Survey of Empirical Fractals: Analysis and Classification (1996-1997):

Publication Numbers: 26,28.

Complex structures are common in nature. Their analysis and
characterization is a difficult task, particularly in the
case of structures which cannot be characterized by one or
a few length scales.
The fractal concepts appear ideal for describing such multi-scale
phenomena and the dynamics that generates them.
For example, in surface growth phenomena one commonly observes
rough surfaces as well as cluster structures which can be characterized
by their fractal dimensions.

Due to the reasons described above,
there has been much experimental effort
in the past fifteen years or so
to explore fractal structures in empirical systems.
Concurrently, models that give rise to a variety of fractal structures
were introduced and studied extensively.
While these models typically exhibit scale-invariant fractal structures,
empirical fractals tend to be bounded by upper and lower cutoffs.
To examine the experimental status of the cutoffs
we performed a survey of experimental studies and found that
experimental reports of fractal behavior are
typically based on a scaling range
which spans less than two decades with very few exceptions.

To obtain insight about this we have analyzed
the growth of diffusion-limited-aggregation-like clusters
on surfaces in molecular beam epitaxy [26].
Using scaling relations for island growth
we obtained the dependence of the upper cutoff on the duration
of the experiment. We obtained indications that in these systems
the limited range may result from the fact that the scaling range
(in decades) increases slowly (logarithmically) with the duration
of the experiment, namely the experimental time increases exponentially
as a function of
Interestingly, this resembles the situation in the theory of
algorithmic complexity, where for intractable problems the
computation time depends exponentially on the input size