To Boldly Determine a Fractal Dimension
Thursday, June 4, 2009 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 4/3 = 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.
Reader Comments (6)
This article raises the issue of "Science in Modern Film and Literature". The reference to the Hausdorrf-Besicovitch dimension could harldly go unnoticed by the committed reader of the Fractalog but may have drawn little attention from todays movie fan. How can this be? We all should be vigilant for an obscure reference.
I was on vacation in Vermont this summer and settled on the Nicholas Cage movie "Knowing". Nicholas Cage plays the part of a Professor whom has discovered that the world will be coming to end (soon) by deciphering an old coded message. A throw away line in the middle of the dialogue mentions the Drake Eqiuation. This is another reference that the committed Fractalog reader would detect immediately. The "long arm" of the Fractalog reached me all the way in the Northern Kingdom.
How about you? How many references are out there? How many have we missed? "The Butterly Effect", with Ashton Kuetcher. The Modern Christian themed, "The Shack" references God's use of Fractals.
So many books,...so many films....so little time...
There is a similar nod to economics in the Star Trek scene of Vulcan schooling, when the camera pan picks up someone reciting the defining characteristics of a public good -- nonrivalry and nonexcludability. A well known economist, Paul Romer, expands on this in the following passage:
Economists studying public finance have identified two fundamental attributes of any economic good: the degree to which it is rivalrous and the degree to which it is excludable. Rivalry is a purely technological attribute. A purely rival good has the property that its use by one firm or person precludes its use by another; a purely nonrival good has the property that its use by one firm or person in no way limits is use by another. Excludability is a function of both the technology and the legal system. A good is excludable if the owner can prevent others from using it. A good such as the code for a computer program can be made excludable by means of a legal system that prohibits copying or by means of encryption and copy protection schemes.
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