Living in New Haven was considered but found impractical because I was mainly at IBM and, later, because IBM continued some perks. It was during these years split between IBM and Yale that IBM Research let go half of its staff and I retired. IBM granted me the title of Fellow Emeritus, the continuing use of my Yorktown office, and a few other benefits that were supposed to last two years but went on for thirteen. So we never moved, and missed much of Yale collegiality—a clear loss.
Commuting by car was tedious, and Aliette went beyond the call of duty by being the driver and enjoying Yale while I was working. The only good story related to commuting is when the architect Philip Johnson invited me to visit him at his famous Glass House. Over coffee, I looked at the rolling estate and observed, “A Connecticut forest is far thicker than that. This looks like the fractal view of Italy as painted by Claude Lorrain. Did you arrange it to fit?” “Of course I did—just look behind you.” I turned and saw—unframed and on an easel—what seemed at first glance to be a genuine Lorrain. Dumbfounded, I forgot to ask whether the priceless painting was permanently placed there. Now that Johnson has died and his estate is a museum, I wonder how they deal with such a masterpiece.
Michael Frame, Friend and Colleague
A special pleasure of my Yale years has been the company of Michael Frame. I met him on a visit to his previous place of work, Union College. Soon I invited him to Yale for a year, during which he gave an immensely popular undergraduate course on fractals. After several further visits, he was made an (indispensable) adjunct professor at Yale.
Michael supervises the mathematics introductory courses but also teaches an elementary and an intermediate course on fractals, for which he has prepared an extensive set of course notes. Moreover, he ran very important summer programs for high school teachers. All his courses are extremely popular, we wrote papers together, and our discussions on mathematics and everything else have been one of the nicest aspects of my time in New Haven.
In 2002, we collaborated on the book Fractals, Graphics, and Mathematics Education, a compilation of articles written by teachers of fractal geometry. These teachers first gathered in December 1997 at a meeting Michael and I held at Yale. As far as I know, this was the first scientific meeting totally dedicated to the teaching of fractals.
Sterling Professor of Mathematical Sciences at Yale
After twelve years as an adjunct professor, I was given tenure as Sterling Professor. An academic’s dream—not only in the United States—is tenure at a great university. But having left the École Normale in 1945, I forgot about academia and moved on. Eventually, I did achieve this dream, but only in the nick of time—in 1999, when I was seventy-five—and on a half-time basis.
“Sterling” is a word with many connotations. It came to matter to me that in the 1920s a grateful Yale alumnus with that family name gave a fortune to Yale. Enough for two buildings to be named after him—a first-rate library and a suitable law school—and professorships chosen by a process whose outcomes ranged from obvious to mysterious. When American academia began to appoint University Professors, Yale merely decided that this ill-defined but always exalted role should be assigned to its Sterling Professors. By a rule inherited from Cambridge or Oxford, Yale must recruit its tenured faculty exclusively from among its own graduates—a requirement harder to amend than to satisfy by a loophole: a special-purpose master of arts degree.
How did this honor affect me? I had often demonstrated the capacity to formulate big dreams that everyone else held to be odd and unreachable—but that I managed to fulfill. A Sterling Professorship of Mathematical Sciences was beyond any such dream, but I was glad to enjoy it as a fitting end to a “march up Mount Parnassus” from such a colorful and crooked path.
Did I perceive the grant of a Sterling as the instant when a maverick cocoon molts into an establishment butterfly? Frankly, I did not. Perhaps because of the studied informality of the event: the president of Yale telephoned, campus mail brought a computer-printed certificate, and the departmental tea served champagne. That was it. Nothing significant changed. I may add that the alumni magazine had planned to feature my arrival in 1987 but did not rush: in fact, it waited long enough to feature together the still-recent Sterling and my forthcoming retirement. The absence of a discontinuity had a deeper reason: I was already at Yale and had no declared enemy.
So Yale was a rousing success where Harvard was not. How to account for this? In terms of awards and membership in academies, the difference between those two mathematics departments was small, but I encountered an altogether different mood. In the 1930s, the Yale mathematics department had been driven by a bitter split between two leading figures: a Norwegian and a Swede, brothers-in-law who became bitter enemies and pushed everyone to choose a side. That dark era was and remains a spur to a strong collegiality.
Isaac Newton Institute
The Isaac Newton Institute for Mathematical Sciences in Cambridge, England, bears some similarities to the Mittag-Leffler Institute in Sweden, but it is larger and of a broader scope. In 1999, from January to April, it held a program on fractals.
The University of Cambridge kindly offered me a visiting Rothschild Professorship, but had to withdraw the offer after finding that I exceeded its retirement age by ten years. However, Gonville and Caius (pronounced “keys”) College made me G. C. Steward Visiting Fellow. Quite an experience! Also, the Cavendish Laboratory made me Scott Lecturer in physics. Between those lectures and many seminars, I had my hands full. The Caius fellowship included a furnished house. I had not biked since Tulle and did not dare try to revive that skill, so I took long walks between the house, the Newton Institute, and Caius College; my doctor was pleased.
(Illustration Credit 27.1)
When I was ready to leave, I was informally told that a long-term visiting fellowship in another college was mine if I was interested. Aliette and I were both extremely tempted, but Yale came up with the Sterling Professorship, which I had no question of turning down, and later grandchildren came and brought us to the other Cambridge, in Massachusetts.
28
Has My Work Founded the First-Ever Broad Theory of Roughness?
HOW CAN IT BE that the same technique applies to the Internet, the weather, and the stock market? Why, without particularly trying, am I touching so many different aspects of so many different things?
An important turn in my life occurred when I realized that something I had long been stating in footnotes should be put on the marquee. I had engaged myself, without realizing it, in undertaking a theory of roughness. Think of color, pitch, heaviness, and hotness. Each is the topic of a branch of physics. Chemistry is filled with acids, sugars, and alcohols; all are concepts derived from sensory perceptions. Roughness is just as important as all those other raw sensations, but was not studied for its own sake.
I started almost from scratch and had to create a new toolbox specifically geared toward the study of forms of roughness that possess certain geometric scaling invariances. Each invariance intrinsically introduces one or more numerical invariants. I reinterpreted one as the first of many quantitative measurements of roughness.
Later, many additional intrinsic measurements were also brought up by fractal and multifractal geometry: it even made a set’s “degree of emptiness” into a concrete and useful notion.
In 1982, a metallurgist named Dann Passoja approached me with his impression that fractal dimension might provide at long last a measure of the roughness of such things as fractures in metals. Experiments confirmed this hunch, and we wrote a paper for Nature in 1984. It brought a big following and actually created a field concerned with the measurement of roughness. I have since moved the contents of that paper to page 1 of every description of my life’s work.
Before my work on roughness, it was either undefined or measured by too many irrelevant quantities. Now it can be measured by one, two, or a few numbers.
The Brownian Coastline Leads Me to the Number 4/3
Weig
ht has long been measured by number, and sensations like color and pitch have long acquired purified forms to which one can attach a well-defined and measurable frequency. But what about roughness? When the great philosopher Plato wrote about sensations, he covered roughness in a mere few lines.
Shortly before I was born, mathematician Felix Hausdorff (1868–1942) assigned to those irregular mathematical shapes called monsters a number he chose to call a “dimension,” a word I have referred to in this book. Having heard of its curious mathematical properties, I wondered whether it was irremediably theoretical or if it could be removed from pure esoterica and reinterpreted into something intuitive, concrete, and even practical. It can indeed!
And the icing on this cake is the story of the Brownian island coastline. Brownian motion’s ups and downs first shined in Bachelier’s ill-inspired but profound model of price variation. Now forget about prices and imagine a point that moves on a piece of paper in such a way that its projections on the left and bottom sides of the sheet are Brownian motions independent of one another. That point is said to perform plane Brownian motion. This concept became widely used in physics and mathematics. But oddly enough, it seemed that no one I ever heard of had examined it on actual samples. So I drew a very long Brownian sample and set myself a challenge. I tried to blend two properties already established by mathematical reasoning and search for new properties that a skilled visual inspection might allow me to observe.
My first efforts were fruitless. Any possible novelty was overwhelmed by a multitude of messy old structures that were begging to be removed.
In particular, my Brownian intervals were deficient—unnatural—from a certain aesthetic viewpoint, one that is familiar in a far simpler shape, as a straight interval’s end positions differ from its middle portion. My first step was to eliminate this complication by creating a loop—as when the straight interval from 0 to 1 is made into the circumference of a circle. Obliging the Brownian motion to end where it had started yielded a distinctive new shape I called a Brownian cluster. But it still had too much irrelevant detail, and the elimination of this extraneous complexity demanded one more step. After many false starts, I separated the picture into two parts by “painting” white for all the points in the plane one can reach from far away, and black for those points “screened” from far away by one or another piece of Brownian motion. The result was astounding.
(Illustration Credit 28.1)
Instantly—but not a second before!—an interesting new island emerged. Automatically, my visual memory recognized some actual islands as well as some islands produced by fractal computer models I had previously devised. The new island’s ragged coastline suggested a new concept, that of the boundary of Brownian motion. In the antivisual world of yesterday, this concept had not occurred to anyone. The picture did not “visualize” any existing question. In this case, the picture had to come first and the question later, as a “caption.”
Extracting a Brownian island’s boundary from Brownian motion achieved something, but less than did the next step, which quantified that resemblance by injecting a numerical measure of roughness. (Contrary to algebraists, who loathe pictures, true geometers accept numbers; we are open-minded to a fault!)
Visually examining the Brownian island’s coastline led me to conjecture that its fractal dimension is 4/3. We promptly measured it and got closer to 4/3 than expected. At that point, I conjectured the value of 4/3 to be mathematically exact.
This experiment was successful in two ways. It confirmed that, even in a hard science, the eye can be retrained to discern new conjectures that might have escaped algebraic analysis forever. It also gave mathematics a new direction to follow. Today, the boundary of Brownian motion might be billed as a natural concept. But this concept could not have progressed to the fractal dimension 4/3 without a careful visual inspection.
My 4/3 Conjecture Spurs a Search for an Elusive Truth
At IBM, where I was working at the time, my friends went on from the Brownian to other clusters. They began with the critical percolation cluster, which is a famous mathematical structure of great interest in statistical physics. For it, an intrinsic complication is that the boundary can be defined in two distinct ways, yielding 4/3, again, and 7/4. Both values were first obtained numerically but by now have been proved theoretically, not by isolated arguments serving no real purpose but in a way that has been found quite useful elsewhere. As this has continued, an enormous range of geometric shapes, so far discussed physically but not rigorously, became attractive in pure mathematics, and the proofs were found to be both very difficult and very interesting.
To prove a purely mathematical conjecture, no number of pictures or examples suffices, but the value 4/3 is so simple that a rigorous proof seemed easy—and indeed wonderfully skilled friends promised to prove the conjecture overnight. They revised this promise to next week, next month, next year—and finally to the twenty-first century and third millennium.
The proof turned out to be extraordinarily tricky, and the success of an eighteen-year worldwide search created an enormous sensation and generated great enthusiasm and activity. The three mathematicians who combined their skills to achieve it won instant acclaim, and in 2006 the youngest of the trio received the prestigious Fields Medal, the award for exceptional mathematical promise. Not only did the difficult proof create its own very active subfield of mathematics, but it affected other, far removed subfields by suddenly settling many seemingly unrelated conjectures.
It was the first Fields in probability theory, but in previous years my key conjecture concerning the Mandelbrot set had already led to two Fields. In an ironic way, my disregard of the customary division of labor has advertised that, in mathematics, the labors of conjecturing and proving may gain by being divided.
My Work Reaches a New Audience
All of this activity has taken me around the world, lecturing, meeting with groups, and showing my pictures.
(Illustration Credit 28.2)
I often hear comments like this from people of all ages: “May I shake your hand? In this country, your fractal geometry is discussed in high school. So we all first heard your name and saw your mathematical pictures several years ago, and we just assumed—without thinking—that you have long been dead. You might have lived shortly after Newton. We can’t believe that we could actually hear you discuss how part of our schoolwork had first come to your mind. To shake your hand would be a strange experience … a big event.”
(Illustration Credit 28.3)
Words from a charming young lady seemingly representing a group of college students who had packed a lecture I had just given. Of course, I was glad to shake that young lady’s hand. Uncanny forms of flattery! Each lifted me to seventh heaven! Truly and deeply, each marked a very sweet day! Let me put it more strongly: occasions like that make my life.
29
Beauty and Roughness: Full Circle
A MEMOIR IS A LESSON IN HUMILITY. I was born in 1924, and it is now 2010. To put my personal achievements in perspective, those dates matter indeed. The Great Depression dominated the earliest world news that I recall, and another depression threatens to dominate my last days. My late adolescence coincided with World War II, which I spent in the impoverished hills of central France. My survival was continually threatened, but my dreams ran free and seeded my future.
Does it matter that I stumbled into IBM Research when its golden age began and stayed until the day it ended? That it is where my wartime dreams finally managed to be realized?
Of those born in the year 1924, I am sure that many became scientists. What made me seek out a role that others missed or spurned? I have always wondered, and I wrote this book in an effort to understand myself.
When I turned thirty-five, I questioned my life. Had I, in my dreams of leaving my mark on science, really “missed the boat”? I am keenly aware that this fear led me to reinvent myself surprisingly late in life, when I did my best-known work. My refoundation of fin
ance was to occur as I neared forty, and the discovery of the Mandelbrot set came at fifty-five. For a scientist, those are unusually—astonishingly—old ages, as many witnesses have noted. And the number of would-be role models I have considered but not followed has been heartbreakingly large.
Had my work on price variation been accepted in the 1960s, I might have settled as a satisfied slave of my creation. Who knows. But events proceeded differently. I was expelled to resume my wandering intellectual life. No official Galilean trial had to punish me for attempting to propagate disallowed ideas. No one was listening, and I had no need to turn my face away to say, very softly, of Earth, “Still it turns.”
What has attracted me to problems that science either had never touched or had long left aside—continually making me feel like a fossil? Perhaps a deficit in regular formal education. My adolescence during the wartime occupation of France was illuminated by obsolete books, ancient problems long abandoned without solution, and timeless interrogations. The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer. In due time, it turned out to be an elusive point where formula and picture meet on even terms, where theory meets the real world, and where mathematics and hard science meet art so that their worth and beauty shine far beyond the narrow world of experts, bringing an element of unity to the worlds of knowing and feeling.
Since I became a scientist, much of my work has consisted of bringing a medley of old issues back to life and triumphant evolution. While they seemed to share little beyond common antiquity, they all eventually revealed themselves as being concerned with roughness in nature and art. Surprisingly, a loop seems to have been established between structures that were first identified for mere decoration, then, much later, introduced by mathematicians for the purpose of pathology. Again, even later, these same structures were used by me for the purpose of science and, unwittingly, as a bonus, for the purpose of creating beauty.
The Fractalist Page 29