Benoit Mandelbrot, who died in 2010, was a brilliant Polish born, French educated mathematician who flourished and became world famous in America. His special genius was his ability to look over disciplinary walls and find a common pattern underlying disparate phenomena. From adolescence he was possessed by an urgent passion to invent a mathematical object that would transform how we look at the world. By most accounts, certainly his own, he was successful, and not shy in saying so: “What shape”, he asks, “is a mountain, a coastline, a river, or a dividing line between 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 recently … by me.” (xi-xii) Mandelbrot’s achievement was to give mathematical form to the feature all these shapes share – the common nature of their intrinsic roughness.
His hero and scientific model, the dedicatee of his memoir, is Johannes Kepler “who brought ancient data and ancient toys together and founded science”. Kepler’s revolutionary toys were the ellipses that replaced circles as the true paths of the planets. Emulating him fueled the search for Mandelbrot’s toys, the mathematical structures he was to name fractals, the founding concept of his science of roughness.
What are fractals? The short answer: mathematical models of self-similarity and self-resemblance; objects whose parts mimic the whole. Natural models abound: a cauliflower is self-similar: it’s made of florets, miniature cauliflowers; and each floret 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 jaggedness, their parts in turn mimic them, and so on. One gets the idea. His memoir tells the long meandering struggle of how he got it.
The memoir opens with a family photograph, a group portrait of a dinner party in June 1930, at the house in the Warsaw ghetto where Benoit was born, every person at which “deeply affected either my blood or my spirit” (4). At the place of honour, sits mathematician Jacques Hadamard “arguably the greatest mathematician in France at that time” (7). Far left is Szolem Mandelbrot, Benoit’s mathematician uncle who was to play an important role in his scientific life. The dinner party is 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, the notable Arnaud Denjoy and Paul Montel who had supervised Szolem’s Ph.D thesis. Others at the dinner table are Benoit’s father (referred to throughout as ‘Father’), Grandfather Szlomo, the white-bearded patriarch of the family who spoke only Yiddish, and cousin Leon, editor of a Polish-language Jewish Daily. At the back, stand aunt Helena Loterman, the only female present, who looked after her father Szlomo, and, her husband; both of whom “vanished in the holocaust”. (8) Not in the photo is the 6-year old Benoit, his younger brother Leon, or their mother (always ‘Mother’).
Shortly after this photograph the depression hit. Father left for Paris to save his wholesale clothing business. But the winds were blowing from Nazi Germany and in 1936 Father decided the family must leave Warsaw and move to Paris, to a two-room flat in Belleville, a slum in the 19th Arrondissement. Within a year, with intense drilling from Mother, he passed the dreaded certificate d’etudes elementaires and entered a Lycée where he was soon “way ahead, reading and dreaming on my own.” (45). Dreaming over the strange shapes in an out-of-date mathematical text book and devouring an obsolete multivolume Larousse-Encyclopedia Father had lugged home. They would go on educational walks across Paris; a sign reading École Polytechnique in faded gold letters prompted the fatherly dream that Benoit might one day attend that illustrious institution (which he did nine years later). By 1939 with the declaration of war and a year after Kristallnacht, Paris was no longer safe for his children and Father moved them to Tulle, to the “dirt-poor hills of unoccupied Vichy France”. (49)
But Tulle became dangerous for Jews when the family’s secret protector, the politician Henri Queuille, lost influence. As a cover, Benoit acquired a false ID, moved around taking odd jobs, became an apprentice in a tool-making shop in Périgeux, and ended up in Lyons. There, studying to pass the exam (the ’Taupe’) to be eligible to enter the grandes écoles, Benoit recognized his mathematical gift. Over the years, the Lycée teachers who coached mathematics for the Taupe had devised speed trials, problems requiring long, complex calculations which students were not expected to finish. Repeatedly Benoit would raise his hand “Monsieur, I see an obvious geometric solution.” (79); and he would be right each time.
August 1944, the Allies land in Southern France and a new life begins with a return to Paris and “exam hell” (85) studying for the prized admission to the École Normale Superior and the École Polytechnique. He succeeds brilliantly in both exams; again the geometrical gift is revealed: nobody but him had finished the triple integral question at the end of the Polytechnique exam. His baffled teacher asks how he did it: “I saw that it is the volume of a sphere. But you must first change the given coordinates to the strange but intrinsic coordinates I thought the underlying geometry suggested.” (87) Benoit could choose: the Normale (pure mathematics/theoretical physics) or the less prestigious Polytechnique (applied mathematics/engineering)? A family war council: Szolem, Father, and a cousin (a distinguished chemist), was convened. For Szolem, the choice was essentially a no-brainer: if Benoit was to be a serious mathematician, it had to be the Normale. The cousin added little; Father, strategic survivor of many upsets, counseled the Polytechnique’s training in science/engineering in a rapidly changing world. Benoit went with Normale. By the end of his first day he knew it was “absolutely the wrong place for me”. (92) The Bourbaki movement led by André Weil, which saw mathematics as the study of abstract sets unconnected to anything outside itself, was about to take over the Normale. Benoit abhorred the movement, called it a “cult”, found Weil-inspired Bourbaki approach to mathematics “positively repellent” (89), and considered Weil (arguably one of the century’s leading mathematicians) his “nemesis”. The next morning he switched to the Polytechnique and graduated two years later.
The years at Polytechnique left him frustrated: his “Keplerian dream remained stuck in a holding Pattern”. (112) What to do next? One of his professors suggested fluid mechanics and Cal Tech as the best place for it. Seeing aeronautics as a possible path to the dream, ignoring Szolem’s disapproval, he applied, was accepted and spent the next two years in Pasadena. Cal Tech was at its height, boasting a slew of world-changing scientists. In particular. Max Delbrück, who was orchestrating the birth of molecular biology, “exhilarating proof that someone with my bent might have a chance after all.” (126) But Cal Tech led nowhere and he returned to France to serve his compulsory year in the military and find a job. Father, scouring newspaper ads, discovered the Dutch electronics firm Philips was looking for an English-speaking grandes école alumnus to work in Paris on the development of colour television. Benoit was a perfect fit; he worked for an enjoyable two years as an electronic research engineer at Philips in Paris.
His mathematical reading, however, was disorganized and going nowhere. A frustrated Szolem gave him “a verbal lashing” (139), demanding he settle on a thesis topic. Benoit would visit him asking each time for something mathematical to read on the long metro ride back to Belleville. One time, Szolem reached into his wastebasket: “Take this reprint. That’s the kind of silly stuff only you can like.” (150) It was a popular science review of a book by George Kingley Zipf, an eccentric Harvard scholar. It featured Zipf’s law concerning word frequencies in any language. For written English the most frequent word (rank 1), ‘the’, will occur approximately twice as often as the second most frequent word (rank 2), ‘of’, three times as often as the third most frequent word, ‘and’, etc. On the journey home he was smitten. Where did such a law come from? Why did it resemble laws in statistical mechanics and information theory? What could possibly link the statistics of word frequencies and the behavior of molecules and information?
Zipf’s law was an example of a kind of law that occurs throughout physics, known as a power law. A power law is a relationship between two quantities, where one quantity varies as a power of the other. For example, a person’s weight varies a cube of their height (power = 3); the gravitational force between two bodies varies as the inverse square of the distance separating them (power = minus 2). And, by Zipf’s law, the frequency of a word varies inversely as its rank (power = minus 1)
The principle applies not just to word frequencies but the sizes of islands, populations of cities, weeks on best-seller lists, and much else; in the 1890s the Italian economist Vilfredo Pareto had observed it for the distribution of personal incomes in Italy. Common to all these distributions, a relatively small number of items – commonest words, richest individuals — account for a large chunk of the total, with the rest diminishing very slowly to nothing: graphically, unlike the familiar Bell curve of random variation from an average, the distribution curve of frequencies has a large hump with a very long tail. The experience was to be “the first of several Kepler moments in my life” (155); the originating one: he was to be obsessed by power laws for more than a decade after this and their ‘long tails’ would resurface in another such moment. The law, transmuted into the Zipf-Mandelbrot law, became his topic. The dissertation, hurriedly assembled, poorly organized, in a field that didn’t yet exist, attracted little attention; but he could now apply immediately for post-doctoral positions.
Fortune smiled. He was to come in close contact with Keplerian giants. A year’s post doc at MIT where Norbert Weiner presided and then a year at Princeton’s Institute of Advanced Study as a post doctoral assistant to John von Neumann. A chance meeting with Oppenheimer led to an invitation to talk about the Zipf-Mandelbrot law at the Institute. The talk fell flat, people fell asleep, and a distinguished historian of mathematics stood up and declared he hadn’t understood a word. Oppenheimer came to the rescue of a paralysed Benoit, succinctly summarizing the lecture’s content, followed by von Neumann adding to its mathematics, converting the potential disaster into a vindication of sorts. After Princeton, a return to France, wooing and marrying Aliete Kagan, a two-year stint in Geneva at the invitation of Jean Piaget, the birth of a son, Laurent, and a teaching job at the university of Lille. But he wasn’t content: chapter heading “An Underachieving and Restless Maverick Pulls up Shallow Roots, 1957-8.” (192) He moves again to America to begin “My Life’s Fruitful Third Stage”.
A summer job in New York at IBM turned into a thirty-eight year sojourn. He was allowed virtually free reign as a research scientist at large. He lectured whenever invited and his ideas attracted enough attention for him to take two years leave from IBM to accept invitations to present his work at MIT and Harvard.
Harvard was the scene of a second, more significant Kepler moment. Invited to speak to about his work on the distribution of personal incomes and popping in to the professor whose group he was to address, he was taken aback: there on a blackboard was the very diagram he was about to draw in his class. But it related to a study of a century of daily cotton prices, a subject remote from Mandelbrot’s topic. Subsequent study revealed the number of spikes greatly exceeded what the only known mathematical model of prices, a 1900 doctoral thesis by Louis Bachelier, predicted. Bachelier assumed two ‘facts’: prices are independent of their predecessors (like coin tosses); and prices varied randomly according to the familiar bell-shaped curve. The distribution of personal incomes and cotton prices had some deep common feature, and once again, the phenomenon of fat tails was in evidence. He set to work on the flaw behind Bachelier’s erroneous assumptions. But interest in his work faded; the moment was against him: mathematical economists were just then re-discovering Bachelier’s thesis, and blithely assuming its ‘facts’; baking them into the present-day machinery of financial capitalism to their models of market behavior, price volatility, risk profiles, portfolio theory, and the Black-Scholes formula for pricing derivatives.
Teaching at MIT and Harvard raised his profile and the possibility of a professorship, but it didn’t happen and, somewhat rebuffed, he returned to the comfortable world of IBM. He went to ground, working for the next ten years on the yet to be crystalized concept of fractals, slowly figuring out its mathematics. He continued to think about natural shapes. How long, he asked, is the coast of Britain? The answer: it depends: the smaller your ruler the longer the coast. Seen from a satellite a bay is smooth; seen from a low flying plane, inlets and promontories in it are revealed; seen from a boat cruising the shoreline each of these inlets is revealed to be made up of smaller formations, and so on. The shape of the coast repeats its characteristic roughness at ever-smaller scales, its length increasing down to the level of individual sand grains. (Once he had the concept, he coukld assign a number — its fractal dimension — to a coastline. The British coastline has a fractal dimension 1.3; the much smoother coast of South Africa about 1.1). Meanwhile, he was piecing the fractal concept together. He created the fractal dimension from a concept formulated 50 years earlier by Felix Hausdorf; and found the required long-tailed probability curves in, of all places, the Polytechnique: they were precisely what his old professor, Paul Levy, worked on. By the early seventies he’d cracked it, and was able to announce the birth of fractals, the concept and the name, in a book, Les objets fractal: forme, hazard et dimension, in 1975, followed by expanded English version Fractals, and by the richly illustrated The Fractal Geometry of Nature. The year 1979-80 at Harvard was his “annus mirabilis” (249). He was invited to teach a course in the mathematics department, began new collaborations in physics and pure mathematics and, the final wonder of that year discovered/invented the extraordinary object known as the Mandelbrot set.
Dubbed “the most complex mathematical object in existence” (251), the Mandelbrot set is a two-dimensional figure whose coils, seahorse shapes, and blobs rimmed by jewel-like clusters of islands defy any coherent description. It is made up of infinitely many resemblances of itself, no two alike, which appear from its depths when one zooms in and magnifies any part of it. The set has diverse admirers from engineers, chaos theorists, and artists, to Platonist mathematicians, and others for whom it serves as a kind of techno-sublime mandala.
Image(s) of the Mandelbrot set if possible
The memoir includes a fragment of it along with a scattering of black and white images, mostly fractal simulacra of mountains, snowflakes, clouds, and so on, as well as a set of colour illustrations of fractal-inspired art and mathematical constructs. The Mandelbrot set has become a visual icon of fractal complexity and chaos theory, an object of deep mathematical research as well as philosophical speculation. (A recent student of mine included a fifteen-page meditation on it in her Ph. D. dissertation). Its infinite complexity and dizzying, ever-changing depths not withstanding, the object results from an astonishingly simple algorithm, which goes as follows. Symbol averse readers might want to skip the next paragraph.
Imagine you could add and multiply points in the plane with the result another point (you can if you identify them with complex numbers). This allows you to form functions such as f(x) = c + x2 where c is a fixed and x a variable point: If one inputs a point p for x, the function will output the point c + p2. One now iterates the function by repeatedly using its output as a new input, starting with x = 0. The result will be a series of points, first c when x is 0, followed by c + c2 when x is now c, then c+ (c + c2)2, and so on, generating an infinite path of points in the plane. If the path spirals off to infinity mark the point c white, otherwise mark it black. Carry out the entire procedure for each point c in the entire plane. The resulting black mage on a white background is the Mandelbrot set. Observe that one cannot humanly draw the image; only a computer can. (The French mathematician Gaston Julia studied iterations of such quadratic functions earlier in the century; if computer graphics had existed then, the object might well bear his name.)
After his year of marvels, Mandelbrot’s achievements were increasingly recognized. His fame as the father of fractals grew; universities, technical institutes and learned societies honoured him with numerous awards and prizes. He was made a professor in the Yale mathematics department, to be ultimately, to his immense pride, promoted to the prestigious Sterling Professorship “the university’s highest rank” (viii). He continued to publish and collaborate through his remaining years on a variety of topics including monographs Fractales, hazard et finance and, at age 78, Fractals and Chaos: the Mandelbrot set and Beyond.
Benoit Mandelbrot died suddenly as he was finishing his memoir; world famous father of fractals, smiling among his grandchildren, having realized his heroic dream of revealing the mathematical order behind the geometry of rivers, stock prices, clouds, word frequency, clusters of galaxies, and coastlines of the world.
In a touching Afterword, Michael Frame, a professor in the Yale mathematics department who collaborated with him, mourns the hole left by Benoit’s absence; he ends with a quotation, an aperçu from Mandelbrot’s last major talk, just before his death: Bottomless wonders spring from simple rules … repeated without end. (307) A profound truth, nowhere more true (or profound) than in mathematics, a subject founded on numbers, whose endless wonders spring from repeating the simplest possible rule: leave a mark.