Testing for scaling in natural forms and observables

Abstract

The general procedure of calculating fractal dimensions or other exponents is based on estimating some quantity as a function of scale and on assessing whether or not this function is a power law. This power law manifests itself in a log (quantity) versus log (scale) plot as a linear region (scaling). It has thus become the practice to estimate dimensions by the slope of some linear region in those log-log plots. When we are dealing with exact fractals (the Koch curve, for example) there are no problems. When, however, we are working with natural forms or observables, problems begin to emerge. In such cases the scaling region is subjectively estimated and often is only the result of the generic property of the quantity to increase monotonically or decrease monotonically as the scale goes to zero irrespective of the geometry of the object. Here we discuss these issues and suggest a procedure to deal with them.