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 Cantor’s
set theory and Peano’s 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
Mandelbrot
Set
Q.
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 screen–they 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 =
-214 -87i
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
portraits–they can break down photographs into fractal
formulas using computers; fractal formulas are used in movie set
design–by 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 eve
rywhere 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
Q.
To which societies do you belong and how have your works been
recognized?
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.
Works Cited
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.
http://www.math.yale.edu/newsite/people/mandelbrot.htm
http://www.fortunecity.com/emachines/e11/86/mandel.html
The on-line
sources are an excellent summary of Mandelbrot’s 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.
