Revelations are not the exclusive province of Catholic saints. One
day in the early 1980s, a colleague of mine rushed into my office and
put a brightly colored image on my desk. - Look! he said, subdued
jubilation in his voice. - Pretty, what is it? - It is the
There was reverence in his usually irreverent voice. My colleague was
something of a free spirit, working for me as a programmer and pursuing
his career as a rock musician, on alternate weeks. It was clear that
he was deeply moved by the image he had put on my desk. He looked as
if something incredible had just happened to him on the way to Damascus.
This was the first time I heard the name Mandelbrot and saw one of
the images that are now so familiar, even in poster art.
Initially, the images may look impressive, but then there are many
examples of artists using computers to generate art. Even children learn
how to make Lissajou figures. The real shock sets in when you learn
that these incredibly intricate pictures are generated by an extremely
simple mathematical formula:
Here, C is a constant and z is a variable that is calculated
iteratively using the formula. In the Mandelbrot set, the initial
value of z, z0 = 0, so z1
= C, z2 = C2 + C
etc. The set is defined simply as those values (or instances) of C
where zn remains limited no matter how many iterations
Clearly, if C = 0, then z will remain 0 no matter how
many iterations we perform, so C = 0 is a member of the Mandelbrot
set. On the other hand, if C = 10 or C = -10, z
will rapidly grow beyond all bounds, so these values do not belong
to the set.
This seems trivial enough, but it is not quite that trivial.
Both zn and C are complex
numbers, each number having a "real" and an "imaginary"
part. Just as "real" numbers are geometrically represented
as points on a line, "complex" numbers are geometrically
represented as points in a plane. Therefore, all possible values of
C can be represented as points in a two-dimensional diagram.
such a diagram, C values that give bounded values of zn
no matter how large n is allowed to become (i. e. no matter
how many iterations are performed) can be assigned a black color,
while C values that lead to unlimited growth of zn
as n grows large can be assigned the background color. The
Mandelbrot set then looks like this:
A rather unremarkable shape with some protrusions. However, if one
zooms in on one of those, similar protrusions at smaller scale appear
at successively larger magnifications. - Interesting, but not enough
to make your jaw drop.
But if, for a given C value, we count the number of iterations
needed for z to surpass a certain limit, and assign a color
to C at that point that corresponds to the number of iterations
needed, amazing things start to happen. This will be especially interesting
along the border between the area where z values remain
limited and the area where z values increase without limit.
This border region is where z is "balancing" between
containment and divergence.
Here is a gallery of some images that have been generated in this
manner. The only parameters that are decided by the programmer/artist
are the size limit of z, the area to be depicted, and the color
coding. This leaves some room for artistic creativity, but the structures
themselves are all defined by the mathematics. They are an inherent
feature of the universe and will look no different to our brothers
in the Andromeda galaxy!
Those swirling patterns continue endlessly at larger magnification.
Similar structures will emerge if we magnify the images by a factor
of a thousand or a trillion. They are self-similar but not repetitive,
i. e. at successively larger magnification, the images resemble each
other but they are never exactly the same.
The shapes may remind us of biological specimens, and this is probably
not just coincidence. Biology, too, exhibits great complexity based
on simple underlying principles. The spiral shapes that occur in snails,
flowers - and in the Mandelbrot set - may be one example of this.
- Today, Mandelbrot's line of research (fractals and self-similarity)
has opened up practical applications in such fields as image compression.
The Mandelbrot set is closely related to Julia sets, studied early
in the last century by the French mathematician Gaston Julia (who
had the misfortune to lose his nose in World War I). Julia was interested
in divergence arising from repeated iterations of polynomials of complex
numbers. His field of research remained obscure until Benoit Mandelbrot
took advantage of the possibilities offered by computers to do experiments
during his research at IBM in the 1970s. The subsequent explosion
of personal computing and computer graphics opened up the field to
thousands of interested amateurs.
Today, any interested aficionado can explore the Mandelbrot set at
home, using one of the many tools that are available on the Internet.
("Mandelbrot" is a good search term. There are not that
many "false positives"!) Here
is one (Java must be installed).
Familiarity breeds contempt, and most scientifically literate laymen
have now gotten accustomed to looking at Mandelbrot images without
giving much thought to the underlying phenomenon of complexity arising
out of simplicity. But somewhere in there, there is room for awe and
- perhaps - for inspiration and hope.