Computers.
These machines are absolutely needed in chaos experamentation. Chaos
calculations are repetitive, boring, and number in the millions. To produce the
Mandelbrot Set on a single screen takes an estimated 6,000,000 calculations.
No human would be patient enough to endure the time and work involved. But a
computer is. Computers are particularly good at mindless repetition. The computer is
our telescope, our microscope, and now, our art gallery. We cannot really explore
chaos without the computer, and we certainly can not produce fractals without them.
Most computer use is based on putting in data, and then instructing the computer to
what output is required. Chaos theory arose as scientists and mathematicians
started to play around with the resulting numbers. To put in numbers and watch as
they careered around the plane, mostly the complex plane, in detailed patterns puzzled
and shocked the early experimentors in chaos. They watched as the computer
produced the numbers, and didn't just wait for the final result. They tried different ways
of plotting and exploring equations, even if it was mainly for the fun of it.
Some of the scientists and their computers produced images which resembles things
found in nature, such as ferns and clouds, mountains and bacteria. They indicated
why we couldn't predict the weather. They seemed to match the behavior of the stock
exchange and populations and chemical reactions all at the same time. Their
investigations suggested answers to questions which had been asked for centuries
about the flow of fluids as they moved from a smooth to irregular flow or about the
formation of snowflakes, about the swing of a pendulum, about tides and heartbeats
and rock formations.
This new theory dealt with a vast range of intellectual domains. Scientists started
plotting the fractals. Some of their plotted fractals again mimicked nature. Some were
stunningly beautiful. Some were just fascinating.
Chaotic Systems are not random, though they may appear to be. All fractals have some
simple defining features:
1. Chaotic systems are deterministic. This says that they have something out there
determining their behavior.
2. Chaotic systems are very sensitive to the initial conditions. A very slight change in
the starting point can lead to enormously different outcomes. This makes the system
fairly unpredictable.
3. Chaotic systems appear to be disorderly, even random, but they are not. Beneath
the random behavior is a sense of order and pattern. Truly random systems are
not chaotic. The orderly systems predicted by classical physics are the exceptions.
In this world of order, chaos rules!
There is a strong link between chaos and fractals. Fractal geometry is the geometry
which describes the chaotic systems we find in nature. Fractals are a language, a way
to describe geometry. Euclidean geometry is a description of lines, circles, triangles,
and so on. Fractal geometry is described in algorithms- a set of instructions on how
to create the fractal. Computers translate the instructions into the magnificent patterns
we see as fractal images.
If you would like to download some fractal images, please go to our download site.