The dimension of fractal objects are not restricted to the numbers 1, 2 or 3 but instead fall somewhere between these dimensions. Perhaps this fact is not so shocking, because a fractal is cloud-like and it is difficult to see where it begins and where it ends.
Very good question.
In the attached pdf file, it is shown that the visible part of a fixed affine line has any dimension between 1 and 2 (see page 8). Another interesting part of the attached paper is the consideration of Cantor sets (see Example 4.13).
Taking my previous post a step further, consider the dimension of a koch snowflake (see attached image). The dimension of the Koch snowflake is
\[
d \approx 1.26185…
\]
For more about this, see
http://www.math.ubc.ca/~cass/courses/m308-03b/projects-03b/skinner/ex-dimension-koch_snowflake.htm
First, suppose we have triangle. second suppose there are 4 functions acting on each side with ratio 1/3 to construct a shape like koh .Third the dimension is Ln(4)/Ln(3)=1.261.
Followers of this thread may find the following downloadable interesting:
A.A. Sonin, The Physical Basis of Dimensional Analysis, 2nd Ed., MIT Press, 1997:
http://web.mit.edu/2.25/www/pdf/DA_unified.pdf
Very interesting question!
I offered an alternative to fractal dimension for characterizing the degree of complexity of fractals or geographic features in particular:
Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541.
https://www.researchgate.net/publication/236627484_Ht-Index_for_Quantifying_the_Fractal_or_Scaling_Structure_of_Geographic_Features
Article Ht-Index for Quantifying the Fractal or Scaling Structure of...
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