From the Publisher
“A heroic story of discovery. . . . Illustrate[s] what it takes for great new science to be created.” —Stephen Wolfram, The Wall Street Journal
“Mandelbrot had the kind of beautiful, buzzing mind that made even gifted fellow scientists feel shabby around the edges. . . . The Fractalist evokes the kinds of deceptively simple questions Mandelbrot asked . . . and the profound answers he supplied.” —The New York Times
“Fascinating and engaging . . . A compelling look at one of the greatest multidisciplinary thinkers of the 21st century.” —Wired.com
“Mandelbrot was a spell-worker who saw connections no one else did and united apparently disparate phenomena. The mathematics of fractals—and pictures of the Mandelbrot set—offered many budding mathematicians their first taste of ‘real’ mathematics, in all its beauty, utility and sheer unexpectedness.” —The Economist
“The delight Mandelbrot took in roughness, brokenness, and complexity, in forms that earlier mathematicians had regarded as ‘monstrous’ or ‘pathological,’ has a distinctly modern flavor. Indeed, with their intricate patterns that recur endlessly on ever tinier scales, Mandelbrot’s fractals call to mind the definition of beauty offered by Baudelaire: C’est l’infini dans le fini.” —New York Review of Books
“If you love fractals, you will love this memoir. . . . Mandelbrot describes his life and times with both introspection and humor.” —New York Journal of Books
“Charmingly written . . . The memoir of a brilliant mathematician who never thought of himself as a mathematician.” —Kirkus Reviews
“Captures the enthusiasm as well as the memories of a visionary who loved nothing better than studying complex multidisciplinary concepts.” —Publishers Weekly
“[Mandelbrot’s] work has spread and impacted so many fields that there’s nobody in the world who is broad enough to appreciate the full impact. . . . [His] mix of gall and genius gave him license to ask the questions no one else did.” —Thomas Theis, director of physical sciences at IBM Research
“Mandelbrot brings us back to the sense of the wonder of things, without giving up the logic.” —John Briggs, author of Fractals: The Patterns of Chaos
“When we talk about the impact inside mathematics, and applications in the sciences, [Mandelbrot] is one of the most important figures of the last 50 years.” —Heinz-Otto Peitgen, professor of mathematics and biomedical sciences at the University of Bremen
Mandelbrot changed the way we look at a wide range of random phenomena from commodity prices to the shapes of mountains, rivers, and coastlines. An “outlier” long before the word became popular, he was born in 1924 to Jewish parents and grew up in Warsaw, Poland, and then Paris, with “a high level of self-confidence” that grounded him throughout his peripatetic life during and after the chaos of WWII. After the war he pursued his scientific dreams at the École Polytechnique in Paris, later at MIT, Princeton, and elsewhere. But the work that led to his great innovation began with his 1958 arrival at the intellectually expansive IBM facility in Yorktown, N.Y. Over the next couple of decades, Mandelbrot discovered patterns in a wide range of phenomena such as price variation and the distribution of galaxies and irregularly shaped objects like clouds that could not be mathematically described. He called his mathematical innovation “fractal geometry.” The memoir captures the enthusiasm as well as the memories of a visionary who loved nothing better than studying complex multidisciplinary concepts. (Mandelbrot died in 2010, after completing this book.) Agent: John Brockman, Brockman Inc. (Oct.)
Mathematician Mandelbrot (Fractals and Chaos: The Mandelbrot Set and Beyond) died in 2010, but not before writing down the story of his life and work. This memoir describes a scientific career marked by unconventional choices and an intuition for overlooked areas of inquiry. His rapidly changing interests brought the mathematician into collaboration with everyone from John von Neumann, an early father of the field of computing, to famed educational psychologist Jean Piaget. These disparate intellectual experiences ultimately come together in Mandelbrot's signal contribution: defining those odd iterative shapes called fractals (the iconic Mandelbrot set being just one example) and using them as models for real-world phenomena like the shapes of mountains and nautilus shells. Though sometimes it struggles under the weight of too much detail, the book is worth reading to learn more about Mandelbrot's unique contributions to the field of mathematics. VERDICT An engaging, nontechnical read for general audiences interested in modern mathematics.—J.J.S. Boyce, formerly with Louis Riel and Pembina Trails Sch. Divisions, Winnipeg
Memoir of a brilliant mathematician who never thought of himself as a mathematician. Part of the reason is that Mandelbrot's work had wide-ranging impact; as his best-known book, The Fractal Geometry of Nature (1982) illustrates, his insights apply across many disciplines. That breadth of interest originated in Mandelbrot's early years, growing up in a Jewish family that managed to dodge the currents of anti-Semitism, moving from Lithuania to Poland to France, where the author spent the World War II years in a provincial town, away from the attention of the occupiers. Early in life, he learned about Johannes Kepler, whose geometric insights changed the nature of astronomy, and Mandelbrot made it one of his goals to achieve a similar breakthrough. After the war, his academic skills got him into the École Polytechnique, an elite training school for military engineers. Then he bounced around from Caltech to the French air force to the University of Paris to the Institute for Advanced Studies. Along the way, he made the acquaintance of an impressive number of scientific giants, acquired a doctorate and a love of music and married Aliette Kagan, with whom he would spend the rest of his life. To this point, his career showed more promise than achievement. Taking a job with IBM, which encouraged basic research with no obvious application to its products, turned out to be his best move. There, he found his interest in "roughness" led to geometric insights that opened doors in a number of fields. The final pages are a summary of accomplishments, publications and recognitions. Interestingly, the narrative deliberately avoids mathematics and therefore gives only the vaguest suggestion of his actual work. That decision undoubtedly makes the book more accessible to general readers, but it also throws the emphasis on the more superficial aspects of his career. Nonetheless, the portraits of his contemporaries and their milieu are worth the read. Charmingly written, but readers interested in the nature of the work that won him his accolades will have to look elsewhere.
Read an Excerpt
Beauty and Roughness
Nearly all common patterns in nature are rough. They have aspects that are exquisitely irregular and fragmented—not merely more elaborate than the marvelous ancient geometry of Euclid but of massively greater complexity. For centuries, the very idea of measuring roughness was an idle dream. This is one of the dreams to which I have devoted my entire scientific life.
Let me introduce myself. A scientific warrior of sorts, and an old man now, I have written a great deal but never acquired a predictable audience. So, in this memoir, please allow me to tell you who I think I am and how I came to labor for so many years on the first-ever theory of roughness and was rewarded by watching it transform itself into an aspect of a theory of beauty.
The broad-minded mathematician Henri Poincaré (1854–1912) remarked that some questions one chooses to ask, while others are “natural” and ask themselves. My life has been filled with such questions: What shape is a mountain, a coastline, a river, or a dividing line between two river watersheds? What shape is a cloud, a flame, or a welding? How dense is the distribution of galaxies in the universe? How can one describe—to be able to act upon—the volatility of prices quoted in financial markets? How to compare and measure the vocabularies of different writers? Numbers measure area and length. Could some other number measure the “overall roughness” of rusted iron, or of broken stone, metal, or glass? Or the complexity of a piece of music or of abstract art? Can geometry deliver what the Greek root of its name seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?
These questions, as well as a host of others, are scattered across a multitude of sciences and have been faced only recently . . . by me. As an adolescent during World War II, I came to worship a major achievement of a mathematician and astronomer of long ago, Johannes Kepler (1571–1630). Kepler combined the ellipses of ancient Greek geometers with a failure of ancient Greek astronomers, who mistakenly believed that persistent “anomalies” existed in the motion of planets. Kepler used his knowledge of two different fields—mathematics and astronomy—to calculate that this motion of the planets was not an anomaly. It was, in fact, an elliptical orbit. To discover something like this became my childhood dream.
A most impractical prospect! Not one leading to a career in any organized profession, nor providing a way of shining in life—a prospect that my uncle Szolem, an eminent mathematician, repeatedly called completely childish. Yet somehow fate did allow me to spend my life pursuing that dream. Through extraordinarily good for- tune, and a long and achingly complicated professional life, it was eventually fulfilled.
In my Keplerian quest I faced many challenges. The good news is that I succeeded. The bad news, or perhaps additional good news, is that my “success” raised a host of new and different problems. More- over, my contributions to seemingly unrelated fields were actually closely related and eventually led to a theory of roughness—a challenge dating back to ancient times. The Greek philosopher Plato had outlined this challenge millennia before our time, but nobody knew how to pursue it. Was I that person?
An acquaintance of mine was a forceful dean at a major university. One day, as our paths crossed in a busy corridor, he stopped to make a comment I never forgot: “You are doing very well, yet you are taking a lonely and hard path. You keep running from field to field, leading an unpredictable life, never settling down to enjoy what you have accomplished. A rolling stone gathers no moss, and—behind your back—people call you completely crazy. But I don’t think you are crazy at all, and you must continue what you are doing. For a thinking person, the most serious mental illness is not being sure of who you are. This is a problem you do not suffer from. You never need to rein- vent yourself to fit changes in circumstances; you just move on. In that respect, you are the sanest person among us.”
Quietly, I responded that I was not running from field to field, but rather working on a theory of roughness. I was not a man with a big hammer to whom every problem looked like a nail. Were his words meant to compliment or merely to reassure? I soon found out: he was promoting me for a major award.
Is mental health compatible with being possessed by barely contained restlessness? In Dante’s Divine Comedy, the deceased sentenced to eternal searching are pushed to the deepest level of the Inferno. But for me, an eternal search across countless scientific fields beyond obvious connection managed to add up to a happy life. A rolling stone perhaps, but not an unresponsive one. Overactive and self-motivated, I loved to roll along, stopping to listen and preach in lay monasteries of all kinds—some splendid and proud, others forsaken and out of the way.
At age twenty, I was one of twenty men who won entry into the most exclusive university in France, the École Normale Supérieure. When I retired at eighty, I was in the mathematics department at Yale as Sterling Professor—one of about twenty people at Yale’s highest rank. I entered and left “active life” under the most exclusive and noncontroversial conditions possible. And along the way I did gather some “moss.”
My life since age thirty-five—a turning point—has been most atypical in different but fruitful ways. It reminds me of that fairy tale in which the hero sees a small thread where none was expected, pulls on it, harder and harder, and unravels a variety of wonders beyond belief . . . all totally unexpected. Examined one by one, these wonders of mine “belonged” to fields of knowledge far removed from one another. One could pursue each on its own, to great benefit, as I did early on in my career. But I later adopted a broader point of view, for which I was well rewarded. All those contributors to different fields were easiest to study when recognized as “peas in a pod,” pearls of all sizes from a very long necklace.
Do those fields seem far removed from one another? Did I scatter my efforts to self-destructive excess? Possibly. Tight and deliberate self-control kept me focused on those rough shapes that had no common name but begged for one. Bringing these separate fields together put me, step by step, in the unexpected, rare, and dangerously exposed position of opening a new field and gaining the right to name it. I called it fractal geometry.
Every key facet of fractal geometry suffers from a quandary that physicists of the early 1900s called a “catastrophe.” The theories of that time predicted an infinitely large value for energy radiated by certain objects. In reality, this was not the case, so something had to give! Solving this quandary was achieved by quantum mechanics, one of the major revolutions of twentieth-century physics and the foundation of much of modern technology, including computers, lasers, and satellites.
What unified all my “peas” was the opposite end of the same quandary. Many domains of science that I dealt with centered around quantities that were assumed to have well-defined finite values, such as lengths of coastlines. However, those finite values resisted being pinned down. Measuring the length of a coastline with shorter measuring rods detects smaller features, leading to longer measurements. The insight that let me study those fields was that one should allow those key quantities to be infinite.
How did this all come to be? Uncle Szolem and I were both born in Warsaw. We each had a good eye and became counted as mathematicians. But the overly interesting times that cursed his teens and later mine, helped shape us into altogether different people. He found fulfillment as a sharply focused establishment insider, while I thrived as a hard-to-pigeonhole maverick.
As an adolescent during World War I, Uncle roamed around a Russia in the throes of revolution and civil war. He was introduced early to a well-defined and nonvisual topic: classical French mathematical analysis. He fell in passionate lifelong love with it and moved to its source. He was soon handed its torch and kept it burning through fair weather and foul.
As an adolescent during World War II, I found shelter in the isolated and impoverished highlands of central France. There I was introduced to a world of images through outdated math books filled with illustrations. After the war, upon acceptance into the École Normale Supérieure, I realized that mathematics cut off from the mysteries of the real world was not for me, so I took a different path.
Half a century before I was born, Georg Cantor (1845–1918) claimed that the essence of mathematics resides in its freedom. His peers went on to invent—or so they thought—a batch of shapes called “monsters,” or “pathologies,” and their study pushed mathematics into a deliberate flight from nature. Helped by computers, I actually drew those shapes and diametrically inverted their original intent. I went on to invent many more, and identified a few as tools that might help handle a host of often ancient concrete problems—“questions once reserved for poets and children.”
Within the purest of mathematics, my unabashed play with abandoned “pathologies” led me to a number of far-flung discoveries. An exquisitely complex shape now known as the Mandelbrot set has been called the most complex object in mathematics. I pioneered the examination of reams of pictures and extracted from them many abstract conjectures that proved to be extremely difficult, motivated a quantity of hard work, and brought high rewards.
Within the sciences of nature, I was a pioneer in the study of familiar shapes, like mountains, coastlines, clouds, turbulent eddies, galaxy clusters, trees, the weather, and others beyond counting.
Within the study of man’s works, I began with a curio: a law for word frequencies. I peaked with an extremely down-to-earth issue: the “misbehaviors” observed in the variation in speculative markets. And I added my grain of salt to the study of visual art.
So where do I really belong? I avoid saying everywhere—which switches all too easily to nowhere. Instead, when pressed, I call myself a fractalist. A challenge I kept encountering—one I never knew quite how to manage—was to do justice to the parts and the whole. In this memoir, I try very hard.
Altogether, plain old-fashioned roughness in science and art is no longer a no-man’s-land. I provided a theory and showed that an astonishing number and variety of questions can now be tackled with powerful new tools. They challenge standard geometry’s conventional view of nature, one that regards rough forms as formless. It appears that, responding to that ancient invitation of Plato, I have extended the scope of rational science to yet another basic sensation of man, one that had for so long remained untamed.
In a life far more interrupted than I would have preferred, basic stability was provided for thirty-five years by IBM Research and then for many years by Yale, and I lived long enough for my work to be appreciated in more grand ways than I ever imagined.
Writing this memoir earlier might have made my professional life a bit easier. But the delay has been fruitful. It has rubbed out some less important details, and my life’s course has become clearer, even to me.