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

by Geoffrey West

Recall that a smooth line has dimension 1, a smooth surface dimension 2, and a volume dimension 3. Thus the South African coast is very close to being a smooth line because its fractal dimension is 1.02, which is very close to 1, whereas Norway is far from it because its fractal dimension of 1.52 is so much greater than 1.

In other words, the behavior of the stock market is a self-similar fractal pattern that repeats itself across all timescales following a power law that can be quantified by its exponent or, equivalently, its fractal dimension.

This has stimulated the development of a new transdisciplinary subfield of finance called econophysics and motivated investment companies to hire physicists, mathematicians, and computer scientists to use these sorts of ideas to develop novel investment strategies.

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A fractal dimension is just one single metric out of many that characterize such systems. It is amazing how much store we put into single metrics such as these. For example, the Dow Jones Industrial Average is almost religiously perceived as the indicator of the overall state of the U.S. economy, just as body temperature is typically used as an indicator of our overall health. Better is to have a suite of such metrics such as you get from an annual physical examination, or what economists generate in order to get a broader picture of the state of the economy.

Natural selection has taken advantage of the fractal nature of space-filling networks to maximize the total effective surface area of these terminal units and thereby maximize metabolic output.

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Now imagine continuously decreasing the size of the animal. Concomitantly, the number of area-preserving branchings where vessels are large enough to support pulsatile waves decreases until a tipping point is reached where the network can support only nonpulsatile DC flow. At that stage even the major arteries become so small and constricted that they are unable to support pulsatile waves. In such vessels, waves become so overdamped due to the viscosity of blood that they can no longer propagate and the flow shifts to becoming entirely steady DC, just like the flow of water in the pipes of your house: pulsatile waves generated by the beating heart are immediately damped as they enter the aorta. This is really weird. Such an animal would have a beating heart but no pulse!

This argument shows that only mammals that are large enough for their circulatory systems to support pulsatile waves through at the very least the first couple of branching levels would have evolved, thereby providing a fundamental reason why there is a minimum size.

as it pumps blood with the same pressure and speed as you do,

Likewise, although a blue whale is a hundred million times (108) heavier than a shrew, the average distance between its capillaries is only about (108)1/12 = 4.6 times larger.

Krogh radius, which is the radius of an imaginary cylinder surrounding the length of a capillary, like a sheath, and which contains all of the cells that can be sustained (just to remind you: a capillary is about half a millimeter long and about five times longer than its diameter). Based on this, one can calculate how large an animal could be before the separation distance between its capillaries gets so large that significant hypoxia develops. This leads to an estimate of about 100 kilograms for the maximum size, roughly equivalent to the largest blue whales, suggesting that they represent the end of the line for the mammalian family.

However, to counterbalance that, it would have to beat only just over a couple of times a minute and sustain a blood pressure similar to ours.

It is this mismatch at the critical interface between capillaries and cells that controls growth and ultimately leads to its cessation: the increase in the number of supply units (the capillaries) cannot keep up with the demands from the increase in the number of customers (the cells).

Because metabolic rate—the rate at which energy is supplied to cells—is the fundamental driver of all biological rates and times, all of the central features of life from gestation and growth to mortality are exponentially sensitive to temperature.

This is huge and therein lies our problem. If global warming induces a temperature increase of around 2°C, which it is on track to do, then the pace of almost all biological life across all scales will increase by a whopping 20 percent to 30 percent.

All of these accomplishments are the result of the fascinating dynamic that was set in motion by the increasing migration of the populace to urban environments and the development of greater social responsibility, with the city as the provider of basic rights and services.

The city as the engine for social change and increasing well-being is one of the truly great triumphs of our amazing ability to form social groups and collectively take advantage of economies of scale.

This is precisely the rule that many decay processes in the physical world follow. Physicists use the term decay rate, rather than mortality rate, to quantify the decay of radioactive material in which “individual” atoms change their state by emitting particles (alpha, beta, or gamma rays) and “die.”

data show that the half-life of publicly traded companies in the United States is only about ten years. So in just fifty years (five half-lives) only (½)5 = 1/32 or about 3 percent are still posting sales. This begs the fascinating question as to whether the same general dynamics underlies the surprising commonality in the mortality of organisms, isotopes, and companies. We will return to speculate about this later.

It’s slightly depressing to see that on average we are physically optimal (100 percent) for only a very few years and that beginning around age twenty it’s literally downhill all the way.

The number of heartbeats in a lifetime is approximately the same for all mammals,

A central feature of how life is sustained is the transportation of metabolic energy through space-filling networks across all scales to service and feed cells, mitochondria, respiratory complexes, genomes, and other functional intracellular units, as symbolized here

Because larger animals metabolize at higher rates following the ¾ power scaling law, they suffer greater production of entropy and therefore greater overall damage, so you might have thought that this would imply that larger animals would have shorter life spans in obvious contradiction to observation. However, we saw in chapter 3 that on a cellular or per unit mass of tissue basis metabolic rate and therefore the rate at which damage is occurring at the cellular and intracellular levels decreases systematically with increasing size of the animal—another expression of economy of scale.

So at the critical cellular level cells suffer systematically less damage at a slower rate the larger the animal, and this results in a correspondingly longer life span.

I am somewhat biased toward them because I believe in the theory and the concept that lowering metabolism decreases damage, slows the aging process, and increases maximum life span.

how long life can be extended by changing body temperature or eating less.

The term Anthropocene has been suggested as the name for this most recent epoch in the history of our planet during which human activities have significantly affected the Earth’s ecosystems.

population that is growing exponentially is defined mathematically as one in which the rate of increase in its size (per minute, per day, or per year, for instance) is directly proportional to the size of the population that’s already there. Thus the growth rate itself grows even faster the bigger the population. So, for example, when the size of a population undergoing exponential growth doubles, the rate at which it is increasing also doubles, which means that it grows faster and faster the bigger it gets, effectively feeding back and running away with itself. Left unchecked, both the population and its growth rate would eventually become infinitely large.

An unfortunate consequence of Malthus’s analyses was that they were interpreted as effectively blaming the poor for their own predicament because they persisted in reproducing too rapidly. It was relatively easy therefore to conclude that it was this rather than exploitation by capitalists that was the origin of their poverty and their generally abysmal condition.

Four companies alone now produce 81 percent of the cattle, 73 percent of the sheep, 57 percent of the pigs, and 50 percent of the chickens consumed in the United States. Globally, 74 percent of the world’s poultry, 43 percent of its beef, and 68 percent of its eggs are produced this way.

Of equal significance is that this huge increase of energy consumption has taken place over an infinitesimally short period of time when measured by evolutionary standards, so any systemic adjustment or adaptation to its impact has been almost impossible to accommodate.

Of the annual world energy consumption, which is up by a factor of almost two from its value in 1980, roughly one third goes to waste. For example, only about 20 percent of the energy in gasoline is actually used to keep a car moving. A major role of innovation is to decrease such inefficiencies by refining extant technologies, inventing new ones, or developing new ways of organizing their uses.

rom a scientific perspective the truly revolutionary character of the Industrial Revolution was the dramatic change from an open system where energy is supplied externally by the sun to a closed system where energy is supplied internally by fossil fuel.

We need to think quantitatively and to develop the underlying science that is needed for addressing these challenges so as to inform rational political decisions.

Preach

In addition, they age, and in the case of companies, almost all of them eventually die, whereas for cities only extremely few ever do, an enigma we shall come to consider

This suggests that despite appearances, cities might also be approximate scaled versions of one another in much the same way that mammals are.

This may seem obvious, but the emphasis of those who think about cities, such as planners, architects, economists, politicians, and policy makers, is primarily focused on their physicality rather than on the people who inhabit them and how they interact with one another. It is all too often forgotten that the whole point of a city is to bring people together, to facilitate interaction, and thereby to create ideas and wealth, to enhance innovative thinking and encourage entrepreneurship and cultural activity by taking advantage of the extraordinary opportunities that the diversity of a great city offers. This is the magic formula that we discovered ten thousand years ago when we inadvertently began the process of urbanization.

Singapore may not be the world’s most exciting city, but its ambience of greenness is palpable.

And most expensive

Here’s the problem: cities do indeed evolve, but they take many decades to change and we simply no longer have the time to wait. It took 150 years for Washington, 100 years for London, and more than 50 years for Brasilia, which is still very much a work in progress.

China has embarked on the daunting task of constructing hundreds of new cities to urbanize 300 million rural residents. Out of expediency, these are being built without any deep understanding of the complexity of cities and their connection to socioeconomic success. Indeed, most commentators report that many of these new cities, like classic suburbs, are soulless ghost towns with little sense of community. Cities have an organic quality. They evolve and physically grow out of interactions between people.

Maybe you can "plant" things in a new city to help it grow organically

Science at its best is the search for commonalities, regularities, principles, and universalities that transcend and underlie the structure and behavior of any particular individual constituent, whether it be a quark, a galaxy, an electron, a cell, an airplane, a computer, a person, or a city. And it is at its very best when it can do that in a quantitative, mathematically computational, predictive framework, as is the case for electrons, airplanes, and computers, for instance.

Theme

Furthermore, the slope of the straight line, which is the exponent of the power law, is about 0.85, a little bit higher than the 0.75 (the famous ¾) we saw for the metabolic rate of organisms (Figure 1). Equally intriguing is that this exponent takes on approximately the same value for how gasoline stations scale across all of the countries shown in the figure.

This value of around 0.85 is smaller than 1, so in the language developed earlier, the scaling is sublinear, indicating a systematic economy of scale, meaning that the bigger the city the fewer the number of gas stations needed on a per capita basis.

only about 85 percent more material infrastructure is needed with every doubling of city size.2 Thus a city of 10 million people typically needs 15 percent less of the same infrastructure compared with two cities of 5 million each, leading to significant savings

To repeat: cities are an emergent self-organizing phenomenon that has resulted from the interaction and communication between human beings exchanging energy, resources, and information.

I have already strongly hinted at the answer: the great commonality is the universality of social network structures across the globe. Cities are people, and to a large extent people are pretty much the same all over the world in how they interact with one another and how they cluster to form groups and communities.

Ants brilliantly self-organized to evolve remarkably robust and hugely successful and sophisticated physical and social structures, but it took them millions of years to do so. Furthermore, they accomplished this more than 50 million years ago and have barely evolved beyond it since.

properties of biological networks that underlie quarter-power allometric scaling are: (1) they are space filling (so every cell of an organism, for instance, must be serviced by the network); (2) the terminal units, such as capillaries or cells, are invariant within a given design (so, for instance, our cells and capillaries are approximately the same as those of mice and whales); and (3) the networks have evolved to be approximately optimal (so, for instance, the energy our hearts have to use to circulate blood and support our cells is minimized in order to maximize the energy available for reproduction and the rearing of offspring).

Thanks for the recap

Nevertheless, it seems that without the motive of self-interest our entrepreneurial free market economy would collapse.

The actual self-similarity of cities more closely reflects the organically evolved hierarchical network structures of transport and utility systems than the rigid hexagonal crystalline structures of Christaller. The city is not a top-down engineered machine dominated by straight lines and classic Euclidean geometry, but rather is much more akin to an organism with its crinkly lines and fractal-like

These arise in cities by much the same process as they do in organisms in that they are the consequence of the generic properties of optimized, space-filling transport networks that constrain how energy and resources are supplied to all parts of the city.