Fractal Geometry

“No one influenced my scientific life more than the uncle I knew best. He was a noted mathematician who reached the top of French academia, the Collège de France, when he was thirty-eight and I was thirteen. So I always knew that science is not just recorded in dusty tomes, but is a flourishing enterprise, and the option of becoming a scientist was familiar to me for as long as I can remember. I might now, following the usual pattern, continue with fond reminiscences of teachers and postdoctoral mentors. But, in fact, I seem to have fled from teachers and mentors, and even existing disciplines. No wonder that my scientific life ran against every pattern. The wonder is that it developed at all.”

“The Mandelbrot set is the modern development of a theory developed independently in 1918 by Gaston Julia and Pierre Fatou. Julia wrote an enormous book—several hundred pages long—and was very hostile to his rival Fatou. That killed the subject for 60 years because nobody had a clue how to go beyond them. My uncle didn’t know either, but he said it was the most beautiful problem imaginable and that it was a shame to neglect it. He insisted that it was important to learn Julia’s work and he pushed me hard to understand how equations behave when you iterate them rather than solve them.”

“I did not do well [on the science schools entrance exams] because of my skills at algebra and complicated integrals—these skills demand training and I had had little formal training—but because of a peculiar inborn gift that revealed itself, quite suddenly in my mid-teens. Faced with some complicated integral, I instantly related it to a familiar shape; usually it was exactly the shape that had motivated this integral. … You might say this was a way of cheating at the exams, but without breaking any written rule.”

“I was cheating but my strange performance never broke any written rule. Everyone else was training towards speed and accuracy in algebra and reduction of complicated integrals; I managed to be examined on the basis of speed and good taste in translating algebra back into geometry and then thinking in terms of geometrical shapes.”

“Anyhow, I think it is a fact that some people think best in formulas, and other people think best in shapes. A hundred years ago, this was almost a platitude among mathematicians, but people who think in formulas now run the show in every branch of science, and for a while they could not tolerate even one person who proclaimed he thinks in shapes.”

“The question I raised in 1967 is, ‘how long is the coast of Britain’ and the correct answer is ‘it all depends.’ It depends on the size of the instrument used to measure length. … It is clear that, as measurement becomes increasingly refined, the measured length will increase. Thus, all the coastlines are of infinite length in a certain sense. But of course some are more infinite than others.”

“It may have become true that people who think best in shapes tend to go into the arts, and that people who go into science or mathematics are those who think in formulas. On these grounds, one might argue that I was misplaced in going into science, but I do not think so. Anyhow, I was lucky to be able—eventually—to devise a private way of combining mathematics, science, philosophy and the arts.”

“Very great minds had tried to tackle turbulence by analytical techniques; they did not succeed, while it seems I succeeded by looking at turbulence via the shapes that it generates.”

“Mathematicians’ lack of perspective can be breathtaking.”

“Science is organized into tight branches, and the only assured way to leave a mark on a branch is to visit it in person, so to speak. This demands adaptability on the part of the visitor, and takes enormous amounts of his time.”

“I looked for ways to apply my gift for shape, and a growing knowledge of various fields, to real, concrete and complex problems. I wanted to keep far from organized physics and mathematics and instead find a degree of order in some area—significant or not—where everyone else saw a lawless mess.”

“I always moved in hot pursuit of a technical problem that was congenial because its ‘taste’ was the same as that of a problem I had met elsewhere and had liked. In due time, of course, the space of topics I like has expanded and eventually it jelled. It is a significant part of a cluster of related investigations that include the ‘search for order in chaos’ and ‘the study of scaling in nature,’ two subjects pursued by many scientists today. During a long earlier period after my 1952 PhD, however, no other individual with skills and energy to spare was giving more than a passing nod to what I was calling the ‘study of erratic natural phenomena’.”

“While a maverick’s story is not in the least an example to follow, it may carry the following useful message: a good sprinkling of diversity is just as indispensable to the good functioning and survival of science as it is to the welfare of society as a whole.”

“Starting that year [1961], I established that this new phenomenon was central to economics. Next, I established that it was central to vital parts of physical science, and moreover that it involved the concrete interpretation of the great counterexamples of analysis. And finally, I found that it had a very important visual aspect. I was back to geometry after years of analytic wilderness!”

“I soon came to distinguish two syndromes in price variation; sudden jumps and non-periodic ‘cycles,’ which I later denoted by the expressions Noah and Joseph Effects.”

“The fate of being called a good physicist by mathematicians and a good mathematician by physicists was one I always feared and fought. And I discount all praise for my work in economics by people other than economists. While it is good to be free of the economists’ ‘before the fact’ censorship, I also want to win ‘after the fact’ understanding and approval of at least a part of their community.”

“Thus, computer graphics allowed the elimination of certain theories simply on the basis of the obvious unreasonableness of the shapes they generate. The same trick worked even better with coastlines and mountains. Graphics techniques gradually became better and better, and we could afford to do some fancy stuff.”

“I started playing with fairly complicated mappings, and was amazed to discover that sets that Julia and Fatou had characterized negatively, as being pathologically complicated, were in fact of extraordinary beauty. As they first emerge on the computer screen, they seem totally strange for a moment, but one soon comes to feel one has always known them.”

“My study showed that this set is an astonishing combination of utter simplicity and mind-bogging complication. At first sight, it is a ‘molecule’ made of bonded ‘atoms,’ one shaped like a cardioid, and the other nearly circular. But a closer look discloses an infinity of smaller molecules shaped like the big one, and linked by what I proposed to call a ‘devil’s polymer.’ Don’t let me go on raving about this set’s beauty.”

“Every discovery I made while at IBM fell well outside the scope of any university department. I did not add new fields because of boredom or coercion, but because of new opportunities.”

“A[n academic] department is like a small specialized business whose principal products are narrowly focused articles, books and new graduates ready to join the corresponding guild. On the other hand, in order to serve the corporation, IBM’s Research Division took a longer and deeper view. Scientifically, it was broadly based and historical circumstances prepared it to risk sheltering a few one-man projects not limited to an established area. It started in the 1950s with none of the glamour of its competitors in academia and industry. By 1993 (when the policy changed), the quality of its staff, most notably the abnormally large numbers of resident mavericks, had made it one of the most respected scientific powerhouses, worldwide.”

“Being at IBM handed me, again and again, marvelous opportunities to make long strings of discoveries that, under different conditions, would have been either discouraged by funding agencies or shared by immediate competition. An unperturbed research environment was a totally unexpected gift.”

“In fact, [IBM] had cornered the market for a certain type of oddball; we never had the slightest feeling of being the establishment.”

“Not knowing how to fit me in an existing peer-review pigeon-hole, that university saw no way to employ me. IBM did.”

“I overturned a horn of plenty in which all kinds of things humanity has always known were located.”