I wrote long ago about a simple way to calculate the fractal dimension for any shape. The central idea behind calculating the fractal dimension (or the any dimension) is the Hausdorff Dimension (applet-ridden page). In very layman terms, the Hausdorff dimension provides a way to calculate the dimension of any space – point, line, plane or even irregular shapes like fractals.
The idea behind the dimension calculation is rather simple – by what factor do the smaller parts fit into the original whole when the dimensions of the original are modified. For example, when the dimensions of a cube are increased by a factor of 2, there are 8 of the original cubes which can be placed in the new cube. Applying the Hausdorff dimension, N = 8, P = 2, p = 1 (the number of increase in units is 8, when the size of the size is doubled). So, the dimension of the cube is log8/ (log (2/1)) = log 8 / log 2 = 3. Of course everyone knows that the cube is 3 dimensioned.
Now the beauty of the equation is that it can help calculate the dimensions for fractals equally easily. This is used in in Season 4, Episode 9 of Numb3rs – Graphic by Charlie Epps . Charlie uses Fractal dimensions to find out if a painting is original or not. The calculation of the fractal dimension is skimmed rather fast. So, let us take it up with the example in the link.
The fractal to use is the Sierpinski triangle. The numbers to use again are simple. For each time there is a change in the dimension of the triangle, the number of triangles increases by 3 fold. For example, if the dimension of the equilateral triangle was 2 initially, the first iteration reduces the dimension to 1. The number of triangles increases from 1 to 3. So, N = 3, P = 2, p= 1. Hence the fractal dimension of the Sierpinski triangle is log 3 / (log (2/1)) = log 3 / log 2 ~ 1.585. The d-dimension for the triangle is 1.585, the just right dimension.
Continuing, if we were to calculate the fractal dimension of the Menger sponge, it is thus. The number of cubes that get generated when the center 1x1x1 cube and the center 1x1x1 cube of each face of a cube are removed is 20. If the original cube was 3x3x3, then each of the new cube’s dimension is 1x1x1. Hence, N=20, P=3, p=1. So, the fractal dimension is log 20 / (log (3/1)) = log 20 / log 3 ~ 2.726 (which is lesser than the dimension of a cube !).
Put very simply, the equation lets one correlate the increase in the number of parts with the space that that shape consumes (hmm..is it the space that the shape consumes or the dimensions that the shape has ?).