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Introduction
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.
 
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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?
 
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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.
 
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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.
 
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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.
 
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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.