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Used with permission. Copyright 1996, James P. Crutchfield. |
Chaos theory
tries to make sense of apparently disordered systems. An early pioneer in
chaos theory was Edward Lorenz (1917- ). In 1960 he was doing
meteorological research trying to develop models for predicting weather. His
formulas started to reveal random patterns which interested him. So he
started to work on formulas that analyzed unpredictable behavior. He
developed the idea that the flapping of a butterfly's wing will create a
chaotic disturbance in the atmosphere called the “butterfly” effect
(click
on picture for animation). A small disturbance can cause a large scale atmospheric
motion, so that the long term behavior becomes impossible to forecast.
Scientists in other fields of science started to see this behavior in other
systems.
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Details of the Koch snowflake |
Helge von Koch
(1870-1924), a mathematician, captured a similar idea earlier. He created
one of the first fractal images now called the Koch snowflake or
star. Imagine an equilateral triangle then keep adding equilateral
triangles that fit. A pattern starts to emerge on this self-duplicating
pattern. If you keep doing this the line around the figure keeps getting
larger and larger. Mathematicians invented “fractal dimensions” or fractals.
Later a scientist, Mitchell Fergenbaum (1944- ) looked at this growth
and noted that they grow at a constant rate of 4.669. The number kept coming
up when he was comparing systems. This was a revolutionary discovery.
Mathematical systems that seemed chaotic were not really. The power of
numbers can now make sense out of a chaotic system!
When you plot fractal equations you
graphical derive fractal images. For example the Mandelbrot set
(equations z = z2 + c) creates a wonderful image. Applications of
fractal images are abundant from chaotic heart beats to charting coastlines as
well as computer art to music.
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Keep adding equilateral triangles
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Mandelbrot image
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