Applications of Fractals
Clouds look very irregular in shape. At some point in your life you probably did look at them wondering how their diverse shapes are capable of resembling many common objects, animals, and people. Yet, for the purpose of this website, the word "irregular" automatically triggers the word "fractal." Yes, indeed, clouds are fractal in shape just like most other objects in nature. Let us first look at experimental evidence that can prove this.
Usually, to prove that something is a fractal it is enough to find its fractal dimension. For something like a cloud it is the best to do it using the geometric method. Obviously, it is not done by measuring the actual cloud, but by measuring its 2D projection, which is the shade. We can make several measurements of the cloudís perimeter using different magnifications. This is achieved by using different sized "yardsticks." If, letís say, our yardstick is 1 kilometer long, the magnification is higher and measurement would be more exact than the one where the yardstick is 10 kilometers long. We also know that in fractals, more detail adds additional irregularities, which adds to the measurement. If we graph log(magnification) against log(perimeter) we should get a line with a positive slope since the perimeter of fractals increases with magnification. Indeed, when graphing it for the clouds, we get something like this:
By adding 1 to the slope (see geometric method) we find the fractal dimension. According to the findings of Lovejoy in 1981, the fractal dimension for most clouds is about 1.164.
Now, having proved that clouds are fractals, it would be good to try using fractals to generate computer models of them. We know for sure that, since clouds are very irregular, we have to use fractals that are random and have Brownian self-similarity. The best ones to use are plasma fractals. To make plasma fractals look like clouds, we can use a color map which uses colors similar to ones on a real cloud photograph. The pictures we can generate using this method are something like this:
We can control how fragmented the clouds are by changing a parameter in plasma fractals called roughness.