Today the search Engine Google celebrates with animated Doodle artwork the 96th birthday of Polish-born, French and American mathematician Benoit Mandelbrot, widely known as the “father of fractal geometry.”
Benoit Mandelbrot was a Polish-born French mathematician best known as the father of fractal geometry. In addition to coining the term “fractal” to describe objects and surfaces which are irregular at various dimensions of scale, he also introduced such concepts as “fractal dimensions” and the particular fractal known as the Mandelbrot set, frequently represented with the mathematical formula z → z2 + c.
Mandelbrot joined IBM’s Thomas J. Watson Research Center in 1958, and remained with the company for the rest of his life, eventually becoming first an IBM Fellow, and then a Fellow Emeritus. He was also a Sterling Professor of Mathematical Sciences Emeritus with Yale University. His best known publications include Les objets fractals (1975) and the The Fractal Geometry of Nature (1982), both translated into several languages. Mandelbrot means “almond bread” in German.
Benoit Mandelbrot was born 20 November 1924, of Lithuanian Jewish descent in Warsaw, Poland. His mother was a doctor and his father, though a scholar from a long line of scholars, earned a living as a clothing wholesaler. His uncle, Szolem Mandelbrojt, a professor of the mathematics at the University of Clermont-Ferrand, moved to Paris in 1929 where he quickly rose to the top ranks of the intellectual elite. The young Mandelbrot and his parents followed in 1936, fleeing the economic depression in Poland.
By the following year Mandelbrot had entered a Lycée, or French secondary school (albeit two years belatedly), and his uncle had become professor at the Collège de France, taking over the post once held by Hadamard. It appeared that things were settling into a pleasant and fruitful routine, with his school lessons supplemented by long talks with his uncle about classical analysis, the iterative work of Pierre Fatou and the equally fascinating Julia Sets generated by Gaston Julia. But it was all ripped asunder by the advent of World War II.
While his father ended up in a prison camp and his uncle found temporary asylum teaching in the United States, the young Benoit sought refuge in the countryside, especially Tulle. After the liberation of Paris in 1944, he was able to take entrance exams for both Ecole Normale Supérieure (Rue d’Ulm) and Ecole Polytechnique. Although Ecole Normale was the desired school for those seeking a career in academia, Mandelbrot eventually fled to Polytechnique to escape the anti-intuitive Bourbaki group (ironically co-founded by Szolem Mandelbrojt) whose influence clashed with classical analysis and ran counter to Mandelbrot’s passion for geometry. Mandelbrot had only passed his entrance exams by an ingenious method of intuiting relationships between geometric forms and mathematical problems.
After graduating from École Polytechnique in 1947, Mandelbrot spent two years in the U.S., studying aeronautics at the California Institute of Technology. He then returned to France where he spent a year fulfilling a compulsory stint in the air force before completing his Ph.D. in Mathematical Sciences at the University of Paris in 1952. Like the geometry that he made famous, both his life and the course of his work was neither linear nor simplistic in shape and form. Mandelbrot said he spent the next two years groping, exploring first one field and then another, without any clear sense of the connecting thread. His far ranging interests included an attempted collaboration with Jean Piaget. He was meanwhile employed by the French National Center for Scientific Research (1949 to 1957) and he spent a year at the Institute for Advanced Study in Princeton.
In 1958 Mandelbrot moved to the U.S. to take up a research position at IBM’s Thomas J. Watson Reasearch Center. It was here that he was asked to tackle the problem of line noise. The reigning joke was that the noise was generated by a “guy with a screwdriver” fiddling with some piece of connected equipment. Meanwhile engineers sought to solve the problem by increasing signal strength to drown out the noise. But Mandelbrot’s work eventually showed that the noise was both consistent and erratic, some kind of inescapable natural feature of the system that did not disappear with increased signal strength. But more remarkably he also showed that every burst of noise also contained within it bursts of clear signal (a situation he conceived of in terms of the Cantor set). Stranger still, he found that the ratio of periods of noise to periods of clean transmission remained constant, regardless of the scale of time used to plot the phenomenon (i.e. months, days, seconds).
In 1961, while working in economics, Mandelbrot traveled to Harvard to give a talk on income distribution. As he entered the lecture hall he found that Hendrik Houthakker, who had invited him, had apparently already written the chart Mandelbrot would use to illustrate his lecture on the chalkboard. But Houthakker explained that the chart was actually a description of the results of his own research into the fluctuations in the cotton market. How was this possible?
Mandelbrot returned home and turned his attention to analyzing cotton prices. Using records dating back to 1900, he began to perceive an astonishing pattern — one that hearkened back to his work on line noise a decade earlier. He discovered that cotton prices followed a pattern that was both erratic and regular. That is, although price changes were erratic in terms of normal distribution and no one could predict the exact amount of any particular price change, the changes themselves followed a symmetrical pattern with regards to scaling. Regardless of whether the scale of time was hourly, daily, or monthly, the curve was the same. And it had remained so for at least 60 years (the length of time covered by his records).
By the late 1960s Mandelbrot’s attention was drawn to the work of aged English Nilologist Harold Edwin Hurst, who had spent countless years analyzing records of the rise and fall of the Nile River. Working from Hurst’s accumulated data, Mandelbrot developed a new model that could mimic the river’s own erratic fluctuations. Later he applied a similar technique for modeling the rise and fall of stock market prices. In both cases, Mandelbrot’s models worked so well that he could even use them to generate bogus data points which, when graphed, could not be distinguished from the genuine erratic phenomenon of the actual stock market, or actual Nile River. But the beauty in Mandelbrot’s models was not that they generated a deceitful randomness, but that they could generate graphed data whose visual pattern accurately mimicked the visual patterns created by real phenomena.
In other words, they were representing the situation geometrically. And just as the youthful Mandelbrot had passed his college entrance exams by translating algebraic problems into geometrical problems, and solving them by intuitively deducing the “perfected” shape, he here realized there was something deeper, something mathematical, behind these strange patterns. That something was self-similar symmetry, or the repetition of self-similar structures from one level of scale to another. Mandelbrot’s ability to uncover this truth was greatly aided by his research position with IBM, where he had access to the tremendous number crunching and iteration power of IBM computers. Unlike mathematicians in earlier periods, including Julia and Fatou, he was not limited by the number of calculations he could churn out by human power alone. What’s more, computers were fast advancing in their ability to render complex visual representations.
All of these factors coalesced when Mandelbrot took it upon himself to address the problem of coastline length, raised previously by British scientist Lewis Richardson. Specifically, the problem, which at first glance reminds one of Zeno’s Paradox, contends that coastlines, like national borders, are far from smooth easily measured lines. In fact, their gnarled twists and turns only grow more complex — and lengthier — at increased orders of magnitude and measurement. The more finely tuned the measuring device — or the close to the object is one’s point of view, the more one sees. From a distance sufficiently far away in space, the coast of France is reduced to a point. From the perspective of a satellite, the same coastline has a substantial, but relatively manageable length. But as one zooms in closer and closer, bays, then coves, then ragged protrusions all loom into view. Closer still and one must include in measurement the dimensions of small tidal pools. Even closer and all the dips and thrusts of the porous tidal rock must be measured and considered — on and on, down to the atomic and subatomic level.
But the question remains: short of crawling along the entire length of the coast with complex measuring equipment in tow, how can one determine the length of the coastline? In pursuit of an answer to this and other eclectic mathematical problems, Mandelbrot hit upon the notion of what he called the “fractal dimension” (derived from the Latin “fractus”, meaning to break). While most of us are used to representing physical objects in terms of one, two, or three dimensions (or four, if one considers time), Mandelbrot came up with a way of representing another “dimension” of an object — that is, its degree of roughness and irregularity.
In 1979 Mandelbrot had again turned his eye to the work of Gaston Julia. Letting computers do the exacting and time consuming work of plotting images and performing seemingly endless iterations, he came up with some unique observations, most notably the now famous Mandelbrot Set fractal that bears his name. As with so much of his previous work, he had uncovered a phenomenon with self-similar symmetry at different levels of scale, yet a tremendous degree of irregularity as well. In fact, a visual exploration of the Mandelbrot Set offers a strange and beautiful excursion that conjures up other adventures in scale now possible through new technologies, vistas of the cosmos and of the super high power microscope.
The concept of the fractal, and of the “fractal dimensions” has inspired scientists in a variety of fields to utilize these new tools to describe, explore, and express the subjects of their particular fields. Scientists studying everything from rock porosity to steel strength to the growth and development of a fetus’ lungs have found fractals useful. Even computer artists trying to render believable still and animated landscapes, plants, and fur have found help from fractals. All of which has prompted a number of creative thinkers, in both the arts and the sciences, to wonder whether the discovery of fractals was not in reality the discovery of the blueprint of all life — or at least of some kind of underlying principle upon which it draws. But despite being the inspiration for such metaphysics, Mandelbrot, when asked if fractals don’t point to a single rule underlying reality, simply stated, “There is no single rule that governs the use of geometry. I don’t think one exists”. In addition, according to Mandelbrot, “The beauty of geometry is that it is a language of extraordinary subtlety that serves many purposes”.
