Interviewer: I am here in the studio with mathematician Benoit Mandelbrot, one of the key figures in the study of fractal geometry. Start out by telling us exactly what a fractal is.
Mandelbrot: A fractal is basically the iterations of an equation forever, the unique thing about them is that they are equally complex at any magnification. If you took a fractal and magnified it by 500x, you would see the same level of detail as you did on the entire thing. Fractals are among the most beautiful of all mathematical forms.
Interviewer: Could you tell us about your formative experience as a mathematician?
Mandelbrot: Why certainly, I was born in Poland during the 1920's into a family that was very focused on academia. I was first introduced to the serious study of mathematics by two of my uncles, especially Szolem Mandelbrot, who was a professor of mathematics at the Collège de France. He introduced me to a particular school of thought that held very strongly the evil of applied mathematics because of its potential use to create weapons. I understand the attitudes of my uncle, but my experience with him actually turned me against pure mathematics.
Interviewer: Fascinating. What was your official education in mathematics? What schools did you attend?
Mandelbrot: To tell the truth, and not to sound pretentious, but circumstances prevented me from acquiring a real college or university education in the traditional sense, so I am primarily self taught. I studied for a while at the Lycée Rolin in Paris, but the onset of World War II caused my family to relocate. I sometimes think that my success can be attributed to my unconventional education. I developed a highly geometric and specialized view of mathematics that has served me very well.
Interviewer: Did you find work easily?
Mandelbrot: I worked at various schools in Europe and the United States, but the job that afforded me the most opportunities was the position I held at IBM.
Interviewer: Such as?
Mandelbrot: What led to my main work with fractals was when I first read a paper by Julia that my uncle had brought to my attention, saying that it was a masterpiece. I read it and disliked it, but then again, by this time the difference in attitudes about mathematics between my uncle and I was widening. Exposure to material such as this allowed me to write my books, "Les objets fractals, forn, hasard et dimension" and "The Fractal Geometry of Nature" in which I elaborated on ideas of mine. It also gave me the chance to work with cutting edge technology, which was critical to some of the discoveries I made.
Interviewer: Tell us about your most visible contribution to mathematics, the Mandelbrot Set.
Mandelbrot: (laughs) Of
course! The Mandelbrot Set is basically just a series of points
on a plane that follow very strict rules. To construct this set
you must fist choose a point, let us designate the point as q0.
Then we must perform several calculations. They are as follows.
To obtain q1,q2,q3,q4,q5 and so on, we do this.
q1= q0^2 + q0
q2= q1^2 + q1
q3= q2^2 +q2
q4= q3^2 + q3
Now if that set (q0 through qinfinity) always remains within 2 of the origin, then the point can be said to be in the Mandelbrot Set. Here is a graphic of the Mandelbrot Set.
Interviewer: It's beautiful.
Mandelbrot: Yes, but this
picture does not represent the complexity of the Set. Here are
some pictures of the set in greater detail.
One can see that hidden deep within the set is a smaller representation of the entire set!
Interviewer: Tell us about your book, The Fractal Geometry.
Mandelbrot: In a practical sense, the book was intended to clarify and expand on my idea's concerning fractals, mostly their relation to natural phenomena. It was my contribution to a mathematical system of thought that could not describe the shape of a mountain, or the graceful curvature of a cloud. It was an attempt to make mathematics slightly less cold and dull. With this book, rather than just filling page after page with equations, I included many illustrations.
Interviewer: The fake landscapes were particularly intriguing.
Mandelbrot: The Landscapes, as well as most of the other illustrations, were done with computers. In order to do fractals of that magnitude, very powerful programs are required. My time with IBM was very helpful in allowing me to gain access to the computers required. Some of the first Imaging Programs were created for this purpose. This brings me to another point. Fractal Geometry is a branch of mathematics that is inextricably linked with technology. A human could never draw the level of detail required to perform these calculations.
Interviewer: Along with the dependence on technology, however, comes a wider range of possible applications, especially artistic ones.
Mandelbrot: Quite true, and I feel blessed to have been one of the individuals who launched the study of them.
Interviewer: Thank you for granting this Interview Dr. Mandelbrot.
Mandelbrot: My pleasure. I leave you with a few of the most beautiful fractals that I have encountered.
In 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'. The following year he received the Franklin Medal. In 1987 he was honored with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993 as well as many other prestigious awards.