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.

Math in the Media

April is Math Awareness Month. Thanks to the Scientific American for an unusually rich selection of mathematical topics this month!

• Space could be finite, merely giving the illusion of infinity, according to Jean-Pierre Luminet, Glenn Starckman and Jeff Weeks. Their article ``Is Space Finite?'' in the April Scientific American explains compact three-dimensional topologies and the possible local geometries consistent with observed cosmic symmetry, with reference to physical theory and experiment. ``Of all the issues in cosmic topology, perhaps the most difficult to grasp is how a hyperbolic space can be finite.'' For a picture of what 3-dimensional hyperbolic space (finite or infinite) would look like from the inside see Charlie Gunn's Hyperbolic Space Tiled with Dodecahedra on the Geometry Center Website.

• Playful thinking, serious mathematics. "Playful thinking can often lead to serious mathematics" is one conclusion drawn from John Conway's career in a profile by Mark Alpert in April's Scientific American. The profile mentions Conway's serious work on finite groups and transfinite numbers, as well as his more playful inventions, including of course the Game of Life, and gives the glimmer of an idea of how these are all connected in Conway's very unusual mind.

• ``Alan Turing's Forgotten Ideas in Computer Science,'' in the April Scientific American is a reminder of the danger of not publishing one's work. A team (B. Jack Copeland and Diane Proudfoot) of philosophy professors have dedicated themselves to the difficult job of retro-dissemination of Turing's fundamental insights into the theory of computation, and in particular the idea of ``oracles:'' currently hypothetical non-computational devices that could hold the key to computing the mathematically uncomputable. The Alan Turing Home Page (maintained by Andrew Hodges) has links to a great wealth of information about the mathematician and his work.

• ``Why does the telephone cord always get twisted?'' Read Ian Stewart's ``Tangling with Topology'' in his Mathematical Recreations column in April's Scientific American and find out. Hint: if you pick up and hang up with the same hand, it probably won't happen. You will also learn the difference between writhing, linking and twisting, and how they are related.

Neckties, a Knotty Mathematical Problem? A piece in Nature (March 4) on necktie knotting was picked up by the New York Times (Henry Fountain, March 9): ``It Takes a Scientist to Tie a Necktie, 85 Different Ways''. Thomas M. A. Fink and Yong Mao of Cambridge University told Fountain that the key to their analysis was making the connection to the idea of a random walk. (On a triangular lattice). For an analysis of this media phenomenon, see Peter Suber's knot page.

Math and populNexkties, a Knotty Mathematical Problem?ar culture. ``Math Emerges Blinking Into the Glare of Popular Culture'' (Paul Lewis, New York Times, March 13) attributes the recent ``boom in sales of popular books about mathematics'' to the attention attracted by Andrew Wiles' proof of Fermat's last theorem and thereby, indirectly, to Andre Weil. As evidence of the boom, Lewis tells us that ``no less than three American writers are working on popular books about the concept of zero.'' Much ado?

The Lagrangian Codes. Not the latest Ludlum but a William Safire column (New York Times, March 22) about ``Joseph Louis Lagrange, the greatest French mathematician,'' whose ``multipliers, weighted with algebraic constraint equations, form composite functions'' which in their augmented form have supposedly been leaked to the Chinese to give them a leg up in the arms race.

The math of slime molds and ant bridges comes to life in a Science Times piece ``Mindless Creatures Acting `Mindfully' '' (George Johnson, March 23). The article describes how cellular automata (Conway's Game of Life may be the best known example) can be bred like livestock to select for specific behaviors. The featured scientists (James Crutchfield and James Hanson) are themselves the fathers of a new field of mathematics: computational mechanics, which ``synthesizes dynamical systems theory and the theory of computation to analyze how patterns, structure, and information processing emerge in spatial systems.'' (from the Princeton University Press blurb for their book on the subject.) For more information on cellular automata visit Moshe Sipper's Brief Introduction to Cellular Automata and check out Juha Haataja's animations.

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