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To Boldly Determine a Fractal Dimension

It happens very quickly, and is very easy to miss, unless one is either an inveterate fractalogist or Vulcan, or both.

In the eminently entertaining Star Trek movie just released there is a scene of young Spock’s school which appears to be a cavernous room with a floor made up of indented hemispherical shells (as if you were on the inside wall of a very large pimple ball). of each “pimple” while a Vulcan student in each “pimple” listening to a lecture, or reciting a lesson while Mathematical expressions are illuminated on the walls

The film takes us for a brief visit to a few of these math-pimples. In one, a pointy-eared student begins his recitation:

The dimensionality equals the log of N...

The statement is not completed, but clearly this is the beginning of the expression for the Hausdorrf-Besicovitch dimension: d = Log(N)/log(s)

This expression has many different variants (I am guessing that this is the one used in Vulcan grade schools), and can be used to easily calculate the dimensions of deterministic fractals. So, e.g., the Cantor set weighs in at a dimension of log(2)/log(3) = 0.6309…, while the Sierpinski Carpet is a more robust log(8)/log(3) = 1.8928…

Combined with clever statistical counting techniques, the dimension of random, and even naturally-occurring fractals can also be determined. The boundary of regular Brownian motion has a dimension of 43 = 1.33, while the surface of the human brain has a dimension of approximately 2.79.

In case you’re wondering what naturally occurring object your brain most closely resembles, note that the dimension of a typical piece of broccoli is 2.66

See this list of fractals sorted by Hausdorff-Besicovitch Dimension for more!

In the lists that regularly appear comparing the mathematical performance of US school kids vs. children in other countries, thank goodness Vulcan is not listed. Not only do they all know about the H-B dimension, they instinctively know the definition of a fractal: an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension.

Live long and prosper, measuring your fractals wherever you find them.