## Math Digest |

- "Is Complexity Simple?",
*Science*, 27 February 2004 - "Virtual plagues get real,"
*Nature*, 26 February 2004 - "Prime Time Entertainment,"
*The Guardian*, 25 February 2004 - "The Not So Random Coin Toss,"
*All Things Considered*, National Public Radio, 24 February 2004 - "Computation's New Leaf,"
*Science News*, 21 February 2004 - "How many M&M's are in this jar? Hint: It's all in the packing,"
*The Baltimore Sun*, 16 February 2004 - "Networking Opportunity,"
*Nature*, 12 February 2004 - "Prisoners of the Dilemma,"
*Nature*, 5 February 2004 - "Incremental Analysis, With Two Yards to Go,"
*The New York Times*, 1 February 2004 - "Pigskin Pythagoras,"
*The Boston Globe*, 1 February 2004 - "Mental Muscles of Steel,"
*Popular Science*, February 2004

** "Is Complexity Simple?" NetWatch, ***Science*, 27 February 2004, page 1267.

Stephen Wolfram's *A New Kind of Science* is now available online. Registration is required to access the 1280-page book. NetWatch notes that the site also has supplementary materials available, such as demo programs and images.

*--- Mike Breen*

** "Virtual plagues get real," by Virginia Gewin. ***Nature*, 26 February 2004.

Gewin reports on reseachers who are incorporating ecological data in mathematical models to fight against infectious diseases. Epidemiologists studying foot-and-mouth disease have used mathematical models to predict the progress of an outbreak and the effectiveness of control strategies. Such models "helped halt the UK foot-and-mouth epidemic by suggesting an appropriate culling strategy." Modelling has also been used to study outbreaks of measles and dengue fever. Impediments to progress in the field are the dependence on incorporating better underlying data, and the lack of dialogue between mathematical modellers and epidemiologists. However, Andrew Dobson (Princeton University) believes that "mathematical modelling will be equally as important as anything that comes from the human genome in terms of its utility in public health."

*--- Annette Emerson*

** "Prime Time Entertainment," by Marcus du Sautoy. ***The Guardian*, 25 February 2004.

In response to a study in Britain on the teaching of mathematics, du Sautoy suggests that the findings should provide good incentive for a rethinking of how mathematics is taught. He says that "mathematics should be as much about its romance and mystery as about sines and cosines," noting that if students studying musical instruments were only taught scales and arpeggios and never exposed to some of the great music they may one day play or compose, those children would be bitter and uninspired. He gives the example of humankind's attempts to understand prime numbers---providing a very brief chronology of discoveries (with an interesting sidelight on how a Wagner musical composition caught the interest of Reimann) and some examples of the usefulness of primes. He also notes how mathematics has become relevant in popular culture---in books, plays and films. He acknowledges that students must learn the technical side that involves hard, tedious work, but advocates that mathematics education include "the great ideas, history and people that make up the true story of mathematics."

*--- Annette Emerson*

** ****"The Not So Random Coin Toss," by David Kestenbaum. All Things Considered, National Public Radio, 24 February 2004. **

Flipping a coin is one of the most often cited examples of a random experiment. Yet Persi Diaconis, Susan Holmes, and Richard Montgomery have found that a coin toss may not yield the assumed 50-50 split. Kestenbaum interviews Diaconis, who says that he and his colleagues have discovered that "coins, when they're tossed by real people, are biased." They found that the probability that a coin will wind up facing the same way as it was when it was tossed is more than .51. Diaconis talks about the particulars of his coin-tossing research and his larger goal of cautioning people when they assume that things are random. The segment concludes with an excerpt from Bill Cosby's classic routine on the football coin toss. (Note: when listening to the interview online, you will first hear a ten-second ad for a tape from NPR.)

See also:

"Toss Out the Toss-Up: Bias in heads-or-tails," by Erica Klarreich,Ê*Science News*, 28 February 2004;

"Heads or Tails?," by Ivars Peterson, *Science News*, 28 February 2004.

*--- Mike Breen*

** "Computation's New Leaf," by Erica Klarreich. ***Science News*, 21 February 2004, pages 123-124.

Klarreich writes that "Plants may perform what scientists call distributed emergent computation," in which there is no central processor, but instead many simple units working together to perform complex "calculations." In the case of plants, the calculations involve problems like how many pores should be open and to what degree. In the article, the behavior of pores is compared to that of cellular automata.

*--- Mike Breen*

** "How many M&M's are in this jar? Hint: It's all in the packing," by Michael Stroh. ***The Baltimore Sun*, 16 February 2004.

Several recent articles have covered a team's research showing that oblate spheroids in the shape of M&M candies pack more effeciently than spheres. Stroh's piece discusses the mathematics behind the story. The lead researcher in the M&M experiements is Salvatore Torquato, professor of chemistry at Princeton University. Stroh notes that mathematicians started working 400 years ago on "the packing problem [which] has played a role in scientific endeavors ranging from the study of human cells to the design of high-speed computer modems." The article gives an overview of the "cannonball question" allegedly posed by Sir Walter Raleigh ("Was it possible to calculate how much ammunition he had in each pile [on the deck of his ship], based solely on its shape and size?"). He got help from a mathematician, who solved that problem in 1591 and then investigated how the cannonballs could be arranged in the ship's hold with the least wasted space. German mathematician Johannes Kepler conjectured a solution that was proven to be correct 387 years later, in 1998, by mathematician Thomas Hales. Mathematicians have studied the efficiency of randomly-packed versus pyramid-stacked spheres as well, and the new research shows that oblate spheroids pack the most densely: "The M&M's filled 68% of the container---four percentage points more than the randomly-packed ball bearings would." This new research, published in *Science*, "could lead to everything from advanced new manufacturing materials to more economical ways of packing and shipping goods."

The article presenting this research is:

"Improving the Density of Jammed Disordered Pakcings Using Ellipsoids," by Aleksander Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, P.M. Chalkin, *Science*, 13 February 2004.

See also the following news reports:

"Packing in the Spheres," by David A. Weitz, *Science*, 13 February 2004;

"Candy Science," by P. Weiss, *Science news*, 14 February 2004

"After Packing M&M's Together, Scientists Like What They See," by Kenneth Chang, *The New York Times*, 13 February 2004;

"Why there's always room for a few extra chocs," *New Scientist*, 21 February 2004, page 15.

*--- Annette Emerson*

** "Networking Opportunity," by Ian Stewart. ***Nature*, 12 February 2004, pages 601-604.

Leeches, network dynamics, and groupoids. Don't see the connection? The leech's heart, which consists of a series of tubes on either side of its body, beats in a very unusual way: on one side, the tubes contract simultaneously, while on the other side, one tube contracts after another, moving from back to front. Approximately every 50 beats, this pattern shifts sides. The network of neurons that cause this phenomenon has been mapped. But a greater mathematical understanding of the occurrence of patterns like synchrony and "phase locking," here and in networks generally, results from the application of groupoid theory.

Stewart offers some illustrations, including an example of three types of networks: a ring, a chain, and a modified chain with feedback, where each node represents a system of differential equations, and each directed edge a "coupling." While only the ring has symmetry, the modified chain with feedback has a "hidden 'rotational' symmetry." Symmetry between specific subsets of the network causes the dynamics in the modified chain to be equivalent to those of the ring. In Stewart's words, "The groupoid formalism makes it possible to approach network dynamics within a coherent theoretical framework. Just as group theory has illuminated pattern formation in symmetric systems, so groupoid theory can illuminate pattern formation in systems with repeated subunits." This application of mathematics to biology is important since nature is filled with networks, and it is an example of the way in which the fields of mathematics and biology are offering new insights and presenting new challenges to each other.

*--- Claudia Clark*

** "Prisoners of the Dilemma," by Martin A. Nowak. ***Nature*, 5 February 2004.

Nowak describes how mathematician Karl Sigmund introduced him to "the prisoner's dilemma," a game in which two players have a choice between cooperation and defection. The author describes how complicated and interesting the gameÊcan become. He introduces the concept of tit-for-tat (TFT) in a single game and in round robins and discusses how researchers in many disciplines have been interested in the prisoner's dilemma. Nowak was a first-year Ph.D. student in biochemistry when he met Sigmund on a holiday, "the turning point that brought together mathematics and biology." He confesses he's "no longer embarrassed to work on games" and is now at the Program for Evolutionary Dynamics at harvard University.

*--- Annette Emerson*

** "Incremental Analysis, With Two Yards to Go," by David Leonhardt. ***The New York Times*, 1 February 2004.

David Romer, an economist at the University of California at Berkeley, published an academic paper, an analysis of whether professional football teams punt more often than is rational. He concluded---using phrases like "Bellman equation" and "dynamic-programming analysis" (gleaned by Leonhardt)---that teams do punt too much. It seems that New England Patriots coach Bill Belichick (who majored in economics at Wesleyan University) responded to a reporter that he had read the paper, and added, "I don't know much of the math involved, but I think I understand the conclusions and he has some valid points." In a game soon after that, Belichick decided to go for a first down instead of "a sure punting situation in the NFL," and made it. Romer surmised that the coach was using analysis instead of gut instinct. Leonhardt's article reported that "Romer's paper is not the only ivory-tower research that has made its way into the coach's head." The Patriots coaching staff wanted statistician Harold Sackrowitz (Rutgers) to critique the team's chart that tells coaches when to go for two points. (Sackrowitz had previously been quoted as saying that "teams try for a two-point conversion too often after scoring a touchdown.") This article's teaser, "The Patriots may or may not engage in a little 'hyperbolic discounting'" at the Super Bowl, appears to have been answered.

*--- Annette Emerson*

** "Pigskin Pythagoras," by Jascha Hoffman. ***The Boston Globe*, 1 February 2004.

The subtitle is "A guy from Framingham [Massachusetts] tries to remake the muddy field of football statistics." FootballOutsiders.com's Aaron Schatz claims that people should "forget touchdowns, total yards, and red-zone efficiency" and "worry about DVOA (defense-adjusted value over average) and line-yards." Hoffman notes that Schatz's "numbers are unique in that they evaluate each play against the league average for plays of its type, adjust for the strength of the opponents' defense, and even try to divide credit for a given play among teammates." The article cites examples, describes why he calls it the Pythagorean theorem of football, and notes that statistician Daryl Morey of STATS, Inc. has adapted and applied the formula to many sports.

*--- Annette Emerson*

** "Mental Muscles of Steel," by Mckenzie Funk. ***Popular Science*, February 2004.

Mathematics comes into play a few times in this article, which is intended to back up the assertion by neuroscientists that people need to "exercise" their brains. The piece includes a science aptitude quiz, information on the brain, and some "exercises" (stimulation)---including how to make and demonstrate a Möbius strip. Mathematician Manjul Bhargava (whom the magazine profiled in "PopSci's Brilliant Ten," November 2002) is quoted: "It takes a computer seconds to multiply two 400-digit numbers. Run the problem backward---ask it to find the two factors---and it'll take millions of years."

*--- Annette Emerson*

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