Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life, in Organisms, Cities, Economies, and Companies

by Geoffrey West

An important lesson from these investigations is that, while it is not generally possible to make detailed predictions about such systems, it is sometimes possible to derive a coarse-grained quantitative description for the average

salient features of the system. For example, although we will never be able to predict precisely when a particular person will die, we ought to be able to predict why the life span of human beings is on the order of one hundred years.

Innovation and wealth creation that fuel social systems, if left unchecked, potentially sow the seeds of their inevitable collapse.

Despite 150 years of intense theoretical and experimental study, a

general understanding of turbulence remains an unsolved problem in physics even though we have learned an enormous amount about it. Indeed the famous physicist Richard Feynman described turbulence as “the most important unsolved problem of classical physics.”

So in terms of the quantities that actually vary, Froude’s number is simply proportional to the velocity squared divided by the length. This ratio plays a central role in all problems involving motion, ranging from speeding bullets and running dinosaurs to flying airplanes and sailing ships.

Consequently, all of these and indeed all of the laws of science must be expressible as

relationships between scale-invariant dimensionless quantities, even though conventionally they are not typically written that way. This was the underlying message of Rayleigh’s seminal paper.

This is characteristic of many scaling arguments: general results can be derived, but details of their mechanistic origins sometimes remain hidden.

This pretty much expresses the credo of modern-day “complexity science,” including even the implication that consciousness is an emergent systemic phenomenon and not a consequence of just the sum of all the “nerve-paths and neurons” in the brain.

This remarkably systematic

repetitive behavior is called scale invariance or self-similarity and is a property inherent to power laws. It is closely related to the concept of a fractal, which will be discussed in detail in the following chapter. To varying degrees, fractality, scale invariance, and self-similarity are ubiquitous across nature from galaxies and clouds to your cells, your brain, the Internet, companies, and cities.

Particularly fascinating is the emergence of the number four in the guise of the ¼ powers that appear in all of these

exponents. It occurs ubiquitously across the entire panoply of life and seems to play a special, fundamental role in determining many of the measurable characteristics of organisms regardless of their evolved design. Viewed through the lens of scaling, a remarkably general universal pattern emerges, strongly suggesting that evolution

has been constrained by other general physical principles beyon...

The fact that this has persisted and remained so robust, resilient, and sustainable for more than a billion years suggests that effective laws that govern their behavior must have emerged at all scales.

Revealing, articulating, and understanding these emergent laws that transcend all of life is the great challenge.

I. Space Filling The idea behind the concept of space filling is simple and intuitive. Roughly speaking, it means that the tentacles of the network have to

extend everywhere throughout the system that it is serving, as is illustrated in the networks here. More specifically: whatever the geometry and topology of the network is, it must service all local biologically active subunits of the organism or subsystem.

II. The Invariance of Terminal Units This simply means that the terminal units of a given network design, such as the capillaries of the circulatory system that we just discussed, all have approximately the

same size and characteristics regardless of the size of the organism.

III. Optimization The final postulate states that the continuous multiple feedback and

fine-tuning mechanisms implicit in the ongoing processes of natural selection and which have been playing out over enormous periods of time have led to the network performance being “optimized.”

When information is dissipated or “reflected,” such as when one side is not listening, it cannot be faithfully or efficiently processed, inevitably

leading to misinterpretation, a process analogous to the loss of energy when impedances are not matched.

It is the mathematical interplay between the cube root scaling law for lengths and the square root scaling law for radii, constrained by the linear scaling of blood volume and the invariance of the terminal units, that leads to quarter-power allometric exponents across organisms. The resulting magic number four emerges as an

effective extension of the usual three dimensions of the volume serviced by the network by an additional dimension resulting from the fractal nature of the network. I

organisms operate as if they were in four dimensions, rather than the canonical three.

It fell to the French mathematician Benoit Mandelbrot to make the crucial insight that, quite to the contrary, crinkliness, discontinuity, roughness, and self-similarity—in a word, fractality—are, in fact, ubiquitous features of the complex world we live in.17 In retrospect it is quite astonishing that this insight had eluded the greatest mathematicians, physicists, and philosophers for more than two thousand years.

Mandelbrot’s insights imply that when viewed through a coarse-grained lens of varying resolution, a hidden simplicity and regularity is revealed underlying the extraordinary complexity and diversity in much of the world around us. Furthermore, the

mathematics that describes self-similarity and its implicit recursive rescaling is identical to the power law scaling discussed in previous chapters. In other words, power law scaling is the mathematical expression of self-similarity and fractality. Consequently, because animals obey power law scaling both within individuals, in

terms of the geometry and dynamics of their internal network structures, as well as across species, they, and therefore all of us, are living ...

Consequently, the frequency distribution of wars follows simple power law scaling indicating that conflicts are approximately self-similar.19 This remarkable result leads to the surprising conclusion that, in a coarse-grained

sense, a large war is just a scaled-up version of a small conflict, analogous to the way that elephants are approximately scaled-up mice. Thus underlying the extraordinary complexity of wars and conflicts seems to be a common dynamic operating across all scales.

Consequently, most physical objects have no absolute objective length, and it is crucial to quote the resolution when

stating the measurement.

As Mandelbrot succinctly put it: “Smooth shapes are very rare in the wild but

extremely important in the ivory tower and the factory.”

You might think that with this knowledge you might soon become rich. Although this certainly gives new insight into hidden regularities in stock markets, unfortunately it has predictive power only in an average coarse-grained sense and does not give specific information about the behavior of individual stocks.

Although there is a mathematical framework for describing and quantifying fractals, no fundamental theory based on underlying physical principles for mechanistically understanding why they arise in general, or for calculating their dimensions, has been developed.

This may have been one of the reasons that his great discovery did not receive quite the appreciation in the physics community and scientific establishment that it perhaps deserved and, as a result, he did not receive the Nobel Prize, despite broad recognition in many quarters and a litany of prestigious awards and prizes.

Almost all of the networks that sustain life are approximately self-similar fractals.

As explained in the previous chapter, a crinkly enough line that is space filling can scale as if it’s an area.

In a similar fashion an area, if crinkly enough, can behave as if it’s a volume, thereby gaining an effective extra dimension: its Euclidean dimension is 2, indicating that it’s an area, but its fractal dimension is 3.

Thus, instead of scaling with classic ⅓ exponents, as would be the case if they were smooth nonfractal Euclidean objects, they scale with ¼ exponents. Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional.

almost none of our man-made engineered artifacts and systems, whether automobiles, houses, washing machines, or television sets, invoke the power of fractals to optimize performance.

As to the possibly less savory parts of his life, he would need to pee about 20,000 liters of urine a day, comparable to the size of a small swimming pool, and poop about 3 tons of feces, a good-size truckload. I shall leave speculations about his sex life to your imagination.

And the theory tells us why: growth is primarily determined by how energy is delivered to cells, and this is constrained by universal properties of networks that transcend design. Among

This is the curse of consciousness. We all know we are going to die. No other organism is burdened with the enormity of the conscious knowledge that it has a finite lifetime and that its individual existence is eventually and inevitably coming to an end. No creature, whether a bacterium, an ant, a rhododendron, or a salmon, “cares” or even “knows” about dying; they live and they die, participating in the continual struggle for existence by propagating their genes into future generations and playing the endless game of the survival of the fittest.

This is truly fantastic. What is so fascinating about this achievement is that it was accomplished without any dedicated global, national, or philanthropic private program to extend life. It just happened all on its own without anyone discovering a magic pill, an elixir of life, or fiddling with anyone’s genes. What has been going on?