Fractal dimension of curves - How Long Is the Coast of Britain?

Benoit Mandelbrot (20 November 1924 – 14 October 2010) wrote a paper in 1967, which advocated a fractal concept to measurements of phenomena in nature such as geographical features. He further developed on the idea and published a book The Fractal Geometry of Nature (readers may wish to click here to buy it). Now, the main focus of this post. Practical applications abound for this concept, which I will go into straightaway.

Given a random curve in 2D space, there are 2 simple ways to measure its length. First method is by varying the length of straight lines used to measure the fractal dimension of a curve. This method is simple to implement if there is only one curve, but the situation gets complicated if there are many random curves. In this case, it is recommended that the other method, the box counting method, is used instead.

The problem is simplified by using a unit circle with centre at origin, in order for ease of understanding this concept. Let n be the number of equal line segments used to calculate the perimeter of this circle. Each segment will have length s. Total length measured in each case is u. Given a self similarity dimension D, these are related by



where b is the intercept of the resulting log-log graph. Note that log can be to any base, and is kept to base 10 in this case. Simply by varying s and calculating u, then plotting a graph, it is possible to get the fractal dimension D by adding 1 to the gradient of the log-log graph. These results are shown in the image below.

As can be seen in the graph above, the gradient is zero, which means self similarity dimension D is 1. This concept can be applied to any curve that has known (measured) coordinates, by varying s and measuring u, then plotting the log-log graph to get its self similarity dimension D.

Now we move on to the box counting method. Using squares of side s in length, define N(s) as the number of squares that contain the curve. This can be illustrated in the picture below, where the domain is split into squares of varying sizes. The red boxes are the boxes that contain the circle. N(s) is simply the sum of the red boxes for each value of s.

Again, by varying the length of s, and calculating respective values of N(s), and plotting a log-log graph, it is possible to find the self similarity dimension D. Note that in this case (see graph below), D is the gradient obtained from the log-log graph.

 From the above graph, the gradient is 1. This means that the self similarity dimension D is 1, which is the same as that found using the line segment method.

That is as simple as it gets. Do leave a comment if you need help using these methods and I will try to get back to you as soon as possible.