There are various reasons to question whether the fractal model is really true, but one of the most important is the validity of the exponent itself—the slope of the line connecting the metabolic rate to the mass. The great merit of the fractal model is that it derives the relationship between metabolic rate and mass from first principles. By considering only the fractal geometry of branching supply networks in three-dimensional bodies, the model predicts that the metabolic rate of animals, plants, fungi, algae, and single celled organisms should all be proportional to their mass to the power of %, or mass075. On the other hand, if the steady accumulation of empirical data shows that the exponent is not 0.75, then the fractal model has a problem. It comes up with an answer that is found empirically not to be true. The empirical failings of a theory may inculcate a fantastic new theory—the failings of the Newtonian universe ushered in relativity—but they also lead, of course, to the demise of the original model. In our case here, fractal geometry can only explain the power laws of biology if the power laws really exist—if the exponent really is a constant, the value of 0.75 genuinely universal.

I mentioned that Alfred Heusner and others have for decades contested the validity of the 3/4 exponent, arguing that Max Rubner's original 2/3 scaling was in fact more accurate. The matter came to a head in 2001 when the physicists Peter Dodds, Dan Rothman, and Joshua Weitz, then all at MIT in Cambridge, Massachusetts, re-examined 'the 3/4 law' of metabolism. They went back to the original data sets of Kleiber and Brody, as well as other seminal publications, to examine how robust the data really were.

As so often happens in science, the apparently solid foundations of a field turned to rubble on closer inspection. Although Kleiber's and Brody's data did indeed support an exponent of 3/4 (or in fact, of 0.73 and 0.72, respectively) their data sets were quite small, Kleiber's containing only 13 mammals. Later data sets, comprising several hundred species, generally failed to support the 3/4 exponent when re-analysed. Birds, for example, scale with an exponent close to 2/3, as do small mammals. Curiously, larger mammals seem to deviate upwards towards a higher exponent. This is in fact the basis of the 3/4 exponent. If a single straight line is drawn through the entire data set, spanning five or six orders of magnitude, then the slope is indeed approximately 3/4. But drawing a single line already makes an assumption that there is a universal scaling law. What if there is not? Then two separate lines, each with a different slope, better approximate the data, so large mammals are simply different from small mammals, for whatever reason.3

This may seem a little messy, but are there any strong empirical reasons to favour a nice crisp universal constant? Hardly. When plotted out reptiles have a steeper slope of about 0.88. Marsupials have a lower slope of 0.60. The frequently cited 1960 data set of A. M. Hemmingsen, which included single-celled organisms (making the 3/4 rule look truly universal) turned out to be a mirage, reforming itself around whichever group of organisms were selected, with slopes varying between 0.60 and 0.75. Dodds, Rothman, and Weitz concurred an earlier re-evaluation, that 'a 3/4 power scaling rule. . . for unicellular organisms generally is not at all persuasive.' They also found that aquatic invertebrates and algae scale with slopes of between 0.30 and 1.0. In short, a single universal constant cannot be supported within any individual phyla, and can only be perceived if we draw a single line through all phyla, incorporating many orders of magnitude. In this case, even though individual phyla don't support the universal constant, the slope of the line is about 0.75.

3 Another re-analysis, published in 2003 by Craig White and Roger Seymour, at the University of Adelaide, came to a similar conclusion.

West and his collaborators argue that it is precisely this higher level of magnification that reveals the universal importance of fractal supply networks—the non-conformity of individual phyla is just irrelevant 'noise', like Galileo's air resistance. They may be right, but one must at least entertain the possibility that the 'universal' scaling law is a statistical artefact produced by drawing a single straight line through different groups, none of which conforms individually to the overall 'rule'. We might still favour a universal law if there was a good theoretical basis for believing it to exist—but it seems the fractal model is also questionable on theoretical grounds.

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