Edward L. Glaeser is an economics professor at Harvard.

“Zipf’s Law ” is one of the great curiosities of urban research. The law claims that the number of people in a city is inversely proportional to the city’s rank among all cities. In other words, the biggest city is about twice the size of the second biggest city, three times the size of the third biggest city, and so forth.

Zipf’s Law is named after the linguist George Kingsley Zipf, who discovered the law when studying the distribution of words: the second most common word in a text typically shows up one-half as often as the most commonly used word. The law has been observed in many other contexts, including firm sizes and income distribution, which follows the closely-connected Pareto Distribution.

In the urban context, the law seems to have been discovered by a German geographer, Felix Auerbach, many decades before Zipf.

Zipf’s Law is intellectually exciting because it appears to be both powerful and mysterious. Why should such a simple relationship hold? Some very smart social scientists have been attracted to explaining why Zipf’s Law emerges. But before discussing their explanations, it is worthwhile descending into the data.

The figure below shows a plot of the population of the logarithm of 2009 metropolitan area populations against population rank. As the plot shows, the biggest and smallest metropolitan areas are somewhat too small to fit Zipf’s Law perfectly. A massive meta-analysis by Volker Nitsch finds that the city-size distributions are typically somewhat too even to be described perfectly by Zipf’s Law.

Edward L. Glaeser

These deviations from Zipf’s Law are modest relative to the challenge to Zipf issued by Thomas Holmes and Sanghoon Lee in the book “Agglomeration Economics,” which I edited and described in last week’s post.

Part of the appeal of Zipf’s Law is that it appears be entirely natural. As Steven Strogatz wrote in The New York Times about one year ago: “No city planner imposed it, and no citizens conspired to make it happen.” But Zipf’s Law seems to be mainly a product of city or metropolitan area boundaries, not the natural distribution of population.

Professors Holmes and Lee ignored political boundaries and split America up using a six-by-six-mile grid. Their cities are squares crafted without any attention to actual boundaries. Using Census Block level data, they calculate the population of each square in the grid. It turns out that Zipf’s Law doesn’t work for these fixed geographic areas.

Professors Holmes and Lee find that “for squares above 1,000 in population, a Zipf’s plot has a piecewise linear shape, with a kink at around a population of 50,000,” and “below the kink the slope is 0.75; above the kink, it is around 2.”

In other words, in dense areas population drops far more quickly with rank than Zipf’s Law would suggest, and in less dense areas, population drops off far too slowly to be compatible with Zipf’s Law. Zipf’s Law is a bust at describing the population levels of areas within fixed boundaries.

But that doesn’t make the law irrelevant. It rather pushes us to understand why Zipf’s Law holds across metropolitan areas, but not in large squares of fixed size. Their work suggests that Zipf’s Law is being powered by sprawl: the spread of population across space within metropolitan areas.

Xavier Gabaix produced a particularly significant contribution to our understanding of Zipf’s Law about 10 years ago. Professor Gabaix follows an old idea that random growth processes could produce Zipf-like outcomes. He shows mathematically that Zipf’s Law will result if the population growth rate of an area is independent of that area’s initial population. Population growth rates are independent of initial population, and this is commonly called Gibrat’s Law, so Professor Gabaix seems to have illuminated the dynamic underpinnings of Zipf’s Law.

But Professor Gabaix’s contribution only pushes the puzzle back one step further. Why should population growth rates be independent of initial levels? Why shouldn’t big cities grow more slowly or small cities catch up?

My own view is that Zipf’s Law is really about the operation of agglomeration — the attraction of people to more people — and sprawl. An initial population attracts more people who live nearby. As long as each person attracts about the same number of new people, then Gibrat’s Law will follow and that gives us Zipf’s Law.

At its heart, this strange mathematical regularity is really once again pointing to the power of agglomeration — the enormous value that human beings place on being near one another.

Update: An earlier version of this post misspelled the name of Steven Strogatz.