The Fractalist

by Benoit Mandelbrot

  Zigzagging Through the First Course Ever on Fractals

  The first course on the topic of fractals—which I improvised from day to day in the spring term of 1980—was closely related to my ongoing research. The course was not for credit, and the auditors ranged from young undergraduates to seasoned Ph.D. candidates.

  To assist the course, I wanted live demonstrations and was advised to hire a senior, Peter Moldave, who turned out to be one of the best programming helpers I ever had. The personal computer had not yet hit the world, and none of the students had a clue about how to use one or about the benefits of its graphics. This, along with Peter’s prowess as programmer, created a sensation. Moreover, Peter was not taking any hard courses during his final term and said he would be delighted to assist me with my research.

  The course limped along until after the spring break, when it morphed into something entirely different: the first-ever public discussion of my discovery of the object that—very late that year—came to be called the Mandelbrot set. Peter’s help was essential to that discovery.

  Ultimately, Harvard did not work out. I was expected to pursue and teach my style of using computers, but computers and their use were not welcome at Harvard. Hence, there was a near-total absence of both equipment and skills among the students and faculty. And because personal computers had not yet become ubiquitous, those who absolutely needed them went elsewhere or had private and well-hidden facilities.

  Having been so outspoken about the sad state of computing at Harvard, I was disappointed that good news did not come until it was too late for me. Out of the blue, David Mumford informed me that the National Science Foundation was responding to critiques like mine by establishing a supercomputer at Geometry Center at the University of Minnesota.

  Wide Wonder, Complexity, and Mystery

  The Mandelbrot set strongly appeals to three very different groups to which I belong: those interested in pictures, complexity, and pure mathematics.

  Pictures

  The complication of the actual pictures obtained when that little quadratic map is repeated very many times—starting with z = 0—is overwhelming. Of course, one could not repeat the formula by hand, only by using a computer. The earliest and “rawest” of these pictures were in black and white—or, more precisely, darkish and whitish grays. The constants are not ordinary (real) numbers, each attached to a point on the line, but complex numbers, each attached to a point in the plane.

  Millions of examples can now be found in hundreds of books and on the Web.

  What about the colors? The defining formula yields a whole number: 1, 2, 3, and so on. To cut down on incomprehensible clutter, I replaced ranges of numbers with shades of gray. Then colors took over. Selected by the programmers, they were completely arbitrary and a reflection of good or bad taste.

  Complexity

  When I set out to study that rule that ends with the word “repeat,” I decided with little reason that nothing of much interest could possibly come from such a simple map.

  Around that time, Andrei Kolmogorov and my IBM colleague Gregory Chaitin had, independently of each other, attempted to measure the complexity of a mathematical structure. They put forward the length of the shortest sentence that could implement that structure. Where does this position the Mandelbrot set? Is it the most complex set in the whole of mathematics, as some have asserted, or is it as simple as its generating formula? I could not decide and concluded that the question begs to be restated in a different way. But given the stark discontinuous contrast between an input and an output that today is nearly instantaneous for the Mandelbrot set, many view it as extremely—miraculously!—complex. I feel exceptionally privileged that my wanderer’s life led me to be the agent of this discovery.

  Pure mathematics

  Had anybody investigated “that set” before I did? No, nobody had. After the fact, extraordinary efforts were made to find predecessors. A claim was put forward on an unmotivated drawing that was too crude to show anything but had been appended to a paper—without comment. Also, someone read through one of Fatou’s long papers and found a mention of “that set” among related ones, but without further discussion or anticipation of any result.

  To my surprise and profound delight, my original paper on the subject was an absolute first. The title is “Fractal Aspects of the Iteration of [Quadratic Maps] for Complex [Parameter and Variable].” It appeared in late 1980 in Annals of the New York Academy of Sciences.

  Inevitable question: Was Annals the worthiest place to publish a groundbreaking paper? Not in the least. But I was embarking on a lecture tour, and a printed text was urgently needed. So I replaced an expository paper I had read at that academy with the more recent work and, wherever I went, carried copies of the proofs. Did this work? This is the paper that led to “that set” being named after me, yet in the early days—when it mattered—hardly anyone quoted it.

  So, ironically, my best-known discovery did not result from the availability of exceptionally good pictures at IBM. It was made at Harvard, where I had to deal with complicated research conditions within a very bad system. The pictures we saw on the first night seemed incomprehensible; the second night, they became more coherent. Within a few days, they had grown completely familiar, as though one had always seen them. Incredible!

  (Illustration Credit 25.7)

  How does the importance of the Mandelbrot set compare to that of fractal finance, which is highly influential in a well-defined community of “practical people”? All my diverse “children of the mind” are equally dear to my heart; they can’t and shouldn’t be compared. In that case, what makes me perceive 1979–80 as an annus mirabilis? My work in 1962–63 made for a wonderful year, but it was a year of a single miracle that developed slowly over time, while the 1979–80 miracle came on like lightning—as miracles should.

  26

  A Word and a Book: “Fractal” and The Fractal Geometry of Nature

  NEVER UNDERESTIMATE THE POWER of a word that appears at the right time and in the right context and—let us not forget—accompanied by the right pictures. The word “fractal” has spread like wildfire to so many minds, books, and dictionaries that it is hard to believe it dates only to 1975. The underlying idea had been written about every so often since time immemorial, and a skeptic may wonder if it was really necessary to invent a word to describe my work.

  How did this word, “fractal,” come into usage? I had to coin it when the French edition of my book was being written—the need for a word had become convincing and I had become confident that it would fly. Did I act like a superstitious parent who names a child only after its birth? Let it be. I also checked in advance that “fractalist” would sound good if a need were to arise for a word to denote me and the followers I hoped to inspire.

  Like my speech in every language, my scientific writing in every field carries a strong foreign accent. Because of this anomaly, several of my papers were rejected and other drafts did not seem worth finishing. Instead, they got filed in some dark corner of my personal archive.

  As a result, a backlog of unfinished drafts began to grow until, at one point, my friend Mark Kac volunteered some unexpected but truly excellent advice. “Most active young scientists know they must publish articles or perish. But your case is different. Unless you stop this avalanche of individual articles and write a book, I shall let you perish.” I am extremely grateful to Mark for this “command.”

  I solved my communications dilemma by publishing a great deal of original work in three books. They arose as successive versions of a broad-based “essay” combining a fractal manifesto and a casebook—that is (using military terms I don’t like but find hard to avoid), a call to arms and stories of successful past campaigns.

  The 1975 “Preview” Book, Les objets fractals

  When my soon-to-come-out book was still tentatively titled, in French, Concrete Objects of Fractional Dimension, the publisher, Flammarion, was horrified and asked for something better. Frien
ds concurred. “You have written about a brand-new idea. You are entitled—in fact, obliged—to give it any name you want. Make it snappy.”

  I could have given a new meaning to some already overloaded old word (think “catastrophe” or “chaos”). But I chose to coin a new word—one not directly evocative of anything in the past. I wanted to convey the idea of a broken stone, something irregular and fragmented. Studying Latin as a youngster taught me that it is a very concrete language. My son Laurent’s Latin dictionary confirmed that the adjective fractus means “broken” or “shattered.” From this adjective, I thought of the word “fractal.”

  We scrounged around IBM and put together tools to produce a camera-ready manuscript. I had been “managing” the book from cover to cover with a tiny “staff”: one constantly changing full-time programmer (and French typist) and one or two part-time associates. This was a powerful moment of triumph against seemingly overwhelming odds. An exhilarating experience for everyone involved, pushing me to the limit and demanding enormous effort from everyone on the project.

  No book is published without some expectation of success, but for the original French preview, the chances of success could not conceivably be forecast. Flammarion had agreed to take the risk of publishing the book only because I had been introduced to the boss by a mutual friend. Sales were slow at first but after a while picked up nicely, and the fourth revised edition is in print today as a popular pocket book. Many years after the first publication, several French mathematicians confided that my book had a great influence on them when they were students. In 1975, however, this bright fate was far in the dim future.

  My slim volume was made part of an illustrious series that had at one point published Henri Poincaré, Jean Perrin, and Louis de Broglie. In 1975, it was barely alive, but it seems that my book revived it.

  When I offered a copy to Szolem, he first congratulated me nicely, then thumbed through it and, seeing it was not a math book, asked, a bit testily, “But what kind of book is it? For whom have you been writing?” My answer: “I don’t know but hope it will create a readership for itself, perhaps even a large one.” My cousin Jacques was present; amused, he asked his father, “In your case, when you write a book, you always know exactly who is going to read it, right?” Szolem responded, “Yes, there are about fifteen people in the world who read everything I write. That is enough. I find that very comforting.”

  A tiny event comes to mind. New books in French were few, but bookstores were numerous and prominent—many occupying locations now taken by travel agents and off-price stores. I came to know personally a lot of the hands-on owners or managers. The manager of the bookstore Offilib was a friend since the time I helped him settle down. He took me aside for a piece of advice: “Your book is marvelous, enchants everyone. But watch out: don’t let yourself be carried away and spend the rest of your life trying to improve it. Go back to something standard and build yourself a reputation that will ease your career.” Advice that—of course—I failed to follow. That bookstore sold so many copies of my 1982 English book that the projected French translation was called off.

  The 1982 Book, The Fractal Geometry of Nature

  Going to Harvard in 1979, I carried with me the computer tape of what I perceived to be the nearly ready third version of the book. But because that year turned out to be an annus mirabilis, the shadowy third version kept being expanded to mention the Mandelbrot set and the first mainstream physics papers. It also kept being reorganized in response to what I learned by teaching that first course on fractals in 1980. Ultimately, I started almost from scratch, and the much-expanded text went smoothly. I succeeded in persuading W. H. Freeman’s top brass to charge a low price for the book and include a sixteen-page color signature (at a time when color was still perceived as expensive) because I felt it would be a good investment. And it was. As feared, the book ran late, but the color signature was available and I took it around to meetings.

  First Fractals Meeting in Courchevel

  In July 1982, while waiting for copies of The Fractal Geometry of Nature to be shipped, I had the pleasure of delivering its content to representatives of the scientific world and watching their reaction. The occasion was the first fractals meeting ever held. The venue was Courchevel, an exclusive ski resort high in the French Alps. The page proofs of The Fractal Geometry of Nature had reached me in Paris, where I spent hours at IBM preparing a photocopy for every participant. From Paris I lugged rock-heavy suitcases. The audience numbered about fifty and was very heterogeneous and not representative of any group, since fractals had no constituency.

  At this meeting, the first I ever organized, nobody could really help me. Almost every author of a contribution to fractals was invited to speak, and IBM branches in different European countries sent a few people whose goodwill seemed valuable. Half of the slots on the program remained empty, and I put my name in each, hoping that somebody in the audience would relieve me from this commitment. Miraculously, the down-to-earth Summer Institutes held before and after mine motivated IBM Europe to provide a computer of good size, by the standards of the day. In addition, my IBM colleagues and close friends Richard Voss and Alan Norton had come along.

  In the absence of core organizers, I wrote very few follow-up and reminder letters, and my travel instructions were complete but without frills. Many participants later confided that they were not quite sure that the meeting was actually going to take place, and reaching the conference hotel gave them a strong sense of accomplishment. Because none of the home institutions of the participants could afford such a machine and such skilled help, the computer room was filled until well past midnight.

  In the summer, ski resorts close or charge little, and the management promised that the hotel would be empty. However, when I arrived a few days before the meeting, the manager expressed profound apologies. The European Youth Orchestra had begged him to rent the vacant half of the hotel, and he had no choice but to agree. He assured me that the musicians would make good neighbors: they would be working so hard that nights would be very, very quiet. Besides, the orchestra promised to allow us to attend the general rehearsal before its job-seeking tour of the music festivals. Also, I would meet its leaders: the Chicago Symphony Orchestra conductor Georg Solti (who could not spend much time visiting with conference attendees because the high altitude sickened him) and Claudio Abbado, the future director of the Berlin Philharmonic. Illustrious past and shining future—not bad at all! When I opened the meeting on Sunday night, I could brag that the usual musical divertissement would be kindly provided by my friends, the maestros Solti and Abbado, live. Of course nobody believed me, but they realized at the concert that I had not been pulling their leg.

  I chaired the entire meeting and channeled the discussion vigorously. Also, even though I eventually found other lecturers, I gave a full quarter of the presentations. I had assumed that many participants would skip Friday afternoon, so I kept the last lecture for myself, and the session before that was taken by a friend who did not mind speaking to an empty room. But to our delight, the room was full until the very end. More surprising was that everybody attended every lecture. The mathematicians were amazed that what they considered to be safe esoterica was in fact part and parcel of nature. The physicists were amazed that many complicated problems could be solved in a simple and transparent way.

  All the Kepler moments of my life to that day had come together.

  Fractals Meetings and Birthday Celebrations

  A multitude of fractals meetings followed this one. Each became increasingly specialized. I expected this to happen, as did other scientists. I recall a meeting in Trieste where a journalist interviewed me with the meeting’s host, Stig Lundquist, then head of the Nobel Committee for Physics. The journalist was astonished to hear that the success of fractals depended on people being familiar with the basic ideas and pushing them in different directions with more specialized topics, resulting in fewer general fractals meetings.

  Se
veral of these meetings doubled as birthday celebrations. In 1989, for my sixty-fifth birthday, my physicist friends Amnon Aharony and Jens Feder, with support from IBM France, organized a marvelous meeting, the Fractals in Physics conference. It was held at Mas d’Artigny in Saint-Paul de Vence, high in the hills above the Riviera.

  After the meeting, Aliette and I stayed on for a day to allow profound exhilaration to cool off and then, still dizzy, took a short vacation. We drove past the place nearby where I had been a horse groom in 1944, and to fulfill an old curiosity, we splurged on a fabulous dinner at Frères Troisgros, the famed four-star restaurant in Roanne. Then we drove on to Tulle, that hollow in the mountains where I had spent several years during the war, which, after all those years, I still consider my true home.

  A bit later, Heinz-Otto Peitgen organized a meeting in Bad Neuenahr. The usual evening talk was replaced by an unexpected treat. My friend, composer György Ligeti, described the deep structure of a piece he had just written. It was part of the series of late piano suites that he did not manage to complete but that became one of the greatest contributions to his repertoire. The score was projected on the screen; his pianist had also been invited and helped the master musician deconstruct, then reconstruct, this very short but unforgettable piece.