Fractal Dimension Computation in Python Code

[Mike Brenner]

Mike Brenner > According to the inventor of fractals (Hausdorff in the
year 1899), you can place the set of 2D points next to a wall and shine
light through them, and the fractal dimension is the percentage of
shadow on the wall.

Alex > Can you give a reference?  This suprises me, because it seems to
me that since a Koch-curve has zero two-dimensional area, it will also
cast a zero-area shadow.  On the other hand, its Hausdorff dimension is
greater than 1, if I recall correctly.  Is Hausdorff dimension the same
as fractal dimension?

Yes, the shadow explanation is approximate. A reference is Federer:
Geometric Measure Theory. 

Yes, fractal dimension is a contraction of fractional dimension and is
the same as the Hausdorff dimension. 

The exact definition of the shadow looks like filling the figure with
circles, then the remainder with smaller circles, and ever smaller
circles, and then add up the areas of all of the circles. 

Later, other mathematicians found out that a simpler definition could be
made with the limit of circles of a single radius approaching zero
instead of all different size circles.

At any given radius of the smaller circles, there is a certain
percentage (THIS IS THE SHADOW) of the figure inside of those circles,
because of the self-similarity, there is a fractal dimension greater
than 1. 

The Koch curve has a Hausdorff Dimension of 1.26, a Fractional Dimension
of 1.26, and a Fractal Dimension of 1.26.