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”