Chaos in Theory and Practice – Fractals in Nature and Technology
The behavior of chaotic systems seems as if no rules applied. However, in reality, when an abstract approach is used, it becomes clear that it is based on hidden deterministic rules that explain the inherent order in chaos.
Physicists like to use diagrams such as the distance-over-time diagram in mechanics or the phase space chart where the location and velocity (or rather the momentum) of one or several particles can be mathematically deduced.
This is exactly where chaotic deterministic systems come into play. A phase space chart gives a complex and structured image of the process of motion. Sometimes, paths of chaotic particles will group along a bent curve without ever actually reaching it or they may fill certain areas but never touch other areas. Mathematical analysis shows that these shapes cannot even be categorized into a dimension such as a line, area, or body.
For example, a circle with an infinite number of small holes has an infinitely small area. Overall it may look like a messy pile of lines, but there is much more to it. Such mathematical objects have a fractal dimension (for example, 2.3 or 1 and 3/7) that can be calculated. A typical trait of chaotic systems is that their trajectory charts form infinitely complex fractal patterns.
Fractals or fractal objects occur in nature. For example, fern leaves or cauliflower, especially the Romanesco variety, grow little copies of themselves on their fern fronds or stalks, which, in turn, grow even smaller copies of themselves and so on. Another good example is how the British coast continually gets more complex the closer you look at it. This is because each magnification adds more detail than the last.
Mathematician Benoit Mandelbrot (b. 1924), known as the father of fractal geometry, calculated that the west coast of Great Britain has the fractal dimension of 1.25. The stock market, in comparison, is an example of temporal fractals. At first sight, it is impossible to distinguish a ten-year chart from a chart of a month, an hour, or even a minute. They all show a similar pattern of sharp spikes and lows.
FRACTALS
Chaotic systems may provide interesting trajectory charts, but perhaps the most beautiful fractal images are designed mathematically. The instantly recognizable appleman set, for instance, is created by repeatedly applying a function to itself.
Changing the original parameters projects new pat- terns onto the computer screen. This concept has many applications. Computer-generated movies, tor example, produce trees and other objects using the same technique; the more fractals are used the more natural an object looks.