## Math Digest |

- "Leaves, Flowers and Garbage Bags: Making Waves,"
*American Scientist*, May-June 2004 - Articles on mathematics and cicada behavior, May and June 2004
- "Math Equation Predicts Musical Reactions,"
*Discovery News*, 25 May 2004 - "Von der Ungleichheit der Knoten im Netzwerk,"
*Neue Zuercher Zeitung,*16 May 2004 - "Glimpses of Genius,"
*Science News*, 15 May 2004 - "WWII code-breakers aim to crack inscription,"
*The Providence Journal*, 13 May 2004 - "The Tug of the Newfangled Slot Machines,"
*The New York Times Magazine*, 9 May 2004 - "Spiral pattern helps cacti deal with stress,"
*New Scientist*, 8 May 2004 - "Calculating Doom," ABCnews.com, 5 May 2004
- "A Numbers Game,"
*Jerusalem Report,*3 May 2004 - "$1.5 million buys book that put world in its place,"
*The Kansas City Star*, 1 May 2004 - "The Cryptography of ... Voting Machines,"
*Discover*, May 2004 - "Gizmos: Enigma-E Electronic Enigma Machine,"
*Discover*, May 2004

** "Leaves, Flowers and Garbage Bags: Making Waves," by Eran Sharon, Michael Marder and Harry L. Swinney. **** American Scientist, May-June 2004, pages 254-261. **

If you've ever noticed that fractals appear in the curving edges of some leaves and flowers, you're not alone. Sharon, Marder, and Swinney explored the causes of this phenomenon by seeing what happens when one tears a plastic sheet imprinted with a grid of dots. They noted that the fractal pattern along the tear is the result of "spontaneous symmetry breaking," partly a result of the stretching near the tear. The authors write that "the distances between the dots on the surface after tearing cannot be met if the sheet is flat. Then, to avoid the expensive compression energy, the sheet happily pays cheap bending energy, as it buckles out of the plane, while trying to generate saddle points everywhere [due to negative Gaussian curvature]."

They also performed experiments with leaves, tubes made of synthetic material that would expand in a chemical solution, and computer models. They noted that while many complex biological systems have complex causes, the reasons behind these fractal patterns are simple. The growth along leaf and flower edges results from a uniform growth law, but---as with the plastic sheets---the geometric limitations of space and the elasticity of the material caused the edges to take on a wavy shape. They conclude, "physics and biology meet at the rippled edges of leaves and flowers to provide one of these rare tractable problems."

*--- Claudia Clark*

** Articles on mathematics and cicada behavior:"Mathematicians explore cicada's mysterious link with primes," by Michael Stroh. **

"Big buzz as cicadas arrive after 17-year gap," by Laura Nelson.

"The 17-Year Itch," by Tabitha M. Powledge.

"Bugs That Count," by Brian Hayes.

Residents in the eastern United States have recently witnessed the arrival of trillions of cicadas. The insects spend most of their time underground, but emerge for six weeks every 13 or 17 years, depending on the species, to reproduce. This year it is the 17-year cicadas that are now above ground. A natural question is, Why are the lengths of the cycles of both species prime numbers? No one knows for sure, but Stroh interviewed Glenn Webb of Vanderbilt University, who said that the prime-numbered lengths make it unlikely that the cicada cycle will coincide with a predator's cycle. The *Nature* article points out that the two different prime-year lengths for the two cicada species also make hybridization of the two species unlikely. The article in *Scientific American* casts doubt on the theories about the reasons for the prime periodicity, noting that "true periodicity is rare in cicadas."

Hayes' article goes into more depth about cicadas' cycles and runs some simulations to analyze population behavior.

*--- Mike Breen*

** "Math Equation Predicts Musical Reactions," by Anna Salleh. ***Discovery News*, 25 May 2004.

Emery Schubert of the University of New South Wales (Australia) has a formula that relates the characteristics of a musical piece with listeners' emotions. Schubert arrived at the formula by monitoring the emotions of 67 volunteers who listened to four pieces of music. Listeners registered their emotions (choosing among negatively aroused, positively aroused, negatively sleepy, and positively sleepy) at each second during 20 minutes of music by moving a mouse over a computer screen. The music characteristics monitored were loudness, tempo, pitch, texture, and brightness. Schubert found that loudness was the most powerful predictor of how arousing a piece of music is, followed by tempo. As might be expected, this is not the last word in composing or enjoying music. Schubert said, "Our emotional response to music is highly complex and has a lot to do with what we bring to the listening experience, such as memory, expectation and conditioning. Before we can compose musical emotions by numbers, we need to convert human experience and cultural knowledge variables into numbers, too. It will be some time before we can do this." He will present his research at the International Conference on Auditory Display in Sydney in July 2004.

*--- Mike Breen*

** "Von der Ungleichheit der Knoten im Netzwerk," by George Szpiro. Neue Zuercher Zeitung, 16 May 2004. **

** "Glimpses of Genius," by Erica Klarreich. ***Science News*, 15 May 2004, pages 314-315.

This article expands considerably on news from December regarding the Stomachion---a fragment of a palimpsest, originally written by Archimedes. The first analysis, done almost 100 years ago, didn't reveal anything beyond what looked like a simple children's puzzle of forming a square out of a collection of 14 triangles, quadrilaterals, and a pentagon. Now it appears that Archimedes was actually doing some combinatorics, in this case counting the number of ways the pieces could be arranged to form a square. Klarreich cites Persi Diaconis, Ron Graham, and Fan Chung, who not only counted the 268 ways to arrange the pieces, but also have looked at properties that the arrangements have. Reviel Netz, a math historian at Stanford, and Nigel Wilson, a classics professor at Oxford, are preparing an article regarding their study of the palimpsest.

*--- Mike Breen*

** "WWII code-breakers aim to crack inscription," Associated Press. ***The Providence Journal*, 13 May 2004.

Code-breakers who worked at Bletchley Park during world War II to crack the Nazi secet codes are now tackling a 10-letter enigma that has perplexed historians and code-breakers for 250 years. The letters---O.U.O.S.V.A.V.V. in a line, with the letters D and M beneath on either end---appear on an 18th-century monument on an English estate. Some think the code reveals the location of the Holy Grail, others think the letters represent a secret message from one person to another. Mathematician Oliver Lawn---who was recruited to Bletchley Park in 1940 while studing mathematics at Cambridge University---is leading the effort.

*--- Annette Emerson*

** "The Tug of the Newfangled Slot Machines," by Gary Rivlin. ***The New York Times Magazine*, 9 May 2004.

Although the article doesn't dwell on the mathematics behind slot machines, it does note that Anthony Baerlocher, chief game designer at International Game Technology of Reno (I.G.T., the country's largest maker of slot machines), was "trained as a mathematician." Slot machines are the number-one revenue generator in casinos and are the country's most profitable form of adult entertainment. "Today's slot machines feature well-choreographed illusions designed to hide a fundamental truth: at heart they're really nothing more than computers whose chips randomly cycle through hundreds of thousands of numbers every second. A player's fate is determined almost the instant play begins. But to simply display a long string a numbers on a computer screen, along with an accounting of money won or lost, would hardly prove entrancing." So Baerlocher and others at the company witness and analyze slot machine addiction and use probability models---"infrequent random reinforcement, or 'intermittent reward'"---to hook and periodically reward slot machine users. Most of I.G.T.'s development costs are devoted to the glitzy sounds, design, graphics and video effects of the machines. The core team that develops a game consists of Baerlocher, a junior mathematician, and computer programmers. As Baerlocher says, "There are two basic elements to any slot machine: the art and the math."

*--- Annette Emerson*

** "Spiral pattern helps cacti deal with stress," by Anil Ananthaswamy. ***New Scientist*, 8 May 2004, page 12.

The relationship between the Fibonacci numbers and spirals that appear in plant growth is well known. But what causes this relationship? This article discusses recent research aimed at answering this question. The researchers, Patrick Shipman and Alan Newell at the University of Arizona, suspected that the spirals emerge because of stresses on the growing plant. In a cactus, these stresses cause sets of ridges to appear on the cactus's outer layer, called the tunica. Shipman and Newell developed a mathematical model of cactus growth that minimizes these stresses. "The number of ridges in each set follows a Fibonacci series which according to Shipman and Newell is exactly the pattern predicted by their mathematical model", the article says.

*--- Allyn Jackson*

** "Calculating Doom," by John Allen Paulos. ****ABCnews.com, 5 May 2004. **

Mathematician Paulos subtitles this month's essay "examining a probabilistic doomsday argument." The essay mentions Bayes' theorem in probability, imagines a cosmic lottery machine to assess overall risk, and recommends *Anthropic Bias*, by Nick Bostrom.

*--- Annette Emerson*

** "A Numbers Game," by George Szpiro. Jerusalem Report, 3 May 2004. **

** "$1.5 million buys book that put world in its place," by Eric Adler. ***The Kansas City Star*, 1 May 2004.

The Linda Hall Library of Science, Engineering and Technology in Kansas City houses first editions of some of the most significant books in science history, and the library recently purchased *Narratio prima*, the full title of which is *First report to Johann Schöner on the Books of the Revolutions of the learned gentleman and distinguished mathematician, the Reverend Doctor Nicolaus Copernicus of Torun, Canon of Warmia, by a certain youth devoted to mathematics*, published in 1540. The Linda Hall Library posts a description and photographs of the rare book.

*--- Annette Emerson*

** "The Cryptography of ... Voting Machines," by Dana Mackenzie. ***Discover*, May 2004.

Mackenzie explains that electronic voting machines are not immune to flaws and fraud. These voting machines look and function much like ATMs, but the author points out that ATMs have built-in safeguards---paper receipts, ID cards, and camera surveillance. Voting machines have no such safeguards, as complete privacy takes priority in elections. Many people were not worried about this until this year when in Broward County Florida (home of the infamous 2000 Presidential paper ballot recounts) no one was able to do a recount of the electronic votes---some of which were blank. This led to a test in which eight computer security experts set out to discover whether electronic voting machines invited potential undetected mischief. Sure enough, weaknesses (in passwords and coding) were detected and exploited. Although some conclude that electronic voting is "relatively safe compared with the alternatives" (dependent upon good programmers and bad hackers), many feel more confident with a paper trail. The magazine lists several resources on voting technology and the controversy.

*--- Annette Emerson*

** "Gizmos: Enigma-E Electronic Enigma Machine," review by William Jacobs. ***Discover*, May 2004.

The original World War Two code-breaking Enigma machines had replaceable rotating wheels that encoders and decoders reconfigured to create and decipher messages. Some of the vintage machines are on display at Bletchley Park in England. Dutch electrical engineers Marc Simons and Paul Reuvers have designed a digital, electronic version that can be purchased and assembled. Jacobs briefly describes the product and informs readers where it can be purchased.

*--- Annette Emerson*

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