TESTING FOR SCALING IN NATURAL FORMS AND OBSERVABLES

The general procedure of calculating fractal dimensions or other expon ents is based on estimating some quantity as a function of scale and o n assessing whether or not this function is a power law. This power la w manifests itself in a log (quantity) versus log (scale) plot as a li near region (scaling). It has thus become the practice to estimate dim ensions by the slope of some linear region in those log-log plots. Whe n we are dealing with exact fractals (the Koch curve, for example) the re are no problems. When, however, we are working with natural forms o r observables, problems begin to emerge. In such cases the scaling reg ion is subjectively estimated and often is only the result of the gene ric property of the quantity to increase monotonically or decrease mon otonically as the scale goes to zero irrespective of the geometry of t he object. Here we discuss these issues and suggest a procedure to dea l with them.