NUMERICAL ESTIMATES OF THE FRACTAL DIMENSION <i>D</i> AND THE LACUNARITY <i>L</i> BY THE MASS RADIUS RELATION

Abstract

The mass radius relation (mrr) allows the estimation of a local fractal dimension Dl, which depends on a chosen site l of a set, taken as center position for the mrr. We find an unexpected wide range of Dl-values for a numerically analyzed Sierpinski triangle. The analysis of a computer simulation of a biological branching pattern shows rather small Dl-values at the border lines to major empty areas and rather high Dl-values in densely grown regions. The local impression by many Dl of a growth pattern can be interpreted biologically as an inverse measure of, for example, the local immune activity in a living object. We also apply the mrr to investigate the lacunarity L of Cantor dusts. Since numerical difficulties to obtain L are based on the finiteness of a set, we analyze projections of fractals on the unit circle, such that a well-defined largest possible gap size is introduced. The first preliminary results imply rather stable numerical values for L and D for many sets, where D is the average over many Dl. L depends on the size of the smallest details in a set. L is larger if many different gap sizes are given in a generator for a fractal curve as opposed to curves, where only one gap size is given. We assume that a totally different approach has to be set up to obtain L numerically of sets of finite size, since projections of sets or the introduction of periodic boundaries seem inadequate.