Anthropologists have developed mathematical models that they say describe development patterns in both modern urban areas and ancient cities settled thousands of years ago.

 Scott Ortman, an assistant professor of anthropology at 
University of Colorado at Boulder, and colleagues developed mathematical models with parameters that help describe how modern cities change as their populations grow. For example, it's known that as a population increases, its settlement area becomes denser, while infrastructure needs per capita decrease and economic production per capita rises. 
 
The variables in their model, such as the cost of moving around, the size of the settled area, the population, and the benefits of people interacting, did not depend on any particular modern technology. 

Maps of the Basin of Mexico. A: Location within Mexico [34]. B: Settlements dating to the Formative period (circle size is proportional to population; colors range from yellow through red to white denoting increases in elevation; gray area shows the extent of Mexico City in 1964) [35]. C: Settlements dating to the Aztec period. During the latter period settlement expanded into the shallows of the lake. Today, settlement covers the entire basin and the lake has been drained. 
Credit: doi:10.1371/journal.pone.0087902

"I realized that if these models are adequate for explaining what's going on in contemporary cities, they should apply to any settlements in any society," said Ortman. "So if these models are on the right track, they should apply to ancient societies too."

To test his idea, Ortman used data that had been collected in the 1960s about 1,500 settlements in central Mexico that spanned from 1,150 years B.C. through the Aztec period, which ended about 500 years ago. The data included the number of dwellings the archaeologists were able to identify, the total settled area and the density of pottery fragments scattered on the surface. Taken together, these artifacts give an indication of the total population numbers and settlement density of the ancient sites.

"We started analyzing the data in the ways we were thinking about with modern cities, and it showed that the models worked," said Ortman. 

The discovery that ancient and modern settlements may develop in similar and predictable ways has implications both for archaeologists and people studying today's urban areas. For example, it's common for archaeologists to assume that population density is constant, no matter how large the settlement area, when estimating the population of ancient cities. The new equations could offer a way for archaeologists to get a more accurate head count, by incorporating the idea that population density tends to grow as total area increases.

In the future, the equations may also guide archaeologists in getting an idea of what they're likely to find within a given settlement based on its size, such as the miles of roads and pathways. The equations could also guide expectations about the number of different activities that took place in a settlement and the division of labor.

"There should be a relationship between the population of settlements and the productivity of labor," Ortman said. "So, for example, we would expect larger social networks to be able to produce more public monuments per capita than smaller settlements."

The findings of the new study may also be useful to studies of modern societies. Because ancient settlements were typically less complex than today's cities, they offer a simple "model system" for testing the equations devised to explain modern cities.

"The archaeological record actually provides surprisingly clear tests of these models, and in some cases it's actually much harder to collect comparable data from contemporary cities," Ortman said.