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An
Introduction to Mathematical Chaos Theory and Fractal Geometry
by
Manus J. Donahue, III |
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 Editor's
note: Manus J. Donahue, III composed this essay in December, 1997 as
a student at Duke University majoring in physics, mathematics and
philosophy. It has been unofficially published in two different
countries, has been cited in The New York Times and has been awarded
technology site of the day. The article has been reprinted here with the
author's permission.
This page has also been selected by the researchers at Lightspan
StudyWeb® to receive their award as one of the best educational
resources on the Web.
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"The stock markets are said to be nonlinear, dynamic
systems. Chaos theory is the mathematics of studying such nonlinear,
dynamic systems."
The world of mathematics
has been confined to the linear world for centuries. That is to say,
mathematicians and physicists have overlooked dynamical systems as
random and unpredictable. The only systems that could be understood in
the past were those that were believed to be linear, that is to say,
systems that follow predictable patterns and arrangements. Linear
equations, linear functions, linear algebra, linear programming, and
linear accelerators are all areas that have been understood and mastered
by the human race. However, the problem arises that we humans do not
live in an even remotely linear world; in fact, our world should indeed
be categorized as nonlinear; hence, proportion and linearity is scarce.
How may one go about pursuing and understanding a nonlinear system in a
world that is confined to the easy, logical linearity of everything?
This is the question that scientists and mathematicians became burdened
with in the 19th Century; hence, a new science and mathematics was
derived: chaos theory.
The acceptable definition of chaos theory states, chaos theory is
the qualitative study of unstable aperiodic behavior in deterministic
nonlinear dynamical systems. A dynamical system may be defined to be a
simplified model for the time-varying behavior of an actual system, and
aperiodic behavior is simply the behavior that occurs when no variable
describing the state of the system undergoes a regular repetition of
values. Aperiodic behavior never repeats and it continues to manifest
the effects of any small perturbation; hence, any prediction of a future
state in a given system that is aperiodic is impossible. Assessing the
idea of aperiodic behavior to a relevant example, one may look at human
history. History is indeed aperiodic since broad patterns in the rise
and fall of civilizations may be sketched; however, no events ever
repeat exactly. What is so incredible about chaos theory is that
unstable aperiodic behavior can be found in mathematically simply
systems. These very simple mathematical systems display behavior so
complex and unpredictable that it is acceptable to merit their
descriptions as random.
An interesting question arises from many skeptics concerning why
chaos has just recently been noticed. If chaotic systems are so
mandatory to our every day life, how come mathematicians have not
studied chaos theory earlier? The answer can be given in one word:
computers. The calculations involved in studying chaos are repetitive,
boring and number in the millions. No human is stupid enough to endure
the boredom; however, a computer is always up to the challenge.
Computers have always been known for their excellence at mindless
repetition; hence, the computer is our telescope when studying chaos.
For, without a doubt, one cannot really explore chaos without a
computer.
Before advancing into the more precocious and advanced areas of chaos,
it is necessary to touch on the basic principle that adequately
describes chaos theory, the Butterfly Effect. The Butterfly Effect was
vaguely understood centuries ago and is still satisfactorily portrayed
in folklore:
For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, the battle was lost;
For want of a battle, the kingdom was lost!
Small variations in initial conditions result in huge, dynamic
transformations in concluding events. That is to say that there was no
nail, and, therefore, the kingdom was lost. The graphs of what seem to
be identical, dynamic systems appear to diverge as time goes on until
all resemblance disappears.
Perhaps the most identifiable symbol linked with the Butterfly Effect is
the famed Lorenz Attractor. Edward Lorenz, a curious meteorologist, was
looking for a way to model the action of the chaotic behavior of a
gaseous system. Hence, he took a few equations from the physics field of
fluid dynamics, simplified them, and got the following three-dimensional
system:
dx/dt=delta*(y-x)
dy/dt=r*x-y-x*z
dz/dt=x*y-b*z
Delta represents the "Prandtl number," the ratio of the fluid
viscosity of a substance to its thermal conductivity; however, one does
not have to know the exact value of this constant; hence, Lorenz simply
used 10. The variable "r" represents the difference in
temperature between the top and bottom of the gaseous system. The
variable "b" is the width to height ratio of the box which is
being used to hold the gas in the gaseous system. Lorenz used 8/3 for
this variable. The resultant x of the equation represents the rate of
rotation of the cylinder, "y" represents the difference in
temperature at opposite sides of the cylinder, and the variable
"z" represents the deviation of the system from a linear,
vertical graphed line representing temperature.
If one were to plot the three differential equations on a
three-dimensional plane, using the help of a computer of course, no
geometric structure or even complex curve would appear; instead, a
weaving object known as the Lorenz Attractor appears. Because the system
never exactly repeats itself, the trajectory never intersects itself.
Instead it loops around forever. I have included a computer animated
Lorenz Attractor which is quite similar to the production of Lorenz
himself. The following Lorenz Attractor was generated by running data
through a 4th-order Runge-Kutta fixed-timestep integrator with a step of
.0001, printing every 100th data point. It ran for 100 seconds, and only
took the last 4096 points. The original parameters were a =16, r =45,
and b = 4 for the following equations (similar to the original Lorenz
equations):
x'=a(y-x)
y'=rx-y-xz
z'=xy-bz
The initial position of the projectory was (8,8,14). When the points
were generated and graphed, the Lorenz Attractor was produced in 3-D.
The attractor will continue weaving back and forth between the two
wings, its motion seemingly random, its very action mirroring the chaos
which drives the process. Lorenz had obviously made an immense
breakthrough in not only chaos theory, but life. Lorenz had proved
that complex, dynamical systems show order, but they never repeat. Since
our world is classified as a dynamical, complex system, our lives, our
weather, and our experiences will never repeat; however, they should
form patterns.
Lorenz, not quite convinced with his results, did a follow-up experiment
in order to support his previous conclusions. Lorenz established an
experiment that was quite simple; it is known today as the Lorenzian
Waterwheel. Lorenz took a waterwheel; it had about eight buckets spaced
evenly around its rim with a small hole at the bottom of each . The
buckets were mounted on swivels, similar to Ferris-wheel seats, so that
the buckets would always point upwards. The entire system was placed
under a waterspout. A slow, constant stream of water was propelled from
the waterspout; hence, the waterwheel began to spin at a fairly constant
rate. Lorenz decided to increase the flow of water, and, as predicted in
his Lorenz Attractor, an interesting phenomena arose. The increased
velocity of the water resulted in a chaotic motion for the waterwheel.
The waterwheel would revolve in one direction as before, but then it
would suddenly jerk about and revolve in the opposite direction. The
filling and emptying of the buckets was no longer synchronized; the
system was now chaotic. Lorenz observed his mysterious waterwheel for
hours, and, no matter how long he recorded the positions and contents of
the buckets, there was never and instance where the waterwheel was in
the same position twice. The waterwheel would continue on in chaotic
behavior without ever repeating any of its previous conditions. A graph
of the waterwheel would resemble the Lorenz Attractor.
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Now it may be
accepted from Lorenz and his comrades that our world is indeed linked
with an eerie form of chaos. Chaos and randomness are no longer ideas
of a hypothetical world; they are quite realistic here in the status
quo. A basis for chaos is established in the Butterfly Effect, the
Lorenz Attractor, and the Lorenz Waterwheel; therefore, there must be an
immense world of chaos beyond the rudimentary fundamentals. This new
form mentioned is highly complex, repetitive, and replete with intrigue. |
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"If
we do discover a complete theory, it should in time be understandable in
broad principle by everyone, not just a few scientists. Then we shall
all, philosophers, scientists, and just ordinary people, be able to take
part in the discussion of the question of why it is that we and the
universe exist."
--- Stephen Hawking
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The extending and folding of
chaotic systems give strange attractors, such as the Lorenz Attractor,
the distinguishing characteristic of a nonintegral dimension. This
nonintegral dimension is most commonly referred to as a fractal
dimension. Fractals appear to be more popular in the status quo for
their aesthetic nature than they are for their mathematics. Everyone
who has seen a fractal has admired the beauty of a colorful, fascinating
image, but what is the formula that makes up this glitzy image? The
classical Euclidean geometry that one learns in school is quite
different than the fractal geometry mainly because fractal geometry
concerns nonlinear, nonintegral systems while Euclidean geometry is
mainly oriented around linear, integral systems. Hence, Euclidean
geometry is a description of lines, ellipses, circles, etc. However,
fractal geometry is a description of algorithms. There are two basic
properties that constitute a fractal. First, is self-similarity,
which is to say that most magnified images of fractals are essentially
indistinguishable from the unmagnified version. A fractal shape will
look almost, or even exactly, the same no matter what size it is viewed
at. This repetitive pattern gives fractals their aesthetic nature.
Second, as mentioned earlier, fractals have non-integer dimensions. This
means that they are entirely different from the graphs of lines and
conic sections that we have learned about in fundamental Euclidean
geometry classes. By taking the midpoints of each side of an equilateral
triangle and connecting them together, one gets an interesting fractal
known as the Sierpenski Triangle. The iterations are repeated an
infinite number or times and eventually a very simple fractal arises:
In addition to the famous Sierpenski Triangle, the Koch Snowflake is
also a well noted, simple fractal image. To construct a Koch Snowflake,
begin with a triangle with sides of length 1. At the middle of each
side, add a new triangle one-third the size; and repeat this process for
an infinite amount of iterations. The length of the boundary is 3 X 4/3
X 4/3 X 4/3...-infinity. However, the area remains less than the area of
a circle drawn around the original triangle. What this means is that an
infinitely long line surrounds a finite area. The end construction of a
Koch Snowflake resembles the coastline of a shore.
The two fundamental fractals that I have included provided a basis for
much more complex, elaborate fractals. Two of the leading researchers in
the field of fractals were Gaston Maurice Julia and Benoit Mandelbrot.
Their discoveries and breakthroughs will be discussed next.
On February 3rd, 1893, Gaston Maurice Julia was born in Sidi Bel Abbes,
Algeria. Julia was injured while fighting in World War I and was forced
to wear a leather strap across his face for the rest of his life in
order to protect and cover his injury. He spent a large majority of his
life in hospitals; therefore, a lot of his mathematical research took
place in the hospital. At the age of 25, Julia published a 199 page
masterpiece entitled "Memoire sur l'iteration des fonctions."
The paper dealt with the iteration of a rational function. With the
publication of this paper came his claim to fame. Julia spent his life
studying the iteration of polynomials and rational functions. If f(x) is
a function, various behaviors arise when "f" is iterated or
repeated. If one were to start with a particular value for x, say x=a,
then the following would result:
a, f(a), f(f(a)), f(f(f(a))), etc.
Repeatedly applying "f" to "a" yields arbitrarily
large values. Hence, the set of numbers is partitioned into two parts,
and the Julia set associated to "f" is the boundary between
the two sets. The filled Julia set includes those numbers x=a for which
the iterates of "f" applied to "a" remain bounded.
The following fractals belong to the Julia set.
Julia became famous around the 1920's; however, upon his demise, he was
essentially forgotten. It was not until 1970 that the work of Gaston
Maurice Julia was revived and popularized by Polish born Benoit
Mandelbrot. |
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Benoit
Mandelbrot was born in Poland in 1924. When he was 12 his family
emigrated to France and his uncle, Szolem Mandelbrot, took
responsibility for his education. It is said that Mandelbrot was not
very successful in his schooling; in fact, he may have never learned his
multiplication tables. When Benoit was 21, his uncle showed him Julia's
important 1918 paper concerning fractals. Benoit was not overly
impressed with Julia's work, and it was not until 1977 that Benoit
became interested in Julia's discoveries. |
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"I
coined 'fractal' from the Latin adjective fractus. The corresponding
Latin verb frangere means "to break": to create irregular
fragments."
--- Benoit Mandelbrot
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Eventually, with the aid
of computer graphics, Mandelbrot was able to show how Julia's work
was a source of some of the most beautiful fractals known today. The
Mandelbrot set is made up of connected points in the complex plane. The
simple equation that is the basis of the Mandelbrot set is included
below.
changing
number + fixed number = Result
In order to calculate points for a Mandelbrot fractal, start with one of
the numbers on the complex plane and put its value in the "Fixed
Number" slot of the equation. In the "Changing number"
slot, start with zero. Next, calculate the equation. Take the number
obtained as the result and plug it into the "Changing number"
slot. Now, repeat (iterate) this operation an infinite number or times.
When iterative equations are applied to points in a certain region of
the complex plane, a fractal from the Mandelbrot set results. A few
fractals from the Mandelbrot set are included below.
Benoit Mandelbrot currently works at IBM's Watson Research Center.
In addition, he is a Professor of the Practice of Mathematics at Harvard
University. He has been awarded the Barnard Medal for Meritorious
Service to Science, the Franklin Medal, the Alexander von Humboldt
Prize, the Nevada Medal, and the Steinmetz Medal. His work with fractals
has truly influenced our world immensely.
It is now established that fractals are quite real and
incredible; however, what do these newly discovered objects have to do
with real life? Fractals make up a large part of the biological world.
Clouds, arteries, veins, nerves, parotid gland ducts, and the bronchial
tree all show some type of fractal organization. In addition, fractals
can be found in regional distribution of pulmonary blood flow, pulmonary
alveolar structure, regional myocardial blood flow heterogeneity,
surfaces of proteins, mammographic parenchymal pattern as a risk for
breast cancer, and in the distribution of arthropod body lengths.
Understanding and mastering the concepts that govern fractals will
undoubtedly lead to breakthroughs in the area of biological
understanding. Fractals are one of the most interesting branches of
chaos theory, and they are beginning to become ever more key in the
world of biology and medicine.
George Cantor, a nineteenth century mathematician, became fascinated by
the infinite number of points on a line segment. Cantor began to wonder
what would happen when an infinite number of line segments were removed
from an initial line interval. Cantor devised an example which portrayed
classical fractals made by iteratively taking away something. His
operation created a "dust" of points; hence, the name Cantor
Dust. In order to understand Cantor Dust, start with a line; remove the
middle third; then remove the middle third of the remaining segments;
and so on. The operation is shown below.
The Cantor set is simply the dust of points that remain. The
number of these points are infinite, but their total length is zero. Mandelbrot
saw the Cantor set as a model for the occurerence of errors in an
electronic transmission line. Engineers saw periods of errorless
transmission, mixed with periods when errors would come in gusts. When
these gusts of errors were analyzed, it was determined that they
contained error-free periods within them. As the transmissions were
analyzed to smaller and smaller degrees, it was determined that such
dusts, as in the Cantor Dust, were indispensable in modeling
intermittency.
The fractals and iterations are fun to look at; the Cantor Dust and Koch
Snowflakes are fun to think about, but what breakthroughs can be made in
terms of discovery? Is chaos theory anything more than a new way of
thinking? The future of chaos theory is unpredictable, but if a
breakthrough is made, it will be huge. However, miniature discoveries
have been made in the field of chaos within the past century or so, and,
as expected, they are mind boggling.
The first consumer product to exploit chaos theory was produced
in 1993 by Goldstar Co. in the form of a revolutionary washing machine.
A chaotic washing machine? The washing machine is based on the principle
that there are identifiable and predictable movements in nonlinear
systems. The new washing machine was designed to produce cleaner and
less tangled clothes. The key to the chaotic cleaning process can be
found in a small pulsator that rises and falls randomly as the main
pulsator rotates. The new machine was surprisingly successful. However,
Daewoo, a competitor of Goldstar claims that they first started
commercializing chaos theory in their "bubble machine" which
was released in 1990. The "bubble machine" was the first to
use the revolutionary "fuzzy logic circuits." These circuits
are capable of making choices between zero and one, and between true and
false. Hence, the "fuzzy logic circuits" are responsible for
controlling the amount of bubbles, the turbulence of the machine, and
even the wobble of the machine. Indeed, chaos theory is very much a
factor in today's consumer world market.
The stock markets are said to be nonlinear, dynamic systems.
Chaos theory is the mathematics of studying such nonlinear, dynamic
systems. Chaoticians have determined that the market prices are highly
random, but with a trend. The stock market is accepted as a self-similar
system in the sense that the individual parts are related to the whole.
Another self-similar system in the area of mathematics are fractals.
Could the stock market be associated with a fractal? Why not? In the
market price action, if one looks at the market monthly, weekly, daily,
and intra day bar charts, the structure has a similar appearance.
However, just like a fractal, the stock market has sensitive dependence
on initial conditions. This factor is what makes dynamic market systems
so difficult to predict. Because we cannot accurately describe the
current situation with the detail necessary, we cannot accurately
predict the state of the system at a future time. Stock market success
can be predicted by chaoticians. Traders can succeed trading from daily
or weekly charts if they follow the trends. A system can be random in
the short-term and deterministic in the long term.
Perhaps even more important than stock market chaos and predictability
is solar system chaos. Astronomers and cosmologists have known for quite
some time that the solar system does not "run with the precision of
a Swiss watch." Inabilities occur in the motions of Saturn's moon
Hyperion, gaps in the asteroid belt between Mars and Jupiter, and in the
orbit of the planets themselves. For centuries astronomers tried to
compare the solar system to a gigantic clock around the sun; however,
they found that their equations never actually predicted the real
planets' movement. It is easy to understand how two bodies will revolve
around a common center of gravity. However, what happens when a third,
fourth, fifth or infinite number of gravitational attractions are
introduced? The vectors become infinite and the system becomes chaotic.
This prevents a definitive analytical solution to the equations of
motion. Even with the advanced computers that we have today, the long
term calculations are far too lengthy. Stephen Hawking once said,
"If we find the answer to that (the universe), it would be the
ultimate triumph of human reason - for then we would know the mind of
God."
The applications of chaos theory are infinite; seemingly random
systems produce patterns of spooky understandable irregularity. From the
Mandelbrot set to turbulence to feedback and strange attractors; chaos
appears to be everywhere. Breakthroughs have been made in the past in
the area of chaos theory, and, in order to achieve any more colossal
accomplishments in the future, they must continue to be made. Understanding
chaos is understanding life as we know it.
Suggested Reading:
"Fractals and Scaling in Finance", 1997; also,
"Fractals", c1977, by Benoit Mandelbrot
"Chaos: Making a New Science", by James Gleick, 1987
"Chaos and Fractals : New Frontiers of Science" by Heinz-Otto
Peitgen et al., 1992
"The Geometry of Fractal Sets (Cambridge Tracts in Mathematics,
85)" by Kenneth J. Falconer, 1986 |
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