Notebook: Explorations of the Application of Fractal Dimension & Ancillary Calculations to Estimate

This book introduces some new fractal dimension concepts to calculate the center point elevation from a 3X3 grid of actual topographic data. The implications are that there is an order to natural systems that has been newly identified.

The material is presented in the form of a notebook study. The innovative fractal-type equations and ancillary concepts are conceptually concise. The calculation framework to obtain the center point values uses the fractal and ancillary equations to construct what are termed "positive" and "negative" criteria, as well as a "position locator." The solution for the center point elevation is arrived at via a mathematical sieve of these three factors.

There are 13 example datasets that are evaluated. In each analysis, only the data from the relevant 3X3 region was used. The topographic data is from the Red Deer River area in Alberta, Canada. The accuracy of the calculated elevations compared to the actual center point data in a notable number of cases is truly amazing, to the point of being essentially perfect.

However, there appear to be limitations to the technique that have yet to be explored to any significant extent. With only 13 datasets investigated, it is expected that additional refinements to the various criteria will be required. A full and final theory will have a more rigorous mathematical foundation to formalize the criteria.

The book includes code snippets of the various calculation methods, so that others may enjoy their own explorations. Ideas for additional studies and improvements are suggested. Predicting values in non-center-point situations would be one of the next steps.

The material is primarily designed to be accessible to individuals with an undergraduate background in mathematics, physics or the sciences.

Applications to other types of data would appear to be possible.