The calculation of the box counting dimension can be greatly simplified to the similarity dimension formula.  The similarity dimension designated by d_s is:

d_s= log (N(r))/log(1/r)

where N is the number of self similar figures composing the original figure and r is the factor by which each piece is shrunk compared with the original.  Knowing these two values we can easily calculate the similarity dimension.  Let’s look at an example.  We can calculate the dimension of the Sierpinski gasket using this formula:

We can see that our original figure was divided into three self similar figures; thus N=3.  Each of these smaller figures has a side that is equal to one half of the side of the original; therefore r=1/2.  Thus:

                                                d_s = log (3)/log (2)

So the dimension of the Sierpinski gasket is approximately 1.585.  The following web page will show more examples of using the similarity dimension formula: http://classes.yale.edu/Fractals/02WorkshopII.html