Overview:
The world of fractal geometry is usually pictured in beautiful images of self-similarity. Although the images are intricate, they are based on relatively simple calculations and replacements.
What Is Self-Similarity?
Self-similarity means that the pattern is the same or nearly the same no matter what scale is examined. The instructions (or equations) to create the pattern are repeated over and over, so that duplicates of the pattern are created at different magnifications. A real-world example of self-similarity is the coastline of a small country such as Great Britain on a world globe. It is irregular. No matter how many times the coastline is magnified, the irregularities show more irregularity.
What Is the Koch Snowflake?
The Koch snowflake is an early example of a geometric figure illustrating the fractal principles of self-similarity. Take an equilateral triangle and subdivide each side into three equal parts, so that the triangle changes into a six-pointed star, and then subdivide each side of the six-pointed star into three equal parts, and on and on. Very soon in the process, it is impossible to tell whether the part of the curve is the original curve or a magnified portion.
What Are Julia Sets and Mandelbrot Sets?
Julia sets are examples of self-similarity the can be calculated by starting with a complex number, squaring it, and adding the number back. When the process is repeated over and over again (called iteration), the resulting figure has an infinitely repeating bounded area. One part of the figure is dark and the other part can be color-coded to represent how quickly the numbers change. The mathematician who developed the equations that described the Julia sets never saw their pictorial representation, because their intricate patterns can only be drawn by computer. Mandelbrot sets are closely related images that were constructed by computer imaging.
What Are Some Applications of Fractal Imaging?
Many processes in the real world can be simulated by fractal imaging. For example, clouds form a fractal structure. Each part of a cloud formation is self-similar to any other part, as are parts of ocean waves. Patterns found by the seismic waves found in earthquakes are fractal, as are the myriad forks of lightning. Most recently, fractal patterns have been found in heart activity, DNA, and blood vessels.
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