Tony Phillips' Take on Math in the Media

This month's topics:

• Math on the Wild Side
• "Intimidatingly awesome" knot theory project
• Beyond Turing machines

Math on the Wild Side

Olivia Judson's weekly New York Times blog "The Wild Side" is being taken over for three weeks by the Steven Strogatz of Cornell, who refers to himself as "(gasp!) a mathematician." Strogatz has posted two columns so far, on May 19 and May 26, 2009.

• May 19: "Math and the City" (available online). Strogatz starts with the observation "One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden." This first column focuses on various studies of scaling, which reveal "Manhattan and a mouse to be variations on a single structural theme." He starts with Zipf's law. George Zipf, a linguist, noticed that if you order the words in a language by the frequency with which they occur (in speech, say), with the most frequent first, then the nth word on the list has 1/n the frequency of the first word, to a good approximation. This distribution, called "Zipf's Law" turns up in many contexts. For example, Zipf "noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows." Now for the scaling: "For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. ... The number of gas stations grows only in proportion to the 0.77 power of population." And here's where the mice -- and elephants -- come in: "on a pound for pound basis, the cells of an elephant consume far less energy than those of a mouse. The relevant law of metabolism, called Kleiber's law, states that the metabolic needs of a mammal grow in proportion to its body weight raised to the 0.74 power." 0.77, 0.74, "Coincidence? Maybe, but probably not. There are theoretical grounds to expect a power close to 3/4." The column ends with a useful set of references.
• May 26: "Loves Me, Loves Me Not (Do the Math)" (available online). Here Strogatz makes his point ("the laws of nature are written as differential equations") by modeling the ups and downs of a tempestuous romantic relationship. He posits a theoretical Romeo and Juliet, with interlocking behavior patterns: "The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she hates him." (The DEs are given in an appendix: dR/dt = aJ, dJ/dt = -bR with a,b > 0, along with the qualitative description of the general solution: "Romeo and Juliet behave like simple harmonic oscillators"). There's a nice riff on the three-body problem at the end involving both Isaac Newton and Strogatz's college girlfriend's old boyfriend. Again, a good set of references.