On June 30, 2019, Mitchell Feigenbaum died in New York City at the age of 74. Mitchell’s most celebrated work was his discovery of the universality of the period-doubling transition to chaos and the associated Feigenbaum constant δ = 4.6692016…, a number as universal as π or e. This result not only gained Mitchell recognition in the scientific community, but also helped bring the subject of chaos to the attention of the popular press, bestselling writers, and even filmmakers.1 Although he was best known for this work, Mitchell’s relentlessly inquiring mind and intense focus enabled him to contribute broadly across many fields.
I first met Mitchell in the fall of 1974 when we joined the Theoretical Division at Los Alamos Scientific Laboratory, newly rejuvenated under the leadership of Peter Carruthers. Our itinerant academic careers often took us in different directions over the ensuing forty-five years, but we remained close colleagues, travel companions, and dear friends. We were, for a time, also brothers-in-law.
To understand Mitchell’s persona and accomplishments, it is essential to recognize that he was not a typical twentieth-century, discipline-oriented researcher. He had much more in common with the eighteenth-century natural philosophers who considered all things in nature within their purview. Mitchell insisted on understanding everything on his own terms and always sought to find the core of a problem. This quest was Faustian in its relentlessness, and thus it is appropriate to refer to Goethe’s Faust to help focus these remarks. Mitchell, as it happens, was fascinated with Goethe, from the great drama Faust to Goethe’s theory of colors.
For Mitchell to consider that he knew something, he had to be able to derive it from first principles. He was never content to start with a shortcut, relying on someone else’s explanation or understanding. This approach often reminded me of a scene early in Faust Part I. While attempting to translate the Bible into German, Faust quickly becomes stuck on the very first line and the most appropriate translation of logos. Normally translated as “the word,” Faust eventually decides logos is better translated as “the deed.”
Mitchell focused on the deed, uncovering the core of a problem and thereby revealing something novel to the world. This drive was apparent in his ferocious powers of concentration. While studying the period-doubling transition to chaos, Mitchell often worked forty-eight hours at a stretch to keep everything in his mind. Although extremely proficient with computing, and computer algorithms in particular, Mitchell was always skeptical of the computer’s ability to reveal dramatic new insights in the absence of a conceptual schema to interpret the results. “It is hard to learn by seeing,” he once observed. “So why should the scales miraculously fall off when one stares at a computer screen?”2 Mitchell began his analysis of the period-doubling sequence using an HP-65 handheld calculator, a device that was relatively slow and had limited accuracy. As a result, Mitchell both had time to think and had to think as he tried to follow the behavior of the seemingly simple problem of the logistic map. The sluggishness of the HP-65 compelled him to carefully examine the parameter values of successive period doublings. Mitchell eventually recognized that they converged geometrically at the rate δ.3 A number of other researchers had previously studied this problem using much more powerful computers. By allowing the computer to do all their thinking for them, they had, in effect, blown right by the period-doubling cascade and missed the core of the problem.
In 1984, Mitchell received a MacArthur Fellowship for his period-doubling discovery. Several years earlier, Albert Libchaber and Jean Maurer had found confirmation for Mitchell’s prediction of universality as part of an experiment on convection in low-temperature helium.4 These results led to Mitchell sharing the Wolf Prize in Physics with Libchaber in 1986.
Many were surprised that Mitchell did not share a Nobel Prize in Physics with Libchaber, especially after having already won both a MacArthur Fellowship and the Wolf Prize. The politics surrounding the Nobel is shrouded in mystery, but it has been suggested that Mitchell may have been overlooked because some sources—not Mitchell himself—oversold the period-doubling discovery as a solution to turbulence. This was not the case. Instead, Mitchell had discovered a new universal constant of nature, δ—a fundamental discovery that was applicable to many fields of science.
In addition to the mathematical maps that made him famous, Mitchell also applied his search for the core of things to the world of geographic maps and mapmaking. He made transformative contributions to cartography by inventing a new form of conformal map projection known as Hammond’s optimal conformal, which provably minimizes distortions in shapes and relative areas of objects in planar maps of the earth’s surface. He also created a chaotic dithering algorithm that allowed for rapid and efficient labeling of maps, markedly improving both their legibility and aesthetic appeal. The benefits of these inventions can be seen in the innovative Hammond World Atlas of 1992, which is truly a work of art.5
Mitchell’s passion for the deed was always accompanied by a doppelgänger passion for the sly misdeed, or practical joke. In the mid-1970s, I obtained a new programmable calculator, an HP-85. Mitchell asked innocently enough whether he could borrow the device to play with it for a while. When I got it back several hours later, every time I turned it on, the calculator played a little ditty and then switched itself off—just as Mitchell had programmed it to do. It took me quite a while to convince him to reverse the changes he had made, so that I could get back to work.
On another occasion, Mitchell joined me in the T-Division office while I was arranging for the secretary, Lois MacFarland, to type up the abstract of a talk I was giving in a few weeks’ time. As is still the case, abstracts of all Los Alamos talks had to be cleared before they could be distributed. I dictated the title “Kink-Antikink Interactions in Lambda Phi Fourth Field Theory” and, without missing a beat, Mitchell added, “and applications to nuclear weapons,” an intervention that guaranteed my abstract would never be cleared. Lois dutifully typed out the newly expanded title, oblivious to Mitchell’s mischief making.
Mitchell only worked on problems that excited him, and his diverse interests took him in many directions. In 1996, he was one of the founders of Numerix, a company that continues to develop novel software algorithms to improve the speed and accuracy of calculations used to price financial derivatives. While serving on a committee of the National Research Council formed to deter currency counterfeiting, Mitchell used his unusual insights into numbers to design patterns that cannot be reproduced, even by high-resolution digital copiers. The design change was as significant as Isaac Newton’s introduction of milled edges on coins to prevent clipping.
In his final studies, Mitchell was investigating visual perception. He was attempting to infer rules associated with vision and our perception, using applications of optics and visual neurophysiology. A summary of his work in this area, entitled “Reflections on a Tube,” remains incomplete and unpublished. It is hoped that interested colleagues will ensure that it is completed in the near future.
Mitchell and I spent many wonderful times together at places ranging from the gorgeous beaches of Woods Hole and St. Bart’s to elegant restaurants in Turin and Berlin, to the rather more spartan environments of Tarusa, Russia, and Kiev, Ukraine. In Tarusa, we viewed the art of a famous nineteenth-century Russian impressionist painter. The artist’s name escapes me because Mitchell insisted on calling him Pajalsta, the Russian for “please” or “you’re welcome,” and that is what has stuck in my memory.
Among his many passions, Mitchell had an enduring love for classical music. He was particularly fond of the great German composers—Beethoven, Mendelssohn, Schubert, Brahms, Wagner, Mahler—and owned an immaculately maintained collection of vinyl LPs. It was only in the last few years that Mitchell was finally willing to admit that the quality of digital recordings could approach that of his treasured LPs.
Mitchell was born in Philadelphia on December 19, 1944. He graduated with a degree in electrical engineering from City College of New York in 1964 at age nineteen. He entered MIT that same year, but soon switched from electrical engineering to physics. Mitchell received his PhD in 1970 after preparing a dissertation on particle physics under the supervision of Francis Low. A subsequent postdoctoral fellowship at Cornell University proved crucial to his career. It was at Cornell that Mitchell first met Carruthers and was introduced to the renormalization group ideas of Kenneth Wilson. In 1974, following another postdoc position at Virginia Polytechnic Institute, he was hired by Carruthers in the Theoretical Division at Los Alamos. Mitchell remained at T-Division until 1982, when he accepted a professorship at Cornell. He relocated again in 1986, taking up his final position at Rockefeller University. In addition to his MacArthur Fellowship and Wolf Prize, Mitchell was a member of the National Academy of Sciences and the American Academy of Arts and Sciences. Although he eschewed administrative responsibilities, Mitchell lent his ideas and prominence to several important ventures, including the founding of the Center for Nonlinear Studies at Los Alamos, the American–Soviet Chaos/XAOC conferences, and Chaos: An Interdisciplinary Journal of Nonlinear Science, published by the American Institute of Physics. In the mid-1990s, Mitchell cofounded the Center for Studies in Physics and Biology at Rockefeller, anticipating efforts elsewhere to integrate modern mathematics and physics into biology.
Mitchell was a truly unique individual. In the words of William Blake, he was able to “see a World in a Grain of Sand, And a Heaven in a Wild Flower.”6 Mitchell was equally at ease discussing the intricacies of the music of Schubert, Beethoven, and Wagner, the art of Rembrandt and Pablo Picasso, the philosophy of Immanuel Kant and Arthur Schopenhauer, and the psychoanalysis of Sigmund Freud. Indeed, Blake’s famous portrait of Newton taking the measure of the world with a compass provides an appropriate image to remember my remarkable friend and colleague.
An earlier version of this remembrance, coauthored with my colleagues Gemunu Gunaratne and Eric Siggia, was published by Physics Today.7 This new version also draws upon remarks I made at a memorial service for Mitchell that was held in October 2019.
