The Fractalist

by Benoit Mandelbrot

  Far more important than tuition, the Caltech course catalog and faculty directory, hot off the press, overwhelmed me with disappointment. All too many of the stars who had made the older catalog so attractive were gone. The specifics varied from case to case. Hugely disappointing, was that the physicist J. Robert Oppenheimer—of wartime Los Alamos fame—had moved to the Institute for Advanced Study in Princeton, New Jersey, where we would meet in 1953. More generally, by Caltech’s past and future standards, theoretical physics was at a low point.

  The Caltech faculty was undergoing a complete overhaul, accompanied by a shift of emphasis. The reason was that the school’s de facto founder and for many years de facto president, Robert A. Millikan (1868–1953), had built Caltech by bringing in many friends in the same age bracket. They were now all retiring or otherwise gone. Freshly retired himself, Millikan was bored and available to students for lunch and chats. These took place at a club called the Athenaeum; one of the big assets of Caltech in that both faculty and graduate students are welcome. Flashing ahead, I was once having lunch there with Millikan when a wan and shabbily dressed gentleman came forward and, very formally, bowed and introduced himself as Laue. I recognized a true giant of physics, Nobel in 1912, whom Wilhelmine Germany made Max von Laue. He had not bent to Hitler, but sad to say, Millikan treated him imperiously nonetheless. I felt and continue to feel that good Germans deserved better.

  The math offerings at Caltech were limited. The one professor with any name recognition was Eric Temple Bell, though it was largely earned for a collection of biographical vignettes of individual masters titled Men of Mathematics. That book was both blamed for historical howlers and credited with enriching the field with many enthusiasts. Intrigued by the man, I often attended his Sunday afternoon open houses just across the street from Caltech in a neighborhood the faculty liked and could afford. A crusty Briton, he kept attacking the much-discussed proposals for federal funding of scientific research through the National Science Foundation. He viewed all federal support as a potential threat to existing local collegiality and the proper decentralization of the U.S. decision-making system—a move toward the horrors of French-style centrally controlled departments but scattered locations. He was farsighted, but I disagreed until much later, when the NSF had consolidated into a bureaucracy from which a solo throwback like me could receive only crumbs.

  Fluid Dynamics in a Period of Maturity

  The worst disappointment of all was that, although the Caltech catalog continued to list Theodore von Kármán as active faculty, he was on leave and had taken up residence in Paris! He had never married, and his sister—also unmarried—managed his household. They had lived in Belgium while he was a professor at Aachen, just across the border in Germany, and she accompanied him to Caltech. But as soon as the war was over, she wanted to return to Europe, where she settled in an elegant hotel in Paris. He dropped by Caltech several times during my stay, then retired.

  Even worse was the replacement of the famed “Kármán circus” by people hardly anyone had (yet) heard of. Fluid mechanics as a whole had become an extremely competitive and “mature” field that was growing slowly and splitting. The rules of fluid motion—called Navier-Stokes equations—are of infamous difficulty. Pure mathematicians and physicists had little to contribute, so it was left to engineers. Another problem: one of their leading textbooks was written nearly a hundred years earlier by an English don, Horace Lamb, before he became Sir Horace.

  In 1947, a key research question was what would happen if an airplane could accelerate enough to reach the sound barrier. Ivory tower theoreticians agonized in one world, and adventurers in another made immense amounts of money to fly unproven rocket-powered contraptions that might or might not take off, fly, or land safely. Kármán’s vaunted combination of theory and practice was no longer a glue that held the worlds together.

  One saving grace was that while the dynamics of smooth flows had matured, the study of turbulence had not. In fact, it was only just beginning to reveal its devilish complication. Story has it that when the great physicist Enrico Fermi (1901–54) was close to death, his friends wanted to know what his first question would be when he met his maker. “What is the cause and nature of turbulence?” was Fermi’s response. In other words, “What is the essence of Navier-Stokes?” Altogether, watching Caltech colleagues at work on turbulence and so-called spectral, or harmonic, analysis was to serve me well in the future.

  Living beings mature, then age and die. However, a science may very well move back from sagging maturity to wild youth. This is what happened to general relativity theory, a mature field in 1950, and to the mathematics beloved by Szolem. Later, the theory of chaos contributed to fluid dynamics and brought me back to it for an important effort: developing a concept called multifractals.

  Tolman, Liepmann, and Other Unforgettable Courses

  Not surprisingly, the Caltech curriculum was a letdown after Carva. A course in elasticity was compulsory, but the one I had at Carva was more advanced; therefore, I dared to cut many lectures. My final exam received an E—instead of solving the problem, I showed that it could not be solved because something was missing. I protested and had (in effect) to teach some delicate points to the lecturer. He relented—but refused me an A.

  The course work reached a high point with the swan song lectures of Richard Chase Tolman (1881–1948) on statistical physics, or thermodynamics, an extraordinarily difficult and subtle topic that brings many seasoned scientists to beg for help, run away, or make dreadful mistakes. Tolman was no technical acrobat and was about to retire, but he started the class with a warning—his course was for those who already knew the subject and was not intended to teach thermodynamics. But he promised to show why it worked. That he did and explained away many of the mysteries that had baffled me at Carva. Learning about this topic from a seasoned master affected my work over much of my life, and helped my thesis and some papers add a few wrinkles to the logical foundations of thermodynamics.

  After Tolman, I learned most from the fluid mechanics course of Hans Wolfgang Liepmann (1914–2009). He did not put the stress on formalism, but on proper understanding of the physics. Once, when he was criticized for his harshness, I heard him say that “simply being Jewish does not prevent me from being a true-blooded Prussian.” He was the only professor I ever feared.

  I also recall fondly two teachers outside of science. The pride Caltech derived from requiring courses in the humanities was well justified. Wallace Sterling, who taught Shakespeare, had been moonlighting from the Huntington Library next door and was also a respected radio columnist. He soon left to lead Stanford University to its present high distinction. Horace Gilbert taught a course on economic institutions. His staunchly conservative stance was not to my taste, but the class was fun and I learned a great deal.

  Airplane Design with Paul MacCready

  On the practical side, I have a fond recollection of a part-time instructor named Klein. A Caltech physicist before the war, he had become a personal technical consultant of sorts for Donald Douglas, the founder and namesake of a then-flourishing aircraft company. This day job made him arrive late and totally unprepared, and he would entertain us with gory stories from the real world. For example, the ads boasted that automobiles were becoming a bit wider every year. Klein told us why: the stamping machines of that day wore out fast. The cheapest solution was to replace the hard surface of the “male” part and gouge the softer “female” side. Klein’s stories kept alive for me Father’s love of machines and gadgets.

  A far more important requirement challenged us: to design a refueling jet tanker—minus the engines. We split into teams of four to deal respectively with the fuselage, wings, tail, and landing gear—which I was in charge of. World War II planes—like the workhorse DC-3—had a small fixed wheel near the tail. It was easy to design, but on the ground made the plane lean back awkwardly. Big retractable wheels in the front—like on the DC-4—made the planes horizontal on the g
round, but still posed many design challenges.

  Our specific design was not memorable. Of course, the aircraft companies’ experts were reading the same books and magazines we were. So when the tanker precursor of the Boeing 707 came out, I recognized “our” approach—and still follow the evolution of its successors. Passenger airplanes have greatly improved in every detail—but not overall. This gives continuing evidence that airplane design has been since 1947 a “mature” field. Spending a lifetime on such details would have been a dreadful experience, and did not tempt me at any point.

  One teammate, Paul MacCready (1925–2007), was unforgettable, and despite our many differences, we were friends. He became an imaginative, tireless, and thoroughly old-fashioned independent inventor whose company, AeroVironment, was motivated by curiosity rather than profit. His fame soared when he designed “airplanes” in which a (well-trained) pilot’s leg power was transferred to a propeller through the pedals and gearbox of a racing bicycle. MacCready’s fascination with bird flight was evidenced by the names he selected for his aircraft. The Gossamer Condor tested the concept, then the Gossamer Albatross crossed the English Channel at low altitude—just above a flotilla ready for any emergency. The later Gossamer Penguin used solar power, as did the Solar Challenger, which crossed the channel at a higher altitude.

  But all that fun came later. At Caltech, he spent his weekends practicing soaring—that is, imitating the birds by flying gliders, light planes with no engine. Only later, when I was serving in the French air force, did I hear from pilots that this soft-spoken man had been the U.S. soaring champion many years in a row before becoming the world champion. A wider community named him engineer of the century. Calling himself an ambivalent Luddite, he advocated unbridled thought. He was a lucky bastard who managed to never leave the sandlot and just kept playing. We always felt in tune but did not see each other often enough. A splendid man.

  The Mathematical Faces of Mechanics

  At Caltech, one of the mathematical faces of mechanics was represented by my glorified master’s thesis. Frank E. Marble assigned a topic in propeller theory, and I worked out the complicated calculations—without either of us becoming too involved. Frank and I remained friends, and he boasts that he helped save me for higher pursuits.

  Another face was personified by the mathematician Paco Axel Lagerstrom (1915–89). A brilliant and cultured Swede, he was strange and mysterious—admired by some students, tolerated by a few, resented by many. We often met socially, so I learned that Paco’s evolving taste had moved him from divinity to philosophy, logic, very pure mathematics, and then a flavor of mathematics he viewed as applied—but I didn’t. On a rare visit, Kármán asked me to describe the topics I was considering for my dissertation. Soon after I began, he broke in to demand which fool could have suggested such an antiphysical topic. I had no choice but to point at Paco, standing beside me. He was called on in turn, and Kármán did not treat him kindly.

  After that incident, my relationship with Paco deteriorated. A course I was taking with him ended with an oral exam. He gave me an A, and then told me, “I think you should not start on a Ph.D. with me because you don’t admire me enough.” He was right and I appreciated his bluntness. Inertia might have led me to try working with him, but the sequel would have been either brief or regrettable. Unfortunately, he was the only professor at Caltech I could ask to supervise my doctoral work. This meant leaving Caltech without a doctorate.

  In a defeated mood, I for a moment yielded to the force of gravity that continued to pull me to pure mathematics. I rejoiced when a top math department, at the University of Chicago, seemed to offer an assistantship. But that assistantship was actually not funded. They said I should register anyhow, because the great mathematician Saunders Mac Lane badly needed a teaching assistant for his algebra course and would find a way to support me. Algebra was (and remains) my least favorite topic in mathematics, and I was not ready to face uncertainty for its sake.

  No Doctorate, but Quality Education and Community

  First and foremost, it is through Caltech that I met Aliette—although this meeting and our marriage did not occur until years later.

  More broadly, the small crowd that chance or necessity had collected at Caltech during my time there was of a quality that I rarely saw anywhere. The intellectual excitement and the feeling of living through uniquely exceptional times were palpable and exhilarating, though burdensome, and persist in my mind to this day. This is also how the world came to feel, since that very small school, over those very few years, produced astounding numbers of Nobel Prize winners and the like. Also, Caltech may be one of a kind in not increasing its permanent faculty.

  (Illustration Credit 10.1)

  I had great fun, enjoyed the Southern California outdoors, and made many lifelong friends. The physicist Donald Glaser and I often went to concerts together, and I followed his career closely. As an experimental high-energy physicist, he invented the bubble chamber, which experts in thermodynamics had declared impossible—against the laws of physics because it contradicted a statement in a book by established physicist Enrico Fermi. That statement was revealed to be incorrect, the chamber became a basic tool, and Glaser earned a Nobel for it. Only then did he reveal that he had “converted” from physics to molecular biology. True to form, he attended the meeting on my seventieth birthday celebrating my versatility and regaled the participants with tales of his.

  Particularly important was Caltech’s Inter-Nations Association, which attracted foreign students and young people from town. The school assisted the INA—possibly even sponsored it. We learned about the New World and described the old one to young Americans who had not yet seen it for themselves.

  One INA regular was Paolo Comba, a math student from a Protestant corner of Italy. Our paths crossed again when we were both at IBM. In retirement he went on to discover many minor planets and marked an old friendship by discreetly naming one after me. As a young scientist, he worked on baby tomatoes, predicting they would move fast from the lab to the grocery store. They did.

  Caltech confirmed my cynicism about an opinion taken for granted at Carva—that a school’s elite status is largely based on attracting elite students. My age cohort included several elite students, but more important were the many with complicated, often heroic backgrounds in wartime.

  Max Delbrück and the Birth of Molecular Biology

  On the small Caltech campus, the burning center of intellectual life was not found in aeronautics. It was a group led by a man of ambition, brilliance, and independence of mind: the great maverick Max Delbrück (1906–81).

  After an inconsequential year in mathematics and aeronautics, I met Gunther Stent (1924–2008), who was at that time a physical chemist. He introduced himself as an incoming postdoc with Delbrück and told me that in a few days another postdoc, the microbiologist Elie Wollman (1917–2008), would arrive from the Pasteur Institute in Paris, along with his wife, Odile. In no time, Gunther and the Wollmans became lifelong friends. Soon I met the phenomenal D. Carleton Gajdusek (1923–2008), an academic superstar. I had moved socially and intellectually over to the in crowd.

  Then and there, despite the reservations and declared hostility of several well-established guilds, Delbrück was orchestrating the birth of a new way of being a biologist. At Caltech at that time, the word “biophysics” was forbidden. But soon their work would become “molecular biology.” And in 1952 this field would come to be known universally, in response to the discovery of that icon of natural geometry—the double helix of DNA. Eventually, molecular biology merged with biochemistry, and genomics took it to an industrial stage. Today’s practitioners complain of it being viewed as a mature field. But in 1949, nothing was further removed from the slow-moving maturity of fluid mechanics.

  A high Prussian aristocrat—a Junker—Delbrück had to leave Germany because he would not swear allegiance to Hitler. One of the would-be assassins of the Führer was a cousin of his. How did he manage h
is unprecedented transition from physics to biology? His early years were unspectacular. As a physicist, he felt that he was hopelessly behind Hans Bethe (1906–2005) and Victor Weisskopf (1908–2002)—his near contemporaries in the entourage of Wolfgang Pauli (1900–58)—and a Delbrück would not settle for second best. The epoch-marking lecture “Light and Life,” which Niels Bohr (1885–1962) gave in 1932, sparked him to become a biologist.

  With Salvador Luria (1912–91), later a Nobel laureate with him, Delbrück wrote a paper that Erwin Schrödinger (1887–1961) noticed and mentioned conspicuously in his book, What Is Life? When the war ended, Caltech went recruiting and gave Delbrück his first real job as a full professor of biology. In time, crowds of physicists followed Delbrück’s new path into biology and the once-spurned biophysics won acceptance. So what did he do after his field became established and ubiquitous? True to his temperament, he left this field for another far less explored one.

  A Belated Delbrück “Treatment”

  Delbrück’s personality would never be described as mild. One day, I noticed that our parties were no longer graced by the presence of a man whom I only recall as Harold. Having asked, I was told that Harold had received the “treatment,” did not do well, and was gone.

  The Delbrück treatment remained a mystery until I ended up receiving it myself many years later. In 1979, the physicist Richard P. Feynman (1918–88) invited me to return to Caltech to give a lecture on fractals (this was right before I discovered the Mandelbrot set). He and Delbrück sat next to each other just under my nose. Throughout, Feynman nodded and smiled approvingly. Delbrück remained stone-faced, and as we walked out, he said casually, “Benoit, won’t you come to my office tomorrow morning at eight.”