Box Counting Approximation Algorithm

When I initially decided to examine EEG’s of the brain, I designed a box-counting approximation algorithm to see if I could get the same results as the authors, but I was unable to find suitable images to test it on.  Nevertheless, here is the algorithm.  It counts the number of q*q boxes that are needed to cover the set, and compares that to 1*1 boxes (pixels), using the formula ln(N(boxes of size 1)/N(boxes of size q))/ln(q).  Black pixels were taken to be in the set, and pixels of all other colours were classed as not being in the set.  The algorithm was tested by creating two iterated function systems that displayed the Sierpinski Gasket and the Sierpinski Carpet.  These have known box-capacity dimensions of ln(3)/ln(2)=1.585 and ln(8)/ln(3)=1.893 respectively.  I found that taking q to be 2, so that boxes of size 2*2 and comparing those to the number of actual pixels in the set gave the best approximation.  This gave dimensions of 1.564 and 1.855 respectively.  Strangely enough, though, when I tested it with the standard Mandelbrot Set and the Mandelbrot Set of f(z) = c*z*(1-z) where z is iterated on 1/2 (that we plotted for assignment 4), these sets resulted in box-counting dimensions of 1.95 and 1.96, implying that they did not have fractional dimension, and that they have a dimension of 2 (taking into consideration that the values are not exact).  Yet zooming in on the standard Mandelbrot set resulted in great variety in the box-counting dimension:  whether this is from the inaccuracy of my program or whether it is a  property of the Mandelbrot set itself is unknown.

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