The City Formula or: It’s all about geometry!

Scaling laws are everywhere

A characteristic feature of complex systems is that when they double in size, many of their parts do not. Typically, some aspects will grow by only about 80 percent, others by about 120 percent. The astonishing uniformity of these two growth rates is known as “scaling laws.” Scaling laws are observed everywhere in the world, from biology to physical systems—and in cities. Yet, while a multitude of examples show their presence, reasons for their emergence are still a matter of debate.

A new publication in the Journal of The Royal Society Interface now provides a simple explanation for urban scaling laws: Carlos Molinero, a researcher at the Hub until last September, and CSH President Stefan Thurner, derive them from the geometry of a city.

Scaling laws in cities

One example of an urban scaling law is the number of gas stations: If a city with 20 gas stations doubles its population say, from 100,000 to 200,000, the number of gas stations does not increase to 40, but only to 36. This growth rate of about 0.80 per doubling – it will always be between 0.75 and 0.85 – applies to much of the infrastructure of a city.

For example, the energy consumption per person or the land coverage of a town rises by only 80 percent with each doubling. Since this growth is slower than what is expected from doubling, it is called sub-linear growth.

On the other hand, cities show more-than-doubling rates in more socially driven aspects. Studies found that people in larger cities earn consistently more money for the same work, make more phone calls, and even walk faster than people in smaller towns. This (super-linear) growth rate is around 120 percent for every doubling.

Remarkably, these two growth rates, 0.8 and 1.2., are showing up over and over again in literally dozens of city-related contexts and applications. But why is that?

It’s all in the geometry

Carlos, who worked extensively on this during his time at the Hub, and Stefan are convinced that these scaling laws can be explained by the spatial geometry of cities. “Cities are always built in a way that infrastructure and people meet,” says Carlos, who is an urban science expert and an architect. “We therefore think that scaling laws must emerge from the interplay between where people live, and the spaces they use to move through a city.”

And Stefan points out what makes the idea so special: “The innovative idea here is how the spatial dimensions of a city relate to each other.”

Fractal Geometry

To come to this conclusion, the researchers first mapped thee-dimensionally where people live. They used open data from every house in more than 4,700 cities in Europe. “We know most of the buildings in 3D, so we can estimate how many floors a building has and how many people live in it,” says Stefan. The scientists assigned a dot to every person living in a building. Together, these dots form sort of a “human cloud” of a city.

Clouds are fractals. Fractals are self-similar, meaning that if you zoom in, their parts look very similar to the whole. Based on the human cloud, the researchers were able to determine the fractal dimension of a city’s population: They got a number that describes the human cloud of each particular city. Similarly, they calculated the fractal dimension of cities’ road networks.

“Although these two numbers vary widely from city to city, we discovered that the ratio between the two is a constant,” Thurner says. The researchers identified this constant as the “sublinear scaling exponent.”

Aside from the mathematical elegance of this explanation, the finding has potential practical value, as the two complexity researchers also point out in their paper. “At first sight the appearance of such a constant looks like magic,” says Stefan. “But it makes perfect sense if one takes a closer look: It’s this scaling exponent that determines how the properties of a city change with its size, and that is relevant because many cities around the world are growing rapidly.”

The formula for sustainable urban planning?

The number of people living in cities worldwide is expected to roughly double in the next 50 to 80 years. “Scaling laws show us what this doubling means in terms of wages, crime, inventiveness or resources needed per person—all this is important information for urban planners,” Stefan points out.

To additionally know the scaling exponent of a city could further help urban planners to keep the gigantic resource demands of urban growth at bay. “We can now think specifically about how to get this number as small as possible, for example through clever architectural solutions and radically different approaches to mobility and infrastructure construction,” Stefan is convinced. After all, the smaller the scaling exponent, the higher the resource efficiency of a city.

Carlos Molinero, Stefan Thurner, How the geometry of cities determines urban scaling laws, Journal of The Royal Society Interface, March 17, 2021

Our Press Release in English.

Our Press Release in German.