Cutting Edge
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Cutting Edge

Cutting Edge

Dr Gregory Farrelly FAITH MAGAZINE May-June 2014

Biological Fractals, Tigers and Trees

Researchers from the University of Maryland and the University of Padua have proposed1 that plants and animals evolved in parallel, driven by the problem of how to use energy efficiently. They have considered a biological formula known as Kleiber’s Law; this states that an organism’s basal metabolic rate (the rate at which it consumes energy at rest) is roughly proportional to its mass raised to the power (exponent) of three-quarters: qa M3/4. A “law” is a strong statement in the natural sciences as it claims to be universally true, a consequence of some underlying symmetry or mathematical property of nature.

In 1999, a team led by Geoffrey West from Los Alamos National Laboratory suggested the reason for this law lies in fractals – mathematical, self-replicating patterns such as the one shown above. West explained that fractal-like shapes are good at maximising surface area, which permits nutrients to be absorbed more efficiently.

In the Faith vision, phenomena such as Kleiber’s law are evidence of the underlying, ordered, mathematical nature of physical reality, rather than special designs or interventions by the Creator.

Almost all organisms exhibit scaling exponents very close to 3/4 for metabolic rate and to 1/4 for biological times and dimensions (eg lifespan, cardiac cycle, aorta length, tree height). The fractal geometry in effect adds an extra (fourth) dimension to the familiar three dimensions of space, accounting for these mathematical exponents. Even in organisms without apparent “fracticality” in their internal forms, Kleiber’s law seems to hold. The more recent researchers state that “the earliest plants and animals had simple and quite different bodies, but natural selection has acted on the two groups so the geometries of modern trees and animals are, remarkably, displaying equivalent energy efficiencies. They are both equally fit. And that is what Kleiber’s Law is showing us.”2

The Maryland-Padua team consider a tree and a tiger. The tree has the easier task – to convert sunlight to energy and move it within a body that is essentially stationary. The tree has evolved a branching (fractal) shape with many surfaces, its leaves, to make this task as efficient as possible. The surface area and the volume of space it occupies are nearly the same numerically. Here, the tree’s nutrients flow at a constant speed, regardless of its size, conforming to Kleiber’s Law. The tiger needs fuel, but “burning” that fuel generates heat that must be dissipated. Because the tiger’s surface area is proportionally smaller than its volume and mass, surface cooling is not efficient enough. As animals get larger in size, their metabolism must increase at a slower rate than their volume. If the surface area were all that mattered, an animal’s metabolism would increase as its size increased, at the rate q0 a M2/3 (two-thirds being the exponent in the surface are to volume relationship: S a V2/3), yet Kleiber’s Law, instead, is obeyed even here, ie qaÊM3/4. The research team have solved this apparent contradiction not with fractals but with the velocity of blood flow, which is proportional to M1/12. In this view, animals adjust the flow of nutrients and heat as their mass changes to maintain the greatest possible energy efficiency. “That is why animals need a pump – a heart – and trees do not.”3

This solution implies “convergent evolution”, the independent evolution of similar features in different species, considered in an earlier edition of this column.4 Here, evolution is driven by the underlying physics and mathematics. Faith readers will recognise the link between the mathematical structure of the physical universe and the Unity-Law of Control and Direction, the metaphysical principle of being and becoming that prompts the questions: Whence comes such a “law” of the universe? And what “being” can cause all other being to exist and hold it in being? This being cannot itself be material since it would then be subject to the contingent laws of being and becoming, typical of the material world. Ultimate being, the ultimate “cause” in Aristotelian philosophy, must therefore be immaterial. This “spiritual” being is, of course, God.

This philosophical and theological view is quite distinct from Intelligent Design arguments. In the Faith vision, phenomena such as Kleiber’s law are evidence of the underlying, ordered, mathematical nature of physical reality, rather than special designs or interventions by the Creator. In this perspective, the physical sciences themselves are evidence for the causality and dependence inherent within matter. It is they that enable a coherent metaphysics to be developed that implies the existence of a spiritual creator and sustainer of the universe – whom we know as God.

The Universe: Merely Mathematics?

In a recent book, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, Max Tegmark, a physics professor at the Massachusetts Institute of Technology, argues that the idea of equivalence means that the universe is a mathematical structure rather than a reality merely describable by mathematics. The false argumentation, convincing to the average reader no doubt, shows the need for physicists to be trained in philosophy and the need for philosophers (and theologians) to be aware of current trends in physics.


1Proceedings of the National Academy of Sciences, Feb. 17, 2014.



4Faith Magazine Jul/Aug 2012.

Faith Magazine

May - June 2014