

To remember and to honor Benoit Mandelbrot, Edge is pleased to present several pieces:
BENOIT MANDELBROT, who died on October 14th, was Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T.J. Watson Research Center. His books include The Fractal Geometry of Nature; Fractals and Scaling in Finance; and (with Richard L. Hudson) The (mis)Behavior of Markets. 

BENOIT MANDELBROT: A REMEMBRANCE We lost an intellectual giant. Benoit Mandelbrot was one of humankind's preeminent mathematicians, yet he was much more than that. Mandelbrot bridged art and science with an ease that seemed unreal, and with a depth matched only by few in history, like da Vinci and Helmholtz. What the physicist Helmholtz did 2 centuries ago for the realm of sound, by combining physics, physiology and musical esthetics and "trusting the ear", Mandelbrot did for the visual realm by "trusting the eye". Trusting the senses was for both of them the beginning, not the end of the discovery process. Both of them were giants of science and took painstaking care in proving their conjectures, which often took years. In the meantime, they were very open about the source of their intuition, which sometimes brought in skeptics — from science or from art. Ultimately, what they did was a true amalgam of art and science, creative in a deepest sense, and permanently changed the way we all perceive the world around us. At the very start of Benoit Mandelbrot's path of discovery was a simple question: "How long is the coast of Britain?" — what happens as you keep zooming in? — and he found that a coastline is essentially infinite, always revealing new features. My conversations with Benoit reminded me of a coastline, a beautiful coastline. — Dimitar Sasselov, Astronomer, Harvard University; 
"WHAT DO YOU BELIEVE IS TRUE EVEN THOUGH YOU CANNOT PROVE IT?" Great minds can sometimes guess the truth before they have either the evidence or arguments for it (Diderot called it having the "esprit de divination"). What do you believe is true even though you cannot prove it?

A
THEORY OF ROUGHNESS [12.20.04] A recent, important turn in my life occurred when I realized that something that I have long been stating in footnotes should be put on the marquee. I have engaged myself, without realizing it, in undertaking a theory of roughness. Think of color, pitch, loudness, heaviness, and hotness. Each is the topic of a branch of physics. Chemistry is filled with acids, sugars, and alcohols — all are concepts derived from sensory perceptions. Roughness is just as important as all those other raw sensations, but was not studied for its own sake. Introduction During the 1980s Benoit Mandelbrot accepted my invitation to give a talk before The Reality Club. The evening was the toughest ticket in the 10year history of live Reality Club events during that decade: it seemed like every artist in New York had heard about it and wanted to attend. It was an exciting, magical evening. I've stayed in touch with Mandelbrot and shared an occasional meal with him every few years, always interested in what he has to say. Recently, we got together prior to his 80th birthday. Mandelbrot is best known as the founder of fractal geometry which impacts mathematics, diverse sciences, and arts, and is best appreciated as being the first broad attempt to investigate quantitatively the ubiquitous notion of roughness. And he continues to push the envelope with his theory of roughness. "There is a joke that your hammer will always find nails to hit," he says. "I find that perfectly acceptable. The hammer I crafted is the first effective tool for all kinds of roughness and nobody will deny that there is at least some roughness everywhere." "My book, The Fractal Geometry of Nature," he says, reproduced Hokusai's print of the Great Wave, the famous picture with Mt. Fuji in the background, and also mentioned other unrecognized examples of fractality in art and engineering. Initially, I viewed them as amusing but not essential. But I soon changed my mind. "Innumerable
readers made me aware of something strange. They made me look around
and recognize fractals in the works of artists since time immemorial.
I now collect such works. An extraordinary amount of arrogance is
present in any claim of having been the first in "inventing" something.
It's an arrogance that some enjoy, and others do not. Now I reach
beyond arrogance when I proclaim that fractals had been pictured
forever but their true role had remained unrecognized and waited
for me to be uncovered." 
A
THEORY OF ROUGHNESS My ambition was not to create a new field, but I would have welcomed a permanent group of people having interests close to mine and therefore breaking the disastrous tendency towards increasingly welldefined fields. Unfortunately, I failed on this essential point, very badly. 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 create a new field. But my work did not give rise to anything like that. One reason is my personality — I don't seek power and do not run around. A second is circumstances — I was in an industrial laboratory because academia found me unsuitable. Besides, creating close organized links between activities which otherwise are very separate might have been beyond any single person's ability. That issue is important to me now, in terms of legacy. Let me elaborate. When I turned seventy, a former postdoc organized a festive meeting in Curaçao. It was superb because of the participation of mathematician friends, physicist friends, engineering friends, economist friends and many others. Geographically, Curaçao is out of the way, hence not everybody could make it, but every field was represented. Several such meetings had been organized since 1982. However, my enjoyment of Curaçao was affected by a very strong feeling that this was going to be the last such common meeting. My efforts over the years had been successful to the extent, to take an example, that fractals made many mathematicians learn a lot about physics, biology, and economics. Unfortunately, most were beginning to feel they had learned enough to last for the rest of their lives. They remained mathematicians, had been changed by considering the new problems I raised, but largely went their own way. Today, various activities united at Curaçao are again quite separate. Notable exceptions persist, to which I shall return in a moment. However, as I was nearing eighty, a Curaçaolike meeting was not considered at all. Instead, the event is being celebrated by more than half a dozen specialized meetings in diverse locations. The most novel and most encouraging one will be limited to very practical applications of fractals, to issues concerning plastics, concrete, the internet, and the like. For many years I had been hearing the comment that fractals make beautiful pictures, but are pretty useless. I was irritated because important applications always take some time to be revealed. For fractals, it turned out that we didn't have to wait very long. In pure science, fads come and go. To influence basic bigbudget industry takes longer, but hopefully also lasts longer. To return to and explain how fractals have influenced pure mathematics, let me say that I am about to spend several weeks at the MittagLeffler Institute at the Swedish Academy of Sciences. Only 25 years ago, I had no reason to set foot there, except to visit the spectacular library. But, as it turned out, my work has inspired three apparently distinct programs at this Institute.
The next MittagLeffler year I inspired came six years ago and focused on my "4/3" conjecture about Brownian motion. Its discovery is characteristic of my research style and my legacy, hence deserves to be retold. Scientists have known Brownian motion for centuries, and the mathematical model provided by Norbert Wiener is a marvelous pillar at the very center of probability theory. Early on, scientists had made pictures both of Brownian motion in nature and of Wiener's model. But this area developed like many others in mathematics and lost all contact with the real world. My attitude has been totally different. I always saw a close kinship between the needs of "pure" mathematics and a certain hero of Greek mythology, Antaeus. The son of Earth, he had to touch the ground every so often in order to reestablish contact with his Mother; otherwise his strength waned. To strangle him, Hercules simply held him off the ground. Back to mathematics. Separation from any downtoearth input could safely be complete for long periods — but not forever. In particular, the mathematical study of Brownian motion deserved a fresh contact with reality. Seeking such a contact, I had my programmer draw a very big sample motion and proceeded to play with it. I was not trying to implement any preconceived idea, simply actively "fishing" for new things. For a long time, nothing new came up. Then I conceived an idea that was less scientific than esthetic. I became bothered by the fact that, when a Brownian motion has been drawn from time 0 to time 1, its two end portions and its middle portion follow different rules. That is, the whole is not homogeneous, exhibits a certain lack of inner symmetry, a deficit of beauty. This triggered the philosophical prejudice that when you seek some unspecified and hidden property, you don't want extraneous complexity to interfere. In order to achieve homogeneity, I decided to make the motion end where it had started. The resulting motion biting its own tail created a distinctive new shape I call Brownian cluster. Next the same purely aesthetic consideration led to further processing. The continuing wish to eliminate extraneous complexity made me combine all the points that cannot be reached from infinity without crossing the Brownian cluster. Painting them in black sufficed, once again, to create something quite new, resembling an island. Instantly, it became apparent that its boundary deserved to be investigated. Just as instantly, my long previous experience with the coastlines of actual islands on Earth came handy and made me suspect that the boundary of Brownian motion has a fractal dimension equal to 4/3. The fractal dimension is a concept that used to belong to wellhidden mathematical esoteric. But in the previous decades I had tamed it into becoming an intrinsic qualitative measure of roughness. Empirical measurement yielded 1.3336 and on this basis, my 1982 book, The Fractal Geometry of Nature, conjectured that the value of 4/3 is exact. Mathematician friends chided me: had I told them before publishing, they could have quickly provided a fully rigorous proof of my conjecture. They were wildly overoptimistic, and a proof turned out to be extraordinarily elusive. A colleague provided a numerical approximation that fitted 4/3 to about 15 decimal places, but an actual proof took 18 years and the joining of contributions of three very different scientists. It was an enormous sensation in the year 2000. Not only the difficult proof created its own very active sub field of mathematics, but it affected other, far removed, sub fields by automatically settling many seemingly unrelated conjectures. An article in Science magazine reported to my great delight a comment made at a major presentation of the results, that this was the most exciting thing in probability theory in 20 years. Amazing things started happening and the MittagLeffler Institute organized a full year to discuss what to do next. Today, after the fact, the boundary of Brownian motion might be billed as a "natural" concept. But yesterday this concept had not occurred to anyone. And even if it had been reached by pure thought, how could anyone have proceeded to the dimension 4/3? To bring this topic to life it was necessary for the Antaeus of Mathematics to be compelled to touch his Mother Earth, if only for one fleeting moment. Within the mathematical community, the MLC and 4/3 conjectures had a profound effect — witnessed recently when the French research council, CNRS, expressed itself as follows. "Mathematics operates in two complementary ways. In the 'visual' one the meaning of a theorem is perceived instantly on a geometric figure. The 'written' one leans on language, on algebra; it operates in time. Hermann Well wrote that 'the angel of geometry and the devil of algebra share the stage, illustrating the difficulties of both.'" I, who took leave from French mathematics at age 20 because of its rage against images, could not have described it better. Great to be alive when these words come from that pen. But don't forget that, in the generations between Hermann Well (18851955) and today — the generations of my middle years — the mood had been totally different. Back to cluster dimension. At IBM, where I was working at the time, my friends went on from the Brownian to other clusters. They began with the critical percolation cluster, which is a famous mathematical structure of great interest in statistical physics. For it, an intrinsic complication is that the boundary can be defined in two distinct ways, yielding 4/3, again, and 7/4. Both values were first obtained numerically but by now have been proven theoretically, not by isolated arguments serving no other purpose, but in a way that has been found very useful elsewhere. As this has continued, an enormous range of geometric shapes, so far discussed physically but not rigorously, became attractive in pure mathematics, and the proofs were found to be very difficult and very interesting. The third meeting that my work inspired at the MittagLeffler Institute of the Swedish Academy, will take place this year. Its primarily concern will be a topic I have already mentioned, the mathematics of the Internet. This may or may not have happened to you, but some nonnegligible proportion of e/mail gets lost. Multiple identical messages are a pest, but the sender is actually playing it safe for the good reason that in engineering everything is finite. There is a very complicated way in which messages get together, separate, and are sorted. Although computer memory is no longer expensive, there's always a finite size buffer somewhere. When a big piece of news arrives, everybody sends a message to everybody else, and the buffer fills. If so, what happens to the messages? They're gone, just flow into the river. At first the experts thought they could use an old theory that had been developed in the 1920s for telephone networks. But as the Internet expanded, it was found that this model won't work. Next they tried one of my inventions from the mid1960s, and it wouldn't work either. Then they tried multi fractals, a mathematical construction that I had introduced in the late 1960s and into the 1970s. Multi fractals are the sort of concept that might have been originated by mathematicians for the pleasure of doing mathematics, but in fact it originated in my study of turbulence and I immediately extended it to finance. To test new internet equipment one examines its performance under multi fractal variability. This is even a fairly big business, from what I understand. ~~ How could it be that the same technique applies to the Internet, the weather and the stock market? Why, without particularly trying, am I touching so many different aspects of many different things? A recent, important turn in my life occurred when I realized that something that I have long been stating in footnotes should be put on the marquee. I have engaged myself, without realizing it, in undertaking a theory of roughness. Think of color, pitch, heaviness, and hotness. Each is the topic of a branch of physics. Chemistry is filled with acids, sugars, and alcohols; all are concepts derived from sensory perceptions. Roughness is just as important as all those other raw sensations, but was not studied for its own sake. In 1982 a metallurgist approached me, with the impression that fractal dimension might provide at long last a measure of the roughness of such things as fractures in metals. Experiments confirmed this hunch, and we wrote a paper for Nature in 1984. It brought a big following and actually created a field concerned with the measurement of roughness. Recently, I have moved the contents of that paper to page 1 of every description of my life's work. Those descriptions have repeatedly changed, because I was not particularly precocious, but I'm particularly longlived and continue to evolve even today. Above a multitude of specialized considerations, I see the bulk of my work as having been directed towards a single overarching goal: to develop a rigorous analysis for roughness. At long last, this theme has given powerful cohesion to my life. Earlier on, since my Ph.D. thesis in 1952, the cohesion had been far more flimsy. It had been based on scaling, that is, on the central role taken by socalled powerlaw relations. For better or worse, none of my acquaintances has or had a similar story to tell. Everybody I have known has been constantly conscious of working in a preexisting field or in one being consciously established. As a notable example, Max Delbrück was first a physicist, and then became the founder of molecular biology, a field he always understood as extending the field of biology. To the contrary, my fate has been that what I undertook was fully understood only after the fact, very late in my life. To appreciate the nature of fractals, recall Galileo's splendid manifesto that "Philosophy is written in the language of mathematics and its characters are triangles, circles and other geometric figures, without which one wanders 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. 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 infinitely great roughness and complexity. However, the infinite sea of complexity includes two islands: one of Euclidean simplicity, and also a second of relative simplicity in which roughness is present, but is the same at all scales. The standard example is the cauliflower. One glance shows that it's made of florets. A single floret, examined after you cut 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's again a cauliflower. A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Many plants are like that. A cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud you don't get something smooth but irregularities at a smaller scale. Smooth shapes are very rare in the wild but extremely important in the ivory tower and the factory, and besides were my love when I was a young man. Cauliflowers exemplify a second area of great simplicity, that of shapes which appear more or less the same as you look at them up close or from far away, as you zoom in and zoom out. Before my work, those shapes had no use, hence no word was needed to denote them. My work created such a need and I coined "fractals." I had studied Latin as a youngster, and was trying to convey the idea of a broken stone, something irregular and fragmented. Latin is a very concrete language, and my son's Latin dictionary confirmed that a stone that was hit and made irregular and broken up, is described in Latin by the adjective "fractus." This adjective made me coin the word fractal, which now is in every dictionary and encyclopedia. It denotes shapes that are the same from close and far away.
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 un smooth, the next mathematical model to try is fractal or multi fractal. A complicated phenomenon need not be fractal, but finding that a phenomenon is "not even fractal" is bad news, because so far nobody has invested anywhere near my effort in identifying and creating new techniques valid beyond fractals. Since roughness is everywhere, fractals — although they do not apply to everything — are present everywhere. And very often the same techniques apply in areas that, by every other account except geometric structure, are separate. To give an example, let me return to the stock market and the weather. It's almost trite to compare them and speak of storms and hurricanes on Wall Street. For a while the market is almost flat, and almost nothing happens. But every so often it hits a little storm, or a hurricane. These are words which practical people use very freely but one may have viewed them as idle metaphors. It turns out, however, that the techniques I developed for studying turbulence — like weather — also apply to the stock market. Qualitative properties like the overall behavior of prices, and many quantitative properties as well, can be obtained by using fractals or multi fractals at an extraordinarily small cost in assumptions. This does not mean that the weather and the financial markets have identical causes — absolutely not. When the weather changes and hurricanes hit, nobody believes that the laws of physics have changed. Similarly, I don't believe that when the stock market goes into terrible gyrations its rules have changed. It's the same stock market with the same mechanisms and the same people. A good side effect of the idea of roughness is that it dissipates the surprise, the irritation, and the unease about the possibility of applying fractal geometry so widely. The fact that it is not going to lack problems anytime soon is comforting. By way of background, a branch of physics that I was working in for many years has lately become much less active. Many problems have been solved and others are so difficult that nobody knows what to do about them. This means that I do much less physics today than 15 years ago. By contrast, fractal tools have plenty to do. There is a joke that your hammer will always find nails to hit. I find that perfectly acceptable. The hammer I crafted is the first effective tool for all kinds of roughness and nobody will deny that there is at last some roughness everywhere. I did not and don't plan any general theory of roughness, because I prefer to work from the bottom up and not from top to bottom. But the problems are there. Again, I didn't try very hard to create a field. But now, long after the fact, I enjoy this enormous unity and emphasize it in every recent publication. The goal to push the envelope further has brought another amazing development, which could have been described as something recent, but isn't. My book, The Fractal Geometry of Nature, reproduced Hokusai's print of the Great Wave, the famous picture with Mt. Fuji in the background, and also mentioned other unrecognized examples of fractality in art and engineering. Initially, I viewed them as amusing but not essential. But I changed my mind as innumerable readers made me aware of something strange. They made me look around and recognize fractals in the works of artists since time immemorial. I now collect such works. An extraordinary amount of arrogance is present in any claim of having been the first in "inventing" something. It's an arrogance that some enjoy, and others do not. Now I reach beyond arrogance when I proclaim that fractals had been pictured forever but their true role remained unrecognized and waited for me to be uncovered. 
WHAT IS YOUR FORMULA? YOUR EQUATION? YOUR ALGORITHM?


BENOIT'S DANGEROUS LIFE I took the above photograph of Benoit Mandelbrot at the unveiling of a centenary plaque at the entrance to John von Neumann's childhood home in Budapest, 16 October 2003. Later, he asked did I know the story of how von Neumann had helped him from beyond the grave? I said no. Benoit explained that this was when he was in deep trouble in his first job at IBM, which "at the time had never fired anybody, but I thought my days had become counted." He told me the story as follows:
—George Dyson, Science Historian; Author, 
THE FATHER OF LONG TAILS [2008] Hans Ulrich Obrist The use of images in mathematics certainly stands completely against the ideology of the 1960s and '70s, when the sciences were sharply classified according to whether images are or are not important. A Germanborn friend of mine, a great biologist and philosopher, went so far as to theorise that progress in science consists in eliminating pictures as much as possible. Mathematics was perfect because it had completely banished pictures… even from elementary textbooks. I put the pictures back. This was received in a very hostile fashion by most of my colleagues. Since then, the opposition to pictures has weakened, simply because they have been so extraordinarily fruitful and because humans are continually changing. HANS ULRICH OBRIST, a Swiss curator, is Codirector of Exhibitions and Programmes and Director of International Projects, of the Serpentine Gallery in London. Hans Ulrich Obrist's Edge Bio Page. 
THE FATHER OF LONG TAILS OBRIST: Like the book you wrote with Michael Frame in 2002, Fractals, Graphics and Mathematics Education? Scientists are not separate from their society and technology. When I was a child, books had no pictures because of economics: pictures were expensive; so books were very dry and grey by design. Russian books, almost until the fall of the Soviet Union, remained extremely grey, whereas now the environment in which everybody is raised is extremely colourful, rich in design. The young mathematicians cannot help being more open to the influence of pictures, but nobody claims that pictures are as important to his or her work as to mine. OBRIST: You were one of the very last students of the mathematician John von Neumann. Can you tell me about this? OBRIST: Albert Hoffman, the inventor of LSD, told me in an interview that chance played a big role in his discovery. How did you discover fractals, and was chance a crucial factor, as in the case of Hoffman? "What would have happened?" Quite possibly, neither phenomenon would have been noticed by anybody; the delay, scientifically speaking, might have cost 40 years in the case of the study of prices, because for 40 years the underlying data was known but not taken seriously, and dismissed. Or my work in this area might have been postponed even further because the conditions which interest so many people might not have changed. The book I mentioned might not have been written at all. An odd thing is that chance has also helped me on many other occasions. Louis Pasteur is credited with the observation that chance can only help the wellprepared mind. I also think that my long string of lucky breaks can be credited to my mode of paying attention: I look at funny things and never hesitate to ask questions. Most people would not have noticed the dirty blackboard, or looked at the article that my uncle gave me because he was not interested. OBRIST: With regard to those two chance events, you remarked in an earlier interview that the connection between them appeared wild to you on the first night, but by the second night, you had become accustomed to it. Can you tell me about that moment? OBRIST: The late artist Alighiero Boetti conceptualised systems of order and disorder in which the order also simultaneously implies a disorder. For instance, he compiled a list of one thousand of the world's longest rivers, published in a book in 1977…Obviously, there is no absolutely fixed length of river, or a single reliable source, there are multiple and varying sources. This project involved immense geographic and scientific measurements, but with a preordained ambiguity in the results. Is this different from your notions of order and disorder? A primitive man or woman saw very few, simple, smooth shapes. For example the full moon is a simple shape, a circle. The pupil and the iris of the eye are circles. Some berries are spherical. But in the wild, almost all the shapes are extremely rough and complicated; there is a sharp distinction between the smooth/simple and the rough/complicated. Historically, geometers concentrated on the properties of a very few smooth shapes and physicists were also significantly devoted to smooth, regular behaviour, with perhaps sometimes a complication of the kind that the French mathematician René Thom theorises as "catastrophe". But trees are not smooth at all, neither are mountains and clouds. A remarkably large number of artists had no vocabulary to express their grasp of the nature of fractals, yet such an understanding comes through very clearly in their work. OBRIST: You have frequently cited the Japanese artist Hokusai as an example of this sensitivity. MANDELBROT: At this point it is mostly in my head… an imaginary museum housing an imaginary collection of great paintings that come from different periods and styles but are linked by the artists' awareness of the splendid totality of fractal structures. The museum would also have empty frames for cultures in which fractality was absent or negligible. For example, I think that the term 'Islamic' art is not useful, in fact is misleading, because Arabic art is not fractal, and Persian art very often is. Shiite and Sunni Muslims differ in many ways, including their art. OBRIST: That whole architectural movement of Max Taut, Hermann Finsterlin, the German expressionist group, was fractal. Not the Bauhaus. OBRIST: In terms of urbanism, can one say that some cities are more fractal than others? On the other hand, I have very strong connections with composers, who inhabit an entirely different world. In particular, György Ligeti came to me and confided 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. 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, the Taj Mahal… and the Mandelbrot Set! That was an extremely strong statement, and I was very pleased to meet him shortly afterwards. We have had very interesting times together, including serious public discussions. OBRIST: I have never met him, but spoke to him once on the phone, some years ago. He had then said we would be meeting for an interview a few years hence because, as he put it, "I still have twenty years of music to write before one thinks of taking stock of all my works!" MANDELBROT: While he was a visiting composer at Yale for one or two weeks, a professor of piano played many of his pieces in his honour. I was dumbfounded by the quality of her playing. Till recently, Ligeti's piano repertoire was recognised as splendid but restricted to a few specialists. But this period was over. The lady was not a famous virtuoso but 'merely' a professor of piano at Yale's very good music school, yet she played Ligeti extremely well. And it was fascinating that Ligeti commented on his own works without the least selfindulgence, in fact, with remarkable ferocity. He said, for example, that to understand a certain piece one had to know that it had been written in the 1950s, during the time he worked in Darmstadt. So many talented musicians worked there that "one would do anything to get noticed!" After he received the prize I mentioned earlier, he felt free to do as he pleased. What he wanted to do at that time was to write for piano; it was then that he really started to compose for this instrument. OBRIST: Ligeti and you are both somewhat in the same 'league', in terms of creativity. MANDELBROT: Thank you. OBRIST: Your great books are quite recent; and your work is ongoing. Yet several mathematicians have told me that the minds in this field produce their best at around 25 years of age… The pressure to produce remarkable findings very quickly is a Damocles' sword hanging over their heads. You are more like a writer or a composer, in the sense that maturity brings about a certain evolution. Some time back, I tried to draw up an informal list of those who have done their best work late in life. Alfred North Whitehead and Bertrand Russell published their celebrated Principia Mathematicae between 1910 and 1913. Therefore, Whitehead was about 50, and before the Principia had not produced anything of significance; Russell was just 30. Yet the cover page lists the coauthors in inverse alphabetic order, meant to emphasise that the senior author was Whitehead. He had been in charge of the mathematical part of the book but Russell was a famous high aristocrat, well known for pacifist views; he went to prison, etc. As a result, Principia Mathematicae is usually considered the work of Russell 'helped' by Whitehead, while the inverse would be more just. OBRIST: Can you describe your current theory of "negative dimension", on which you are planning to write a book? Increasingly negative dimensions characterise objects that become increasingly empty. At first glance this may appear as a kind of bad science fiction, but it is very practical and makes possible to attach a number to the notion of a "progressive emptiness", which elaborates on the common notion of simple emptiness. It is a concept one should not romanticise, so I have recently decided to write of negative prodimension. It promises to become a new domain. Small or big…? The future will tell. After retiring from IBM, I taught mathematics at Yale for 18 years. What was easy at IBM was difficult at Yale, and vice versa. I am keen to finish this work on negative dimensions; it has certainly started well. Also, befitting my age, I should not defer too long the completion of my memoirs. It must not be done too early, but one should not wait too long either. OBRIST: Time is everything… There is also the question of transdisciplinarity, carried out with true contemporary geniuses. I have mentioned Ligeti; and there is another composer, Charles Wuorinen, with whom I did an extraordinary show titled "Music and Fractals" at the Guggenheim Museum in 1990. It is fascinating to see how two people from such different cultures can collaborate, if they desire to do so. OBRIST: The notion of risk explored in your book on markets leads me to ask if it is possible to have a 'fractal' view of the art market. MANDELBROT: Absolutely. The inequality of prices on the art market is astounding: from tens of millions of dollars to only a few dollars. I became deeply interested in the paintings of Frantisek Kupka, the first avantgarde Czech painter, because a certain period of his work was clearly 'fractal'. A major part of Kupka's paintings belonged to a Jewish banker in Prague who had financially supported Kupka, disregarding the fact that the artist was antiSemitic. Those paintings, first confiscated by the Nazis and then by the Communists, have returned to his heirs. But export restrictions affect the market, and another patron, a mysterious German lady, also owns many of his paintings and can clearly influence their value. MANDELBROT: Not at all! To my disappointment, this film nowhere credited my scientific publications. Star Trek 2 was made by Lucas Films; they simply bought my 1977 book, Fractals: Form, Chance, and Dimension, which was not an extravagant investment for them! [Laughs]. The mountains in the film were made using a variant of my method. When the film was first released in California, I did not go to the prerelease show, simply because I did not think that it was worth the travel. However, when it came to my neighbourhood, something quite surprising took place. One of my assistants who had seen the film conveyed the bad news that the special effects created with fractals had been edited out. The next day my wife and I went to the film, and the fractals were staring at both of us. My assistant had been misled by the realistic treatment; he had not seen the fractals, as he was still not used to them. Every graphic design class teaches fractals, and in commercials it is a commonplace technique, an unnoticed daily application… MANDELBROT: Arrabal has written many articles on me, based on his own observations and on interviews that he has read. In addition, many of his recent novels have realistic characters who seem to be inspired by my personality. I do not read all the books that have a character who resembles me. I prefer novels that feature my mathematical work, which usually means the Mandelbrot Set. In 1990, the wellknown science fiction author Arthur C. Clarke published a book based on the Mandelbrot Set. Titled The Ghost from the Grand Banks, it is about an expedition to find the wreck of the Titanic. The cover shows a Mandelbrot Set with the Titanic sinking in the centre. Very nice. Hudson was probably implying that it is important in scientific work to take a view and a presentation which is 'active'; and nothing is more active than an opera! Many scientific articles are completely flat because they are written for people who do not have to be convinced. They are part of a small circle within a wellestablished domain; they write for each other, know more or less everybody, or are introduced by their thesis supervisors or mentors. As a result, style is a very secondary and unimportant thing for them. In my case, the fact that I write for an unknown public necessarily influences and shapes my style. Whether it is opera or Greek drama, one must know how to enter into a subject quickly because one cannot assume that the public will wait to understand. One has to be able to speak to people in their style, motivate and perhaps amuse the reader a little. OBRIST: Paraphrasing the title of Rainer Maria Rilke's magnificent 1903 text Letters to a Young Poet, do you, in 2005, have advice for a young researcher or a young mathematician? One has to be very flexible, more than before. My only advice would be to always keep your options open, because it is possible that you might have to change domains… OBRIST: There are many anecdotal stories about wellknown mathematicians and scientists such as Jacques Hadamard and Geoffrey Harold Hardy, whose critical ideas came to them intuitively, in unexpected circumstances. This has led to discussions about the nature and sources of our knowledge. My scientist friend Israel Rosenfield asks about your view of knowledge, how the brain works, your ideas about memory, perception and consciousness… Is your approach to this reflected in your own work? MANDELBROT: I've skimmed through Hadamard's book, but never read it properly; anyway, it was written late in his life, with much help from his daughter. I've also read The Mathematician's Apology by Hardy. Indeed, the question of chance is very disturbing. If I not had paid attention to those pictures, my science would have been very different, and I might have become a different person. Perhaps — but this is not for me to judge — science would have progressed differently. Hadamard was a wise and balanced man. Hardy was, an ideologue with ideas about 'pure mathematics' that I consider ridiculous; yet these ideas keep being revived, then collapse, and so on. The canonical example Hardy gave of 'pure mathematics', with no applications outside of the selfinterest of studying the topic, was number theory in general, and the study of prime numbers in particular. Number theorists such as Hardy took pride in doing work that had absolutely no military significance. However, this vision was shattered when in the 1970s prime numbers became the basis for the creation of public key cryptography algorithms. This is a use that Hardy, a pacifist, would have hated. When discovering the Mandelbrot Set, I had absolutely no impression of inventing it. While nobody had seen it before, I had a very strong feeling that it existed but remained hidden because nobody had the insight to identify it. All that is actually a great mystery. Mathematics and music are crucial achievements of humanity, and continue to expand in directions quite different from what the ideologues expect. OBRIST: Alighiero Boetti, the artist we discussed earlier, repeatedly turned to your work; for him, it was an example of resistance against the homogenising forces of globalisation, since your perspectives foreground variety, diversity, complexity. Today we are in a very different moment, of a rapidly expanding and intensifying globalisation. What is your point of view here, in relation to your study of markets? MANDELBROT: This question is far too complicated to be answered in a few words, so I prefer not to try. I don't think globalisation necessarily decreases variety, but this is a tentative opinion. I enjoy diversity enormously. I favoured it in my choice of topics to pursue, and I still do my best to help it increase rather than vanish. OBRIST: A wonderful conclusion. Thank you very much. Translated from the French by Aruna Popuri. 

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