Tutorial Chapter 6
Geometric Method
WHAT’S WRONG WITH SIMILARITY METHOD?The similarity method for calculating fractal dimension is great if you have a fractal composed of a certain number of identical versions of itself. However, try using it for the coast of Britain. That’s impossible because all lines there have different sizes and require different magnifications. And we wouldn’t suggest counting them either!
A WAY OUT
There is a simple way out of this. We know that a true fractal has an infinite amount of detail. This means that magnifying it adds additional detail, which increases the overall size. In non-fractals, however, the size always stays the same. For example, look at the diagram below, where we graphed sizes of some non-fractals with different magnifications. If you graph log(size) against log(magnification) you get straight horizontal lines. It shows that the sizes don’t change, which means that the figures are not fractals.
Now take some fractals and do the same. You will no longer get horizontal lines since the sizes increase with magnification. This proves the figures to be fractals.
We can now easily calculate fractal dimensions using the slopes of these lines. This is done using a simple formula:
fractal dimension = slope + 1
HOW IS IT USEFUL?
The geometric method can be used very efficiently for natural irregular shapes that exhibit Brownian self-similarity. It was used to calculate dimensions of coasts, borderlines and clouds.
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