Determination of fractal dimensions for geometrical multifractals

Two independent approaches, the box counting and the sand box methods are used for the determination of the generalized dimensions (Dq) associated with the geometrical structure of growing deterministic fractals. We find that the multifractal nature of the geometry results in an unusually slow convergence of the numerically calculated Dq's to their true values. Our study demonstrates that the above-mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres. In this case the latter approach provides better estimates of the generalized dimensions.