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geometries” are not related to each other essentially, either in the mathematical or physiological domain, but are often made vaguely equivalent on the basis of their lexical similarity.

An experimentally meaningful relationship between fractal statistics (hazard), dynamical fractals (dimension) and fractal geometries (form), has to be proven on a case by case basis and not assumed from their common designation. Among the informal attempts to do this have been those that involve the branching pattern of nerves and the associated reductions in their diameter-dependent characteristic conduction velocities yielding a multiplicity of “arrival times.” There is, however, a more central idea common to these concatenated meanings of fractal: the statistical, dynamical and geometric expressions of “scaling,” a word which is not mentioned in Mandelbrot’s book titles. The cluster of theories, theorems and methods associated with the idea of scaling (and renormalization) have led to Nobel Prizes for Flory (1971), Wilson (1975) and de Gennes (1979) and the (equivalent mathematical) Field’s Medal for McMullen (1994). There is speculation that the last two awards were supported by the inspiration and interest given their research by Mandelbrot’s intuitions and books.

Scaling laws take the place of (unknown causal) physical laws by indicating the proportion by which observables of a system can be changed in relationship to each other such that some statement about them, “this varies with that,” still holds. In a cross species comparison, as the average weight of a mammalian body, called lb, increases, the skeletal weight, called w, increases at an exponentially greater rate: w goes like lb 1.08 where lb 1.0 would indicate that they grew across species at the same rate. Plotting log (lb) on the x axis and log (w) on the y axis in a log-log plot results in a straignt line with a slope that indicates the power law scaling relationship between body weight and skeletal weight across mammals. The slope of the scaling exponent of 1.08 is a little over 45° = 1. In contrast, the metabolic rate, r, goes like (lb) 0.75, r ≈ (lb) 0.75. Larger animals (relative to their weight) have lower basal metabolic rates (Schmidt-Nielsen, 1984). We don’t completely know the chain of intervening mechanisms that relate these variables to each other but we do know

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