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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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