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He Conceived the Mathematics of Roughness

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Hank Morgan/Time Life Pictures/Getty Images
Benoit Mandelbrot, 1982. Behind him is an attempted computer simulation of a crater field. Crater fields, such as those occurring on the moon, are formed by the cumulative impact of meteorites. They have a fractal structure, one that can be mimicked by computer methods. But the program that generated this not very plausible lunar landscape contained an error, leading Mandelbrot to dub the image ‘the computer bug as artist.’

Benoit Mandelbrot, the brilliant Polish-French-American mathematician who died in 2010, had a poet’s taste for complexity and strangeness. His genius for noticing deep links among far-flung phenomena led him to create a new branch of geometry, one that has deepened our understanding of both natural forms and patterns of human behavior. The key to it is a simple yet elusive idea, that of self-similarity.

To see what self-similarity means, consider a homely example: the cauliflower. Take a head of this vegetable and observe its form—the way it is composed of florets. Pull off one of those florets. What does it look like? It looks like a little head of cauliflower, with its own subflorets. Now pull off one of those subflorets. What does that look like? A still tinier cauliflower. If you continue this process—and you may soon need a magnifying glass—you’ll find that the smaller and smaller pieces all resemble the head you started with. The cauliflower is thus said to be self-similar. Each of its parts echoes the whole.

Other self-similar phenomena, each with its distinctive form, include clouds, coastlines, bolts of lightning, clusters of galaxies, the network of blood vessels in our bodies, and, quite possibly, the pattern of ups and downs in financial markets. The closer you look at a coastline, the more you find it is jagged, not smooth, and each jagged segment contains smaller, similarly jagged segments that can be described by Mandelbrot’s methods. Because of the essential roughness of self-similar forms, classical mathematics is ill-equipped to deal with them. Its methods, from the Greeks on down to the last century, have been better suited to smooth forms, like circles. (Note that a circle is not self-similar: if you cut it up into smaller and smaller segments, those segments become nearly straight.)

Only in the last few decades has a mathematics of roughness emerged, one that can get a grip on self-similarity and kindred matters like turbulence, noise, clustering, and chaos. And Mandelbrot was the prime mover behind it. He had a peripatetic career, but he spent much of it as a researcher for IBM in upstate New York. In the late 1970s he became famous for popularizing the idea of self-similarity, and for coining the word “fractal” (from the Latin fractus, meaning broken) to designate self-similar forms. In 1980 he discovered the “Mandelbrot set,” whose shape—it looks a bit like a warty snowman or beetle—came to represent the newly fashionable science of chaos. What is perhaps less well known about Mandelbrot is the subversive work he did in economics. The financial models he created, based on his fractal ideas, implied that stock and currency markets were far riskier than the reigning consensus in business schools and investment banks supposed, and that wild gyrations—like the 777-point plunge in the Dow on September 29, 2008—were inevitable.

I was familiar with these aspects of Mandelbrot’s career before I read this memoir, a draft of which he completed shortly before his death at the age of eighty-five. I knew of his reputation as a “maverick” and “trouble-maker”—labels that, despite his years with IBM, seemed well merited. What I wasn’t prepared for was the dazzling range of people he intersected with in the course of his career. Consider this partial listing of the figures that crop up in his memoir: Margaret Mead, Valéry Giscard d’Estaing, Claude Lévi-Strauss, Noam Chomsky, Robert Oppenheimer, Jean Piaget, Fernand Braudel, Claudio Abbado, Roman Jakobson, George Shultz, György Ligeti, Stephen Jay Gould, Philip Johnson, and the Empress of Japan.

Nor did I realize that Mandelbrot’s casually anarchic ways at IBM were at least partly responsible for the advent of that bane of modern life, the computer password. What struck me most, though, was the singularity of Mandelbrot’s intuition. Time and again, he found simplicity and even beauty where others saw irredeemable messiness. His secret? A penchant for playing with pictures, a reliance on visual insight: “When I seek, I look, look, look….”

Mandelbrot was born in 1924 into a Jewish family that lived in the Warsaw ghetto. Neither of his parents was mathematical. His father sold ladies’ hosiery, and his mother was a dentist—adept, thanks to her “strong right hand and powerful biceps,” at pulling teeth. His uncle Szolem, however, was a mathematician of international rank who trained in Paris and became a professor at the Collège de France. “No one would influence my scientific life as much as Szolem,” Mandelbrot tells us—though the nature of his uncle’s influence would turn out to be rather peculiar.

Describing his Warsaw childhood, he vividly recalls, for example, the manure-like stench attaching to one of his mother’s dental patients, who defrayed the cost of repairing a mouthful of rotten teeth by bringing the family fresh meat from the slaughterhouse where the patient worked. With the Depression his father’s business collapsed, and eventually the family left Poland for Paris, traveling across Nazi Germany in a padlocked train. “Of the people we knew, we alone moved to France and survived,” Mandelbrot writes, adding that many of their neighbors in the Warsaw ghetto “had been detained by their precious china, or inability to sell their Bösendorfer concert grand piano….”

Paris enchanted the young Mandelbrot. His family set up housekeeping in a cold-water flat in the then-slummy neighborhood of Belleville, near the Buttes Chaumont, but the boy avidly explored the city at large—the Louvre, the old science museum on the rue St.-Martin, the Latin Quarter. In school, Mandelbrot distinguished himself as un crack—slang for high achiever—and even un taupin: “linguistically,” he tells us, “an extreme form of the American ‘nerd’” (the word derives from the French word taupe, meaning “mole”).

What gave him an edge was his ability to “geometrize” a problem. Instead of shuffling formulas like his fellow students, he used his prodigious visual memory to see how a complicated equation might harbor a simple shape in disguise. In a nationwide competitive exam, he tells us, he was the only student in France who managed to solve one especially fiendish problem. “How did you manage?” asked his incredulous teacher, a certain Monsieur Pons. “No human could resolve that triple integral in the time allowed!” Mandelbrot informed his teacher that he simply changed the coordinates in which the problem was stated so its geometrical essence, that of a sphere, was revealed—whereupon M. Pons walked away muttering, “But of course, of course, of course!”

Mandelbrot was fourteen when World War II broke out. With the fall of Paris, he and his family sought refuge in Vichy France, where, as Jews of foreign origin, they lived in constant fear of denunciation and soon had to split up. Using an assumed name and furnished with fake papers, Mandelbrot pretended to be an apprentice toolmaker in a hardscrabble village in Limousin (where a trace of the rural accent was added to his mix of slum Parisian and correct French). After a close brush with arrest, he made his way to Lyon, where, under the nose of Klaus Barbie, he refined his geometrical gift with the help of an inspired teacher at the local lycée.

It was during this time that he conceived what he calls his “Keplerian quest.” Three centuries earlier, Johannes Kepler had made sense of the seemingly irregular motions of the planets by a single geometrical insight: he posited that their orbits, instead of being circular as had been supposed since ancient times, took the form of an ellipse. As a teenager, Mandelbrot “came to worship” Kepler’s achievement and aspired to do something similar—to impose order on an inchoate area of science through a bold geometrical stroke.

It was in postwar Paris that Mandelbrot began this quest in earnest. Uncle Szolem urged him to attend the École Normale Supérieure, France’s most rarefied institution of higher learning, where Mandelbrot had earned entry at the age of twenty (one of only twenty Frenchmen to do so). But the aridly abstract style of mathematics practiced there was uncongenial to him. At the time, the École Normale—dite normale, prétendue supérieure, says the wag—was dominated in mathematics by a semisecret cabal called Bourbaki. (The name “Bourbaki” was jocularly taken from a hapless nineteenth-century French general who once tried to shoot himself in the head but missed.) Its leader was André Weil, one of the supreme mathematicians of the twentieth century (and the brother of Simone Weil).

The aim of Bourbaki was to purify mathematics, to rebuild it on perfectly logical foundations untainted by physical or geometrical intuition. Mandelbrot found the Bourbaki cult, and Weil in particular, “positively repellent.” The Bourbakistes seemed to cut off mathematics from natural science, to make it into a sort of logical theology. They regarded geometry, so integral to Mandelbrot’s Keplerian dream, as a dead branch of mathematics, fit for children at best. So, on his second day at the École Normale, Mandelbrot resigned. His uncle was disgusted by his decision, but this only fortified his resolve. Whereas Szolem was a “prudent conformist who promptly joined the soon-to-be- powerful Bourbaki,” Mandelbrot saw himself—somewhat megalomaniacally, he concedes—as a “dissenter” who would overturn its orthodoxy.

Groping his way toward this goal, Mandelbrot first enrolled in another of France’s grandes écoles, the École Polytechnique, then traveled to the United States to pursue, abortively, the study of aeronautics at Caltech. On his return to France, he found himself drafted into the French military. After a somewhat comical stint in the French air force (during which he seems to have spent most of his time indulging a newfound passion for classical music at Parisian concerts), he became a “not-so-young” grad student at the University of Paris, “then at a low point in its long and often glorious history.”

It was in casting about for a thesis topic that he had his first Keplerian glimmer. One day Uncle Szolem—who by now had written off Mandelbrot as a loss to mathematics—disdainfully pulled from a wastebasket and handed to him a reprint about something called Zipf’s law. The brainchild of an eccentric Harvard linguist named George Kingsley Zipf, this law concerns the frequency with which different words occur in written texts—newspaper articles, books, and so on. The most frequently occurring word in written English is “the,” followed by “of” and then “and.” Zipf ranked all the words in this way, and then plotted their frequency of usage. The resulting curve had an odd shape. Instead of falling gradually from the most common word to the least common, as one might expect, it plunged sharply at first and then leveled off into a long and gently sloping tail—rather like the path of a ski jumper. This shape indicates extreme inequality: a few hundred top-ranked words do almost all the work, while the large majority languish in desuetude. (If anything, Zipf underestimated this linguistic inequality: he was using James Joyce’s Ulysses, rich in esoteric words, as one of his main data sources.) The “law” Zipf came up with was a simple yet precise numerical relation between a word’s rank and its frequency of usage.

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