 | Benoit Mandelbrot is pictured here inside a fractal frame. The concept of finding order in seemingly irregular systems has been applied to everything from music to the stockmarket. |
Many years ago, Benoit Mandelbrot, the Sterling Professor of Mathematical Sciences, observed order and simplicity in complex systems that seemingly had none -- systems such as coastlines, clusters of galaxies and blood vessels.
His realization led to a new paradigm in mathematics and his new persona as the "father of fractals." Mandelbrot's work has influenced statistical physics and fields as diverse as graphic design, astronomy, meteorology and computer science. He also is credited with reintroducing the eye and experimentation to the study of mathematics.
In recognition of his work, Mandelbrot will be awarded the Japan Prize by The Science and Technology Foundation of Japan in April. The prize, which honors "original and outstanding achievements that contribute to the progress of science and technology and the promotion of peace and prosperity of mankind," will be presented at a ceremony hosted by the Emperor of Japan.
Born in Poland, Mandelbrot moved with his family to France and studied at the Ecole Polytechnique in Paris.He had a long association with IBM's research laboratories in New York, where he is IBM Fellow Emeritus of the T.J. Watson Research Center.
Mandelbrot spoke with the Yale Bulletin & Calendar recently about fractal geometry, math prodigies and his own childhood, among other subjects. The following is an edited transcript of that discussion.
Fractal geometry is a geometry of roughness. Fractals are geometric shapes that have about the same form whether they are examined from close up or far away.
A piece of music is an organized "whole" that obeys diverse structures. Many have long been identified by musicians, investigated very explicitly and taught to students.
For example, scales were known to musicians long before they were understood by science. However, one structure was left for teachers to enforce informally. "Wallpaper music" and "musak" can be repetitive, but real music cannot. For example, it is important that the movements of a sonata are distinguished by different speeds -- such as fast, slow and fast. Moreover, features other than speed vary within each movement and also within smaller portions of a movement. Besides, the "degrees of variability" on the small, medium and large sections of music must "balance" one another.
Curiously, musicians did not know how to express this notion of balance in useful formal fashion. But, once they saw fractal pictures, several prominent musicians immediately commented that balance is a form of self-affinity, one aspect of fractality. This "analysis" can be continued by a "synthesis," insofar as suitably chosen random "noises" are perceived as (bad) music.
Graphic design often seeks to disturb. Hence, it does not necessarily seek balance. But when it does, it inevitably injects fractality.
Business cycles are oscillations that take a few years to go up, then a few years to go down. But economists must also consider long cycles, as well as faster fluctuations that are not called cycles, but might have been.
There is an old adage on the stock market that price variation is exactly as rough on small scales as it is on large scales. This notion was not taken seriously, but in the 1960s I took it literally and developed its many implications. It is now widely called "scaling." A striking fact is that it allows, and almost demands, sudden crashes. This led me to propose models of price variation that, compared to alternative models, are not only far more accurate, but also simpler and more "efficient" in terms of consequences drawn from a small number of assumptions.
No. I wish I could prevent its being used for that, but others claim that they do use it successfully. A problem with most workers in finance is that they insist on instant applicability and do not take time to develop the models. I try to be more careful.
Yes and no. I found mathematics easy but for many years was more concerned with languages and history. My gift for mathematics revealed itself partly during the last year of high school and fully when I was 19. But this gift and my interests were and continue to be largely focused on geometry rather than algebra.
I was fascinated by the geometry of Caucasian rugs. Also, my father was fascinated with maps, and they filled our apartment. I can't remember a time when I could not read them fluently.
When I was 13, my father's younger brother became professor of mathematics at the College de France in Paris. Therefore, I always knew that mathematics can be a fulfilling career. My uncle and I talked endlessly but were very different. Nevertheless, he influenced me profoundly.
It used to be considered a truism that professional mathematicians sharply subdivide into geometers and algebraists. But one does not become a professional mathematician without being selected and approved. For many decades, the selection process favored the algebraists and did not let the geometers through, unless they agreed to change. This may explain why most living mathematicians are less drawn to shapes than used to be the case historically -- either to shapes or things. They only deal with already established abstractions for which the connections with shapes and things are often remote.
In my case, because of the impact of World War II on my education, I went through the selection process without having been changed by schooling. I think that in the future geometers will again receive a fairer share.
Proponents of the "old math" wanted to teach the subject in the order that it was developed by humanity. Proponents of "new math" wanted to teach it in the style that prevails in research mathematics. Nobody asked the "customers" to express an opinion. But the fractals in my books immediately attracted the attention of young programmers. This spread to other students and attracted their attention and passion. A grassroots movement exploded and Michael Frame and I, and also others, are trying to ride this wild horse.
Two efforts at Yale use fractals in education. "Math 190" ["Fractal Geometry"] is a rarity: a very large course in mathematics. It is offered to undergraduates in all fields except math, hard sciences and engineering. Many were discouraged from math by advisers who told them that any grade less than "A" might prevent them from being admitted at Yale.
In addition, summer workshops on fractals are offered to college and high school teachers.
The utilitarian answer is that mathematics is essential in many life activities. Unfortunately, the teaching of mathematics has nearly always started by concentrating on the aspects that are among the most boring and leave the students' imagination unchallenged. The exact opposite is true of fractals, especially as one sees them being drawn on the computer, then redrawn after the construction has been changed a little.
Mathematics has a great strength that also leads to a great weakness. The monumental strength is that established mathematics can be taught from top down in a way that allows no doubt about the validity of the result. In this approach, the principles are simple -- but also very far from real -- shapes and things. They are not inspiring in themselves and can be called dull, dry, cold. Students must learn something very unnatural: to dominate boredom and recall those principles very exactly. Gratification is endlessly postponed and cannot be found in the end product as much as in quirks of the proofs that lead to that end product.
In limited but significant ways, fractal geometry is the opposite. Of the mathematical topics that can be taught in schools, fractals are the only concept that was developed by someone who is still alive. Moreover, fractals have the romance of being beautiful and involving two forms of drama: the drama that is provided by recent resistance to their acceptance and the drama that comes with the fact that near-beginning students can actually understand what great living mathematicians try -- and somtimes still fail -- to prove.
Two major factors are in favor of fractals. One results from their effectiveness in the sciences. Some bring to mind real things like mountains and clouds or stock market charts. Others are extravagant and totally new, but unconsciously bring to mind all kinds of decorative patterns that humanity has known since time immemorial. The second factor is that gratification comes quickly. The path from silly formulas to impossibly difficult problems is much shorter than is usual in mathematics.
In swimming, tennis, gymnastics, or the piano, the best performers must start very young and must train every waking hour. There is no way to start late and catch up. In mathematics the best students do the assignments in no time, and many examples suggest that where there is a will to catch up, there is always a way. [Carl Friedrich] Gauss, "the Mozart of mathematics," and a small number of other former child prodigies are far from being representative. Few child prodigies become leading mathematicians, while many leading mathematicians spend time on other things until age 17 or even 20. Some have been forced to take a selection exam twice. Others shine as undergraduates and then sink.
There is a strong argument against taking too much math too soon. Many young children can go through all the motions without real appreciation.
At this very moment, I struggle with a paper that promises to be one of the most striking I ever wrote. I wait for three fat papers to come off the press. Mostly, I struggle with what one can call my "legacy," in the form of several books, some of them with co-authors or assistants.
One such book is a memoir of an eventful and turbulent life: gambles that led to cliffhangers but ended well; stories that are fun to tell and to hear. As I had hoped when I was young, my life wove together, perhaps only for a fleeting moment, many separate strands that belong to altogether different fields of knowing and feeling.
-- By Jacqueline Weaver
T H I S
'Father of fractals' discusses
'essential' role of math in life
Could you please define fractal geometry in the simplest terms for the uninitiated?
How does fractal geometry apply to music and art?
Is graphic design based on fractal geometry?
How does fractal geometry apply to economics in general and the stock market in particular?
Have you used fractal geometry in charting stocks for investors?
Were you interested in mathematics as a child?
How did your childhood hobbies et cetera reflect this interest in mathematics?
Was anyone else in your family a mathematician?
You say that at a young age you began to see mathematical problems as shapes. Is this common among other mathematicians?
Tell us about the programs that you have to encourage the teaching of fractal geometry in schools.
How will knowing about fractal geometry help children in school and in life?
Does fractal geometry make mathematics more interesting and less intimidating to children?
How can parents recognize if their child is gifted in mathematics and how would you suggest that they nurture that talent?
What are you working on now?
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