The Fractalist

by Benoit Mandelbrot

  In a 1965 publication, I showed that while Hurst had no clue about what he had discovered, his formula indeed holds—and has unexpectedly far-reaching consequences. To a scientist it means that the span of dependence in the flow of the Nile is infinite—while for the Rhine it is finite, and even short. What a joy to quote the Bible as a (pure) scientific reference! But did all that matter in practice? I heard that the Aswan Dam’s engineers, instead of following Hurst, had followed the cold war international political imperatives.

  The study of rivers brought me to the distinction between two kinds of fractals: the self-similar (shapes scaled by the same amount in every direction, like coastlines) and the self-affine (shapes scaled by different amounts in different directions, such as turbulence).

  My explanation of Hurst’s formula was another Kepler moment. After it was published, I pursued the mathematical aspects with mathematician John W. Van Ness. Then I wrote a long series of papers with hydrologist James R. Wallis. IBM Research prided itself on bringing the two of us together. It seems that many big dams are built in China. I wonder whether they are Hurst-Mandelbrots.

  Distribution of Galaxies

  That the Milky Way is one of a number of similar “objects” in the sky is a surprisingly recent notion: it dates from the decade when I was born. So is the galaxy. Incredible but true, however, long before any evidence became available, the concepts of a galaxy and clusters of galaxies had been repeatedly invented and forgotten. Also, the natural assumption that faraway shining objects are uniformly distributed in space was analyzed and shown to lead to the embarrassing Olbers paradox, which argues that the sky must be uniformly and infinitely bright. A way to avoid this paradox was proposed by a science fiction writer, Edmund Fournier d’Albe, and developed by astronomer Carl Charlier. But the profession never took it seriously, largely because it required the universe to have a well-defined “cluster” and because relativity theory demands a well-defined overall density of mass. Somehow, I found out about this bit of esoterica, instantly identified Fournier d’Albe’s model as a primitive fractal, and proposed one, then another, less primitive models.

  The title of a draft of my first paper on galaxy clusters implied that clustering is an illusion. In duller words, this is the way data are spontaneously interpreted by the human eye, but not necessarily a property of the problem at hand. “Tell me if I understand correctly what you have been telling us. We astronomers take it for granted that galaxy clusters are real things out there,” the man said, pointing his finger to the sky (well, the ceiling). “What you propose is that they may very well be right here.” He pointed to his temple. “Is that right?”

  Sometime around 1990, I was in a Tyrolean resort hosting a meeting on the large-scale structure of the universe. The questioner was a person I did not recognize and never saw again. I felt very good. At long last, a key aspect of my fractal model of galactic intermittence—one I had patiently described in each of my “Fractals” essays—was receiving a reasonably serious hearing.

  The audience might have cursed me for being a troublemaker: I was bringing new tools to a corner of astronomy that had been placid; I was sowing doubt and creating new problems. Observers used to take what they saw for granted: galaxies are organized into clusters, which are themselves organized into superclusters—a splendid new use, as I saw it, of Ptolemy’s classic model of the motion of planets. For the “reductionists”—theoreticians whose business it is to “reduce” everything to a field’s basic principles—the task is to explain why galaxies cluster and in doing so to predict the cluster sizes.

  My alternative to Ptolemy is far more parsimonious and suitably Keplerian: I claim that the distribution of galaxies is fractal. The point is that in some fractals, clusters are completely real because they have been included by construction; in other fractals, no clusters have been included by construction but the mind sees them anyhow. Fractality and hierarchy manifest a peculiar consonance. Below are two images of galaxies: on the left, a real galaxy cluster from the Center for Astrophysics and Space Sciences at the University of California, San Diego, and, on the right, a computer fractal model of galaxies.

  My analysis led me to conclude that up to a certain depth in the universe, galaxies had not a uniform, but a fractal, distribution and were easy to construct. With half a line of formula, I got all this clustering of galaxies—superclustering—out of it. That is, my model automatically reduced the overwhelming complication of reality to a single basic principle—a principle at the core of science, one which tries to duplicate the complication of reality by using very simple rules.

  (Illustration Credit 23.1)

  Helping Lady Luck Through Telephones

  Do you recall that my testing of cotton prices began with a mysterious diagram on a blackboard? Well, Lady Luck struck again when I was asked to help with some troublesome noise on data transmittal telephones, and I found a way I liked of thriving as a jack-of-all-trades.

  An odd thing is that chance has helped me on many occasions. Louis Pasteur is credited with the observation that chance favors the prepared mind. I think that my long string of lucky breaks can be credited to my always paying attention. I look at funny things and never hesitate to ask questions. Most people would not have noticed the dirty blackboard or looked at the article that Szolem pulled from his wastebasket for me to read.

  That 1951 reprint and that diagram on the blackboard are both examples of what are now called long-tailed or fat-tailed distributions. These episodes made me the first well-trained mathematician to take those tails seriously. As a result, I have sometimes been called the father of long tails. Whether long or fat, those tails are an intimate part of the fractal family. So it makes perfect sense that I have since been called the father of fractals.

  In the spring of 1962, when my friends at IBM heard about my going to Harvard, they pushed me to give a seminar—just to explain how someone like me had managed to land “this plum” (their words). I obliged, convinced them that I could not help them manage their savings, and found myself in the flattering yet burdensome position of a man who could work miracles … and perhaps assist them in their IBM jobs. The questions I was asked were unpromising, except for one that looked interesting, but was a long shot. Linking computers to telephone lines was proving harder than expected, and a friend at IBM, Jay M. Berger, had been assigned a problem concerning the distribution of errors on those lines. He and his assistants were supposed to find out why they clustered this way. None of the textbook laws of averages seemed to apply. To paraphrase Julius Caesar, I came, saw, and was immediately hooked. Once again, I brought together a problem from one world and a tool from a far-removed other world. A second major Kepler moment within a year.

  The reports from Berger’s group satisfied his managers. A paper I wrote with him on error clustering in telephone circuits provoked a tempest in a small but very important teapot. The experts caught on and soon my work became standard material. I was invited to the epicenter of expertise, Bell Laboratories. In time, they stopped sending me their papers, and I stopped asking them questions. But the seed had been planted.

  Of Galileo’s many gifts to scientific knowledge, here is an essential one that requires no formula. His world believed that the heavens were orderly, while everything on Earth was a mess. To the contrary, Galileo found plentiful messes on the moon—its craters. He also found order on Earth—the falling of stones pulled down by gravity. In this sense, George Kingsley Zipf—whom we met when I told the story of my Ph.D. thesis—was solidly pre-Galilean. He believed that in the physical sciences, randomness follows the distribution called normal, Gaussian, or bell-shaped, while in the social sciences—word frequencies, personal income—the distribution is the so-called hyperbolic.

  Finance was far from filling my time during the academic year 1962–63. Every week—in addition to teaching economics—I was compensating for academic deprivation at the bucolic IBM Research. When not asked, I shamelessly volunteered to speak at one s
eminar or another at Harvard, MIT, or elsewhere. Also, I managed to attend innumerable seminars on countless topics—a form of continuing education that my overworked local friends could only dream of. The substance of my talks, prepared at IBM, concerned the first (Pareto-Lévy-Mandelbrot) of my three successively improved models of financial prices. The creative aspect involved new input that triggered late work in hydrology and the second (Hölder-Hurst-Mandelbrot) model in finance. My Harvard years in applied sciences were a direct outcome of Hölder-Hurst-Mandelbrot, but soon involved another new input that led to work on turbulence and the third financial model.

  To an incredible degree, the incessant wild motion of that year has left a deep trace throughout my life. My schedule was so packed that my self-inflicted wounds at the University of Chicago soon began to heal.

  After one repeat of my standard talk on price variation to a group of noneconomists, an auditor—to whom I am greatly indebted—spoke up. He observed that some aspects of what I said reminded him vaguely of something he had heard about the variability observed in the discharges of rivers. I became very excited. This was about the time when the Berger-Mandelbrot paper on telephone errors was published. Also, economics had led me to worry about oil fields. So I knew that two examples of scaling in the physical world had to be added to Pareto’s law of income distribution and my work on prices. The river discharges promised to add a third, extremely different one, so I rushed to visit the Harvard hydrologist Harold Thomas. He referred me to the work of the hydrologist Harold E. Hurst. Solving the Hurst puzzle tested my skills but took little time.

  Snatched Up by Harvard Applied Sciences

  I had no ongoing contact with the people who were involved in this episode of my life, so my memory of what followed has faded. First, I was asked to return to Harvard the next year and give a few lectures on my findings. I did not find this appealing. Then I was introduced to a physicist who was the dean of applied sciences. He proposed to bring me back to Harvard in 1964–65—but a move back to IBM and another to Cambridge would have been a logistical nightmare and did not please Aliette.

  We settled on an alternative: continue at Harvard in 1963–64, but take a few steps north, from economics to applied sciences. A few steps across Harvard changed the environment. In economics, when I asked for stationery, the chairman’s office provided a starter supply and added that I would surely want to use my own letterhead. In applied sciences, there was a proper stationery office, open stack, like at IBM. My economics office had its own telephone. In applied sciences, I shared a telephone line with three or four regular professors, including a Nobel Prize winner. An additional visitor who did not mind hogging the phone forced them to add a new shared telephone.

  We rented the house of noted MIT physicist Victor Weisskopf, who was on leave directing CERN in Geneva. The attic contained piles of French comic books like Tintin. Aliette read them to young Laurent, then suggested he look at them by himself. He did and in the process learned to read French. Later, Didier followed the same path.

  Weisskopf was a charming and cultured man. When I saw him last—in Alpbach in his native Austria—he was eighty-four. At a festival, a band of vacationers heard me lecture on mathematics and then him on physics. During the discussion, I pointed out that my lecture was presented as an idiosyncratic view of my field, and his as a general talk on physics. At lunch, he complained about how hard it was for him to finish his memoir, and urged me not to write mine too early—certainly not as long as I still could do science. I promised, and now can only hope that my wait has not been too long.

  Teaching at Harvard Applied Sciences

  A one-term course on the Hurst puzzle of persistence in hydrology was mentioned in private conversations with the dean. But the public announcement listed a perennial title, Topics in Applied Mathematics, adding that the instructor for the 1963 fall term was going to be me.

  The first day repeated my feat of the fall of 1962. An exceptionally large class had assembled. The reason the dean was so agreeable to my coming became clear: the division offered too few courses.

  Second surprise: not one hydrologist attended, so I wrote a brief note in French to claim credit for a breakthrough but put developing it on hold—where it stayed for several years. That large class had assembled because of my paper with Berger on errors in telephone channels. Some other students had exhausted Harvard’s slim pickings in electrical engineering. And there were a number of postdocs and senior researchers.

  (Illustration Credit 23.2)

  My material on telephone errors was a bit skimpy for one term, but a good soldier (or good actor, if you prefer) does not say, “I can’t.” So at many class meetings, I reported on progress since the preceding meeting—often made the morning before the class. Terrifying stimulation, but very effective. Only once was I forced to sneak into the classroom minutes ahead to write on the blackboard, “Due to unforeseen circumstances, today’s meeting is canceled.” On another occasion, I started by asking the class to forget the previous week’s two meetings because ten-minute-long substitutes I had developed over the weekend were easier and also went further.

  One student, a naval officer, had to cut short an assignment at Harvard and report in a few months for submarine duty. To satisfy the rules, he needed one more credit but had already taken every other Harvard offering he might consider. Though unprepared, he asked to be accepted as a charity case. I agreed and reassured him that visiting lecturers were not supposed to fail anybody.

  Later, when he brought his term paper, he asked for permission to make a brief statement. “Sure, go ahead!” “Sir, as I had told you, I was absolutely unprepared for your course. My paper is not good at all—it’s all right. Whatever grade you give me will not affect my career. The main thing I wanted to say is that in your course I did learn something very important. I had been told that science was created by humans, but in all my other courses it seemed created by creaky machines. Your course made me watch science being created. Thank you, sir. It was a great experience, sir. Good-bye, sir.” With that, he clicked his heels and exited my life.

  Extremely moved and convinced he was not faking, I thought of my uncle Szolem’s visceral dislike of “elegant lecturers.” They make everything seem crystal clear, but looking at your notes in the evening, you realize that some detail had been forgotten and without that detail everything collapses. Szolem preferred—and practiced himself—the very different style of the man who taught him French mathematical analysis in Kharkov during the civil war that followed the Bolshevik Revolution. Serge Bernstein corrected himself constantly, as if reinventing or at least fully appreciating for the first time the material he was teaching. Szolem would say, “As he went on, he seemed to painfully tear mathematics from his body.”

  The year 1963–64 marked my crossing back from the social to the physical sciences. Trifling issues known only by a few specialists and understood by nobody—so that they were called “anomalous” or went under many other empty names—led me into the midst of a key scientific topic: intermittence in turbulence.

  No Permanent Position at Harvard

  Friends apparently took it for granted that Harvard would offer me a permanent position in applied sciences in a context broader than had Chicago. Aliette and I fondly hoped that would be the case. Rumors grew, then stopped. I inquired and found that I had indeed been considered, but my overly optimistic friends had not weighed in as firmly as needed. One likely opponent was the eminent expert in fluid mechanics, George Carrier (1918–2002). After I explained to him an early form of my multifractal description of turbulence, he responded that if that direction were to prevail, the study of turbulence would cease to interest him.

  Ultimately, my interests and achievements were viewed in Chicago as absurdly broad, and at Harvard as absurdly narrow! Unfortunately, I had to agree that those opinions were not entirely unreasonable. I did not fulfill Chicago’s specific needs but was readying to move through many other sciences.

&nbs
p; In Voltaire’s Candide, the ever-optimistic Dr. Pangloss claims that everything turns out for the best in the best of all possible worlds. Given that my fate was to conceive and develop fractal geometry during the following ten years, Pangloss could argue that neither Chicago nor Harvard would have provided me with the right environment. This tired argument would praise torture as a way of enhancing sainthood. And as Harvard concluded, the swath I was about to cut was to be comparatively narrow. Not expected by them was its being ubiquitous, highly visible, and widely influential.

  After 1964, I stopped worrying about the suitability of IBM and set to work. What I accomplished during the decade mirabilis of the sixties was to culminate with an annus mirabilis at Harvard in 1979–80.

  A Rare Institute Lecturer at MIT

  Harvard was out, but—as was said—Aliette and I had grown enchanted with life in Cambridge. In hindsight, remaining there would not have been the best choice, but at that time we would have much preferred to stay. Therefore, I took the short path to MIT worn by generations of Harvard rejects and refreshed an old relationship with Jerome Wiesner. Jerry had sailed with John F. Kennedy and had been his widely acclaimed science adviser, a post he kept briefly under Lyndon Johnson. He had returned to MIT and was at that point dean of science.

  In no time, he arranged for me to become Institute Professor at MIT. Peter Elias (1923–2001), an old friend of mine and Jerry’s successor as head of electrical engineering, took care of the paperwork. That this should have been possible was a tribute to Jerry’s skills and the level of institutional flexibility he maintained. Institute Professorships were originally meant for scholars crossing many fields, but they soon began to be granted to former administrators or became the most senior chairs in traditional departments.