Special Interests
Like most people, I have several special interests that impact my mind
and sometimes my art. These
include: Chaos
Theory, Golf, Runes, Creative Writing, and Christianity, but there are others. I regularly read, contemplate, and sometimes practice
various topics of special interest to me.
CHAOS THEORY
This introduction to Chaos Theory is taken in part from Greg
Rae’s Home Page: http://www.imho.com/grae/chaos/index.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)
Lorenz searched for a
simpler system that had sensitive dependence on initial conditions. His first
discovery had twelve equations. He took the equations for convection, and
stripped them down, making them unrealistically simple. The system no longer
had anything to do with convection, but it did have sensitive dependence on its
initial conditions, and there were only three equations this time. Later, it
was discovered that his equations precisely described a water wheel.
The
equations for this system also seemed to give rise to entirely random behavior.
However, when he graphed it, a surprising thing happened. The output always
stayed on a curve, a double spiral. There were only two kinds of order
previously known: a steady state, in which the variables never change, and
periodic behavior, in which the system goes into a loop, repeating
indefinitely. Lorenz's equations were definitely ordered - they always followed
a spiral. They never settled down to a single point, but since they never
repeated exactly, they weren't periodic either. He called the image he got when
he graphed the equations the Lorenz attractor.
In 1963, Lorenz
published a paper of his discovery. He included the unpredictability of the
weather, and discussed the types of equations that caused this type of
behavior. The journal he was able to publish in was a meteorological journal.
Lorenz's discoveries weren't acknowledged until years later, when they were
rediscovered. Lorenz had discovered something revolutionary; now he had to wait
for someone to discover him.
An employee of IBM, Benoit Mandelbrot was a mathematician studying this self-similarity. One of the areas he was studying was cotton price fluctuations. No matter how the data on cotton prices was analyzed, the results did not fit the normal distribution. Mandelbrot eventually obtained all of the available data on cotton prices, dating back to 1900. When he analyzed the data with IBM's computers, he noticed an astonishing fact:
The numbers that produced aberrations from the point
of view of normal distribution produced symmetry from the point of view of
scaling. Each particular price change was random and unpredictable. But the
sequence of changes was independent on scale: curves for daily price changes
and monthly price changes matched perfectly. Incredibly, analyzed Mandelbrot's
way, the degree of variation had remained constant over a tumultuous sixty-year
period that saw two World Wars and a depression. (James Gleick,
Chaos
- Making a New Science, pg.
86)
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. ![[Figure 4]](interest_files/image001.gif)
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.
The Koch curve brings up
an interesting paradox. Each time new triangles are added to the figure, the
length of the line gets longer. However, the inner area of the Koch curve
remains less than the area of a circle drawn around the original triangle.
Essentially, it is a line of infinite length surrounding a finite area.
To get around this
difficulty, mathematicians invented fractal dimensions. Fractal comes from the
word fractional. Fractal has come to mean any image that displays the attribute
of self-similarity. The bifurcation diagram of the population equation is
fractal. The Lorenz Attractor is fractal. The Koch curve is fractal.
Later, a scientist
by the name of Feigenbaum was looking at the
bifurcation diagram again. He was looking at how fast the bifurcations come. He
discovered that they come at a constant rate. He calculated it as 4.669. In
other words, he discovered the exact scale at which it was self-similar. This
was a revolutionary discovery. He had found that a whole class of mathematical
functions behaved in the same, predictable way. This universality would help
other scientists easily analyze chaotic equations. Universality gave scientists
the first tools to analyze a chaotic system. Now they could use a simple
equation to predict the outcome of a more complex equation.
Many
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
www.cs.ucr.edu/
~ddreier/mandelbrot.jpg
Chaos even has applications outside of science. 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,
There are other aspects of chaos theory in mathematics that are interesting to contemplate. Consider, for example, fractional dimensions or the measurement and division of various lengths and widths of objects. To read some more about this go to Greg Drae’s discussion of this on his page “Fractional Dementia”
GOLF
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.
RUNES
. My watercolor and acrylic painting explores
repetitive patterns and relationships between colors and shapes and their
connection to the natural world.
On a trip to
RUNES are an
alphabetic script used by peoples of
Sometime
around the fifth century AD, the Anglo-Saxon invasions of northern
Runes were used by the ancient Celtic cultures for reading the conditions or elements in the present. Each Rune is associated with a particular tree. Trees were to the Anglo-Saxon Druids, what the crucifix is today for Christians, the symbol of where they came from (above/branches), who they were (physical/trunk), and where they were going (below/roots).
Today, Runes have been rediscovered as a symbolic system and have gained popularity as a means of divination. But, they are more than a curious alternative to Tarot cards and crystal balls. They provide a key to understanding the lives and beliefs of the ancient people who created them, and have much to teach us about a way of life that may have been more intimately connected to the natural world, and, perhaps, to the realm of spirit, than our own.
CREATIVE WRITING
The last couple
of years I was on the faculty at
CHRISTIANITY
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