Fractal Dimension

The concept of a fractal dimension is based on the idea of measuring things with differently sized rulers, as already mentioned in the article on the Mandelbrot set. Mathematically (but very loosely), the fractal dimension is defined as: if I cover an object with open balls (the “rulers”) of a certain size, and I reduce that size by a certain factor, how does the number of balls that are needed increase? Here, ball is meant in the mathematical sense, as the inside of a sphere (without the sphere itself if it’s an open ball). In two dimensions, the ball is a disk. In one dimension, it’s a line segment. The relation between the number of balls \(n\), the scaling factor \(s\), and the dimension \(d\), is

\[n=\frac{1}{s^d}.\]

This formula works perfectly for normal (non-fractal) objects: In one dimension, the number of balls (line segments in that case) doubles if you halve the size (\(s=1/2\) and \(d=1\), so \(n=2\)). This is exactly what you’d expect: say that you need 10 rulers of 1 m long to cover a line; if you then switch to 0.5 m rulers, you’ll need 20. In two dimensions, the number of balls (circles) is squared (\(s=1/2\) and \(d=2\), so \(n=4\)). And so on…

What if we look at this the other way around? We could determine the (possibly non-integer) fractal dimension of an object by observing \(n\), given \(s\), and then calculating \(d\) by rewriting the above equation as

\[d=-\frac{\log{n}}{\log{s}}.\]

In Practice

Let’s look at an example of an object that has a fractal dimension that’s not an integer, the Koch curve. This curve was named after Helge von Koch, a Swedish mathematician. Constructing the curve is easy: start from a line segment, divide that into three equal parts, and replace the middle part by (two sides of) an equilateral triangle. Repeat this ad infinitum. Note that the Koch curve is just the limit of this process, since the intermediate steps are only there to show the construction. The animation below (made with Fractint) shows the first six steps of the construction process.

Koch curve

Although it seems that the Koch curve is just a “wiggly” version of the original line segment, something strange is going on. For starters, the length of the curve is infinite. This follows from its construction. At each step, the length increases by a factor of 4/3, since each interval in the curve is replaced by the structure with the triangle, which has four sides of length one-third of the interval. So, if the original interval has length 1, the total length of the curve is

\[\lim_{x\to\infty}{\left(\frac{4}{3}\right)}^x=\infty.\]

So, the Koch curve is a line of infinite length, “folded” into a finite area. The fractal dimension is a way of expressing exactly how “folded” it is.

What is the fractal dimension of the Koch curve? The computation of the length already gave it away. In each step, the length of the curve increases by a factor of 4/3. This means that, if you divide the length of the ruler by three (\(s=1/3\)), you will need four rulers (\(n=4\)) in the next step of the construction. This makes the fractal dimension of the Koch curve about 1.26, calculated as

\[d=-\frac{\log{4}}{\log{\frac{1}{3}}}=1.2618595\ldots\]

The fractal dimension of the border of the Mandelbrot set is, amazingly, two! This means that it’s so “wiggly” that it fills (part of) the plane. This also means that the Mandelbrot set is an object of which the border has the same fractal dimension as the object itself. Try that with a square!