Today we elaborate on our previous discussions here and here about the work of Geoffrey West, the lessons of universal scaling laws, and key considerations for building things that we hope will grow.
There was a time back in the 1990s when everywhere Geoffrey West looked, he saw fractions. Not just any fractions, but the same ones. The fraction 3/4 and its complementary fraction 1/4 seemed especially pervasive. They described how quickly animals grew relative to their mass and how their metabolic rates changed at each size interval. They described the rates of oxygen consumption, blood flow, daily food requirements, and other biological functions.
West wasn’t the first to notice these things. In the 1930s, the Swiss agricultural biologist Max Kleiber had formulated Kleiber’s Law, which quantified how an animal’s metabolic rate is proportional to its mass raised to the power of three-fourths. The law applied to tiny animals and giant animals and all the animals imaginable, to all the animals on Noah’s Ark and even to Noah himself.
West’s curiosity caused him to look for applications of Kleiber’s law beyond the animal kingdom. He found there are similar laws, including those with the same power fractions, that govern the attributes of cities, including GDP, wages, population density, infrastructure, patents, crime, and disease. If you told him the size of your city, he could predict its attributes. You could use West’s information like a multiplication table to determine if your city was above average or below average, and by how much. Very useful!
West identified that the same predictable scaling laws governed the corporate world too. You could look at the number of employees in your company and be able to identify the company’s expected growth rate, profitability, likelihood of going out of business, and other characteristics. Very helpful for corporate execs.
So what was it about these reappearing fractions, in particular 3/4 and 1/4? Why did they appear in the biological world as well as in the man-made worlds of city planning and corporate affairs? Was there something special about them? And, if so, what lessons could we learn?
In his work with scientists James Brown and Brian Enquist, West developed a theory to explain the quarter-power scaling phenomenon in nature. They alleged the internal networks that distribute resources throughout an animal’s body are structured with space-filling efficiencies that lead to predictable scaling relationships. As an animal grows larger, occupying more and more three-dimensional space, its internal networks grow in observance of what West described as a fourth dimension, a fractal dimension. That is why the quarter-power scaling laws are so widespread.
What is a fractal? A fractal is a complex geometrical shape that exhibits self-similarity. It is a shape that repeats itself at different scales. Fractals can be seen all through the physical world. A tree is a fractal, with each branch appearing like a small version of the tree, and each leaf comprised of the same tree-like structure. A cauliflower is a fractal, with each bulb appearing to be a smaller version of the whole. Rivers exhibit fractal properties, as do shorelines, snowflakes, ice crystals, and lightning bolts. The songs of birds and the chirps of insects follow fractal-like rhythms, as does the crackling of fire and the crashing of ocean waves.
So why would West have the audacity to propose that fractals be considered as their own separate dimension of space? After all, isn’t the physical size of the cauliflower just that, the three-dimensional size of the cauliflower? What’s the deal with this fourth dimension? If you were hired to measure the surface area of the cauliflower, and you had the right tools and made sure you were careful enough, couldn’t you just take your measurements, submit your answer, and collect your paycheck?
Well, as it turns out, the surface of the cauliflower actually cannot be measured exactly. The surface of the cauliflower can only be measured to within a certain degree of approximation. The measurement you get is related to the size of the ruler you use to take the measurements.
Now I don’t want to take us down a fractal rabbit hole, but let’s consider one more example of the measurement dilemma. Let’s say you wanted to measure the coastline of North Carolina. You could start by taking a photo from a satellite, calculate the scale, measure the orderly line that you see, and come up with your answer. The simplified answer is 301 miles, which is the straight-line measurement of the state across the Atlantic Ocean. A more detailed measurement of the many twists and turns of the coast, estuaries, and tidal areas would yield a measurement of 3,375 miles.
Of course, you could get more granular by taking many photos from a low-flying plane. These pictures would give you more detail than the single satellite photo, and you would see there were smaller variations in the coastline that you only see when looking more closely, and these variations would add some area to your initial calculation.
You could repeat the process again, this time walking the shoreline itself, observing more details that you didn’t see in the pics, adding more area to your measurement. In fact, this would show that in addition to twisting and turning in two-dimensional space, as seen from your aerial photos, the coastline also rises and falls in three-dimensional space, and this too adds area to your number.
It becomes pretty clear that as you look more and more closely, right down to the individual rocks and sandbars, pebbles and twigs and roots that collectively comprise the coastline, the calculated surface area keeps growing. You have now entered the fractal dimension.
Normally when you think about spatial dimensions — to the extent that you think about spatial dimensions at all — you think about reassuringly solid objects like Rubik’s cubes, billiard balls, mountains, planets, or whatever. Those things have three dimensions that are well understood.
If you are inclined towards abstract thinking, and perhaps if you know a little about physics, and especially if you ever saw episodes of Carl Sagan’s Cosmos series from the 80s (and if you didn’t, you should stop whatever you are doing and go binge-watch them right now!), you may be familiar with the concept of spacetime. Einstein developed the theoretical and mathematical frameworks for spacetime, which essentially categorize time as a coordinate that is intertwined with the three coordinates of space: length, width, and height.
What West et al proposed was different. They proposed that there is space within three-dimensional space — a fourth physical dimension — and this is the reason that the number four appears so pervasively in the power laws that explain the world around us.
So what’s the lesson?
Well, different people may come away with different lessons. Some may focus on the fractal aspect, recognize the futility of trying to measure things with precision. Since your measurement accuracy is dependent on the accuracy of your ruler, you have to be practical and settle for some level of approximation. “Don’t let perfect be the enemy of good.”
Some may see a metaphorical lesson, that nature demonstrates structures that work, structures that have been developed over millions of years of evolution, and you would do well to emulate them. In other words, “You don’t have to reinvent the wheel.”
Others may focus on the efficiency aspect, that you have to optimize resources relative to your size if you want to keep growing. That idea resonates for me, and it hints at the real lesson, but it still seems incomplete.
For me the real lesson is, if you’re going to build something that you will want to grow, then consider how its growth potential is related to the way you build it. Whether it’s a biological system, a company, a city, a software platform, or a social movement, the structure you build determines how efficiently resources flow through it and therefore how much it can grow.
There’s a reason the e-commerce company Etsy’s market cap is $6 billion and Amazon’s market cap is $2 trillion. Etsy specializes in handmade, vintage, and custom products for a niche market of customers that appreciate niche offerings. Even though Amazon also distributes many niche products, the majority of its business relates to mass-produced products for mass-market consumers. No matter how large Etsy grows, it will be constrained by some inherent growth inhibitors that don’t constrain Amazon.
Not that there’s anything wrong with being small. There isn’t. It’s just that small things cannot grow large if their internal structures aren’t scalable. To grow large, you need a structure designed for large growth.