The systematic study of cities dates back at least to the Greek historian Herodotus. In the early 20th century, scientific disciplines emerged around specific aspects of urban development: zoning theory, public health and sanitation, transit and traffic engineering. By the 1960s, the urban-planning writers Jane Jacobs and William H. Whyte used New York as their laboratory to study the street life of neighborhoods, the walking patterns of Midtown pedestrians, the way people gathered and sat in open spaces. But their judgments were generally aesthetic and intuitive … “*They had fascinating ideas*,” says Luís Bettencourt, a researcher at the Santa Fe Institute, a think tank better known for its contributions to theoretical physics, “*but where is the science?*” Bettencourt, a physicist, practices a discipline that shares a deep affinity with quantitative urbanism. Both require understanding complex interactions among large numbers of entities: the 20 million people in the New York metropolitan area, or the countless subatomic particles in a nuclear reaction.

The birth of this new field can be dated to 2003, when researchers at SFI convened a workshop on ways to “model”—in the scientific sense of reducing to equations—aspects of human society. One of the leaders was Geoffrey West, who was also a theoretical physicist, but had strayed into biology, exploring how the properties of organisms relate to their mass.

An elephant is not just a bigger version of a mouse, but many of its measurable characteristics, such as metabolism and life span, are governed by mathematical laws that apply all up and down the scale of sizes. The bigger the animal, the longer but the slower it lives: **A mouse heart rate is around 500 beats per minute; an elephant’s pulse is 28.** If you plotted those points on a logarithmic graph, comparing size with pulse, every mammal would fall on or near the same line. West suggested that the same principles might be at work in human institutions.

Out of that meeting emerged a collaboration that produced the seminal paper in the field: “*Growth, Innovation, Scaling, and the Pace of Life in Cities*.” In six pages dense with equations and graphs, West, Lobo and Bettencourt, along with two researchers from the Dresden University of Technology, laid out a theory about how cities vary according to size.

The relationship is captured by an equation in which a given parameter—employment, say—varies exponentially with population. In some cases, the exponent (power) is **1**, meaning whatever is being measured increases linearly, at the same rate as population. Household water or electrical use, for example, shows this pattern; as a city grows bigger its residents don’t use their appliances more. Some exponents are greater than 1, a relationship described as “superlinear scaling.” Most measures of economic activity fall into this category; among the highest exponents the scholars found were for “private [research and development] employment,” **1.34**; “new patents,” **1.27**; and gross domestic product, in a range of **1.13 to 1.26**. If the population of a city doubles over time, or comparing one big city with two cities each half the size, gross domestic product more than doubles. Each individual becomes, on average, 15 percent more productive. Bettencourt describes the effect as “*slightly magical*,” although he and his colleagues are beginning to understand the synergies that make it possible. Physical proximity promotes collaboration and innovation, which is one reason the new CEO of Yahoo recently reversed the company’s policy of letting almost anyone work from home.

Lastly, some measures show an exponent of less than 1, meaning they increase more slowly than population. These are typically measures of infrastructure, characterized by economies of scale that result from increasing size and density. New York doesn’t need four times as many gas stations as Houston, for instance; gas stations scale at **0.77**; total surface area of roads, **0.83**; and total length of wiring in the electrical grid, **0.87**.

Remarkably, this phenomenon applies to cities all over the world, of different sizes, regardless of their particular history, culture or geography. Mumbai is different from Shanghai is different from Houston, obviously, but in relation to their own pasts, and to other cities in India, China or the U.S., they follow these laws. “*Give me the size of a city in the United States and I can tell you how many police it has, how many patents, how many AIDS cases*,” says West, “*just as you can calculate the life span of a mammal from its body mass.*”

Whitney walks swiftly through Madison Square Park to the Shake Shack, a hamburger stand famous for its food and its lines. He points out the two service windows, one for customers who can be served quickly, the other for more complicated orders. This distinction is supported by a branch of mathematics called queuing theory, whose fundamental principle can be stated as “*the shortest aggregate waiting time for all customers is achieved when the person with the shortest expected wait time is served first, provided the guy who wants four hamburgers with different toppings doesn’t go berserk when he keeps getting sent to the back of the line.*”

At the Times Square subway station, Whitney buys a fare card, in an amount he has calculated to take advantage of the bonus for paying in advance and come out with an even number of rides, with no money left unspent. On the platform, as passengers rush back and forth between trains, he talks about the mathematics of running a transit system … The calculation, simplified, is this:

*Multiply the number of people on the express train by the number of seconds they will be kept waiting while it idles in the station. Now estimate how many people on the arriving local will transfer, and multiply that by the average time they will save by taking the express to their destination rather than the local.*

This can lead to the potential savings, in person-seconds, for comparison. The principle is the same at any scale, but it is only above a certain size of population that the investment in dual-track subway lines or two-window hamburger stands makes sense. Whitney boards the local, heading downtown toward the (Math) museum.