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(Illustration by Traci Daberko)

Life in the City Is Essentially One Giant Math Problem

Experts in the emerging field of quantitative urbanism believe that many aspects of modern cities can be reduced to mathematical formulas

Whitney heads uptown, to Central Park, where he walks on a path that for the most part skirts the hills and declivities created by the most recent glaciation and improved by Olmsted and Vaux. On a certain class of continuous surfaces—of which parkland is one—you can always find a path that stays on one level. From various points in Midtown, the Empire State Building appears and disappears behind the interposing structures. This brings to mind a theory Whitney has about the height of skyscrapers. Obviously big cities have more tall buildings than small cities, but the height of the tallest building in a metropolis doesn’t bear a strong relationship to its population; based on a sample of 46 metropolitan areas around the world, Whitney has found that it tracks the economy of the region, approximating the equation H=134 + 0.5(G), where H is the height of the tallest building in meters, and G is the Gross Regional Product, in billions of dollars. But building heights are constrained by engineering , while there’s no limit to how big a pile you can make out of money, so there are two very rich cities whose tallest towers are lower than the formula would predict. They are New York and Tokyo. Also, his equation has no term for “national pride,” so there are a few outliers in the other direction, cities whose reach toward the sky exceeds their grasp of GDP: Dubai, Kuala Lumpur.

No city exists in pure Euclidean space; geometry always interacts with geography and climate, and with social, economic and political factors. In Sunbelt metropolises such as Phoenix, other things being equal the more desirable suburbs are to the east of downtown, where you can commute both ways with the sun behind you as you drive. But where there is a prevailing wind, the best place to live is (or was, in the era before pollution controls) upwind of the city center, which in London means to the west. Deep mathematical principles underlie even such seemingly random and historically contingent facts as the distribution of the sizes of cities within a country. There is, typically, one largest city, whose population is twice that of the second-largest, and three times the third-largest, and increasing numbers of smaller cities whose sizes also fall into a predictable pattern. This principle is known as Zipf’s law, which applies across a wide range of phenomena. (Among other unrelated phenomena, it predicts how incomes are distributed across the economy and the frequency of the appearance of words in a book.) And the rule holds true even though individual cities move up and down in the rankings all the time—St. Louis, Cleveland and Baltimore, all in the top 10 a century ago, making way for San Diego, Houston and Phoenix.

As West and his colleagues are well aware, this research takes place against the background of a huge demographic shift, the predicted movement of literally billions of people to cities in the developing world over the next half century. Many of them are going to end up in slums—a word that describes, without judgment, informal settlements on the outskirts of cities, generally inhabited by squatters with limited or no government services. “No one has done a serious scientific study of these communities,” West says. “How many people live in how many structures of how many square feet? What is their economy? The data we do have, from governments, is often worthless. In the first set we got from China, they reported no murders. So you throw that out, but what are you left with?”

To answer those questions, the Santa Fe Institute, with backing from the Gates Foundation, has begun a partnership with Slum Dwellers International, a network of community organizations based in Cape Town, South Africa. The plan is to analyze the data gathered from 7,000 settlements in cities such as Mumbai, Nairobi and Bangalore, and begin the work of developing a mathematical model for these places, and a path toward integrating them into the modern economy. “For a long time, policy makers have assumed it’s a bad thing for cities to keep getting larger,” says Lobo. “You hear things like, ‘Mexico City has grown like a cancer.’ A lot of money and effort has been devoted to stemming this, and by and large it has failed miserably. Mexico City is bigger than it was ten years ago. So we think policy makers should worry instead about making those cities more livable. Without glorifying the conditions in these places, we think they’re here to stay and we think they hold opportunities for the people who live there.”

And one had better hope he is right, if Batty is correct in predicting that by the end of the century, practically the entire population of the world will live in what amounts to “a completely global entity...in which it will be impossible to consider any individual city separately from its neighbors...indeed perhaps from any other city.” We are seeing now, in Bettencourt’s words, “the last big wave of urbanization that we will experience on Earth.” Urbanization gave the world Athens and Paris, but also the chaos of Mumbai and the poverty of Dickens’ London. If there’s a formula for assuring that we are headed for one rather than the other, West, Koonin, Batty and their colleagues are hoping to be the ones to find it.

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