Q. What is
your current position?
A. Professor of Mathematics at Yale University and Physicist at the Thomas J. Watson Research Center of IBM in Yorktown Heights, NY.
Q. What has been your greatest achievement and when did it take place?
A. The discovery of fractal geometry in 1975.
Q. What does the word fractal mean?
A. I coined the name from the Latin adjective fractus, which means fragmented and irregular.
Q. Would you briefly describe what fractal geometry is?
A. It is math that defines many of the irregular and fragmented patterns of nature around us, and shows how these complicated aspects of nature have a mathematical basis. It leads to full-fledged theories by identifying a family of shapes I call fractals. Shapes previously described as grainy, tangled, tortuous, hydralike, or pimply, can be approached in a rigorous and quantitative fashion.
Q. What gave you the inspiration for your research?
A. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern math began with Cantors set theory and Peanos space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. With the availability of the computer, I was able to make great strides in my research, where it would have taken pre-computer mathematicians a lifetime to get the same results.
Q. Can you give some examples of irregular and complicated phenomena that cannot be described by Newton or by Euclidian geometry?
A. Some examples are the shapes of mountains and clouds, how galaxies are distributed in the universe, and the way prices fluctuate in the financial markets. One way of obtaining such a description is to seek a model. I had to invent or identify mathematical rules that can produce mechanical or computer forgeries of some part of the reality.
Q. What are some of the characteristics of these geometric shapes called fractals?
A. Contrary to the shapes defined by Euclid, they are not regular. First, they are irregular all over. Secondly, they have the same degree of irregularity on all scales. A fractal object looks the same when examined from far away or nearby. It is, therefore, self-similar. The small pieces of the whole are well-defined objects whose shape is roughly that of the previously examined whole. In nature, examples of fractals are ferns, cauliflower, and broccoli, where each branch is very like the whole. The rules that govern growth ensure that small-scale features are translated into large-scale ones.
Other fractals, however, deviate from linear self-similarity. Some describe general randomness, while others can describe chaotic, or nonlinear, systems- where the factors affecting the way the system behaves are not proportional to the effects they produce.
Fractals also have fractal dimension, which is not a whole number. The dimension of a fractal falls between dimensions. Line dimension is 1, square dimension is 2, cube dimension is 3, but fractal dimension equals 1.2345... A way to illustrate what is meant by fractal dimension is to take a sheet of paper, which will be considered a plane of 2 dimensions. If you crumple it up, the object created is neither a plane nor a sphere, but something folded in between the second and third dimensions. In fractal geometry, this paper has a dimension of about 2.5.
Q. Can you briefly describe the Mandelbrot set and explain its importance?
A. The Mandelbrot set is my contribution to the area of nonlinear fractals. The set results from iterating a simple equation on the computer. It produces extraordinary, complex graphics. Some people call it the icon for nonlinear fractal geometry. Examining the Mandelbrot set has led to many conjectures that were simple to state but hard to prove. Studying them has produced a lot of interesting side results. Many related fractals lead to beautiful and intriguing graphics. In fact, several shapes known today as fractals were discovered many years ago, some by Gaston Julia, Poincare, and Fatou between 1875 and 1925, but nobody realized their importance as visual descriptive tools and their relevance to the physics of the real world.
One model where random fractals describe the real world is a form of random growth called diffusion limited aggregation, or DLA. DLA produces complex, tree-like shapes which can be used to model how ash forms, how water seeps through rock, how cracks spread in a solid, and how lightening discharges.
DLA Random Fractal Set
Can you explain how a fractal image is developed?
A. It starts with a geometric figure called the initiator and a mathematical rule called an algorithm. The resulting figure after one application of the rule is called the generator. Iteration uses the results of one generation for the initial condition of the next. The computer uses a slightly different process in generating fractal images.
Q. Can you explain how a computer generates fractal images?
A. Instead of having a geometric figure as an initiator, the images require an iterating function with a numerical initiator. Their complexity is limited by the number of units on the computer screenthey are colored pixel by pixel.
To begin, picture the computer screen as an invisible grid which is a complex plane that uses complex numbers(real numbers added or subtracted to imaginary numbers) such as 5 + 2i. On the complex plane, the x-axis is the real axis and the y-axis is the imaginary axis. A complex number can be plotted on this plane. The letter z is used to represent a complex number, so z = x + yi. This is called a complex variable. Next, a function is a sort of mathematical machine. The function f(x) = 2x + 4 tells you to double a number and add 4. The variable x is whatever number you choose to put into this function. In generating fractals, you use the functions of a complex variable, which is represented by z, such as the function f(z) = z2 + c where c is a complex number constant which means it does not change throughout the entire fractal calculation. It remains constant throughout the entire two hundred million possible calculations of each fractal image.
Taking the iterative function f(z) = z2 + c on the complex plane, where z and c represent complex numbers, this is what happens:
Let z = 0 Let c = 2 + 3i
f(0) = 02 + 2 + 3i = 2+3i
f(2 + 3i) = (2 + 3i)2 + 2 + 3i =
(2 + 3i)(2 + 3i) + 2 + 3i =
4 + 12i + 9i2 + 2 + 3i =
4 + 12i -9 + 2 + 3i = -3 + 15i
Iterating again, f(-3 + 15i) = (-3 + 15i)2 + 2 + 3i =
The computer can do these calculations at a very rapid pace.
When the iterative process is continued, the values in each generation travel farther and farther from the origin. The points obtained in each generation provide the framework for the orbit of the z, called the seed, in this case 0. For most values of c, the orbit of zero will shoot out to infinity, but for some, the orbit will settle close to a particular number or will alternate back and forth between values. The complex numbers can be separated according to what types of orbits zero has. This separation forms the basis for an exclusive set. This select group is called the Mandelbrot set. In summary, a complex number, c, belongs to the Mandelbrot set if the orbit of zero does not fly off to infinity.
Q. In what fields are fractals most useful?
A. They have been useful in many fields of study. Fractal formulas help musical composers write new songs; photographers have found fractals useful in reprinting portraitsthey can break down photographs into fractal formulas using computers; fractal formulas are used in movie set designby programing computers, alien worlds are created in movies such as Star Wars. Fractals are important in the fields of science and medicine. Fractals have been discovered in DNA patterns in chromosomes, and everywhere in the human body, including the circulatory system and lungs. The AIDS virus and cancer cells show fractal characteristics, and by studying them in the context of fractal geometry, researchers hope to discover how they form. In the study of the universe, fractal patterns have been noted in shapes, features, and placement of heavenly bodies. Modern problems such as static in a telephone or an oil spill can be solved through the use of fractal geometry. Mathematicians now have the ability to create models that describe many complex problems and therefore have the means to try to solve them.
Computer Generated Fractal Landscape
To which societies do you belong and how have your works been
A. I belong to the National Academy of Sciences, the American Academy of Arts and Sciences, and have received the 1985 Barnard Medal for Meritorious Service to Science, the 1986 Franklin Medal for Original and Eminent Service in Science, the Steinmetz and Richardson Medals, the Wolf, Harvey, Humboldt, Nevada, Honda, Science for Art and Proctor Prizes, and the Caltech Distinguished Service and Scott Awards.
Q. How would you sum up your research?
A. It has concentrated on extreme and unpredictable irregularity in natural phenomena in the physical, social, and biological sciences. I hope that the Mandelbrot set and other fractal graphics which appear on posters and even T-shirts, will help to give people a feeling for the beauty and eloquence of mathematics, and its profound relationship with the real world.
Mandelbrot, Benoit. The Fractal Geometry of Nature. New York: W.H.Freeman and Company, 1977.
The Fractal Geometry of Nature is superb because it is filled with color and black- and- white illustrations from many fields of knowledge and gives interesting details about the people who contributed to the subject. Since it is written entirely by Mandelbrot, one can get a sense of who he is and how he thinks. His emphasis is on applying mathematics to our natural world. His work is revolutionary and he writes so that non-mathematicians can understand what he is saying, which is truly remarkable.
Pteigen, Heinz-Otto and Dietmar Saupe, editors. The Science of Fractal Images . New York: Springer-Verlag, 1988. Connecticut: Greenwood Press, 1998
The Science of Fractal Images is also an amazing book. The forward of the book is written by Mandelbrot, so you get a good feel for how he views the research in his field. He writes in a casual, friendly tone, and in a very understandable way. There are a lot of interesting diagrams and impressive photographs in this book. Mandelbrot has an essay on fractal landscapes in this book.
The on-line sources are an excellent summary of Mandelbrots work. It is a good starting point if you have never researched fractals before. You can contact him by telephone and e-mail and get the latest updates of his research.