Conclusions And Recommendations

The purpose of this introductory chapter was to present some of the key concepts of fractal geometry and multifractal measures, which will be put to use in later chapters of this book. One of the principal take-home messages of this introduction is that extreme caution needs to be exercised when applying 'fractals' to natural systems. Indeed, unlike with most other mathematical theories that have been used to describe natural systems, starting with Euclidian geometry, there has historically been a considerable level of vagueness and ambiguity associated with the definition of most fractal concepts. As yet, unresolved operational issues complicate the evaluation of fractal characteristics for many systems. These challenges lead to a high risk of using the same terms to mean different things, of using different terminologies to cover the same basic reality, or of having different observers end up with very different perspectives on the fractality of given systems. The literature is replete with examples of confusion that can be traced back to these inherent ambiguities in the theory. Only if one is very clear about precisely what is meant by each term, and about how given parameters are evaluated, will one be truly able to assess the usefulness of fractal geometry for the description of natural systems.

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