The Limited Scaling Range of Empirical Fractals

A notion which has been advanced intensively in the past two decades, is that fractal geometry describes well the irregular face of Nature. We were prompted by Marder’s recent article in Science [1], to comment here on the applicability of this wide-spread notion. Marder summarizes a simulation study of fractured silicon nitride by Kalia et. al [2] which successfully mimics experimental data, and emphasizes the role of fractal geometry in describing complex-geometry physical structures, in general. Specifically, the results of Kalia’s et. al were interpreted as “showing that this mechanism... leads to fractal fracture surfaces”. However, upon examining Kalia’s results (Fig. 4 in Ref. [2]) one finds that Marder’s statement is based on four exponents, all of which hold over less than one order of magnitude. We recall that a fractal object, in the purely mathematical sense, requires infinitely many orders of magnitude of the power law scaling, and that a consequent interpretation of experimental results as indicating fractality requires, “many” orders of magnitude. We also recall that, for instance, in the celebrated fractal Koch flake, one order of magnitude means about two iterations in the construction and that such two-iterations Koch curve is not a fractal object. It is our feeling that Marder, like many others in the scientific community, may have been swayed by the wide spread image and belief that many-orders fractality abounds in experimental documentation. In a recent detailed statistical data analysis we have shown that this is not the case, at