This straightforward observation has some surprising implications. It suggests, for instance, that modern cities are the real centers of sustainability. According to the data, people who live in densely populated places require less heat in the winter and need fewer miles of asphalt per capita. (A recent analysis by economists at Harvard and U.C.L.A. demonstrated that the average Manhattanite emits 14,127 fewer pounds of carbon dioxide annually than someone living in the New York suburbs.) Small communities might look green, but they consume a disproportionate amount of everything. As a result, West argues, creating a more sustainable society will require our big cities to get even bigger. We need more megalopolises.

But a city is not just a frugal elephant; biological equations can’t entirely explain the growth of urban areas. While the first settlements in Mesopotamia might have helped people conserve scarce resources — irrigation networks meant more water for everyone — the concept of the city spread for an entirely different reason. “In retrospect, I was quite stupid,” West says. He was so excited by the parallels between cities and living things that he “didn’t pay enough attention to the ways in which urban areas and organisms are completely different.”

What Bettencourt and West failed to appreciate, at least at first, was that the value of modern cities has little to do with energy efficiency. As West puts it, “Nobody moves to New York to save money on their gas bill.” Why, then, do we put up with the indignities of the city? Why do we accept the failing schools and overpriced apartments, the bedbugs and the traffic?

In essence, they arrive at the sensible conclusion that cities are valuable because they facilitate human interactions, as people crammed into a few square miles exchange ideas and start collaborations. “If you ask people why they move to the city, they always give the same reasons,” West says. “They’ve come to get a job or follow their friends or to be at the center of a scene. That’s why we pay the high rent. Cities are all about the people, not the infrastructure.”

It’s when West switches the conversation from infrastructure to people that he brings up the work of Jane Jacobs, the urban activist and author of “The Death and Life of Great American Cities.” Jacobs was a fierce advocate for the preservation of small-scale neighborhoods, like Greenwich Village and the North End in Boston. The value of such urban areas, she said, is that they facilitate the free flow of information between city dwellers. To illustrate her point, Jacobs described her local stretch of Hudson Street in the Village. She compared the crowded sidewalk to a spontaneous “ballet,” filled with people from different walks of life. School kids on the stoops, gossiping homemakers, “business lunchers” on their way back to the office. While urban planners had long derided such neighborhoods for their inefficiencies — that’s why Robert Moses, the “master builder” of New York, wanted to build an eight-lane elevated highway through SoHo and the Village — Jacobs insisted that these casual exchanges were essential. She saw the city not as a mass of buildings but rather as a vessel of empty spaces, in which people interacted with other people. The city wasn’t a skyline — it was a dance.

If West’s basic idea was familiar, however, the evidence he provided for it was anything but. The challenge for Bettencourt and West was finding a way to quantify urban interactions. As usual, they began with reams of statistics. The first data set they analyzed was on the economic productivity of American cities, and it quickly became clear that their working hypothesis — like elephants, cities become more efficient as they get bigger — was profoundly incomplete. According to the data, whenever a city doubles in size, every measure of economic activity, from construction spending to the amount of bank deposits, increases by approximately 15 percent per capita. It doesn’t matter how big the city is; the law remains the same. “This remarkable equation is why people move to the big city,” West says. “Because you can take the same person, and if you just move them to a city that’s twice as big, then all of a sudden they’ll do 15 percent more of everything that we can measure.” While Jacobs could only speculate on the value of our urban interactions, West insists that he has found a way to “scientifically confirm” her conjectures. “One of my favorite compliments is when people come up to me and say, ‘You have done what Jane Jacobs would have done, if only she could do mathematics,’ ” West says. “What the data clearly shows, and what she was clever enough to anticipate, is that when people come together, they become much more productive.”

West illustrates the same concept by describing the Santa Fe Institute, an interdisciplinary research organization, where he and Bettencourt work. The institute itself is a sprawl of common areas, old couches and tiny offices; the coffee room is always the most crowded place. “S.F.I. is all about the chance encounters,” West says. “There are few planned meetings, just lots of unplanned conversations. It’s like a little city that way.” The previous evening, West and I ran into the novelist Cormac McCarthy at the institute, where McCarthy often works. The physicist and the novelist ended up talking about Antarctic icefish, the editing process and convergent evolution for 45 minutes. Of course, these interpersonal collisions — the human friction of a crowded space — can also feel unpleasant. We don’t always want to talk with strangers on the subway or jostle with people on the sidewalk. West admits that all successful cities are a little uncomfortable. He describes the purpose of urban planning as finding a way to minimize our distress while maximizing our interactions. The residents of Hudson Street, after all, didn’t seem to mind mingling with one another on the sidewalk. As Jacobs pointed out, the layout of her Manhattan neighborhood — the short blocks, the mixed-use zoning, the density of brownstones — made it easier to cope with the strain of the metropolis. It’s fitting that it’s called the Village.