| The non-graphic applications I've heard of:
1. Finance. Mandelbrot's original claim to fame was actually in
finance, where he showed a fractal-like distribution of cotton
prices over time. The original article he wrote is:
"The Variation of Certain Speculative Prices"
Benoit Mandelbrot
Journal of Business
vol. 36, 1963
pp. 394 - 419
I have a paper copy and would be glad to send copies to those
interested, but be warned: Mandelbrot, in my opinion, is one
of the world's worst writers. In the same journal issue is
a comment on Mandelbrot's findings by Eugene Fama, which I
found more accessible (I have a copy of this as well). Another
source is in "The Fractal Geometry of Nature", which contains
a version of the theory as well as corrected versions of the
original graphs.
In a nutshell, Mandelbrot claims that for a long time series of
prices, the distribution of differences between logs of prices
is the same, no matter how far apart the price differences are
measured. Thus:
For series of prices P(1) through P(n)
For a constant "k", Plot the cumulative distribution of
values for all D, where:
D = log(P(m+k)) - log(P(m))
The shape will be sort of sigmoidal looking when plotted
on double log paper (see "The Fractal Geometry of Nature").
Mandelbrot's claim is that the distribution will look the
same regardless of what "k" is chosen, with just a shift
to the right or left on the graph. In other words, looking
at monthly price differences is similar to looking at daily
differences. This is analogous to zooming in on the
Mandelbrot set. More specifically, Mandelbrot claimed that
the distribution is "stable Paretian".
If this claim is true, it means that there may be intrinsic
limits to statistical analysis of these time series. Why?
Because the stable Paretian distribution has an infinite
variance, which thoroughly messes up the ability to apply
the usual statistical tests (these generally assume a
finite variance) in predicting level of risk, etc.
2. High-speed data-comm. There was an article in the last few months
in New Scientist (I think) about a new ultra high speed modem which
uses fractals for data encoding/compression.
3. Tactile sensation. There was a report somewhere (sorry, I've
completely forgotten), in which experimenters asked subjects to
feel surfaces with different amounts of roughness and to make a
naive numerical judgement of "degree of roughness". They then
correlated this with the fractal dimension of the surface, and
got very good results (R or R-squared >= 0.9, as I recall).
I don't know if anyone did anything with this, but you could
imagine a use as e.g. an ergonomic contribution to material design.
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