3/4-power law

Interesting guest column in the New York Times on Tuesday, May 19, titled "Math and the City" by Steven Strogatz. In addition to reviewing Zipf’s law, where the population of a city roughly inversely proportional to its rank according to their populations, the column reviews some relatively new mathematical discoveries that have to do with how city size affects markets and operations.

From the column:

For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.

and

There are theoretical grounds to expect a power close to 3/4. Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you’d expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes — precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.