Most people don’t think of New Mexico as a large state. Texas, California, Montana and Alaska are all significantly larger, and a number of other Western states are about the same size. But almost two-thirds of the U.S. population lives in a combined area about the size of New Mexico. That’s because most Americans live in or near cities.

And if you’ve been reading political commentary over the last decade or so, you might think that cities are about to take over (or have already taken over) our national politics. The urban population in the United States is expanding, and much of the economic growth related to technology and other new industries seems to be concentrated in cities and the surrounding areas.

But last month Sean Trende and I looked at the data and found that the largest urban centers in the United States hadn’t gained much political power over the last three decades. Moreover, Donald Trump won the presidency despite losing badly to Hillary Clinton in the largest cities. So it’s worth asking: What happened? Why hasn’t megalopolis America, with its increasing economic strength and growing population, taken over politics?

I looked into the data and found that while the largest American cities are growing, their relative power has been held down by similar levels of population growth in towns and smaller cities. And if we dive just a bit more deeply into these numbers, we’ll see that a simple, little-known but eerily accurate mathematical law might have predicted it all.

Big Cities Are Growing -- But So Are Small Cities and Large Towns

Over the past few decades, major cities like New York, Chicago, Los Angeles, Dallas, Philadelphia, Houston and Washington, D.C., have gained millions of residents (the exact definition of a city is detailed later in this article). And if you look at the raw data, it’s easy to see why much of the commentary has focused on a few dozen large or fast-growing cities.

This graphic shows the trend in raw population levels in the 100 largest metro areas in the U.S. from 1950 to 2010. There are too many cities to label, but the pattern is easy to see. Cities like New York (the top line), Los Angeles (the second highest) and Chicago (the third) have a much larger population and have seen much bigger raw increases in population than the mass of smaller cities crowded into the bottom of the graph. So it would be easy to look at the raw growth and population of a city like Houston or Washington, compare it to a place like York, Pa., and conclude that the megalopolises will own the future of American politics.

But that sort of thinking would miss two key facts: smaller cities and large towns have an aggregate rate of growth close to that of the largest cities, and there are a lot more York-sized cities than New York-sized ones.

I used “CBSA Divisions” to examine these trends. The basic idea is simple -- we used metro areas (divided up by the Office of Management and Budget as Core-Based Statistical Areas) and population estimates from the Census Bureau to place every county in the United States into one of six categories: mega-city, large city, small city, large town, small town or rural. The basic idea is that cities and towns of all shapes and sizes are more than just their downtown or densely packed core -- the surrounding areas, suburbs and city itself come together to form an economic and cultural unit. So Washington shouldn’t just be thought of as the National Mall or even what’s within the District of Columbia. Suburban and urbanized parts of Northern Virginia and Maryland are part of D.C., as is a bit of exurban West Virginia. These CBSA divisions also pick up on the difference between small towns like my hometown of Parkersburg, W.Va. (an area with a little under 100,000 people, one mall and no Starbucks), from completely rural areas in Alaska, the Great Plains or even the southeastern part of West Virginia.

The technical details (skip this paragraph if you’re not interested in methodology) are slightly more complicated. We used the latest OMB CBSA Divisions and 2015 population estimates from the American Community Survey to place each county in a category. Mega-city counties were part of CBSAs with 5 million or more people, large cities had 1 million to 5 million people, small cities had 500,000 to 1 million people, large towns had between 100,000 and 500,000 people and small towns had less than 100,000 people in the CBSA that contained them. Rural areas weren’t part of any CBSA. Hard divisions like these always involve judgment calls. For example, the Phoenix-Mesa-Scottsdale area is a “large city” with well over 4 million people. One could reasonably argue that Phoenix is better classified as a mega city and bears more resemblance to Los Angeles than to a city of just 1 million people. Objections like this are reasonable – there’s more than one way to skin a cat. But our categories are intuitive, informative and simple so we stuck to them.

If we look at the population growth in each of these divisions, we can see that while the largest cities have been growing at a rapid pace, large towns and small cities have been keeping up.

The top two lines show large cities (places like Minneapolis, Denver, Salt Lake City or Pittsburgh) and mega cities (New York, Los Angeles, Philadelphia and a few others) have been growing rapidly. But so have “Large Towns” (places like Sioux City, Iowa, or Bangor, Maine) and “Small Cities” (similar in size to the Toledo, Ohio, area) have been growing at similar rates. The growth of these large towns and small cities might escape popular knowledge because it’s less physically visible than mega-urban growth is. Anyone who lives in or visits Washington can see the cranes and watch new suburbs being built. But if a similar rate of growth were spread out over every Springfield, Mo., and Bangor, Maine, in the country, the change would be more subtle.

Small cities and large towns have, by keeping up with the mega-cities, kept the urban/rural division of power relatively stable for the past few decades.

While rural areas and small towns (think of areas with a population less than 100,000) lost influence over the past half-century, the larger divisions have all made up a relatively stable percentage of the overall population since the 1980s. That’s why a candidate like Trump, who ran extremely well in rural areas and held his own outside large cities and mega-cities, can still build an Electoral College majority despite losing big in the most heavily urban areas.

The Weirdly Effective, Extremely Simple Math That Could Explain It All

When Sean Trende and I initially started digging into these patterns in our Election Review series, we thought it was a little weird. Neither of us could come up with a reason why the distribution of the population should be so constant over multiple decades of social and economic change. But then a reader pointed us to a little-known mathematical mystery called Zipf’s Law.

The exact statement of Zipf’s Law involves more formal math than we usually explain here, but it basically states that if you take a specific country, count the number of cities at a certain population level and then count the number of cities with twice that population, the former number will often be close to double the latter.

For example, the 2010 population figures show that there were 51 metro areas with a population of over 1 million residents. Given that, Zipf’s Law might predict that there would be 102 metro areas with half that population -- 500,000. There were 104. Similarly, there were nine cities with a population over 5 million, so Zipf’s law would predict that there would be 18 cities with 2.5 million or more residents. There were 21. And it works for other factors – not just doubles and halves. In 2010 there were 17 metro areas with a population of 3 million or more, so Zipf’s Law would predict that there would be 51 cities with 1 million or more people (because 1 million is a third of 3 million, we multiply 17 by 3 and get 51). There were exactly 51 such cities. More rigorous analysis has shown that Zipf’s Law holds for the population data I’ve used in this piece.

It’s important to note that Zipf’s Law isn’t like the laws you might learn in an introductory physics class. It involves randomness. It has constraints (e.g. it only works for cities of a certain size, so it doesn’t apply to shrinking rural and small town areas shown above). So calculations like the ones done above won’t always yield perfect results. But it works much more effectively than almost all other economic or political “laws.”

Zipf’s Law itself is interesting and important, but the mechanics behind it are what really help explain the persistent split we see in mega-urban/rest-of-the-U.S. population. Harvard Professor Xavier Gabaix has written that Zipf’s Law will hold if every city, irrespective of its size, follows roughly the same random growth processes. In a random process like this, some cities will shrink (e.g. Rust Belt cities that have fallen on hard times), others will expand (e.g. Western cities benefiting from a tech boom), but if Gabaix is right then small cities and large cities shouldn’t grow at different rates because of their differences in size.

If this explanation of Zipf’s Law is correct (and it might not be – the law is still a live academic issue and I haven’t read all the literature on it), then we have an explanation for the stability we saw in the previous graph. Zipf’s Law holds for that data, so maybe New York City, Peoria, Illinois, Toledo, Ohio and Boulder, Colo., all by their nature undergo somewhat similar growth processes – regardless of their size. That would be consistent both with Zipf’s Law and my analysis of the population data.

It’s also worth noting that the accuracy and usefulness of Zipf’s Law in this case isn’t its most unusual feature.

Zipf’s Law is eerie because it’s universal. It holds for cities in Germany, Brazil, China, India, Indonesia, Nigeria and Russia as well as the United States. Data from India in 1911, Argentina in 1860, China in the mid-19th century and the United States as early as 1790 show that Zipf’s Law also holds across eras. In other words, Zipf’s Law might be illuminating something universal about how all humans – despite being in different eras and on different continents – tend to build modern cities.

It gets weirder. The specific mathematical form of Zipf’s Law wasn’t discovered by urbanists or political scientists – it comes from linguistics. The law uses the exact same math to show that in most languages, the most used word (“the” in English) appears about twice as often as the second most frequently used word (“of”), three times as frequently as the third most frequent word, etc. Zipf’s Law also appears in economics, biology and a whole host of other disciplines.

So maybe the persistent hyper-urban vs less-urban population divisions in the U.S. can’t be solely traced back to extreme partisanship, racial resentment, economic anxiety, cultural fights, ideological self-sorting or the other depressing concepts that seem to explain everything in today’s politics. Maybe this unexpected stability is a product of natural human behavior and eerily universal math.