Scale: The Universal Laws of Life and Death in Organisms, Cities and Companies

by Geoffrey West

strength of pillars, beams, and limbs is determined by the size of their cross-sectional areas and not by how long they are.

Thus a post whose rectangular cross-section is 2 inches by 4 inches (= 8 sq. in.) can support four times the weight of a similar post of the same material whose cross-sectional dimensions are only half as big, namely 1 inch by 2 inches (= 2 sq. in.), regardless of the length of either post.

That’s why builders, architects, and engineers involved in construction classify wood by its cro...

Now, as we scale up a building or an animal, their weights increase in direct proportion to their volumes provided, of course, that the materials they’re made of don’t change so that their densities remain the same: so doubling the volume doubles the weight.

Thus, the weight being supported by a pillar or a limb increases much faster than the corresponding increase in strength, because weight (like volume) scales as the cube of the linear dimensions whereas strength increases only as the square.

if the size of the structure, whatever it is, is arbitrarily increased it will eventually collapse under its own weight. There are limits to size and growth.

relative strength becomes progressively weaker as size increases.

“the smaller the body the greater its rel...

Galileo, relative strength systematically increases as size decreases.

for a single order of magnitude increase in area, volumes increase by 3/2 (that

the weight is increased by a single order of magnitude, then the strength only increases by 2⁄3 of an order of magnitude.

The Richter scale actually measures the “shaking” amplitude of the earthquake recorded on a seismometer. The

corresponding amount of energy released scales nonlinearly with this amplitude in such a way that for every order of magnitude increase in the measured amplitude the energy released increases by one and a half (that is 3⁄2) orders of magnitude.

factors involved, metabolic rate plays an important role. Drugs, like metabolites and oxygen, are typically transported across surface membranes, sometimes via diffusion and sometimes through network systems.

As a result, the dose-determining factor is to a significant degree constrained by the scaling of surface areas rather than the total volume or weight of an organism, and these scale nonlinearly with weight.

simple calculation using the 2⁄3 scaling rule for areas as a function of weight shows that a more appropriate dose for elephants should be closer to a few milligrams of LSD rather than the several hundred that were actually admi...

laws, mathematical analyses, and mechanistic explanations. Quetelet’s body-mass index is defined as your body weight divided by the square of your height

Quetelet’s body-mass index is defined as your body weight divided by the square of your height and is therefore expressed in terms of pounds per square inch

The idea behind it is that weights of healthy individuals, and in particular those with a “normal” body shape and proportion of body fat, are presumed ...

BMI is defined as body weight divided by the cube of the height;

innovation, must occur in order to continue growing and avoid collapse.

Growth and the continual need to be adapting to the challenges of new or changing environments, often in the form of “improvement” or increasing efficiency, are major drivers of innovation.

chemical composition, as it changes size. Consequently, in order to build larger structures or evolve larger organisms beyond the limits set by the scaling laws, innovations must occur that either change the material composition of the system or its structural design, or both.

A simple example of the first kind of innovation is to use a stronger material such as steel in place of wood for bridges or buildings, while a simple example of the second kind is to use arches, vaults, or domes in their construction rather than just horizontal beams and vertical pillars.

Given the limited tensile strength of wood there is clearly a limit as to how long a span can be traversed in this way. This was solved by the simple design innovation of introducing stone support piers in the middle of the river that effectively extended the bridge to be a succession of individual beam bridges.

This way of thinking about innovation, which relates it to the drive or need to grow bigger, to expand horizons and compete in ever-larger markets with its inevitable confrontation with potential limitations

Failure and catastrophe can provide a huge impetus and opportunity in stimulating innovation, new ideas, and inventions whether in science, engineering, finance, politics, or one’s personal life.

Isambard Kingdom Brunel, and why is he famous? Many consider him the greatest engineer of the nineteenth century, a man whose vision and innovations, particularly concerning transport, helped make Britain the most powerful and richest nation in the world.

However, it was widely believed that a ship powered purely by steam would not be able to carry enough fuel for the trip and still have room for sufficient commercial cargo to be economically viable.

He realized that the volume of cargo a ship could carry increases as the cube of its dimensions (like its weight), whereas the strength of the drag forces it experiences as it travels through water increases as the cross-sectional area of its hull and therefore only as the square of its dimensions. This is just like Galileo’s conclusions for how the strength of beams and limbs scale with body weight. In both cases the strength increases more slowly than the corresponding weight following a 2⁄3 power scaling law. Thus the strength of the hydrodynamic drag forces on a ship relative to the weight of the cargo it can carry decreases in direct proportion to the length of the ship. Or to put it the other way around: the weight of its cargo relative to the drag forces its engines need to overcome systematically increases the bigger the ship. In other words, a larger ship requires proportionately less fuel to transport each ton of cargo than a smaller ship.

Bigger ships are therefore more energy efficient and cost effective than smaller ones—another great example of an economy of scale and one that had enormous consequences for the development of world trade and commerce.12

the most salient being in the development of steam engines where understanding the relationship between pressure, temperature, and volume of steam helped advance the design of very large, efficient boilers, the kind that allowed engineers to contemplate building great ships the size of the Great Eastern that could sail across the globe. More significant, investigations into understanding

and it is very likely that you’ve never heard of the Navier-Stokes equation, but it has played, and continues to play, a major role in almost all aspects of your life. Among many other things, it underlies the design of airplanes, automobiles, hydroelectric power stations, and artificial hearts, the understanding of blood flow in your circulatory system, and the hydrology of rivers and water supply systems.

It is fundamental to understanding and predicting the weather, ocean currents, and the analysis of pollution and is consequently a key ingredient in the science of climate change and the predictions of global warming.

the nonlinearity arises from feedback mechanisms in which water interacts with itself.

square of the ship’s velocity divided by its length multiplied by the acceleration due to gravity.

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.

most famous dimensionless number is pi (π), the ratio of the circumference of a circle to its diameter.

as a ship moves through water it continually creates wakes and surface waves whose behavior is constrained by the pull of gravity—in fact, the waves you are familiar with on oceans and lakes are technically called gravity waves. So indirectly gravity plays an important role in the motion of ships.

intensity of light waves scattered by small particles must decrease with the fourth power of their wavelength. Thus, when sunlight, which is a combination of all of the colors of the rainbow, scatters from microscopic particles suspended in the atmosphere, the shortest wavelengths, corresponding to blue light, dominate.

life uses essentially the same basic building blocks and processes to create an amazing variety of forms, functions, and dynamical behaviors. This is a profound testament to the power of natural selection and evolutionary dynamics. All of life functions by transforming energy from physical

chemical sources into organic molecules that are metabolized to build, maintain, and reproduce complex, highly organized systems. This is accomplished by the operation of two distinct but closely interacting systems: the genetic code, which stores and processes the information and “instructions” to build and maintain the organism, and the metabolic system, which acquires, transforms, and allocates energy and materials for maintenance, growth, and reproduction.

The search for fundamental principles that govern how the complexity of life emerges from its underlying simplicity is one of the grand challenges of twenty-first-century science.

at every organizational level average idealized biological systems can be constructed whose general properties are calculable.

Thus we ought to be able to calculate the average and maximum life span of human beings even if we’ll never be able to calculate our own. This provides a point of departure or baseline for quantitatively understanding actual biosystems, which can be viewed as variations or perturbations around idealized norms due to local environmental conditions or historical evolutionary divergence.

“high energy physics” is the name of the subfield of physics concerned with fundamental questions about the elementary particles, their interactions and cosmological implications.

It’s life’s change agent. It clears out the old to make way for the new.

biologist Sir D’Arcy Wentworth Thompson in his classic book On Growth and Form, published in

and “die,” much like cars and washing machines. However, to understand how something ages and dies, whether an animal, an automobile, a company, or a civilization, one first needs to understand what the processes and mechanisms are that are keeping it alive, and then discern how these become degraded with time.

Metabolism is the fire of life . . . and food, the fuel of life. Neither the neurons in your brain nor the molecules of your genes could function without being supplied by metabolic energy extracted from the food you eat. You could not walk, think, or even sleep without being supplied by metabolic energy. It supplies the power organisms need for maintenance, growth, and reproduction, and for specific processes such as circulation, muscle contraction, and nerve conduction.