This introduction to Chaos Theory is taken in part from Greg Rae’s Home Page: http://www.imho.com/grae/chaos/chaos.html
chaos theory" comes from the fact that systems described by the theory are apparently disordered. But chaos theory is really about finding the underlying order in apparently random data. The first experimenter in chaos was Edward Lorenz. In 1960 he had a computer set up with equations to model what the weather might be.
One day in 1961 when he
wanted to see a particular sequence again he started in the middle of the
sequence instead of the beginning. He used the number off his printout and left
to let the computer run. When he returned, the sequence had evolved
differently. Instead of the pattern as before, it diverged, ending up wildly
different from the original. Eventually he figured out what happened. The
computer stored the numbers to six decimal places in its memory. To save paper,
he only had it print out three decimal places. In the original sequence, the
number was .506127, and he had only typed the first three digits, .506.
By all conventional ideas
of the time, it should have worked. He should have gotten a sequence very close
to the original sequence. A scientist considers it lucky if measurements have
accuracy to three decimal places. Surely the fourth and fifth, impossible to
measure using reasonable methods, can't have a huge effect on the outcome of
the experiment. Lorenz proved this idea wrong.
The flapping of a single butterfly's wing today
produces a tiny change in the state of the atmosphere. Over a period of time,
what the atmosphere actually does diverges from what it would have done. So, in
a month's time, a tornado that would have devastated the Indonesian coast
doesn't happen. Or maybe one that wasn't going to happen,
does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos
, pg. 141)
One mathematician, Helge von Koch, captured this idea in a mathematical construction called the Koch curve. To create a Koch curve, imagine an equilateral triangle. To the middle third of each side, add another equilateral triangle. Keep on adding new triangles to the middle part of each side, and the result is a Koch curve. A magnification of the Koch curve looks exactly the same as the original. It is another self-similar figure.
scientists were exploring equations that created fractal equations. The most
famous fractal image is also one of the most simple. It is known as the
Mandelbrot set. The equation is simple:
z=z2+c. Fractal structures have been noticed in many real-world
areas, as well as in mathematician's minds. Blood vessels branching out further
and further, the branches of a tree, the internal structure of the lungs,
graphs of stock market data, and many other real-world systems all have
something in common: they are all self-similar.
Mandelbrot Set Image
Computer art has become more realistic through the use of chaos and fractals. Now, with a simple formula, a computer can create a beautiful, and realistic tree. Instead of following a regular pattern, the bark of a tree can be created according to a formula that almost, but not quite, repeats itself.
Music can be created using fractals as well. Using the Lorenz attractor,
Diana S. Dabby, a graduate student in electrical
engineering at the Massachusetts Institute of Technology, has created
variations of musical themes. ("Bach to Chaos: Chaotic Variations on a
Classical Theme", Science News,
I’ve been playing golf since about age 12. As Mike Linder states in his book “Golf and the Spiritual Life, there just is no other game like it. Only in golf are the players expected to keep the rules on themselves. Of course, some pay less attention to rules than others, but by and large the scorekeeping and assessing of penalties are done by each player. It occurs to me that Golf is very much like life. People set specific goals to reach and follow certain rules in getting there. Honesty is a strong value in life and in Golf. In football, no player ever tells the official that “No, I didn’t catch the ball – I trapped it against the ground”, to the contrary football players claim catches and look for advantages regardless of the rules. But if a golfer inadvertently moves the ball before hitting, the player will call the penalty even if no one is watching. In Golf, I’m always just competing against myself and the golf course. It is the only game I can think of where an opponent will cheer when you make a good play.
I’ve always been interested in birds and their place in our environment. My home has a waterway behind it and my yard is frequented by White Egrets, Great Blue Herons, Little Green Herons, and a whole assortment of the usual backyard birds at various times of the year such as Cardinals, Crows, Hawks, Sparrows, Warblers, Bluebirds, and Painted Bunting. I began to include these beautiful creatures in my otherwise abstract paintings. I've had a great deal of fun working to resolve this real and abstract contradiction.
. My watercolor and acrylic painting sometimes explores
repetitive patterns and relationships between colors and shapes and their
connection to the natural world.
On a trip to
The last couple
of years I was on the faculty at
I make no
apologies for my religion. In this age
of political correctness people sometimes seem to bend over backwards to keep from offending
anyone of any religious group or anti-religious group, except for
Christians. I teach a Sunday School class and have learned a great deal as a result of
reading the Holy Bible. The Book is a
documentary of the relationship of an entire people with God and how they were
affected by that relationship. Being the
chosen people was not an easy task for