Introduction to Fractal Theory

What is a fractal? In the most generalized terms, a fractal demostrates a limit. Fractals model complex physical processes and dynamical systems. The underlying principle of fractals is that a simple process that goes through infinitely many iterations becomes a very complex process. Fractals attempt to model the complex process by searching for the simple process underneath.

Most fractals operate on the principle of a feedback loop. A simple operation is carried out on a piece of data and then fed back in again. This process is repeated infinitely many times. The limit of the process produced is the fractal.

Almost all fractals are at least partially self-similar. This means that a part of the fractal is identical to the entire fractal itself except smaller.

Fractals can look very complicated. Yet, usually they are very simple processes that produce complicated results. And this property transfers over to Chaos Theory. If something has complicated results, it does not necessarilly mean that it had a complicated input. Chaos may have crept in (in something as simple as round-off error for a calculation), producing complicated results.

Fractal Dimensions are used to measure the complexity of objects. We now have ways of measuring things that were traditionally meaningless or impossible to measure.

Finally, Fractal research is a fairly new field of interest. Thanks to computers, we can now generate and decode fractals with graphical representations. One of the hot areas of research today seems to be Fractal Image Compression. Many web sites devote themselves to discussions of it. The main disadvantage with Fractal Image Compression and Fractals in general is the computational power needed to encode and at times decode them. As personal computers become faster, we may begin to see mainstream programs that will fractally compress images.

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Copyright Eric Green, 1998