# V.12:1 (38-41): Polarized Fractal Efficiency by Hans Hannula, Ph.D., C.T.A.

Item# \V12\C01\POLARIZ.PDF
\$3.95
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## Product Description

Polarized Fractal Efficiency By Hans Hannula, Ph.D., C.T.A.

What is fractal geometry, anyway, and how do you use it? Well, you'll find out here. Hans Hannula of MicroMedia describes the construction and use of an indicator derived from fractal geometry, the mathematics that describe chaotic systems.

Most chaotic systems produce some form of graphic representation. For example, turbulent flow in a stream produces swirls, eddies and vortices. Early chaos researchers found that the triangles, squares, lines and cubes of Euclidean geometry simply did not help in describing, studying or understanding their research problems.

Fortunately, mathematician Benoit Mandelbrot recognized this problem and solved it by describing fractal geometry. As he studied various problems being researched, he realized that many of these problems had in common graphic representations of a very squiggly line. So he asked himself the profound question, "What is the dimension of a squiggly line?"

The problem can be represented as shown in Figure 1. A straight line has a dimension of one. A plane surface has a dimension of two. A squiggly line has a dimension between one and two, depending on how much it squiggles. The dimension of the line is not an integer like 1, 2 or 3 but can be a fraction, leading to the term fractional or fractal dimension. Mandelbrot discovered that many chaotic systems had a constant fractal dimension. Others discovered that systems with the same fractal dimension had other properties in common. Thus, fractal dimension became an important tool in chaos work.

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