Sunday, November 07, 2010

Looking for fractals in all the wrong or right places - I

Here is the first of several attempts at documenting fractal structures from science to art.

First, however, a little bit about technique: The magnitude of a particular data set, whatever its source, is ranked from greatest to least. Then, fractal binning is applied over a range of dimensions. The maximum dimension is D(max) = 2*n, in which the value of n produces the smallest value larger than the maximum magnitude of the data set. Each dimension is then determined by D(I) = D(maximum)/M, in which M = 2*N/M, M =1, M*. Each bin I for dimension D(I) is occupied if there is one or more values such that bin I = integer (magnitude/D(i)). Then for each dimension, the number of occupied bins is totaled.
First example: Number of performances of Broadway musicals, ranked from most to least, for the top 100 (not counting some which are still running).
The longest running musicals (Andrew Lloyd Webber's Cats is the current leader among closed shows; however his Phantom of the Opera is still running).

Note that a true fractal would display a linear trend on a log-log plot. However a simple linear trend is apparent for only the top two bins, not one extended over a longer range of scales.

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