6.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Benoit Mandelbrot | 2/4 | https://en.wikipedia.org/wiki/Benoit_Mandelbrot | reference | science, encyclopedia | 2026-05-05T18:09:35.179387+00:00 | kb-cron |
=== Randomness and fractals in financial markets === Mandelbrot saw financial markets as an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations. In his early work, he found that the price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter. The latter work from the early 60s was done with daily data of cotton prices from 1900, long before he introduced the word 'fractal'. In later years, after the concept of fractals had matured, the study of financial markets in the context of fractals became possible only after the availability of high frequency data in finance. In the late 1980s, Mandelbrot used intra-daily tick data supplied by Olsen & Associates in Zurich to apply fractal theory to market microstructure. This cooperation led to the publication of the first comprehensive papers on scaling law in finance. This law shows similar properties at different time scales, confirming Mandelbrot's insight of the fractal nature of market microstructure. Mandelbrot's own research in this area is presented in his books Fractals and Scaling in Finance and The (Mis)behavior of Markets.
=== Developing "fractal geometry" and the Mandelbrot set === As a visiting professor at Harvard University, Mandelbrot began to study mathematical objects called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set which he began visualizing in late 1979 and first published on in 1980.
In 1975, Mandelbrot coined the term fractal to describe these structures and first published his ideas in the French book Les Objets Fractals: Forme, Hasard et Dimension, later translated in 1977 as Fractals: Form, Chance and Dimension. According to computer scientist and physicist Stephen Wolfram, the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before". Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals":
Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space. Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. Fern leaves and Romanesque broccoli are two examples from nature." He points out an unexpected conclusion:
One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot. Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphics computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s psychedelic art with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist Johannes Kepler, who calculated and described the orbits of the planets.
Mandelbrot, however, never felt he was inventing a new idea. He described his feelings in a documentary with science writer Arthur C. Clarke:
Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science. According to Clarke, "the Mandelbrot set is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity?" Clarke also notes an "odd coincidence":
the name Mandelbrot, and the word "mandala"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas. In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature. This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts". In 1987, Mandelbrot joined the Department of Mathematics at Yale, while continuing as an IBM Fellow until his fellowship ended in 1993. He joined the Department of Mathematics at Yale, and obtained his first tenured post in 1999, at the age of 75. He invited his colleague Michael Frame to work at Yale and co-published various articles with him. At the time of Mandelbrot’s retirement in 2005, he was Sterling Professor of Mathematical Sciences.