Since its inception, science relied on predictability and order. The true
beauty of science was its uncanny ability to find patterns and regularity in seemingly
random systems. For centuries the human mind as easily grasped and mastered the
concepts of linearity. Physics illustrated the magnificent order to which the
natural world obeyed. If there is a God he is indeed mathematical. Until the
19th century Physics explained the processes of the natural world successfully,
for the most part. There were still many facets of the universe that were an
enigma to physicists. Mathematicians could indeed illustrate patterns in nature
but there were many aspects of Mother Nature that remained a mystery to
Physicists and Mathematicians alike. Mathematics is an integral part of
physics. It provides an order and a guide to thinking; it shows the
relationship between many physical phenomenons. The error in mathematics until
that point was linearity. “Clouds are not spheres, mountains are not cones,
bark is not smooth, nor does lightning travel in a straight line.” - Benoit
Mandlebrot. Was it not beyond reason that a process, which is dictated by that
regularity, could master a world that shows almost no predictability
whatsoever? A new science and a new kind of mathematics were developed that
could show the universe’s idiosyncrasies. This new amalgam of mathematics and
physics takes the order of linearity and shows how it relates to the
unpredictability of the world around us. It is called Chaos Theory.
The secular definition of chaos can be misleading when the word is used in a
scientific context. As defined by Webster’s dictionary chaos is total disorder.
That may lead one to believe that chaos theory is indeed the study of total
disorder, which it truly is not. In 1986 at a prestigious conference on Chaos
another definition for chaos was introduced. It is stochastic behavior
occurring in a deterministic set. This definition of chaos was hesitantly
brought forth. The scientists, mathematicians and intellectuals present were
hesitant to define a concept they did not truly understand yet. They left the
scientific community with a rather cryptic and oxymoronic definition of chaos.
Deterministic sets behave by precise unbreakable law. Stochastic behavior is
the opposite of deterministic it has no finite laws, it is totally dependant
upon chance. The dissected definition of chaos is lawless behavior that is
ruled entirely by law. (Stewart 16-17)
The principles of Chaos Theory are complex and abstract. Perhaps the simplest
and most essential ideas behind chaos theory are embodied in the aphorism known
as the Butterfly Effect. The butterfly effect states that the flapping of a butterfly’s
wings in Hong Kong can change the weather in New York. It means that a
miniscule change in the initial conditions of a system, in this case the
weather, is magnified greatly in the end conditions of that same system. The
ultra sensitivity to the initial conditions of a system was not a new and
striking discovery. In fact it was shown in ancient 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 the want of a battle, the kingdom was lost!”
The smallest variation in the initial conditions of a system can result in huge
differences in concluding events. There was no nail, and because of this
seemingly insignificant detail in the initial condition, the kingdom was lost.
Another example of the butterfly affect is two pieces of wood floating on a
river. Place those two logs at nearly the same point on the river and let them
go. It is absolutely impossible to predict where those logs will be later
downstream. When those logs are set on the water a slight breeze, a fish that
swims underneath one of them, or even a single droplet of additional water in
the initial stage can totally change the end result until no resemblance between
the two is seen. (Briggs, Peat 49) There is a definite correlation between that
small butterfly and a storm in New York, as well as the two logs. Chaos Theory
states that within the unpredictability that makes those changes there is
indeed a specific order. Chaos works in order and within all order there is
chaos.
The butterfly effect as well as the two logs depends solely on iteration.
Iteration is feedback that continually reabsorbs its predecessors. Iteration is
a very common process, which can appear in fields as diverse as artificial
intelligence or the cycling replacement of cells in the human body. (Briggs
Peat 66) Iteration provides a sort of self-reference. For example the word
“time” is defined with words such as “period” or “instant”. Look up the
definition of those words and it will eventually lead back to the word “time”.
(Briggs Peat 68)
MIT meteorologist Edward Lorenz has the distinction of being the first person
to show how iteration creates chaos. In 1960 he was solving non linear equations
on his computer that would show a model for the earth’s atmosphere. He repeated
a certain forecast to check his data and when he substituted the numbers in the
second time he rounded off the figures to three decimal places instead of the
six he received initially. He plugged in these numbers and left the computer.
He returned to a surprise. The forecast before him was not a double check on
his previous information, it was a totally new forecast altogether! That three
decimal place difference between the two sets of numbers had been magnified
greatly in the process of solving those equations. (Briggs Peat 68-69)
Just as the butterfly effect embodies the principles of Chaos Theory, a single
image has become an emblem for the early pioneers of chaos. The Lorenz
attractor (Figure 1) is a magical image that resembles an owl’s mask or a
butterfly’s wings. (Gleick 29) Fig. 1
Lorenz then tried to model the chaos of a gaseous system, like the earth’s
atmosphere. He used his knowledge in the physics field of fluid dynamics to
simplify three equations to invent the following three-dimensional system of
equations:
dx/dt=delta*(y-x)
dy/dt=r*x-y-x*z
dz/dt=x*y-b*z
Where delta is an inconsequential constant for which Lorenz used a value of
ten. The variable r is the difference in temperature between the top and the
bottom of the gaseous system. The variable b is the width to height ratio of
the box, which contains the gaseous system; Lorenz used 8:3. When a gas is
heated form below it tends to organize itself into a cylindrical form. Hot
fluid rises to the top, loses heat and falls to the bottom otherwise known as
convection. As the temperature increases the cylinder becomes wavy and then
become wild and chaotic. (Gleick 25) The resulting x in the equation is the
rate of rotation of the cylinder, y is the difference in temperature at
opposite sides of the cylinder, and the variable z represents the difference of
the gaseous system from a line, which represents temperature. When Lorenz
plotted these three equations no geometrical shape or curve appeared, but the
weaving object known as the Lorenz Attractor. The system never repeats itself,
so the diagram never intersects. It loops around and around forever. The motion
of the attractor is theoretical but it accurately conveys the action of the
real system.
The dimensions seen in everyday life are rather straightforward and comforting;
zero, one, two, or three. Chaos theory speculates that the world may not be all
that cut and dry. Consider the dimension of a ball of string. From a great
distance the ball is a point and had no dimension. From a few feet away it
looks normal and has three dimensions. From a minute distance a single thread
is seen as a weaving line with one dimension. As an even lesser distance the
line turns into columns of definite thickness, it has three dimensions. Closer
still the thread is lost to individual hairs the ball is again one-dimensional.
(Briggs Peat 94) The twisting and turning of the ball of yarn very closely
resembles the contortion of the Lorenz attractor. Both figures have a
non-integral dimension, the defining trait of a fractal dimension. The
irregularity and detail of these two objects illustrate fractal geometry.
(Briggs Peat 95)
Fractal Geometry was developed by Beniot Mandlebrot, a polish mathematician who
was influenced the work of Gaston Julia. During World War I Julia started
sketching fractal shapes, which were unexplainable through the methods of
Euclidian geometry. (Gleick 221) Fractals are defined by infinite detail; infinite
length, no slope, a fractional dimension, self-similarity, and they can be
generated by iteration. (Briggs Peat 95) An example of a fractal shape is the
Koch Curve, or the Koch snowflake (Figure 2). (Gleick 93)
Fig. 2
It starts off as an equilateral triangle, adding to each side another triangle
in the middle. This process is repeated to infinity. The length of the boundary
created by this fractal is infinite yet the area of the curve is less than the
circumscribed circle around the original triangle. (Gleick 99) “An infinitely
long line surrounds a finite area.” – James Gleick
The two concepts of fractals and attractors are intimately linked. Through
fractal geometry it is found that attracters are indeed fractal curves.
Wherever there is chaos there must also be its visual representation, fractal
geometry. This suggests a connection between every chaotic process. The
formation of branches of the human lung and the motion of a fast flowing river
can now be seen as nearly identical. Both chaotic processes emerge from a
fractal order. Fractals are another amazing contradiction in Chaos Theory.
Fractals are both complex and simple. They are complex because of their
infinite detail and structure and as unique as the human fingerprint, no two
fractals are the same, yet they are simple because they are formed by the
successive applications of simple iteration. (Briggs Peat 95-97)
Benoit Mandlebrot was an intellectual renaissance man. He was a very gifted man
with an amazing brain and an ego to match. He was one invited to speak at
Harvard University. He entered Harvard’s Littauer Center only to find the
diagram he was going to use already on the blackboard. He jokingly asked the
hosting professor how his information arrived before he did. It turns out that
the diagram on the board was eight years of cotton prices. Mandlebrot diagram
was that of income distribution in an economy. Two unrelated topics, which
showed the same trends. (Gleick 83-84) This is an example of self-similarity.
It manifests itself in many other ways. Fractals are self-similar; in that case
at higher and higher magnification the fractal image resembles the original.
(Figure 3)
Fig. 3
The stock market is indeed chaotic and also self similar. It is truly random,
but shows an orderly trend. It is highly dependant upon initial conditions, but
because it is nearly impossible to describe those initial conditions it is
impossible to predict the action of the market. Short term trading is random
and futile. Long term trading however is not random at all. (Gleick 85) A
deterministic order comes from chaos over time.
Chaos Theory has made quite an impact on the modern world. Even in its infancy
it has been a powerful tool in shaping popular thought of the natural world.
Once dismissed as a theoretical science with no practical application, chaos
theory has blossomed into an intricate and beautiful pattern, much like the
fractals it deals with. Chaos theory is a complex combination of math and
physics, but with its mastery comes a new era in the human understanding of the
world around us.
A. A violent order is disorder: and
B. A great disorder is an order.
These two things are one
Wallace Stevens
“Connoisseur of Chaos”