Applications of Fractals
Coastlines and Borderlines
Benoit Mandelbrot, the founder of fractals has first noticed the properties of fractals on the coast of Britain. He realized that no matter how small a piece of the coast is, it will still have its own bays, harbors, and capes. Basing himself on Richardsonís data, he was able to prove that many coasts as well as borderlines are fractal.
Richardson searched many encyclopedias to find data about the lengths of certain borderlines. He found enormous differences in data from different countries. For example, Portugal claimed its border with Spain to be 1214 km, while Spain claimed it to be 987 km. Portugal, as a smaller country would definitely measure its border more accurately. Thus, we know that the increase of accuracy increased the measurement... which is one of the properties of fractals! This happens because fractals are figures with an infinite amount of detail, and measuring more accurately adds more of these details, which adds to the overall size. Mandelbrot claimed that the difference in the two measurements were due to the fact that Spain used a "yardstick" that was bigger than Portugalís. If, for example, Spain measured the border with a 2 kilometer yardstick, its measurement would be less exact than Portugal, which used a 1 kilometer yardstick. If we graph log(total length) against log(length of yardstick), we get lines with negative slopes since the total length decreases with the increase of the size of the yardstick:
Using this graph, we can find fractal dimensions of the coasts and borders by using a modified version of the geometric method. Since the magnification, used in the geometric method is equal to 1 / size of yardstick, the identity log (1/x) = Ė log (x) will tell you that the slopes of the line above are the negated slopes of the lines in geometric method. A little bit of algebra will tell you that in order to get fractal dimension, you need to subtract the slope from 1. Note that in the above diagram, the lines of more irregular lines, such as the coast of Britain are more steep than the lines of more smooth lines, such as the coast of South Africa. This is due to the fact that the more irregular a curve is, the higher its fractal dimension.
Simple models of coasts can be made with base-motif fractals that use polygons for the bases. Such fractals are also called Koch Islands. Below are three examples of the Koch Snowflake which uses a triangle, a Quadric Koch Island which uses a square, and the Gosper Island which uses a hexagon:
These pictures, obviously, are terrible in modeling real-life coats since they are too perfectly symmetrical and self-similar. The solution to this problem is to use fractals with Brownian self-similarity, such as the plasma fractals. This gives the coasts randomness, which makes them more realistic. Below is an example of a coastline made using a plasma fractal: