Chaologists contend that not only in certain mathematical constructs, but also in the real world, there exist orderly systems in disguise, or chaos. Because the definition of chaos is so unwieldy, I have tried to summarize its main characteristics in the following table:

what it isn't	what it is
is not erratic	is erratic-looking, but is in fact ordered
is not dependent on external variables	is entirely self-generated
is not the result of error	is dependent upon the initial conditions, or "control parameter"
is not predictable in the long term	is fairly accurately predictable in the short term
is noninvertible, i.e., one cannot determine a chaotic system's prior history	is the result of a deterministic process, i.e., can be expressed as a mathematical equation with a given initial condition
is not found in linear systems, i.e., the plotted equation is a straight line	is found only in nonlinear systems, i.e., the plotted equation is not a straight line
	is found in feedback systems where the past affects the present and the present affects the future

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Mathematically, chaos can be achieved by the simple iteration of certain equations. The recipe for chaos in the real world, however, is still in theory stage. Chaologists propose several possible causes of chaos:

• The value of a control parameter is increased to a point where chaotic behavior sets in (see next section, below).
• The nonlinear interaction of two or more separate physical operations.
• Ever-present environmental noise affecting otherwise regular motion.

A control parameter is a constant in an equation for which we get to choose the value. Set the parameter's value at a certain number, and a system will evolve in a certain way. Set the parameter's value at a different number, and all chaos may break loose. Thus, the parameter is king, and dictates whether or not a system travels the chaotic route.

One of the main characteristics of chaos is sensitivity to initial conditions. In a chaotic system, a tiny historical accident can lead to a major and unexpected long-term outcome. Every other piece of chaos literature will cite the "butterfly effect" example; I offer another:

Suppose that war is a chaotic system. Given this, the eventual fall of the powerful Roman Empire (big significant outcome) could be attributable to whether a single Roman soldier, at a certain point in a single battle, turned his head to the right or to the left (tiny, seemingly insignificant event).

Although extreme sensitivity to initial conditions alone does not necessarily lead to chaos, it is a very important feature of chaos, for it is the factor that makes long-term predictions meaningless.
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Somewhere between our favorite dimensions of length, width, and height, the natural phenomenon of the fractal takes place. A fractal is a pattern of self-similarity; any part of the pattern can be extruded and magnified and will look like the whole. Look around and you will likely see a fractal: I see them in the stucco that decorates my house, and in bolts of lightning. Where fractals occur, chaos often does, too.

Benoit Mandelbrot wrote the formula that allowed us to see the art of chaos via fractals. To generate a fractal, apply a simple feedback formula or algorithm to the numbers located on the "Mandelbrot Set", a region of the complex number plane. Plot the result. Any fractalnaut can create stunning organic-looking art in this manner.

Omnis ars naturae imitio est. (Seneca)
All art is but imitation of nature.

My Chaos Art Exhibit features photographs and canvas paintings of self-similar images or fractals found in nature. I have not included any computer-generated fractals on this site, as they occur in great quantity elsewhere on the Web. The following sites should satisfy your craving for fractal art:

• Cheshire Cat Fractals by Alice Kelley
• Ultra Fractal by Janet Preslar
• Sylvie Gallet's Fractal Gallery by Sylvie Gallet
• QMi - The Fractal Gallery by many authors
• Fractal eXstreme: Mandel Louvre by many authors

If you are lucky enough to find chaos in the system you are studying, you have power. The fact that chaos exists can:

• Lead you to the discovery of a scientific law, and greater understanding.
• Increase the accuracy of short term predictions.
• Help you determine reliably when your predictions will be accurate and precisely when they will be worthless.
• Help you create a model of how a process works.

Did you scroll down the page looking for an image to click on? If you are interested in chaos theory, but hate reading (who are you?), I highly recommend Darren Aronofsky's Pi (1997), a psychological thriller about faith in chaos. Even the script is mathematical, which is why it is one of my ten favorite movies of all time.

Try Pi, then buy Pi on VHS or DVD.
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My recommendations:

• Chaos Theory Tamed, Garnett P. Williams, 1997
• Fractal Geometry of Nature, Benoit B. Mandelbrot, 1988
• The Web of Life, Fritjof Capra, 1997

Brush up on your chaos theory:

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