Tutorial Chapter 7
Box-counting MethodWHY NEED A THIRD METHOD?
Both, the similarity method and the geometric method of calculating fractal dimension require you to measure the size. For many fractals itís not hard, but try measuring the diffusion fractal!
AND WHAT IS IT?
Instead of doing something as crazy as that, we can use a simpler method. Consider putting the fractal on a sheet of graph paper, where the side of each box is size h. Instead of finding the exact size of the fractal we count the number of boxes that are not empty. Let this number be N. Making the boxes smaller gives you more detail, which is the same as increasing the magnification. In fact, the magnification, e, is equal to 1/h. In the lesson on similarity method we found that the formula for fractal dimension is D = log N / log e. With this method we can change it to:
D = log N / log (1/h)
Making h smaller will make the dimension more accurate. For 3-D fractals you can do the same with cubes instead of squares, and for 1-D fractals you can use line segments.
For example, letís calculate the fractal dimension of the Box Fractal.
Using box-counting method, we put the fractal on a sheet of graph paper. For this fractal, we use boxes with sizes 1/3 and 1/9.
In the first case, 5 boxes are not empty. In the second case, there are 25 boxes and in the third there are 125. Using these numbers, we find that in the first case D = log 5 / log [1/(1/3)] = 1.46. If you do it for the second case, you will find that the answer is the same, which means that our dimension is accurate.
Box-counting method is very useful for natural shapes that are hard to measure, especially bacteria cultures.