our major conversations revolve around. It is mostly matters literary and aesthetic, or historical gossip about people of extraordinary intellectual refinement. I mean refinement, not achievement. Mandelbrot could tell stories about the phenomenal array of hotshots he has worked with over the past century, but somehow I am programmed to consider scientists’ personae far less interesting than those of colorful erudites. Like me, Mandelbrot takes an interest in urbane individuals who combine traits generally thought not to coexist together. One person he often mentions is Baron Pierre Jean de Menasce, whom he met at Princeton in the 1950s, where de Menasce was the roommate of the physicist Oppenheimer. De Menasce was exactly the kind of person I am interested in, the embodiment of a Black Swan. He came from an opulent Alexandrian Jewish merchant family, French and Italian- speaking like all sophisticated Levantines. His forebears had taken a Venetian spelling for their Arabic name, added a Hungarian noble title along the way, and socialized with royalty. De Menasce not only converted to Christianity, but became a Dominican priest and a great scholar of Semitic and Persian languages. Mandelbrot kept questioning me about Alexandria, since he was always looking for such characters.
True, intellectually sophisticated characters were exactly what I looked for in life. My erudite and polymathic father – who, were he still alive, would have only been two weeks older than Benoit M. – liked the company of extremely cultured Jesuit priests. I remember these Jesuit visitors occupying my chair at the dining table. I recall that one had a medical degree and a PhD in physics, yet taught Aramaic to locals in Beirut’s Institute of Eastern Languages. His previous assignment could have been teaching high school physics, and the one before that was perhaps in the medical school. This kind of erudition impressed my father far more than scientific assembly-line work. I may have something in my genes driving me away from
Although Mandelbrot often expressed amazement at the temperament of high-flying erudites and remarkable but not-so-famous scientists, such as his old friend Carleton Gajdusek, a man who impressed him with his ability to uncover the causes of tropical diseases, he did not seem eager to trumpet his association with those we consider great scientists. It took me a while to discover that he had worked with an impressive list of scientists in seemingly every field, something a name-dropper would have brought up continuously. Although I have been working with him for a few years now, only the other day, as I was chatting with his wife, did I discover that he spent two years as the mathematical collaborator of the psychologist Jean Piaget. Another shock came when I discovered that he had also worked with the great historian Fernand Braudel, but Mandelbrot did not seem to be interested in Braudel. He did not care to discuss John von Neuman with whom he had worked as a postdoctoral fellow. His scale was inverted. I asked him once about Charles Tresser, an unknown physicist I met at a party who wrote papers on chaos theory and supplemented his researcher’s income by making pastry for a shop he ran near New York City. He was emphatic:
THE PLATONICITY OF TRIANGLES
Now, why am I calling this business Mandelbrotian, or fractal, randomness? Every single bit and piece of the puzzle has been previously mentioned by someone else, such as Pareto, Yule, and Zipf, but it was Mandelbrot who a) connected the dots, b) linked randomness to geometry (and a special brand at that), and c) took the subject to its natural conclusion. Indeed many mathematicians are famous today partly because he dug out their works to back up his claims – the strategy I am following here in this book. “I had to invent my predecessors, so people take me seriously”, he once told me, and he used the credibility of big guns as a rhetorical device. One can almost always ferret out predecessors for any thought. You can always find someone who worked on a part of your argument and use his contribution as your backup. The scientific association with a big idea, the “brand name”, goes to the one who connects the dots, not the one who makes a casual observation – even Charles Darwin, who uncultured scientists claim “invented” the survival of the fittest, was not the first to mention it. He wrote in the introduction of
Triangles, squares, circles, and the other geometric concepts that made many of us yawn in the classroom may be beautiful and pure notions, but they seem more present in the minds of architects, design artists, modern art buildings, and schoolteachers than in nature itself. That’s fine, except that most of us aren’t aware of this. Mountains are not triangles or pyramids; trees are not circles; straight lines are almost never seen anywhere. Mother Nature did not attend high school geometry courses or read the books of Euclid of Alexandria. Her geometry is jagged, but with a logic of its own and one that is easy to understand.
I have said that we seem naturally inclined to Platonify, and to think exclusively in terms of studied material: nobody, whether a bricklayer or a natural philosopher, can easily escape the enslavement of such conditioning. Consider that the great Galileo, otherwise a debunker of falsehoods, wrote the following:
The great book of Nature lies ever open before our eyes and the true philosophy is written in it. … But we cannot read it unless we have first learned the language and the characters in which it is written. … It is written in mathematical language and the characters are triangles, circles and other geometric figures.
Was Galileo legally blind? Even the great Galileo, with all his alleged independence of mind, was not capable of taking a clean look at Mother Nature. I am confident that he had windows in his house and that he ventured outside from time to time: he should have known that triangles are not easily found in nature. We are so easily brainwashed.
We are either blind, or illiterate, or both. That nature’s geometry is not Euclid’s was so obvious, and nobody, almost nobody, saw it.
This (physical) blindness is identical to the ludic fallacy that makes us think casinos represent randomness.
But first, a description of fractals. Then we will show how they link to what we call power laws, or scalable laws.
The veins in leaves look like branches; branches look like trees; rocks look like small mountains. There is no qualitative change when an object changes size. If you look at the coast of Britain from an airplane, it resembles what you see when you look at it with a magnifying glass. This character of self-affinity implies that one deceptively short and simple rule of iteration can be used, either by a computer or, more randomly, by Mother Nature, to build shapes of seemingly great complexity. This can come in handy for computer graphics, but, more important, it is how nature works. Mandelbrot designed the mathematical object now known as the Mandelbrot set, the most famous object in the history of mathematics. It became popular with followers of chaos theory because it generates pictures of ever increasing complexity by using a deceptively minuscule recursive rule;
These objects play a role in aesthetics. Consider the following applications:
