So, here's a thing. If you double the size of a city, then you don't quite double its infrastructure. If a city of 1m expands to become one of 2m, then the number of, say, petrol stations it has per person will fall by approximately 15 per cent.

This rule of thumb carries over to all sorts of other stuff: the size of the road network, the length of its electrical cables, the size of its sewer system... The more people you have, the less of any given resource each of them will need.

You can probably get your head round this fairly easily: we're all familiar with economies of scale. But bear with me, because it's about to get weirder.

That's because there are other things that increase as a city grows – and the magic number this time is exactly the same. If you double the size of a city, the average wages will increase by approximately 15 per cent. So will the number of patents logged. So will crime. So will energy consumption. So will the number of AIDS cases. For a city, it turns out, growth is kind of a good news/bad news deal.

“The problem of the 21st century is urbanisastion”

All this is the contention of Professor Geoffrey West: an English-born theoretical physicist, now working at the Santa Fe Institute in New Mexico. A few years ago he turned his attention to the urban world, set up a team to gather numbers on cities all over the world, and started crunching the hell out of them. His goal, he explained in a 2011 TED talk, was nothing less than universally applicable, grand unified theory:

"My provocative statement is that we desperately need a serious scientific theory of cities. [That] means quantifiable, relying on underlying generic principles that can be made into a predictive framework. That's the quest."

Most of West's research has involved elementary particles, and even though his attention has long since moved onto subjects as diverse as cities, corporations and biology, he says that he still thinks of himself as a physicist: he just wants to bring that paradigm to more complex systems. Physics, he tells me, “is about simplicity. You can encapsulate enormous amounts of data and concepts – the origins of the universe, a theory of everything – in a single equation".

A slide from West's Ted talk, showing how employment and wages increase with population.

Cities are a bit more complicated than particles of course, subject to social and political forces that are entirely independent of straightforward metrics like population size. (Even the latter is contested: West himself bemoans the fact that there isn't a single, universal definition of how to define their limits.) As a result, any two cities of similar size, but which have had radically different histories – London and Moscow, say, or New York and Jakarta – will look radically different, too.

But West's contention was that it ought to be possible to understand and predict the characteristics of a city in at least broad strokes. Even if there’s no way to predict when any given individual will die, "we ought to be able to understand why the scale of a human lifespan is 100 years". By the same token, while you can't look at London and predict the characteristics of Moscow, you should be able to use the two cities’ relative sizes to make some educated guesses about Manchester.

When he started this line of research, one of West’s questions was whether cities behaved according to biological rules: “Are cities part of biology?" he asked in his TED talk. “Is London a great big whale?"

As it turns out, though, things are rather more complicated than that. Animals obey what are known as “sub linear relationships”: if you double the size of an organism, you only need 75 per cent more energy to run it. Pound for pound, elephants are more efficient than mice, so have lower metabolic rates and longer lifespans. (West has studied this, too.)

Cities obey these rules when it comes to their infrastructure. But they invert them when it comes to social factors like wages, patents, crime and so forth, where instead they obey super linear relationships: the bigger something is, the more per capita it has. "The question you have to ask,” West says, “is why should that be when urban systems evolved independently.”

His answer is that it’s because of the thing they have in common: human beings, and the network effect you get from putting a lot of them in one place. "The bigger you are, the more interactions take place, the more we talk, and the more we can create more wealth, new ideas, and so on.”

West's research has come in for criticism. Not every animal has been kind enough to obey his laws (the crayfish stubbornly refuses). The same is true of cities, and it isn't always a matter of diverse national histories, either. Over the last couple of decades, the US has seen a lot of growth shift to the suburbs, with much of the real economic innovation coming not from highly networked and high energy places like New York or LA, but from leafy low density areas like Silicon Valley.

West’s response is that his work is simply a first draft, something he expects others to refine and build on. "Overall the response has been very positive. The only serious negative response has come from economists. But I don't think of economics as a serious science, I think they were just threatened."

Anyway, he says, it’s better to have a flawed theory than no theory at all – and coming up with one is a matter of some urgency. The one thing that stops a superlinear growth curve – and the reason we wouldn’t be better off living in one planet-wide city of 8bn people – is the limitation of natural resources. The bigger cities get, the higher living standards are, and the more we consume. (There is some evidence contradicting this in specific contexts - New Yorkers burn less petrol than the residents of Phoenix, Arizona – but it does seem, globally, to hold true.) That means that, once in a while, we run into a cliff edge: you simply can't get any bigger at current resource levels.

Historically we've got around that through technological innovation. But the more energy hungry we get, the faster those innovations have to come to keep things moving. In other words, West told TED, "We're not only on a treadmill that's going faster: we have to change the treadmill faster and faster. The question is, can we as socio-economic beings avoid a heart attack?"

The grand unified theory of cities isn't going to solve that all by itself. But it could make a contribution, by offering some clues as to how we maximise growth while minimising energy usage, West says. "The purpose of urban planning is finding a way to minimize our distress while maximising our interactions."

Previous attempts to begin new cities from scratch – from Canberra to Milton Keynes to Brasilia – have initially been a failure. It’s only after half a century or more of “iteration and organic growth that they began to take on a certain well-being,” West notes. Today, though, we don't have 50 years: around a million people are estimated to be moving to cities every week, and China is aiming to build hundreds of new settlements over the next couple of decades.

"The problem of the 21st century is urbanisation," West concludes. "Climate change is actually a supplementary problem to understanding cities.” You wouldn’t build an aeroplane without doing some science, he notes – but when it comes to our urban habitats, “that is pretty much what we've done in the past". If we're to survive the 21st century, it might be time for a change in approach.

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