Au22


Universal Translation

Saturday, February 21, 2009

Wisdom Knowledge...Chaos Mathematics

Chaos theory attempts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions. A common example of this is known as the Butterfly Effect. It states that, in theory, the flutter of a butterfly's wings in China could, in fact, actually effect weather patterns in New York City, thousands of miles away. In other words, it is possible that a very small occurance can produce unpredictable and sometimes drastic results by triggering a series of increasingly significant events.




CHAOS MATHEMATICS


The natural world has always had a chaotic way about it. The mathematical world has always been amazing complex. So why has Chaos theory just evolved as such a critical part of science, mathematics, and art?


Simple. Computers.


The calculations involved are repetitive, boring, and number in the millions. To produce the Mandelbrot Set on a single screen takes an estimated 6,000,000 calculations. Nu human would be stupid enough to endure the boredom. But a computer will. 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 it, and we certainly can't produce fractals unaided.

However, it is necessary to use the computer as an investigative tool. Most computer use is based on putting in data and instructing the computer on what output is required. Chaos Theory arose as scientists and mathematicians started to play. To put in numbers and watch as they careered around the plane, mostly the complex plane, in detailed patterns. They watched as the computer produced the numbers, and didn't just wait for the final result. And they tried different ways of plotting and exploring equations- mostly for the fun of it.

Playing with mathematicians, science and computer programming produced images which looked like nature. Ferns and clouds and 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, about the formation of snowflakes, about the swing of a pendulum, about tides and heartbeats and cauliflower and rock formations.

This new theory dealt with a vast range of intellectual domains. And they started plotting the fractals. Some mimicked nature. Some were stunningly beautiful. And some were just fascinating.


Chaotic Systems are not random. They may appear to be. They have some simple defining features:

1. Chaotic systems are deterministic. This means they have something 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.

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.

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Knowledge of Self: A Collection of Wisdom on the Science of Everything in Life

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