A physicist explains why we've never been able to come up with a proper metaphor.

Practically since the dawn of cities, people have been trying to define them. Cities are like machines that can be optimized! Or insect colonies in social structure! Or ecosystems in biology! When humans cluster together in dense settlements, it’s clear that we collectively create some kind of dynamic thing that is capable – apart from all of our individual effort – of evolving and producing creative and economic outputs that simply aren’t found on the farm. But how do we conceptualize that thing?

“People have thought about cities just about every possible way since the Greeks,” says Luis Bettencourt, a physicist with the Santa Fe Institute, the scientific research and education center. “It’s an incredibly rich topic, and I think the curious thing is that we’ve tried to throw every analogy at it. But there’s a sense in which most of them fail somewhere. And to me, what that always meant was that we needed to discover better what cities really are.”

You may by now be familiar with some of the mind-bending findings that have come out of the Santa Fe Institute, from Bettencourt and his colleague Geoffrey West and others. Over the last several years, they’ve been processing data that never existed before on all kinds of quantifiable characteristics of cities: the length of road networks, the average income of inhabitants, the number of patents per capita. And they’ve found that all of these characteristics scale with city size: As population grows, so do all of these other factors, in remarkably predictable ways across the globe and throughout time.

Pre-Columbian cities in the Basin of Mexico exhibited these exact same patterns 2,000 years ago. And older European cities that developed before the advent of urban planning did as well. Those cities, Bettencourt says, remain universally beautiful to us today because they evolved according to the same basic principles that underlie urban development today.

But if complex cities everywhere – even if they look different, from Phoenix to Montreal to Rome – seem to mathematically obey some fairly simple and universal parameters, how do we define that common thing that they are, if it's neither a machine nor an insect colony nor an ecosystem?

Bettencourt, in a paper published today in the journal Science, finally offers up an answer that borrows a bit from physics, economics, sociology, biology and a handful of other disparate reaches of science. We can never get the analogy quite right, he says, because cities are a thing that is found nowhere else in nature.

“We tend to look at things by the way they look, by form,” he says. And this is why most of our existing metaphors fail. “All the successful theories of science are not about form at all – they’re about function. They’re about how things develop, how things change. They’re about process.”

If we take that point seriously, he says, we have to ask what cities do, not what they look like, or even how they grow. At their most fundamental, cities are not really agglomerations of people; they’re agglomerations of connections between people. All of their other properties – the roads we build to reach each other, the density required to do that, the economic products and ideas we create together – derive from this fact.

Cities, Bettencourt has concluded, are a “special kind of social reactor.” And, as such, they all evolve according to a small set of basic principles that can be used to predict the average social, spatial and infrastructure properties of any metropolitan place. If you click through to the Science paper, Bettencourt casts all of these principles in mathematical equations (“for these ideas to be science,” he says, “it is important that they can be written in math”).

We’ll spare you the sigma signs and exponents. But Bettencourt is basically describing interconnected relationships between the population growth of a city; the incremental expansion of the infrastructure networks that more people require; the socioeconomic outputs that come from our social interaction; and the density that necessarily develops over time so that we can still benefit from ever-more social connections without spending ever-more energy to reach each other.

As cities grow, Bettencourt says, the city comes to you. This is a high-minded way of talking about infill development. If cities continued to grow but only grew outward, you would never get any benefits out of knowing or working with new people, since you'd have to sit in traffic for two hours to reach them. Density, however, allows us to reap the benefits of more social connections without adding too many costs in congestion and energy (like gas). All of this enables the amazing growth and benefits of cities to be open-ended.

Here, Bettencourt graphs the GDP of several actual cities compared to their population, to show what this open-ended relationship look like (GDP grows faster than population):

"The Origins of Scaling in Cities," by L.M.A. Bettencourt in Science. Bridgeport, CT (green circle), Riverside, CA (yellow circle), and Brownsville, TX (red circle) are shown in the inset.

And here is what the total length of road infrastructure looks like compared to population (new infrastructure grows slower than population):

"The Origins of Scaling in Cities," by L.M.A. Bettencourt in Science.

Bettencourt’s theoretical framework suggests that a kind of optimal city exists when we have the most social interaction – and social and economic output coming from it – with the least cost of connecting people and goods and ideas to each other. A sprawling city, for instance, isn’t reaching the full potential it could achieve if more people moved into town in denser development. Likewise, a dense but congested city loses some of the potential it could achieve with better transportation.

"The Origins of Scaling in Cities," by L.M.A. Bettencourt in Science.

The idea that cities are governed by some universal rules of math may make it sound like the urban planner has little control. But, in fact, Bettencourt sees the planner’s job to try to steer cities toward that optimal point (G*) on the above graph. Beyond that point, the number of social interactions in a city can still grow, but the cost of them rises faster than the benefit.

Ideally, as cities grow, all of this means that they should become even more productive, even more powerful. And in this way, at least, they are like one thing in nature. As stars compress matter, they burn brighter and faster the bigger they are. But stars can eventually run out of energy. They’re not open-ended. And they’re isolated systems, where cities rely on food and other resources from beyond their borders.

If the idea of a city as “social reactor” is still a bit too abstract for you, perhaps try this hybrid: “It’s part star, part network,” Bettencourt says. “But it’s really it’s own new thing, for which we don’t have a strict analogy anywhere else in nature.”

Top image: Gunnar Assmy/

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