What does the borough of Manhattan have in common with a mouse? As Steven Srogatz
explains it, they are "variations on a single structural theme.”
In a guest column in The New York Times, "Math and the City,”
Strogatz describes similar patterns mathematicians have seen in social and biological systems. Strogatz is a professor of applied mathematics at Cornell University, His books include Sync, The Emerging Science of Spontaneous Order
, and The Calculus of Friendship: What a Teacher and Student Learned about Life While Corresponding about Math, to be published in August. Melanie Mitchell
, in her book Complexity: A Guided Tour
, offers a lucid discussion of power laws, fractals and scaling, and why a tiny mammal and big city can operate on the basis of similar underlying principles.
Fifty years ago, Strogatz writes, George Zipf, a Harvard linguist who had studied word distribution in various pieces of writing, looked at size distribution of cities and found that in any country, the biggest city is always about twice as big as the second largest, and about three times as big as the third, and that the size and rank pattern is one that continues. Strogatz adds that, amazingly, the law has held in different countries, different cultures, and in different time periods. Xavier Gabaix wrote an in depth discussion of Zipf’s law for cities
in the August 1999 issue of The Quarterly Journal of Economics.
Scientists don’t agree on why Zipf’s law seems to work, but new research has been done this decade on the mathematics of cities. Mathematicians have been looking at how size affects infrastructure. Studies have been done, for example,. on the number of gas stations, miles of roadways, and length of electrical cable. It turns out that bigger is greener, and that all of these resources decrease, on a per person basis, as city size increases. Strogatz says these resources all grow in proportion to a power of the population that is pretty close to three fourths.
Now comes another amazing fact that Strogatz thinks is not likely to be coincidental. The metabolic needs of animals grow in proportion to body weight raised to the 0.74 power. Mitchell explains in her book how three scientists, James Brown, an ecologist at the University of New Mexico, Brian Enquist, a biology graduate student, and Geoffrey West, a theoretical physicist, collaborated to probe the mystery of ¾ power scaling. They suspected that in living creatures, the answer might lie in the branching networks of blood vessels that carry nutrients to cells and the branching structures of the bronchi in the lungs that carry oxygen to blood vessels. Mitchell explains, with elegance and detail, that fractal structure is one way to generate a power law distribution. She says the three scientists developed a mathematical model of blood vessels and bronchi as "space-filling” fractals, and discovered that, as body mass rises, metabolism decreases, and that the metabolic rate scales with body mass to the 3/4 power. In other words, a mouse will consume more energy, per pound, than a person or an elephant, and there is a mathematical formula that calculates how much more. Further, fractal networks of living creatures and resources in cities may generate similar power law distributions. Mitchell’s stories of decades of biological research andmathematical reasoning are fascinating, and she discussed the intricate math in her chapter notes.
In a cover note for Complexity: A Guided Tour, Strogatz, wrote, "Finally! For years people have been asking me where they can learn the basics of complexity theory. Now I’ve got the answer. Read Melanie Mitchell’s book.”