What are the problems with using the box counting method to determine a fractal dimension?
I am currently looking at the fractal dimension of a Barnsley fern. By differing the number of points used to create the fern, different answers are obtained for the dimension. This is certainly what is expected based on the logic - with only 10 points the pattern doesn't look at all like a Barnsley fern. However what is the actual explanation behind this?
Or in other words how do you get the most accurate measurement for the dimension using the box count method? Is it density related, the more points in a given area the more accurate the measure?
After plotting the graph and determining the gradient for 10, 100, 1000 and 10,000 points it does seem to follow that more points give a more accurate measurement but what is the science/maths behind this?