Roughness in Painting and Music
My earliest fractal “forgeries” of trees and mountains made me wonder: If nature’s real trees and mountains are indeed fractal, should not the same be true of their representations by painters? Think of Leonardo da Vinci’s celebrated drawing A Deluge, reproduced in The Fractal Geometry of Nature. Unquestionably, it is fractal.
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Skilled artists must find arrangements, like mixtures of eddies of all sizes, that look balanced; does not that mean that elements of all sizes are distributed in a natural—that is, fractal—way?
The Fractal Geometry of Nature reproduced Hokusai’s print The Great Wave, the famous picture with Mount Fuji in the background. Hokusai was at his peak around 1800, but history provides examples of many earlier painters or philosophers who were aware of complicated shapes with fractal structure. Claude Lorrain, a French painter who worked mostly in Italy, painted landscapes that claim to be realistic, but in fact are extraordinarily simplified and easily interpreted in fractal terms. Historically, painters have always seen the possibilities of fractal structure, but it did not develop into a geometry since very few wrote about it and probably none read about it.
The Russian painter Wassily Kandinsky (1866–1944) was filmed as he worked on a sheet of paper about three feet square. He began with a slash across the whole and then added shorter slashes. When the film stopped, he was at work on many even shorter slashes, confirming a feeling I had looking at Kandinsky’s paintings: he understood fractality—perhaps not explicitly, but intuitively.
Initially, I viewed these works of art as amusing, though not essential. But I soon changed my mind as innumerable readers made me aware of something strange. I began to recognize fractals in the works of artists since time immemorial. A remarkably large number of artists had no vocabulary to express their grasp of the nature of fractals, yet such understanding comes through clearly in their work.
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One mathematical structure I called the Sierpiński gasket, made of several identical parts, turns out to be very common in decoration in Italian churches, either in mosaics on pavement or in paintings on the roof and ceiling. Other fractal structures were found in Persian and Indian art of different periods.
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I have strong connections with composers, who inhabit an entirely different world. In particular, György Ligeti confided to me that until he saw my pictures, he had not understood an important aspect of music: it is not free to do as it pleases, because it must be fractal. The schools of music never taught how to distinguish music from noise. When Ligeti received a prize in New York, a major article appeared in which he listed the greatest designs ever. The list included the Book of Kells and the Taj Mahal … and the Mandelbrot set! That was an extremely strong statement, and I was pleased to meet him shortly afterward. We have had interesting times together, including serious public discussions. Charles Wuorinen is another widely known contemporary composer who understands fractality. He liked to say he had used a fractal approach to composition for some time. He was well aware that much of Western music exhibits similar structures over different time scales. Wuorinen and I did an extraordinary show at the Guggenheim in 1990 called Music and Fractals. It is fascinating to see how two people from such different cultures can collaborate, if they have the desire to do so.
Unrelated Deeds or a Unified Fractal Approach to Roughness?
Isaiah Berlin (1909–97), a British philosopher and man of action—whom I met—has written about the distinction the ancient Greek writer Archilochus drew between the fox, who knows many things, and the hedgehog, who knows one big thing. Once, colleagues assigned to introduce me before a lecture kept asking whether I viewed myself as a fox or a hedgehog. The point is that they all saw me as having two faces.
The marble sculpture, below, represents Janus, the Roman god of doorways and bridges. He was believed to have two opposite and contrasting faces—one to judge and perhaps to repel, the other to welcome and attract.
For better or worse, two faces are also an appropriate metaphor for one of mankind’s greatest accomplishments—and my field of study—which is broadly interpreted as the mathematical sciences. Its judging face is that of a purist, a specialist taking pride in thoughts that the bulk of humanity views as dry and cold. Its welcoming face is actually something of a blur of many roles that mathematics plays in the labors and pleasures of daily life: architecture, engineering, and the arts. Symbolically, they look simultaneously into the future and the past.
Being myself a faithful—though by reputation a turbulent—servant of mathematics, I have continually rebelled against those two faces looking in opposite directions. I rejoiced in learning that, many centuries ago, the two faces had been turned in the same direction, peace prevailed, and splendid fruit came forth.
Fractal geometry is one of those concepts which at first invites disbelief but on second thought becomes so natural that one wonders why it has only recently been developed.
Real Roughness Is Often Fractal and Can Be Measured
The foremost measure of roughness is fractal dimension. The simplest form of fractal dimension is the similarity dimension, and the earliest illustration of this is a curve provided by Helge von Koch. Because its length is infinite, the Koch curve began as one of those monsters—or toys, as I refer to them. Fractal geometry brought out the wonder by setting it to the task of describing coastlines and then mastering nature. An even more monstrous monster appeared when Giuseppe Peano constructed a “curve” that visits every point of the plane. It created a storm among mathematicians and a deep split between purist extremists and those who care about the real world. A universally held opinion was that the Peano curve was totally nonintuitive and extravagant. These were words not of disappointment but of great pride on the part of pure mathematicians. The illustration below combines the Koch curve (the outline, or coastlines) and the Peano curve (the rivers, or blood networks, across the surface of the plane).
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Unimaginable privilege, I participated in a truly rare event: pure thought fleeing from reality was caught, tamed, and teamed with a reality that everyone recognized as familiar. Monsters were made into servants—in the manner that Kepler pioneered when he showed that planets’ orbits fitted ellipses, which had been the ancient Greeks’ playthings.
Roughness is ubiquitous in nature and culture—found in the distribution of galaxies and in the shapes of coastlines, mountains, clouds, trees, and the various ducts in the lungs; also in stock-price charts, paintings, music, and several mathematical constructions (well-known ones and those I fathered). Less familiar but worth a mention: the roughness of clusters in the physics of disorder, turbulent flows, chaotic dynamical systems, and anomalous diffusions and noises. These are typical of the many topics I studied.
Like smooth shapes exemplified by the ideal circle, mathematical fractals are described by absolutely precise formulas that the computer can implement, as closely as one wishes, with very concrete objects: pictures. Each picture led me to specific insights into a specific area of science and art. Some pictures proved to have a profound and durable impact and were expanded by several very focused investigators.
The set of color illustrations in this book are varied in many ways. Some are natural, some are works of art, but most are purely mathematical constructs drawn by computer with the help of appropriately chosen formulas. Those formulas share an essential property that I spent my whole active life investigating from all sides: roughness. They are not drawn merely to be pretty, but to serve all kinds of purposes in all kinds of sciences. This is why several seem realistic, reminding us of shapes in nature. They are all fractal, which is why, when asked what I do, I call myself a fractalist.
To appreciate the nature of fractals, recall Galileo’s splendid 1632 manifesto: [Philosophy] is written in the language of mathematics, and its characters are triangle
s, circles, and other geometric figures, without which … one is wandering about in a dark labyrinth. Observe that circles, ellipses, and parabolas are very smooth shapes and that a triangle has a small number of points of irregularity. These shapes were my love when I was a young man, but are very rare in the wild. Galileo was absolutely right to assert that in science those shapes are necessary. But they have turned out not to be sufficient, “merely” because most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales.
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The cauliflower is the standard example of shapes that appear more or less the same at all scales. One glance shows that it is made of florets. A single floret, examined after you cut away everything else, looks like a small cauliflower. If you strip that floret of everything except one floret of a floret—very soon you must take out your magnifying glass—it is again a cauliflower. A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Another example of this repeated roughness is the cloud. A cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud, you get not something smooth but irregularities on a smaller scale.
Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. What I’m asserting, very strongly, is that when some real thing is found to be unsmooth, the next mathematical model to try is fractal or multifractal. Since roughness is everywhere, fractals are present everywhere. And very often the same techniques apply in areas that, except for geometric structure, seem completely independent.
Fractals Have Been Here Forever and Now Have a Home
For the most part, there was no place where the things I wanted to investigate were of interest to anyone. So I spent much of my life as an outsider, moving from field to field. Now that I look back, I realize with wistful pleasure that on many occasions I was ten, twenty, forty, even fifty years ahead of my time. Until a few years ago, the topics in my Ph.D. dissertation were unfashionable, but they are very popular today.
My ambition was not to create a new field, but I would have welcomed a permanent group of people with interests close to mine and therefore breaking the disastrous tendency toward increasingly narrowly defined fields. Unfortunately, I failed on this essential point. Order doesn’t come by itself. In my youth, I was a student at Caltech while molecular biology was being created by Max Delbrück, so I saw what it means to bring a new field into existence. But my work did not give rise to anything like that. One reason is my personality—I don’t seek power or run around asking for favors. A second is circumstances—I was in an industrial laboratory because academia found me unsuitable. Besides, establishing close, organized links between activities that otherwise are very separate might have been beyond any single person’s ability.
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I did not plan any general theory of roughness, because I prefer to work from the bottom up, not from the top down. So even though I didn’t try to create a field, now, long after the fact, I am enjoying this enormous unity and emphasize it in every new publication.
I reach beyond arrogance when I proclaim that fractals had been pictured forever but their true role remained unrecognized and waited to be uncovered by me.
As my wandering life fades away, I keep thinking of the wild ambition to survive and shine that has pushed me since adolescence. Each partial success aroused some old expectation or some old hunger. Ironically, this same pattern is one I have often dealt with in my research. Even at this late stage, I suffer when some event reawakens an old fleeting hope I had to leave untested. In the words of George Bernard Shaw:
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.
Only late in life did I see this quote, and—to help knowledge and reason advance—I had been quite unreasonable all along.
You have now heard my story. Does not the distribution of my personal experiences remind one of the central topic of my scientific work—namely, extreme fractal unevenness? All counted, I have known few minutes of boredom. It has been great fun, and to some extent the fun continues. What else could one have asked for?
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To be close to my grandchildren, I have retired from Yale, closed my IBM office near New York, and endured the agony of downsizing from a big house to a Boston apartment. As I have always known, uprooting can be rational but is never sweet.
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An old man by now, past two times forty, I see myself continuing in some ways to mature. And to remain embattled. How come? Perhaps by fluke, but I think mostly for a reason.
Afterword
Michael Frame, professor of mathematics, Yale University
BENOIT MANDELBROT died shortly before he could make final revisions to this memoir. Aliette, his wife of many years, asked me to write this afterword. I hope I can offer a slightly different perspective on how Benoit’s work fits into the worlds of science and culture.
I met Benoit twenty years ago when he hired me to join his group at Yale. I think he brought me into his world because, in a specific sense, we both were little kids. Benoit would call with a question or an idea and we’d be off. Then I’d glance at the clock and an hour or two would have passed. We’d go our separate ways for a week or so, working out some details of what we’d seen. He’d call and we’d be off again. I think I shared his sense of innocent wonder.
Benoit loved complicated things, the roughness of coastlines and price graphs, the music of Charles Wuorinen and György Ligeti, the paintings of Augusto Giacometti and prints of Hokusai. What he saw in all of these, what may have helped guide his thinking through the wanderings of his long (but still too short) career, was a sense that there were common features to all these examples. Patterns that kept recurring as he looked ever closer. To be sure, many scientists and artists had noticed this, and the examples of continuous, nowhere differentiable curves were familiar from basic real analysis courses. But Benoit saw much more, a way to quantify these recurring patterns so that complicated shapes might be easily understood dynamically, as processes, not just as objects.
The power of this paradigm is immense, and still persists. In September 2010, I had the pleasure of watching the eighty students in my fractal geometry course enter the classroom not knowing how to generate fractal images, follow some simple steps, and by the end of the class period be able to tell me how to generate these fractals just by looking at the images.
I pointed out just how much their understanding had grown that day. Their looks of surprise gave way to grins and “cools.” (Then I warned them not to expect such miracles every day.) This is what Benoit gave the mathematical world. If anyone doubts the power of his gift, compare a standard geometry class lesson on plane transformations with this day in a fractals class.
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What makes this effective is the visual complexity of the images—reflecting Benoit’s inspiration—together with the ability to decode these images into a few simple rules. The other point is that once a student learns how to see these patterns, the solution can be tested in seconds using basic software. Visual experimentation implemented by computers—another of Benoit’s initially unpopular causes—is now so commonplace that it warrants no remark.
Outside of science, Benoit probably spent the most time on fractals in finance. Bachelier’s 1900 model exhibited three properties: scaling, independent jumps, and short tails (large jumps are rare indeed). The first of these properties fit into, and perhaps helped form, Benoit’s view of the fractality of financial series. The other two properties do not agree so well with observations. Benoit’s 1960s st
udies, fractional Brownian motion and Lévy stable processes, also are scaling. The first has dependent jumps but still has short tails; the second has long tails but independent jumps. In the 1990s, Benoit developed an extraordinarily simple and elegant approach, the multifractal cartoon. These cartoons are scaling, have dependent jumps and long tails, and can be fine-tuned easily. Much of Benoit’s work was based on a simple idea—scaling, iteration, and dimension—applied with great finesse in new settings.
By far, the biggest surprise is the Mandelbrot set. In class we set up the simple formula and describe the iteration process and how to color-code the result. Then we run the program and wait for the shock to spread across the room. “This formula produces that picture??? Are you kidding me?” “Just wait, you haven’t seen anything yet. Let’s magnify a bit and see what we find.” “You mean those complicated twirls and swirls still come from the same little formula?” “Absolutely.” This miracle needs no further discussion. Look at the pictures, remember the simple formula, think, and be amazed.
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Benoit wrote and talked about fractal patterns in art, architecture, music, poetry, and literature. He was overjoyed to find fractal aspects—surely discovered thanks to a careful eye for the subtle patterns of nature—in the art of Hokusai and Dalí, the architecture of Eiffel, the music of Ligeti and Wuorinen, the verses of Stevens, the plays of Stoppard. Every new instance of fractals pleased him. Of course, Benoit was clear, often blunt, in his criticism of misapplications of his ideas. Given the range of these mistakes, his ire was understandable.
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