CODEWITHSUNDEEP
wolfram-mathematicafractals
user1748343
user1748343

Reputation:

Validating fractal dimension computation in Mathematica

I've written an implementation of the standard box-counting algorithm for determining the fractal dimension of an image or a set in Mathematica, and I'm trying to validate it. I've generated a Sierpinski triangle matrix using the CellularAutomaton function, and computed its fractal dimension to be 1.58496 with a statistical error of about 10^-15. This matches the expected value of log(3)/log(2) = 1.58496 incredibly well.

The problem arises when I try to test my algorithm against a randomly-generated matrix. The fractal dimension in this case should be exactly 2, but I get about 1.994, with a statistical error of about 0.004. Hence, my box-counting algorithm seems to work perfectly fine for the Sierpinski triangle, but not quite so well for the random distribution. Any ideas why not?

Code below:

sierpinski512 = CellularAutomaton[90, {{1}, 0}, 512];
ArrayPlot[%]
d512 = FractalDimension[sierpinski512, {512, 256, 128, 64, 32, 16, 8, 4, 2}]

rtable = Table[Round[RandomReal[]], {i, 1, 512}, {j, 1, 1024}];
ArrayPlot[%]
drand = FractalDimension[rtable, {512, 256, 128, 64, 32, 16, 8, 4, 2}]

I can post the FractalDimension code if anybody really needs it, but I think the solution (if any) is not to do with the FractalDimension algorithm, but the rtable I'm generating above.

Upvotes: 0

Views: 1411

Answers (1)

Clifford T. Brown
Clifford T. Brown

Reputation: 11

I have studied this problem a little empirically in consultation with a well-known physicist, and we believe that the fractal dimension of a random point process goes to 2 (in the limit, I think) as the number of points grows large. I can't provide an exact definition of "large" but it can't be less than a few thousand points. So, I think you should expect to get D < 2 unless the number of points is quite large, theoretically, large enough to tile the plane.

I would be grateful for your FractalDimension code!

Upvotes: 1

Related Questions