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Fractional Dimensions
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Question:
I have heard that fractals can be described using units raised
fractionally. For example, the two-dimensional Mendelbrot set has infinite area,
described as m^2, yet zero volume, described as m^3. Some fractional power
sy> (say m^2.7) could be used to describe the blank "space" in the set.
While fascinating, it seems purely mathematical. Does m^2.7 represent an
object existing in 2.7 dimensions; and could we exist and pass through
fractional dimensions? Or is 2.7 just a number, unusable by practical
physicists?
Replies:
I have not been working with fractals, but I think that I can give a
broad answer to your question. Yes, "fractal dimension" does have a
meaningful definition. The dimension tells you, in a sense, just how
"fractal" the boundary is. For a two dimensional structure, the fractal
dimension tells you how fast the boundary length grows with the area of
the structure,
Jack L. Uretsky
In other words, "it is just a number". It has been pointed out,
however, that nature seems to like fractals. Many plants
and trees seem to follow a fractal branching pattern, and you
can imagine that the coastline of a country is really
a fractal with no well-defined length. The practical consequence
is that you have to say something like "smoothing the coastline
on a length scale of 1 mile" when give a length number,
because the coastline smoothed to 1 mile is considerably shorter
than the coastline smoothed to 1 foot, and considerably
longer than the coastline smoothed to 100 miles.
Arthur Smith
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Update: June 2012
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