The Fractalist

by Benoit Mandelbrot

  Fortunately for me, my corner of French education had little faith in classroom teaching and great faith in its ability to select the best among self-motivated individuals. If a dissertation is to be written solo, no special arrangements are needed. Among bad alternatives, an incoherent or indifferent environment is best.

  Take my Ph.D. chairman, de Broglie. His dissertation—which became one of the two sources of quantum mechanics—was also written with no help. Fifty years after it was defended, the question arose: How did it manage to be accepted? A committee member who was still alive confirmed the rumor that the great physicist Paul Langevin had found the thesis incomprehensible. But he could imagine no harm in accepting it because de Broglie would surely follow the example of his elder brother, Duc Maurice de Broglie, and never apply for a job. However, he sent the thesis to Einstein, and the rest is history.

  The LEP Laboratory of Philips Electronics

  While I was a Ph.D. student, the graduate fellowship from the National Center for Scientific Research (CNRS) would have been extraordinarily meager. I preferred to lead a parallel life. Having finally satisfied the air force, I first followed inertia; that is, I simply resumed the line of thought that had already led me to aeronautics at Caltech in 1947 and sought something on the interface of mathematics and flight. At ONERA (the French counterpart to NASA), a potential supervisor confessed that he had not enough ideas for himself and could not supervise me. Nevertheless, I met the proper authorities, interviewers expressed great enthusiasm for my qualifications, and I was introduced to the big boss. He assured me that his official approval would come in a few days, but he also held other jobs, micromanaged everything, and clearly spread himself too thin. Days and weeks passed with no letter, only telephoned reassurances that the paperwork was on the boss’s desk waiting to be signed. Father’s chronic worry proved true, and he saw additional evidence of how dreadful it was to be employed by a state agency. Aeronautics had become less and less attractive to him. Quietly, he started scanning the newspapers for openings for technical positions.

  One job listing that he really liked gave an address but no name. He found out that it was from Philips SA, the closely held French branch of the Dutch-based electronics multinational. More precisely, it came from a new research division, LEP. Not the obvious Laboratoire d’Electronique Philips, but Laboratoire d’Électronique et Physique Appliquées. Father saw a seamless move from aeronautics to electronics within a big international company. Even if a revolution broke out in one country, they would reassign you to some other subsidiary.

  Philips was seeking an alumnus of a grande école fluent in English and well informed of a technique called spectral analysis. I was a polytechnicien with very high rank, had spent two years at illustrious Caltech with top grades, and knew about spectra in two ways—Caltech friends made practical use of them in turbulence research, and I had “inherited” elements of the theory from Szolem.

  The job was a perfect fit for both sides. Within days, I was called by the Dutchman who ran Philips France from the elegant avenue Montaigne near the Champs-Élysées. Soon came a signed and stamped job offer, with a salary higher than the one ONERA had promised.

  Unlike Father, I mostly favored Philips because it seemed compatible with writing or even inspiring a doctoral thesis. But what made Philips so interested in spectra? The hope was that I would fulfill a genuine and basic need. Reluctantly but feverishly, the TV industry was then preparing for color, and it faced a little-known technical quirk—letting white light go through a prism. Newton had analyzed white light into a “spectral” image made of colors ranging from red to violet. Similarly, a sound can be analyzed into pure sounds of all frequencies, and in the simplest example, into a fundamental and its harmonics. Hence the alternative terms “spectral” and “harmonic” analysis.

  This analysis inspired a color TV system, perfected by RCA and GE engineers, called NTSC, after the National Television Standards Committee. It’s now ancient and obsolete but—like QWERTY typewriters—still in use. As they were learning the ropes, the engineers at the French subsidiary of Philips needed a theoretician to hold their hand.

  Several countries in Europe developed better systems. Poor old NTSC came to be reinterpreted as meaning “Never the Same Color.” To ensure compatibility with the existing black-and-white receivers, the signal included a detailed image that old TVs would interpret as black but color TVs would interpret as green. The full-color image was achieved by adding red and blue images that were much fuzzier than the green one.

  Philips colleagues were struggling to improve on the black-and-white iconoscope by designing and then building an exquisitely complicated device called a supericonoscope. Failure upon failure—then success. Shortly after those engineers had achieved consistent success, they were transferred to a distant factory and began mass-producing that contraption.

  My role model at Philips was the distinguished physicist Hendrik Casimir (1909–2000). He was the technical director of the Philips Research Laboratories in Eindhoven—arguably close in quality to the famed Bell Laboratories. On a dime, he could turn between science, technology, management, and corporate or national policy, and he instantly got the point of every presentation. His visits to check on us were irregular but rumored not to be random. Eindhoven was a provincial one-company city, and educated Dutchmen of his time were fluent in French, so successful new shows invariably brought Casimir back to combine business with pleasure in Paris.

  Philips sported an extraordinarily revealing history. The family of the founder, as it happens—hard to believe—were close relatives of Karl Marx! They owned a tannery in Eindhoven, so the nascent company enjoyed low labor costs without having to move. The Netherlands took until after 1900 to sign the international agreement on copyrights—as revealed by those old French books that I read as a child in Paris. The same was the case for patents.

  After postdocs at MIT and the Institute for Advanced Study, I found that Philips no longer had use for me. The TV group had advanced from research to development. My stint at Philips was short, but I learned many things. In a way, working for industry was a rehearsal for a far longer stint at IBM, and working experience with spectra was very useful indeed.

  Father Dies in 1951

  Shepherding me toward Philips became the last in the long series of gifts from Father. I don’t recall either him or Mother being sick in bed or visiting a doctor. Their explanation was that less fortunate persons would have perished early in one of the catastrophes that they had managed to sail through.

  But cancer struck. First a successful kidney operation brought several years of good enough health. Then lung cancer struck. Father refused a second operation, and his doctor advised heavy doses of radiation, the outcome of which would be quick—either way. We later found that every encyclopedia in the house had bookmarks under “cancer.” Besides, every day the newspaper brought a detailed update on fellow sufferer King George VI of Great Britain.

  When my first paper came out, Father was so ill that I could not wait for reprints to be made, so I borrowed a library copy. It was not clear if he fully understood what I was showing him. He died a short time later.

  Chronic lack of money, hopeless overwork, and Father’s exhausting business travel—and then illness—meant that my parents could rarely entertain. So we expected few mourners, but a small crowd joined Mother and her two sons at Father’s burial. Szolem had been traveling, and the ceremony was held up until he came back. Before leaving, he stopped by and ended by saying, “À bientôt donc.” With all of Szolem’s travels, we wondered if this “See you soon” was meant to be understood as “if fate allows.”

  Unexpectedly, Mother insisted on a religious funeral. The rabbi in Brive during the war—who surely had sent that Angel to protect us—had conveniently moved to Paris, was found, and agreed to officiate. His eulogy was neither corny nor canned. He recalled, with surprisingly many personal details and in very warm terms, that though the war had led him
to meet many parents ready for any personal sacrifice for the sake of their children, none had come close to Father.

  14

  First Kepler Moment: The Zipf-Mandelbrot Distribution of Word Frequencies, 1951

  “TAKE THIS REPRINT. That’s the kind of silly stuff only you can like.”

  These words of Uncle Szolem—ending a visit—opened a door. The words in this reprint first seemed narrow and undistinguished, then profoundly flawed. But I figured out how to correct this flaw and—endless surprise—in the hour that followed experienced my first Kepler moment.

  I allowed my finger to be touched by a complicated set of gears that soon grabbed my body—and never let go. In a different analogy, I found myself in the position of that child in a story who noticed a bit of string and—out of curiosity—pulled on it to discover that it was just the tip of a very long and increasingly thick string … and kept bringing out wonders beyond reckoning.

  Oddly but almost ineluctably, that string, that reprint, ended up directing me to some of the main themes of my scientific life: unevenness, inequality, roughness, and the concept of (as well as the word) fractality. On many occasions, I was to feel that the topic was nearly exhausted, that little else remained to be said—but it kept reappearing from a totally unexpected direction.

  A Fateful Metro Ride

  At the end of a day spent near the Sorbonne, it was not much of a detour—before taking the metro home—to stop at Szolem’s flat. A chat in his study often turned to debate.

  The quote opening this chapter was Szolem’s response to my routine request for reading material for the long ride home. That day, he pulled out of his wastebasket a reprint he had recently received from the Harvard mathematician Joseph L. Walsh (1895–1973), president of the American Mathematical Society. This reprint was a friendly review in the popular monthly magazine Scientific American of a book titled Human Behavior and the Principle of Least Effort, written by George Kingsley Zipf (1902–50). Independently wealthy, this academic character was a university-wide lecturer at Harvard in a self-invented field he called statistical human ecology. His topic was the oddest imaginable—an absurdly simple mathematical formula that claimed to be a universally valid summary of a mass of empirical observations on how the words in ordinary writing are distributed between common and rare.

  I became hooked: first deeply mystified, next totally incredulous, and then hopelessly smitten … to this day. I saw right away that, as stated, Zipf’s formula could not conceivably be exact. But the metro ride was long and I had nothing else to do. By its end, I had derived a more general version I could explain and was dying to confront it with data. I soon decided to pursue this strange avenue, all the way to a Ph.D. It is known today as the Zipf-Mandelbrot law.

  Everybody—above all, Szolem and Marcel-Paul “Marco” Schützenberger (1920–96), a man I had recently befriended—was aghast. They saw Zipf as a crank! Counting words was neither real math, nor real science, nor real anything. Nobody with even minimal technical skill was interested. It would never lead to a proper job. No professorship. Marco located Zipf’s book and made me take a look. On the whole, it was indeed dreadful. But if you could ignore the text and believe the graphs, they covered many fields and were fascinating. They contradicted Zipf’s claim about word frequencies that Walsh had accepted—but confirmed the Zipf-Mandelbrot formula-to-be!

  So I could respond to my friends by broadening Plutarch’s advice: to admire part of a man’s works, you need not admire everything the man claimed. To my rational side, the fact that science’s central casting office considered Zipf an oddball was not sufficient reason to disregard him. To my herd-averse, rebellious side, it may even have been a plus.

  In short time, the Zipf-Mandelbrot formula became part of my Ph.D. dissertation. Then other graphs in Zipf’s book filled several years with interesting developments. I then left Zipf behind and allowed my path to be guided by logical necessity, pure chance, or unabashed play. Eventually, all of that led to fractals.

  Inequality and Unevenness Are Everywhere

  How long does a book on the best-seller list remain there? Most stay for a few weeks, but a few may remain for a hundred weeks or even more. This extreme inequality is basic publishers’ folklore.

  Type a name into an Internet search engine. Some names draw a blank, many have a small number of hits, but a few draw millions of hits. Think about the geographical areas of islands. Greenland and Madagascar are huge, while a countless number are tiny. What about the inequality of sizes of the states in the United States? What about the even greater inequality of areas of the French provinces before the Revolution cut them into near-equal departments, of Soviet republics as contrived by Stalin, or of the parts of present-day Russia?

  Extreme inequality is a familiar pattern in nature and in the works of humans. Such distributions are called long-tailed distributions. For them, no value is typical, and the contrast between short and long tails came to play a central role in my work.

  Most long-tailed distributions have important consequences, but the papers and books written on this topic over the years were disappointing. My luck was to begin with the distribution of word frequencies—a thoroughly atypical example without any important consequences, and uniquely easy to handle.

  Incidentally, in 1952, my first involvement with long tails involved no computers. I first saw a computer in 1953 and first used one in 1958, after I went to IBM.

  Zipf’s Universal Power Law for Words

  In written text or in speech, some words, such as “the” or “this,” have a well-defined frequency. Other words are so rarely used that they have no defined frequency. Here was Zipf’s game: Pick a text and count how many times each word appears in it. Then give each word a rank: 1 for the most common word, 2 for the second most common word, and so on. Statisticians rarely use this method, but there is nothing wrong with it. Finally, graph the frequency of each word against its rank.

  An odd and hard-to-read pattern emerges. The curve does not fall gradually from most common to least common word. At first, it plunges vertiginously, then drops more gradually, continuing in a long tail that declines very slowly—like an exaggerated profile of a ski jumper leaping into space, to land and coast down the gentler slope below. By the very definition of rank, frequency varies inversely with rank. Zipf claimed something far stronger: it is about one-tenth of the inverse of rank. So the product of a word’s frequency and its rank is approximately equal to one-tenth. The curve almost merges with the coordinate axes—making it near impossible to read.

  To compare such curves, it is best to replot them more legibly by replacing both the rank and the frequency with their logarithms. While it may be a bit scary, this word denotes something quite innocuous. A number’s decimal logarithm is roughly its length when it is written using the standard digits from 0 to 9. More precisely, it is smaller—by at most 1—than that number of decimal digits. Thus, the logarithms of numbers from 100 to 1,000 grow from 2 to 3. Taking Zipf’s claim that each word’s frequency is exactly one-tenth the inverse of its rank, on a doubly logarithmic graph, it follows that the data fall along a straight line with a slope of –1, one that decreases vertically by 1 for every horizontal increase by 1.

  The language—English, French, Latin, whatever—does not matter. Neither—quite oddly—does the writer’s degree of literacy. This is an example of what physicists were soon to call a universal relationship. Another notion in physics, called scaling, is one that underlies fractals. Zipf, eyeballing his charts and fitting a curve to the data, devised a formula for it. Walsh featured that formula and observed that it baffled everybody who looked at it. Inspecting those graphs cautiously and critically was a practice that physics came to adopt around 1900, then revived in the 1970s and 1980s—and one I have espoused since the early 1950s.

  Unfortunately, Zipf’s assumption yields conclusions that are simply impossible. For example, it implies that, as a text unfolds, roughly every tenth word has not been used
before. One would expect new words to enter at a gradually decreasing rate. Worse: by the definition of frequency, the different words’ percentages must add up to 100 percent—but Zipf’s formula contradicts this absolute mathematical requirement. One facile way out is to “truncate”: to assume that new words stop being added as soon as the total number of different words has reached 22,000 (the exponential of 10). How could such a universal limitation apply to both James Joyce and an illiterate? In fancy words used by physicists around 1900, Zipf’s original law suffers from a “divergence difficulty” called an “ultraviolet catastrophe,” making his claims mathematically self-defeating.

  Might this be the reason that everyone who looked closely dismissed the whole silly business? Zipf’s claims seemed admirably objective but actually hid the fact that on Zipf’s graphs the product of frequency and rank is not the universal constant one-tenth. It varies! However, let me confess that I also did not immediately pay attention. I recall accepting, for the sake of argument, that the original formula represented the data to some degree, and attempted to reduce it to some basic principle—free of any “catastrophe” that might account for James Joyce, illiterates, and others in between.

  The fact that it applies to all languages—is universal—implies that Zipf’s law is irrelevant to the core of linguistics, which is grammar. In one of the very few clear-cut eureka moments of my life, 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. Those “details” had eluded not only Zipf—not trained as a scientist or mathematician—but also Walsh. Anyhow, appreciating the history of ideas does not make a street-smart scientific explorer. My good fortune resided in an unfair advantage. I was to be the first—and for an interminable time, the only—trained mathematical scientist to take Zipf’s law seriously.