Fractals
When viewed from space, the coastline of an island such as Britain, which can be seen it its entirety at a glance, presents a random, jagged appearance. As you approach the island, the large-scale sense of the coastline disappears beyond the horizon allowing you to see finer details of the places where rock meets the sea. But at this close distance, the impression is still the same as the one from space; the coastline looks random and jagged. If you come right up to the rock, nose to nose with the water splashing at your face, you see the same thing; a jagged rock sitting in the water. In nature and the abstract world of geometry, the general appearance of some objects never changes no matter how close or far from the object you are when you view it. An object with this property is called self-similar or fractal.
Fractals and their properties were first brought to the attention of the world by the work of Benoit Mandelbrot in the 1970s. You can make a fractal shape by using a simple technique. Begin with a line segment. Erase the middle third of the line segment and put in its place a large, upside-down "V" so that the ends of the "V" connect to the remaining line segments and each side of the "V" has the same length as the removed segment. You have just "grown" a new shape that now has four line segments instead of just one.
Now, on each of the four new line segments, repeat this procedure. Erase the middle third of one line segment and put in its place an upside-down "V" as before. Do this for each line segment. Continue repeating the process indefinitely. This iterative construction creates a seemingly random, complex shape that is called the Koch curve. But because the shape was made according to a strict formula in which a line segment was altered after erasing a third of it, you will find that as you look at the shape closely (after magnifying it three times in this case), it will appear exactly the same as it did when viewed on a larger scale. Each magnification of three times will present the same shape. This idea is somewhat contrary to our intuition, which insists that a complex shape should somehow become simpler as we examine it more closely.
The magnification factor and the number of new units that are introduced with each change of shape are important properties that establish the dimensionality of the fractal. Unlike the traditional Euclidean dimension for lines, planes, and space, a fractal dimension may be noninteger. For example, the fractal dimension of the Koch curve is (log 4)/(log 3), or approximately 1.26. Dimensionality provides a way of classifying and studying fractals.
Fractals have found a place in the field of statistical physics by helping to analyze complex, random-seeming events such as phase transitions and the diffusion of gases. Strong links between the dimensionality of fractal shapes and chaotic motions have helped to create a new branch of physics called chaos theory. Mathematicians and computer scientists have used fractals to compress computer images for data storage and for fast transmission over computer networks. But perhaps the most immediately appreciable and well-known of the applications of fractals is in visual art, where computers have used fractal generating programs to create amazingly beautiful designs and images that seem to open up into an entire universe within themselves. Fractal algorithms have also been used by graphic artists to draw artificial landscapes in nature, such as mountain ranges, trees and forests, that are remarkably realistic in appearance.
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