#74 from R&D Innovator Volume 3, Number 1          January 1994

Fractals—Keeping the Rough Edges
by Benoit B. Mandelbrot, Ph.D.

Dr. Mandelbrot, who recently retired as IBM Fellow of the IBM Thomas J. Watson Research Center, is Abraham Robinson Professor of Mathematical Sciences at Yale University.  He has been Visiting Professor of Economics, Applied Mathematics, and Mathematics at Harvard; Visiting Professor of Engineering at Yale; and Visiting Professor of Physiology at the Albert Einstein College of Medicine.  Among his many honors is the 1993 Wolf Prize for Physics.

The shapes of mountains, trees, clouds, island coastlines and broken stones are too complicated to be described by Euclidean geometry:  Mountains no more resemble cones than clouds do spheres.  Through fractals, we now have a geometry that describes the shapes of mountains, clouds, and other irregular objects.

Broadly speaking, fractals are shapes whose roughness and fragmentation neither tend to vanish nor to increase, but rather remain essentially unchanged as one zooms into a structure (as with a telescope or microscope).  These structures have a “self-similarity” at many spatial scales.

I’ve been fortunate to see fractal geometry, which I originated in 1975, permeate many fields, including economics, biology, mathematics, art, physics, and engineering.  Even the entertainment industry used fractal modeling to form a new planet in the movie, Star Trek II:  The Wrath of Khan. 

What especially pleases me is that fractal structures strike almost everyone in a friendly but forceful, almost sensual manner.  The artist, the child, and the “man in the street” never seem to get enough of fractals.  They certainly never expected to receive anything so pleasurable from mathematics!

Chess, Not Multiplication Tables

I was born in Poland in 1924.  My mother, a physician, was worried about epidemics and didn't want me to be around lots of children, so I skipped the first two grades of elementary school and studied with my unemployed uncle, Loterman.  He encouraged me to train my mind by reading, and my memory by paying attention to miscellaneous facts.  He disliked rote learning and avoided teaching me the alphabet and multiplication tables (even today they give me trouble).  Most of the time we played chess and read maps. 

I always knew that some people could make a living creating new mathematics, since another uncle was a successful mathematics professor in Paris.  When I was 12, our family moved to France.  I was fascinated by shapes, and remembered forever their properties and peculiarities, but my Parisian uncle thought geometry was a dead subject and never ceased to wonder what had "gone wrong” with me for choosing such an interest.

In 1942 and 1943, the German occupation of France was tightening, and my family decided it would be safer for me to leave school—so I hid, moving from place to place and doing odd jobs.  Once, as an apprentice toolmaker on the railroad, I got to apply my interest in shapes:  The war had caused a shortage of new parts and I was good at making locomotive components from scrap steel.

When Paris was liberated in 1944, I took the entrance exams of the leading science schools.  I had little training in algebra and complicated integrals, while the other students had benefited from an uninterrupted education.  But I could usually find geometric counterparts for almost any analytic problem; for example, when faced with a complicated integral, I instantly related it to a familiar shape.  It turned out that because of this odd skill, I ranked near the top.  You might call this a legal way of “cheating.” 

My uncle took it for granted that I would follow in his footsteps; instead, I went to L’Ecole Polytechnique.  Its best graduates were guaranteed very good jobs, but some administrative rule disqualified me.  Since my rank no longer mattered, I put more attention to my many interests, rather than focusing on classes and aiming for the highest grades.

Multidisciplinary Interests

I won a two-year scholarship to Caltech but found no one I wished to emulate, so I returned to France and defended my thesis in 1952.  Titled “Games of Communication,” it overlapped several disciplines, including statistics, thermodynamics, linguistics, and mathematics. 

John von Neumann sponsored me for a year at the Institute for Advanced Study in Princeton, but it turned out that he didn’t give me significant direction since he was usually away from campus.  As I followed my own instincts, I must have appeared quite undisciplined, frequently dropping a topic in the middle of writing about it to pursue a new interest in a totally foreign field.

Not until I joined the IBM research unit in 1958 did I find a suitable environment to pursue my broad interests.  But even there, I was often viewed as a stranger with the inconvenient habit of wandering across boundaries.  Luckily, the findings I developed could not be ignored, and I began acquiring a kind of fame.  I was popular as a visiting professor in diverse departments, but no major university wanted a permanent professor with such unpredictable interests.  On the other hand, Ralph Gomory, who headed IBM research, sheltered my work, gave me a small staff, and made me an "IBM Fellow," which allowed me tremendous freedom.

In 1961, it was clear to me that I had identified a new phenomenon present in many aspects of nature, something I called “erratic behavior.”  (Others gave it a better name, “chaos.”)  I established that this phenomenon was fundamental to economics, then showed it was central to the physical sciences.  Self-similarity as a description of "chaotic" events became my major interest.  Fortunately for IBM, this field helped solve practical problems about noise and turbulence that the company was interested in.

Still, I remained an outsider in every field I worked in, and just couldn't get my interdisciplinary and philosophical views accepted.  For instance, while working on economics, I was dying to mention that my methods were also pertinent to physics, but the referees of my papers told me to remove this broader philosophy.  Later, when I studied turbulence (which, because of its unpredictability, resembled the stock market), my broader comments were again removed, and many papers were totally rejected.

Finally, in 1973, I was invited to present a major talk in France, which I saw as a golden opportunity to present a general manifesto and explain how my interests fit together.  The lecture was a great success.  Soon, to denote my unified approach, I coined the term fractal.  Within a few years fractal geometry became accepted.

Discovery of Fractals

Much of my work involved “roughness,” which I measure by a quantity called fractal dimension.  In 1975, while working on fractal dimension, I coded some structures for computer graphics representation and was surprised at what I found.  The rough geometric structures were visually rich and thus presumably would require complex rules, but the algorithms to generate them are so extraordinarily short as to look positively dumb!  This means they must be called simple. 

The simplicity of the concept makes it relatively easy for people in diverse fields to apply fractal geometry to a vast number of previously intractable problems.  The discovery of fractals is a fulfillment of what seemed nothing but a dream:  the hope of describing “chaotic” nature as the accumulation of many simple steps.

On occasion, people describe me as a man of great vision.  This term is flattering and seems appropriate, but is not.  I had no over-reaching idea, no star I would see and follow.  Perhaps, my success is better described in reference to smell.  I wonder what I would have achieved if I had a “standard” education, which may not have allowed me the freedom to wander, to follow “scents” that pleased me.