One implication is that, like the elephant and the mouse, “big cities are not just bigger small cities,” says Michael Batty, who runs the Centre for Advanced Spatial Analysis at University College London. “If you think of cities in terms of potential interactions [among individuals], as they get bigger you get more opportunities for that, which amounts to a qualitative change.” Consider the New York Stock Exchange as a microcosm of a metropolis. In its early years, investors were few and trades sporadic, Whitney says. Hence “specialists” were needed, intermediaries who kept an inventory of stock in certain companies, and would “make a market” in the shares, pocketing the margin between their selling and buying price. But over time, as more participants joined the market, buyers and sellers could find one another more easily, and the need for specialists—and their profits, which amounted to a small tax on everyone else—diminished. There is a point, Whitney says, at which a system—a market, or a city—undergoes a phase shift and reorganizes itself in a more efficient and productive way.
Whitney, who has a slight build and a meticulous manner, 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.” (This assumes that the line closes at a certain time so everyone gets served eventually. The equations can’t handle the concept of an infinite wait.) That idea “seems intuitive,” says Whitney, “but it had to be proved.” In the real world, queuing theory is used for designing communications networks, in deciding which packet of data gets sent first.
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. You might think, he says, that an express should always leave as soon as it’s ready, but there are times when it makes sense to hold it in the station—to make a connection with an incoming local. 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. (You’ll have to model how far passengers who bother to switch are going.) 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 museum.
It also can be readily seen that the more data you have on transit usage (or hamburger orders), the more detailed and accurate you can make these calculations. If Bettencourt and West are building a theoretical science of urbanism, then Steven Koonin, the first director of New York University’s newly created Center for Urban Science and Progress, intends to be in the forefront of applying it to real-world problems. Koonin, as it happens, is also a physicist, a former Cal Tech professor and assistant secretary of the Department of Energy. He describes his ideal student, when CUSP begins its first academic year this fall, as “someone who helped find the Higgs boson and now wants to do something with her life that will make society better.” Koonin is a believer in what is sometimes called Big Data, the bigger the better. Only in the past decade has the ability to collect and analyze information about the movement of people begun to catch up to the size and complexity of the modern metropolis itself. Around the time he took the job at CUSP, Koonin read a paper about the ebb and flow of population in Manhattan’s business district, based on an exhaustive analysis of published data on employment, transit and traffic patterns. It was a great piece of research, Koonin says, but in the future, that’s not how it will be done. “People carry tracking devices in their pockets all day long,” he says. “They’re called cellphones. You don’t need to wait for some agency to publish statistics from two years ago. You can get this data almost in real time, block by block, hour by hour.
“We have acquired the technology to know virtually anything that goes on in an urban society,” he adds, “so the question is, how can we leverage that to do good? Make the city run better, enhance security and safety and promote the private sector?” Here’s a simple example of what Koonin envisions, in the near future. If you are, say, deciding whether to drive or take the subway from Brooklyn to Yankee Stadium, you can consult a website for real-time transit data, and another for traffic. Then you can make a choice based on intuition, and your personal feelings about the trade-offs among speed, economy and convenience. This by itself would have seemed miraculous even a few years ago. Now imagine a single app that would have access to that data (plus GPS locations of taxis and buses along the route, cameras surveying the stadium’s parking lots and Twitter feeds from people stuck on FDR Drive), factor in your preferences and tell you instantly: Stay home and watch the game on TV.
Or some slightly less simple examples of how Big Data can be used. At a lecture last year Koonin presented an image of a large swath of Lower Manhattan, showing the windows of some 50,000 offices and apartments. It was taken with an infrared camera, and so could be used for environmental surveillance, identifying buildings, or even individual units, that were leaking heat and wasting energy. Another example: As you move around the city, your cellphone tracks your location and that of everyone you come into contact with. Koonin asks: How would you like to get a text message telling you that yesterday you were in a room with someone who just checked into the emergency room with the flu?
Inside the Museum of Mathematics, kids and the occasional adult manipulate various solids on a series of screens, rotating them, extending or compressing or twisting them into fantastical shapes, then extruding them in plastic on a 3-D printer. They sit inside a tall cylinder whose base is a rotating platform and whose sides are defined by vertical strings; as they twist the platform, the cylinder deforms into a hyperboloid, a curved surface that somehow is created out of straight lines. Or they demonstrate how it is possible to have a smooth ride on a square-wheeled tricycle, if you contour the track beneath it to keep the axle level. Geometry, unlike formal logic, which was Whitney’s field before he went to Wall Street, lends itself particularly well to hands-on experiment and demonstration—although there are also exhibits touching on fields he identifies as “calculus, calculus of variations, differential equations, combinatorics, graph theory, mathematical optics, symmetry and group theory, statistics and probability, algebra, matrix analysis—and arithmetic.” It troubled Whitney that in a world with museums devoted to ramen noodles, ventriloquism, lawn mowers and pencils, “most of the world has never seen the raw beauty and adventure that is the world of mathematics.” That’s what he set out to remedy.
As Whitney points out on the popular math tours he runs, the city has a distinctive geometry, which can be described as occupying two-and-a-half dimensions. Two of these are those you see on the map. He describes the half-dimension as the network of elevated and underground walkways, roads and tunnels that can only be accessed at specific points, like the High Line, an abandoned railroad trestle that has been turned into an elevated linear park. This space is analogous to an electronic printed-circuit board, in which, as mathematicians have shown, certain configurations cannot be achieved in a single plane. The proof is in the famous “three-utilities puzzle,” a demonstration of the impossibility of routing gas, water and electric service to three houses without any of the lines crossing. (You can see this for yourself by drawing three boxes and three circles, and trying to connect each circle to each box with nine lines that do not intersect.) In a circuit board, for conductors to cross without touching, one of them sometimes must leave the plane. Just so, in the city, sometimes you have to climb up or down to get to where you are going.