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Various exact algorithms and defining formulas have been devised for the calculation of parameters called 'fractal dimensions'. Practical applications of FD's are evaluation, comparison and classification of an amazingly wide range of natural phenomena.

Yet, while a FD is an exact number, I can find no calculation - of any directly measurable physical or mathematical quantity - that has a FD as an input.

How can it be that something so deep, simple, exact and powerful is... a dead end?

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  • $\begingroup$ This question (v1) seems like a list question. $\endgroup$ – Qmechanic Feb 10 '15 at 20:44
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The fractal dimension property of a material or artifact gives us a scale invariant measure of its structure complexity. My own Master's thesis worked (partially) on quantifying fractal dimension of soot/ash agglomerate particles because, via empirical relationships, it can be used to put a bound on aerodynamic properties, and optical properties such as complex refractive index and absorption/scattering coefficients.

In material science, it can, for example, be used to determine porosity and permeability of solids, quantify and analyse dielectric breakdown, determine microscopic features of mechanical stress, and analyse fractures.

Applications outside materials include being able to to compress important information and pick out useful attributes stored in large (even multimedia) datasets, and even being used to analyse electroencephalograms to identify characteristics of otherwise-too-short traces.

I also nearly did my PhD on the relationship between river fractal geometry and flood risk.

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    $\begingroup$ Thanks for the extensive answer. Interesting stuff! At first glance most calculations border on empirically finding correlations between fractal dimensions and physical properties. Real theoretical calculations seem to be rare, as I suspected. I'll dig through the material later, then comment. $\endgroup$ – Zaaikort Feb 10 '15 at 22:45
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    $\begingroup$ One of the (freely accessible) linked articles,"Fractal analysis of permeabilities for porous media" onlinelibrary.wiley.com/doi/10.1002/aic.10004/abstract, contains a theoretical FD-based prediction of a physical property. So the answer to my question is probably: yes. I'd like to find more than just one example though... $\endgroup$ – Zaaikort Feb 16 '15 at 18:44
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Fractal Dimension is used in my own research on EEG as an input in a long array of input ranging from first and second order statistics to various distance metrics or quantities in information theory in order to categorize different types of EEG response.

This is very interesting stuff because fractal dimension roughly tells you the self-similarity of a signal. During drowsiness for example, your brain's EEG signal display a waveform that has a distinct periodic, self similar behavior. Fractal dimension is calculated using a brief segment (roughly 1 second) worth of EEG data, which may consists of 256 or even more points. Imagine this waveform as the Koch Snowflake expanded to a line. This technique can used to measure an airplane pilot's mental state of drowsiness, and alerts the command center when fractal dimension reaches a certain threshold.

This is one engineering aspect of fractal dimension.

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