution. In 1939, Father moved the family to Tulle in the South-West, in the ‘dirt-poor hills of unoccupied Vichy France’.