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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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Merry Morse/Przemyslaw Prusinkiewicz, 1986
A real fern, left, and a fractal fern created using a recursive mathematical system

Zipf’s law, which has been shown to hold for all languages, may seem a trifle. But the same basic principle turns out to be valid for a great variety of phenomena, including the size of islands, the populations of cities, the amount of time a book spends on the best-seller list, the number of links to a given website, and—as the Italian economist Vilfredo Pareto had discovered in the 1890s—a country’s distribution of income and wealth. All of these are examples of “power law” distributions.* Power laws apply, in nature or society, where there is extreme inequality or unevenness: where a high peak (corresponding to a handful of huge cities, or frequently used words, or very rich people) is followed by a low “long tail” (corresponding to a multitude of small towns, or rare words, or wage slaves). In such cases, the notion of “average” is meaningless.

Mandelbrot absorbed Zipf’s law on the metro ride home from his uncle’s. “In one of the very few clear-cut eureka moments of my life,” he tells us, “I saw that it might be deeply linked to information theory and hence to statistical thermodynamics—and became hooked on power law distributions for life.” He proceeded to write his Ph.D. thesis on Zipf’s law. Neither his Uncle Szolem nor his dissertation committee (headed by Prince Louis de Broglie, one of the founders of quantum theory) paid much heed to his effort to explain the significance of power laws, and for a long time thereafter, Mandelbrot was the only mathematician to take such laws and their long tails seriously—which is why, when their importance was finally appreciated a half-century later, he became known as “the father of long tails.”

Having launched himself with his offbeat thesis as a “solo scientist,” Mandelbrot sought out other similarly innovative mathematicians. One such was Norbert Wiener, the founder (and coiner) of “cybernetics,” the science of how systems ranging from telephone switchboards to the human brain are controlled by feedback loops. Another was John von Neumann, the creator of game theory (and much else). To Mandelbrot, these two men were “made of stardust.” He served as postdoc to both: first to Wiener at MIT, and then to von Neumann at the Institute for Advanced Study in Princeton—where he had a nightmarish experience. Delivering a lecture on the deep links between physics and linguistics, he watched as one after another of the famous figures in the audience nodded off and snored. When he finished, the renowned historian of mathematics Otto Neugebauer woke the sleepers by shouting, “I must protest! This is the worst lecture I ever heard!” Mandelbrot was by now paralyzed with fear, but happily a formidable pair came to his rescue: first Robert Oppenheimer, who flawlessly conveyed the intended gist of the lecture in one of his legendary “Oppie talks”; and then von Neumann, who did the same in one of his equally celebrated “Johnny talks.” The audience was transfixed, and the event ended in triumph.

Returning to Europe, Mandelbrot, now newly married, spent a couple of blissful years with his bride in Geneva. There the psychologist Jean Piaget, impressed by his work in linguistics, tried to take him on as a mathematical collaborator. Mandelbrot declined the offer, despite his (qualified) respect for the great man: “While Piaget could be vague or wrong, he was not a phony….” Fernand Braudel invited him to set up a research center in Paris near the Luxembourg Gardens to promote the sort of quantitative history favored by the Annales school. But Mandelbrot continued to feel oppressed by France’s purist mathematical establishment. “I saw no compatibility between a university position in France and my still-burning wild ambition,” he writes. So, spurred by the return to power in 1958 of Charles de Gaulle (for whom Mandelbrot seems to have had a special loathing), he accepted the offer of a summer job at IBM in Yorktown Heights, north of New York City. There he found his scientific home.

As a large and somewhat bureaucratic corporation, IBM would hardly seem a suitable playground for a self-styled maverick. The late 1950s, though, were the beginning of a golden age of pure research at IBM. “We can easily afford a few great scientists doing their own thing,” the director of research told Mandelbrot on his arrival. Best of all, he could use IBM’s computers to make geometrical pictures. Programming back then was a laborious business that involved transporting punch cards from one facility to another in the backs of station wagons. When his son’s high school teacher sought help for a computer class, Mandelbrot obliged—only to find that soon students all over Westchester County were tapping into IBM’s computers by using his name. “At that point, the computing center staff had to assign passwords,” he says. “So I can boast—if that’s the right term—of having been at the origin of the police intrusion that this change represented.”

It was chance again that led to Mandelbrot’s next breakthrough. Visiting Harvard to give a lecture on power laws and the distribution of wealth, he was struck by a diagram that he happened to see on a chalkboard in the office of an economics professor there. The diagram was almost identical in shape to the one Mandelbrot was about to present in his lecture—yet it concerned not wealth distribution, but rather price jumps on the New York Cotton Exchange. Why should the pattern of ups and downs in the market for cotton bear such a striking resemblance to the wildly unequal way wealth was spread through society? This was certainly not consistent with the orthodox model of financial markets, which was originally proposed in 1900 by a French mathematician named Louis Bachelier (who had copied it from the physics of a gas in equilibrium). According to the Bachelier model, price variation in a stock or commodity market is supposed to be smooth and mild; fluctuations in price, arranged by size, should line up nicely in a classic bell curve. This is the basis of what became known as the “efficient market hypothesis.”

But Mandelbrot, returning to IBM and sifting with the aid of its computers through a century of data from the New York Cotton Exchange, found a far more volatile pattern, one dominated by a small number of extreme swings. A power law seemed to be at work. Moreover, financial markets behaved roughly the same on all time scales. When Mandelbrot took a price chart and zoomed in from a year to a month to a single day, the wiggliness of the line did not change. In other words, price histories were self-similar—like the cauliflower. “The very heart of finance,” Mandelbrot concluded, “is fractal.”

The fractal model of financial markets that Mandelbrot went on to develop has never caught on with finance professors, who still by and large cling to the efficient market hypothesis. If Mandelbrot’s analysis is right, reliance on orthodox models is dangerous. And so it has proved, on more than one occasion. In the summer of 1998, for example, Long-Term Capital Management—a hedge fund founded by two economists who had won Nobel Prizes for their work in portfolio theory and staffed with twenty-five Ph.D.s—blew up and nearly took down the world’s banking system when an unforeseen Russian financial crisis foiled its models.

Mandelbrot resented being “pushed out of the economic mainstream.” He recounts, with some bitterness, how a job offer from the University of Chicago’s business school, a bastion of efficient market orthodoxy, was extended to him and then rescinded by its dean (and Reagan’s future secretary of state), George Shultz. Harvard, too, declined to offer a permanent position to the visiting Mandelbrot after initially expressing interest. He took these seeming snubs in stride. Upon returning to IBM, he experienced “the warm feeling of coming home to the delights of old-fashioned collegiality in a community far more open and ‘academic’ than Harvard.” Indeed, Mandelbrot was to remain based at IBM until 1987, when the corporation saw fit to end its support for pure research. He was thereupon invited to teach at Yale, where in 1999, at the age of seventy-five, he finally got academic tenure—“in the nick of time.”

It was at Harvard in 1980 that Mandelbrot made the most momentous discovery of his career. Through his friend Stephen Jay Gould—a fellow champion of the idea of discontinuity—Mandelbrot was invited to teach a course showing how fractal ideas could shed light on classical mathematics. This led him to take up “complex dynamics,” an abstract approach to the study of chaos. Complex dynamics had flourished in Parisian mathematical circles in the early twentieth century, but it soon led to geometrical forms that were far too complicated to be visualized, and the subject became frozen in time.

Mandelbrot saw a way to unfreeze it—through the power of the computer. At the time, computers were pretty well disdained by mathematicians, who “shuddered at the very thought that a machine might defile the pristine ‘purity’ of their field.” But Mandelbrot, never a purist, got his hands on a brand-new Vax supermini in the basement of Harvard’s science center. Using its graphic capabilities as a sort of microscope, he set out to investigate a certain geometrical figure generated by a very simple formula (the only formula, by the way, he allowed in this memoir).

What he found, as the computer produced increasingly detailed pictures of this figure, was utterly unexpected: a wondrous world of beetle-shaped blobs surrounded by exploding buds, tendrils, curlicues, stylized seahorses, and dragon-like creatures, all bound together by a skein of rarefied filaments. At first he suspected that the geometrical riot he was seeing was due to faulty equipment. But the more the computer zoomed in, the more precise (and fantastic) the pattern became; indeed, it could be seen to contain an infinite number of copies of itself, on smaller and smaller scales, each fringed with its own rococo decorations. This was what came to be dubbed the “Mandelbrot set.”

Mandelbrot justly deems the set named in his honor to be a thing of “infinite beauty.” Its detailed geometry, still far from fully understood, encodes an infinite bestiary of chaotic processes. How could such an endlessly complex object—the most complex, it has been claimed, in all mathematics—arise from such a simple formula? For the mathematical physicist Sir Roger Penrose, this uncovenanted richness is a striking example of the timeless Platonic reality of mathematics. “The Mandelbrot set is not an invention of the human mind: it was a discovery,” Penrose has written. “Like Mount Everest, the Mandelbrot set is just there!”

The Fractalist is not a flowing memoir; indeed, it has a fractal roughness of its own. No doubt it would have been more polished had the author lived longer: his passion for revising and tinkering with sloppy drafts, he tells us, rivaled Balzac’s. Occasionally there are jarring notes of hauteur (“I reach beyond arrogance when I proclaim…”) and injured merit (“I don’t seek power or run around asking for favors…. Academia found me unsuitable”). Little attempt is made to explain to the lay reader the author’s mathematical innovations, like his use of dimension to measure fractal roughness. (The coastline of Britain, for example, is so wiggly that it has a fractal dimension of 1.25; it thus falls somewhere between a smooth curve, which has dimension 1, and a smooth surface, which has dimension 2.)

Mandelbrot can be pardoned for not belaboring such technicalities in this book. His tone as a memoirist is more philosophical. The world we live in, he writes, is an “infinite sea of complexity.” Yet it contains two “islands of simplicity.” One of these, the Euclidean simplicity of smooth forms, was discovered by the ancients. The other, the fractal simplicity of self-similar roughness, was largely discovered by Mandelbrot himself. His geometric intuition enabled him to detect a new Platonic essence, one shared by an unlikely assortment of particulars, ranging from the humble cauliflower to the sublime Mandelbrot set. The delight he took in roughness, brokenness, and complexity, in forms that earlier mathematicians had regarded as “monstrous” or “pathological,” has a distinctly modern flavor. Indeed, with their intricate patterns that recur endlessly on ever tinier scales, Mandelbrot’s fractals call to mind the definition of beauty offered by Baudelaire: C’est l’infini dans le fini.

Letters

A Random Walk with Louis Bachelier October 24, 2013

  1. *

    The word “power” here refers not to the political or electrical kind, but to the mathematical exponent that determines the precise shape of a given distribution. 

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