Measuring Fractal Dimension ?
Steven D'Aprano
steve at REMOVETHIS.cybersource.com.au
Sun Jun 14 16:00:56 CEST 2009
Lawrence D'Oliveiro wrote:
> In message <slrnh33j2b.4bu.peter at box8.pjb.com.au>, Peter Billam wrote:
>
>> Are there any modules, packages, whatever, that will
>> measure the fractal dimensions of a dataset, e.g. a time-series ?
>
> I don't think any countable set, even a countably-infinite set, can have a
> fractal dimension. It's got to be uncountably infinite, and therefore
> uncomputable.
Incorrect. Koch's snowflake, for example, has a fractal dimension of log
4/log 3 ≈ 1.26, a finite area of 8/5 times that of the initial triangle,
and a perimeter given by lim n->inf (4/3)**n. Although the perimeter is
infinite, it is countably infinite and computable.
Strictly speaking, there's not one definition of "fractal dimension", there
are a number of them. One of the more useful is the "Hausdorf dimension",
which relates to the idea of how your measurement of the size of a thing
increases as you decrease the size of your yard-stick. The Hausdorf
dimension can be statistically estimated for finite objects, e.g. the
fractal dimension of the coast of Great Britain is approximately 1.25 while
that of Norway is 1.52; cauliflower has a fractal dimension of 2.33 and
crumpled balls of paper of 2.5; the surface of the human brain and lungs
have fractal dimensions of 2.79 and 2.97.
--
Steven
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