At first the experts thought they could use an old theory developed in the 1920s for telephone networks. But as the Internet expanded, it was found that this model would not work. Next they tried one of my inventions from the mid-1960s, and it wouldn’t work either. Then they tried multifractals, a mathematical construction I had introduced in the late 1960s and into the 1970s. Multifractals are the sort of concept that might have been created by mathematicians for the pleasure of doing mathematics, but in fact it originated in my study of turbulence. To test new Internet equipment, one examines its performance under multifractal variability. This is even a fairly big business, from what I understand.
25
Annus Mirabilis at Harvard: The Mandelbrot Set and Other Forays into Pure Mathematics, 1979–80
WHAT I ACHIEVED OR STARTED during the spring of 1980 went well beyond the wildest dreams of my adolescence under foreign occupation.
“I see you carry another batch of computer pictures. Are they your latest? May I have a look? Hmm … To me, they tell absolutely nothing. How can one possibly extract any kind of mathematics from such squiggles? Can this game really concern that ancient theory of Pierre Fatou and Gaston Julia on the iteration of rational functions? Their time seems long past.”
When and where did I first hear comments and questions like those, and what motivated them? It is true that I have heard many such utterances throughout my life, but I heard them with special intensity during the 1980 spring term whenever I approached the pigeonholes that held the professors’ incoming mail.
(Illustration Credit 25.1)
It was my third yearlong visit to Harvard, but my first in the mathematics department—a visit quite different from the earlier ones. A new world was being revealed to mathematicians … or perhaps an older world was being painfully revived.
Day after day, colleagues, students, and passersby witnessed a slowly unfolding process—one I had never lived through, and one the Harvard community of pure mathematics had not experienced for generations and did not in the least expect. For me, the process was intoxicating. For the mathematicians, it was baffling at best, and in many cases unwelcome or worse. This process was a step-by-step transmutation that began with almost meaningless ink smudges that were transformed first into rough observations, then into increasingly more precise ones, and finally—insofar as I am concerned—into fully phrased mathematical conjectures. The resulting pictures were amazing.
These pictures were intriguing objects I then called lambda and mu-ma—alternative ways of representing a fundamental new mathematical structure that became known as the Mandelbrot set. It has been called the most complex object in mathematics, has become a topic of folklore, and remains my best and most widely known contribution to knowledge.
(Illustration Credit 25.2)
At first there was one “island,” then more. As these “offshore” islands began to appear, they were hard to differentiate from specks of dirt.
(Illustration Credit 25.3)
I could only prove the simplest conjectures. I knew I would be unable to prove the harder ones, so I had to abandon them, complaining all the while and loudly calling for a full and rigorous proof. Skilled mathematicians at Harvard and in Paris were informed and soon gathered; shortly after, they proved several of my conjectures and many of their own. As decades passed, numerous additional conjectures joined mine, and many have been proved in exquisite ways. My first key conjecture has been rephrased, yet has survived multiple expert searches for proof and remains proudly open.
Today—thirty years after those heady events—the branch of mathematics that my conjectures revived continues to shine brightly.
A Luncheon That Changed a Life
How did all this come to be? In the mid-1970s, I often saw Stephen Jay Gould (1941–2002), a lively paleontologist with multiple appointments at Harvard. Quite independently, we had become two very visible champions of discontinuity—he in paleontology and I in the variation of financial prices. Early in 1977, I was visiting Boston and called him to see if he was free for lunch while I was in town. He was, and a date was set.
He came with a mathematician friend from Harvard, the number theorist Barry Mazur. Barry often visited Paris, was fluent in French, and had read my 1975 Les objets fractals with enthusiasm. I showed him the brand-new, expanded English version, the 1977 Fractals. A spirited conversation was cut off by our respective schedules. It was a Friday, and Barry invited me to brunch at his house the next day. Who would refuse?
At brunch, he pressed me on two topics. One was the original papers and books of the early days of real analysis, the period around 1900 when it was viewed as a collection of diverse mathematical “pathologies”—toys, in my thinking. The second topic was the already substantial number of cases where I had transmuted such a toy into a tool. As we talked, Barry said, “You know, this would make a wonderful course in our department. The current course in real analysis has become so fast moving and streamlined that the concepts seem to come from out of the blue, unconnected to any motivation. I had thought of a supplemental course to fill in the gaps, but I don’t know the history that well and could not come up with even one real application. No one else could. Would you be interested in trying?”
Indeed, I was! Arranging for me to be invited, Barry could not have imagined what he was setting in motion. However, my younger son, Didier (a toddler on my previous visits), was to be a high school senior in 1978–79, so we could not leave that year. Instead, we settled on 1979–80. As it turned out, Didier went on to Harvard, so we all moved to Cambridge together.
As the fall of 1979 approached, the contract for my leave from IBM was not yet finalized. But I was told not to worry, because during the fall term I would just be visiting Harvard privately and would not teach until the following term, in the spring of 1980.
Physics in Broken Dimension
Having time to pursue research that fall allowed me to begin what became a long-term collaboration with Amnon Aharony, a physicist from Tel Aviv University who had been visiting IBM in Yorktown.
I listened to several of the talks he gave. After one of them, I made some comments. “You know what,” he responded, “you may be right. That crazy mathematical idea of shapes of fractional dimension may well have a useful bearing on my kind of physics. We must work together and take a close look.”
So we did and became deeply involved, first at Harvard during that year of miracles, and later in many places over many years. We investigated the use of shapes of fractional dimension. Most of our papers concerned spaces where dimension is not 1, 2, 3, or higher but a fraction, and brought fractals toward the mainstream of statistical physics.
This peculiar notion entered mathematics and physics independently, and each discipline responded differently. Mathematicians offered many definitions, while physicists proceeded heuristically, essentially asking if the calculation had predicted what was observed—the proof being in the pudding. In this collaboration, everybody’s skills were essential, and the result “smelled good”—though it was not final. This led me to put forward a bold conjecture: that solving the usual partial differential equations of physics can yield either familiar and expected smoothness, or fractality.
A Grand Old Problem Frozen in Time
How did the Mandelbrot set arise and provoke such strong reaction? Basically, from a challenge that I “inherited” from Uncle Szolem when I was a student in the 1940s.
“One of the oldest, simplest, and greatest problems in all of pure mathematics reached a peak decades ago with Pierre Fatou and your teacher Gaston Julia. Then—for lack of new questions—their work screeched to a halt. It must not be allowed to remain frozen. I tried myself, very hard, several times, to revive it, but always failed. Over a quarter century, everyone who tried also failed. Go and see what you can do. Here are reprints of their old papers. Hold on to them because they are rare and quite valuable.”
I took Szolem’s advice and the reprints, hoping to report b
ack that I went, saw, and won. I went, but saw nothing I could advance. Just as Szolem and everyone else had, I looked for questions combining sufficient novelty with a sufficiently good chance of being answered. I failed.
Early in life, I learned that for a scholar, nirvana is to take an unsolved problem that had been stated long before and solve it. I also learned that a mathematical problem could be well stated yet remain unsolved for a long time, even centuries, while a whole field develops around it. And I understood from readings and course material that a field might simply die for lack of manageable and interesting unsolved questions. All this brought faint solace.
As I see it now, the thoroughness of the failure to further the work of Julia and Fatou implied that the missing ingredient could not simply have been a more powerful or ingenious angle on mathematics as it existed in their time. For one thing, between the 1910s and 1950, power had drifted to the friends of André Weil, the members of Bourbaki, who quite deliberately focused on entirely different problems.
Then, in 1953–54, when I was at the Institute for Advanced Study in Princeton, another key ingredient began to take hold. My sponsor, John von Neumann, was trying to interest his colleagues in the equations of the weather. Only a handful of mathematical equations—already known in the eighteenth century—had solutions given by explicit mathematical formulas. In all other cases, including those for von Neumann’s study of the weather, no such solutions were worth dreaming of. To Johnny, this indicated that one should seek answers from numerical simulations using the computer. But Johnny died before convincing anybody.
An overwhelming majority of the mathematicians of that day shuddered at the very thought that a machine might defile the pristine “purity” of their field and deliberately erase the past. Starting with my work on prices, I immediately understood the power of the computer, even though I never learned to program one. When I was at Harvard, a colleague reported being astonished that a computer could help one of the Ph.D. candidates achieve control of a difficult mathematical problem. In no way did any mathematician expect that a great ancient problem could be revived by the computer. A revival does not happen all by itself—never has. In this case, it would happen because of a combination of chance events in my life.
One was a lengthy obituary of Henri Poincaré written by Jacques Hadamard, Szolem’s predecessor and sponsor at the Collège de France. After Hadamard died, Szolem put together his collected works, and he gave me a set. The obit dwelled on a dry-sounding mathematical topic called limit sets of Kleinian groups, which I knew about from books for advanced high school students. I was reminded of the time Szolem persuaded me to revive Julia and Fatou, and my interest was sparked once again. Having a Princeton math student as a “visiting assistant” for the holidays, I sought and formed a construction of Kleinian group limit sets.
A Turning Point in Mathematics
This story hinges on a very plain formula—the only one to be allowed in this memoir. “Allowed” does not mean having to be understood, appreciated, or acted on. It suffices to observe that the formula is very short:
Pick a constant c and let the original z be at the origin of the plane; replace z by z times z; add the constant c; repeat.
In mathematical notation, this instruction would reduce to three letters and three symbols. In mathematical lingo, this is a quadratic map, something close to an ancient curve called a parabola. But in the Mandelbrot set, z denotes a point in the plane, and the formula expresses how a point’s position at some instant in time defines its position at the next instant. Again, in mathematical lingo, this formula defines the very simplest form of dynamics in discrete time—a form called quadratic dynamics. Fine, the formula is indeed breathtakingly simple. So why bother?
This formula is then iterated—that is, repeated with no end—defining with increasing refinement a shape that can be approximated using a very simple computer program.
To wide surprise, this shape is both overwhelmingly rich in detail and minutely subtle, and it continues to provide a common and fertile ground for exploration: from Brahman mathematicians to students and those in the earthy lower castes, from artists to the merely curious. Its infinite beauty, appreciated by so many, was completely unexpected and brought forth countless challenges, which mathematics and philosophy have not yet exhausted. Immediately and with no prompting, zooming in on a point on this set’s boundary fascinates all eyes, young and old.
Needless to say, I don’t feel I “invented” the Mandelbrot set: like all of mathematics, it has always been there, but a peculiar life orbit made me the right person at the right place at the right time to be the first to inspect this object, to begin to ask many questions about it, and to conjecture many answers. Though it had not been seen before, I had a very strong feeling that it existed but remained hidden because nobody had the insight to identify it.
(Illustration Credit 25.4)
Zoom toward a point on the circumference of a circle and the curvature gradually “irons out,” yielding an increasingly straight line. But zoom instead toward a boundary point on the Mandelbrot set and what you see becomes ever more beautiful, wild, baroque, and complex in many distinct ways, which the set of color images in this book illustrate. I have heard it described as “pretty—yet pretty useless.” Important applications of new discoveries take time to be revealed, and we have seen that the Mandelbrot set has powerful redeeming features. Thought wanders to Napoleon’s saying that a good sketch, in all its complexity, is worth a thousand words, or even to the biblical Let there be light. By now, I should have become blasé, but I hope I never will. With due humility, these magic words of Charles Darwin apply:
From so simple a beginning, endless forms most beautiful and most wonderful have been, and are being, evolved.
More or less actively, I have lived with this set for over thirty years—and would have been thrilled to live with it far longer, were it not that success invites too many other seekers.
Preview of the Mandelbrot Set at the New York Academy of Sciences
The mathematical theory of chaos was a hot topic in the late 1970s and the focus of a big conference on nonlinear dynamics held at the New York Academy of Sciences in 1979. I had not yet discovered the Mandelbrot set, but I spoke about my work on iteration as it stood just before that key period and could not resist giving an idea of the slide show spectacular that I was beginning to carry around the world.
The audience was overwhelmed, and there were few questions. But there was a follow-up not to be forgotten. That was the day’s last session, and—to my delighted surprise—my IBM colleague and friend Martin Gutzwiller asked me to show those pictures again. Most of the audience stayed for the encore.
The proceedings of that conference became a major reference book. When the time came to turn in my section, I submitted instead the first announcement of the key facts about the Mandelbrot set. The announcement included several of those early pictures. Worried that the printer would think the pictures were ink smudges, I added this instruction: “Do not clean off the dust specks. These are real and important.”
I gave the same talk at Harvard, where a form of mathematical physics was a topic of broad and growing interest. The last of many questions came from David Mumford (b. 1937), an algebraic geometer and laureate of the Fields Medal, a professor of mathematics at Harvard, a colleague, and a warm host I wish to thank here. “Couldn’t the same approach devise a fast algorithm for Kleinian limit sets?” For a hundred years, this had been a goal of many mathematicians, including great ones, and probably also of countless amateurs. Astonishing but true (and, given the simplicity of the “trophy,” perhaps almost embarrassing), all those seekers had failed.
David asked if I could look at Kleinian limit sets. Delighted, I responded that—at least for one important special case—I had indeed discovered a construction and showed him the draft of my paper. He marveled, then observed that the tools I had used were ancient, utterly elementary, and certainly intimately familiar
to Poincaré, Robert Fricke, and Felix Klein—skilled men who had first raised the question a hundred years earlier. The self-inverse limit set on the left led me to the more general Kleinian group limit set on the right.
(Illustration Credit 25.6)
(Illustration Credit 25.5)
David wondered aloud what made me succeed where those seekers and so many others had failed. My answer distilled—once again—the already told story of my scientific life: when I seek, I look, look, look, and play with pictures. One look at a picture is like one reading on a scientific instrument. One is never enough.
At that point in history, Kleinian groups were in a holding pattern. Towering figures—including Lars Ahlfors (1907–96) at Harvard and Lipman Bers (1914–93) at Columbia—had made great strides. But the impression prevailed that their act was hard to follow, and hardly anyone was interested in my algorithm. But one day, the overly thin walls of my Harvard office allowed me to overhear the words “Kleinian group.” The speaker, who turned out to be a colleague, S. J. Patterson, confirmed that there was little interest in the topic. I convinced him that this perceived lack of interest deserved to be tested, and a seminar was organized. We had about thirty people at the first meeting!
Mumford naturally attended, and he became very supportive of my work. In record time, my assistants taught him computer programming. I also introduced him to David Wright, the student I overheard talking with Patterson. Mumford admitted that he held summer jobs as a programmer but thought that, as a Harvard graduate student in mathematics, he should not advertise this heresy. I assured him that before long it might cease to be one. Indeed, many mathematicians—though surely not all!—soon became enthusiastic about the power of the computer, and Mumford moved away from algebraic geometry, a field in which he was a major figure. He experimented on the computer with Kleinian groups richer in structure than those I had looked at. One of his earliest illustrations, implemented on a visit to IBM, first appeared in The Fractal Geometry of Nature. His interests have now moved on to a computer-based theory of vision.
The Fractalist Page 26