Fractals — A Beginner’s Guide
What are Fractals?
Fractals are complex, endless geometric patterns with infinite complexity. As a result of this property, most fractals theoretically have an infinite perimeter. Another property which many fractals possess is being self-similar. When sections of a self-similar fractal’s perimeter are magnified, the perimeter appears to repeat itself in a similar pattern an infinite number of times.
The Coastline Paradox
Here’s an interesting real life application of fractals. How would you measure the UK’s coastline. With a meter rule? A micrometer? Or perhaps with straight lines spanning several kilometres? The truth is, with each method of measurement, your answer would change drastically. For instance, if the UK’s coastline was measured with 50km rulers, it would be around 3400km long. The Ordanance Survey, however, state the UK’s coastline to be 17820km long. This contrast in values is derived from a difference in measurement equipment, with the OS using equipment with a higher resolution. Could this measurement get more accurate if finer equipment was used? Of course. Imagine how much the value of 17820km would increase if the UK’s coastline was measured entirely with 30cm rulers. The increase would be astronomical. As the resolution increases to subatomic level, the UK’s coastline would converge on infinity. This is the coastline paradox. This idea of an infinite coastline means that the UK is essentially a geometric pattern with infinite complexity (at a subatomic level) so therefore, the UK’s outline can be modelled as a fractal.
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Dimensions in Fractals
As you very well know, a line has one dimension; a plane has 2 dimensions and space has 3 dimensions. But how many dimensions would a pattern such as the Sierpinski triangle have? The answer to this question is, in fact, a fraction, and below is the explanation of how to calculate the dimension of self-similar fractals.
Calculating the Dimension of a self-similar 2D fractal
If you were to double the length of a line, how many original lines would the enlarged line be comprised of? 2, which can be written as 2 to the power of 1, therefore proving that a line is one-dimensional. Similarly, if the length of a square was doubled, four (or 2 to the power of 2) original squares are needed to fill the enlarged square, therefore proving that a square is 2 dimensional. Furthermore, if the length of a cube was doubled, eight (or 2 to the power of 3) original cubes are needed to fill the enlarged cube, therefore proving that a cube is 3 dimensional. If the length the famous fractal Sierpinski’s triangle was doubled, how many original triangles would this new enlarged triangle be comprised of? Three, because a the Sierpinski triangle is made up of 3 smaller copies of itself. So 2 to the power of what equals 3? Well, the answer is clearly not an integer, so logarithms must be used to calculate the fractional dimension of Sierpinski’s triangle. This means that the logarithm base 2 of 3 (~1.585) is equal to the dimension of Sierpinski’s triangle. In conclusion, an equation to calculate the dimension of fractals is r to the power of D = N, where r is the scaling, N is the number of copies needed to fill the new shape and D is the dimension.
Calculating the Dimensions of any 2D fractal
This trick is very useful, but it has one major drawback: not all fractals are self-similar, such as the UK’s coastline. Imagine you have a grid of very small squares. Lay the UK’s coastline over this grid and count how many squares the UK’s area occupies. Now, increase the UK’s size by a scaling factor of 2 and count the squares inside this new area. If you divide the number of new squares by the number of old squares, the answer will be a number (N) which can be expressed as 2 to the power of D (the dimension). Therefore, r (which is, in this case, 2) to the power of D equals N. This means that the dimension can be approximated by taking the logarithm base r of N. As the squares’ area approaches 0, this approximation of the fractal’s dimension converges on the true answer.
Why are fractals useful?
You may be thinking that this area of mathematical theory is far too abstract but it has its uses in the real world. For example, fractals can help to predict the pattern of growth for mutated, cancerous cells so that doctors know the subsequent cells to target in cancer treatment. Fractals can also be used to predict the trajectory of life-threatening meteorites before they strike Earth.
