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## Math Digest |

- "Proof Reading": Review of
*The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching*edited by Chris Pritchard,*American Scientist*, January-February, 2004 - Review of
*Strange Curves, Counting Rabbits, and Other Mathematical Explorations,*by Keith Ball, New Scientist, 31 January 2004 - "Dead Right": Review of
*Alan Turing,*edited by Christof Teuscher,*New Scientist*, 24 January 2004 - "Number of the beasts,"
*New Scientist*, 24 January 2004 - "Dancing the quantum dream,"
*New Scientist*, 24 January 2004 - "Formula and Function,"
*Science*, 23 January 2004 - "A Different String,"
*Science*, 23 January 2004 - "Gardening by numbers," Nature, 22 January 2004
- "Pennies Falling from Heaven: Getting Inspiration---and Fun---With a Spin From a Master,"
*LocalTechWire.com*, 20 January 2004 - "Know thy neighbor,"
*New Scientist*, 17 January 2004 - "Maths muddle,"
*New Scientist*, 17 January 2004 - "Infinite Confusion": Review of
*Everything and More: A Compact History of Infinity*by David Foster Wallace,*Science*, 16 January 2004 - "Splitting Terrorist Cells,"
*Science News*, 10 January 2004 - "Destined for destruction,"
*New Scientist*, 10 January 2004 - "Ocean devotion,"
*The Age*, 9 January 2004 - "Cake-cutting perfected,"
*Nature Science Update*, 7 January 2004 - "Former IU and Olympic swim coach dies at 83,"
*The New York Times*, 5 January 2004 - "Arithmetic and the Candidates," abcnews.com, 4 January 2004
- "The Curious History of the First Pocket Calculator,"
*Scientific American*, January 2004 - "Funny Songs About Math and Furniture,"
*Popular Science*, January 2004 - "2003: Mathematicians Face Uncertainty,"
*Discover*, January 2004

** "Proof Reading": Review of The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching edited by Chris Pritchard. Reviewed by Peter Hilton and Jean Pedersen. **

This book, a joint venture of the Mathematical Association (UK) and the Mathematical Association of America (US), is a collection of articles about geometry. The reviewers "strongly recommend this book to all lovers of mathematics." An interesting feature is "Desert Island Theorems," which appear at the end of the first sections of the book and are essays about particular theorems in geometry that the essay authors could not do without (if they were stranded on a deserted island). The reviewers wish that there was more post-secondary geometry in the book, but write that "This book is a source of intellectual delight and can easily be dipped into during one's leisure moments."

*--- Mike Breen*

** "Bunny loops the loop": Review of Strange Curves, Counting Rabbits, and Other Mathematical Explorations, by Keith Ball. Reviewed by Ben Longstaff. **

This enthusiastic review praises the book for showing that mathematics "is deeply and gloriously satisfying". In ten chapters the book explores a divers range of mathematical topics in a way that is accessible to "anyone with a bit of calculus." The book does not just present polished results; as the reviewer puts it, "This is maths with the hood up and the engine running."

*--- Allyn Jackson*

** "Dead Right": Review of Alan Turing, edited by Christof Teuscher. Reviewed by Mike Holderness. **

This book is a festschrift from a 2002 celebration of the 90th anniversary of the birth of the mathematician Alan Turing. The book discusses Turing's best known work, on the foundations of computing, as well as his lesser known forays into neural networks and genetics. The book also contains reminiscences about Turing's time breaking codes at Bletchley Park. The reviewer calls the book "difficult" but seems to have found it valuable.

*--- Allyn Jackson*

** "Number of the beasts," by Emily Sohn. ***New Scientist*, 24 January 2004, pages 38-41.

This article discusses recent research showing how animals---including human babies---have an innate sense of number. This research could help to improve teaching strategies, unravel basic brain mechanisms, and even shed light on certain brain disorders.

*--- Allyn Jackson*

** "Dancing the quantum dream," by Paul Parsons. ***New Scientist*, 24 January 2004, pages 30-34.

This article discusses intriguing research that could prove to be the basis for quantum computers. No one has yet figured out how to build a quantum computer, but the research being done today demonstrates just how powerful such devices would be and has begun to lay the theoretical foundations for their development. In 1989, physicist Edward Witten made a crucial connection between the ground-breaking knot theory work of Vaughan Jones and the quantum world. Witten showed that the paths quantum particles take weave together into a braid that is amenable to analysis by the techniques Jones developed. This kind of braiding affects the quantum states of the particles, and those quantum states are exactly what encode the results of computations done in a quantum computer. But there is a catch: As Parsons puts it in the article, "braiding doesn't affect the observable properties of most particles, which means you can't tell anything about the braid---and thus the computation---from examining them." A possible way around this difficulty is to make use of a new type of particle called a "nonabelian anyon". Theoretical researchers who work in this area, such as Fields Medalist Michael Freedman, are convinced that nonabelian anyons must exist, but they have not yet been found. Some physicists are now on the hunt to find them.

*--- Allyn Jackson*

** "Formula and Function," in NetWatch, ***Science*, 23 January 2004, page 443.

The Wolfram Functions Site has over 80,000 functions and formulas, including Bessel functions and elliptic integrals. The site also has graphs and animations. Site co-creator Michael Trott says that there are plans to add more information to the site, such as how certain functions got their names.

*--- Mike Breen*

** "A Different String," in Random Samples, ***Science*, 23 January 2004, page 463.

Mathematicians Claude Gauthier and Samuel Gaudet of the University of Moncton (Canada) have invented a new musical instrument. Their Y-shaped guitar, called a tritare (or tritar), resulted from their research into *p*-series. Each string has three ends, which creates "unpredictable overtones" and colors. This short piece has a picture of Gauthier and Gaudet with their creation. News of the tritare also appeared in the 28 November issue of The Chronicle of Higher Education, noted in the January 2004 edition of Math in the Media , by Tony Phillips.

*--- Mike Breen*

** "Gardening by numbers," by John D. Barrow. ****Nature, 22 January 2004. **

Architect Charles Jencks has created a 12-hectare Garden of Cosmic Speculation in Scotland. The garden is not yet open to the public, but Barrow (Department of Applied Mathematics and Theoretical Physics, Cambridge University) walks the reader through the profusely-illustrated book of the same name (published by Frances Lincoln, 2003). Jencks's home features logistic mapping of panels, non-periodic Penrose tiling on carpets, asymmetrical room designs, spirals and more. Outside, the garden is displayed in crescents and strips of water that "can be seen to trace the characteristic patterns of the phase space of trajectories in the famous Henon strange attractor." Barrow describes wave designs in wrought-iron gates, Möbius twists, fractals, and declares the garden inspriational.

*--- Annette Emerson*

** "Pennies Falling from Heaven: Getting Inspiration---and Fun---With a Spin From a Master," by Eric Jackson. ***LocalTechWire.com*, 20 January 2004.

Jackson turns an encounter with Princeton mathematician John H. Conway into a lesson in innovation for people in businesses large and small. Aside from Conway's ability to demonstrate how pennies standing on end will fall---heads up---Jackson was struck by the traits that made Conway such an interesting and successful innovator. The behaviors that can be found in innovators in all fields are: playing, paying attention to the details others may miss, experimenting based on observations, and going out to garner the reward, "whether it be having some fun with their colleagues, achieving recognition for their work, or introducing a new profit stream in their business."

*--- Annette Emerson*

** "Know thy neighbor," by Mark Buchanan. ***New Scientist*, 17 January 2004, pages 32-35.

This article discusses the concept of "small worlds" in network theory. This concept is often described using the "six degrees of separation" experiment, which was carried out by psychologist Stanley Milgram in the 1960s. In the experiment, people were given letters to be sent to a recipient unknown to them. The people sent the letters to friends whom they thought might be socially closer to the recipient; those friends sent the letters to their friends, and so on. Surprisingly, most of the letters reached the recipient in just 6 steps---hence the phrase "six degrees of separation". In the 1990s, mathematicians Steven Strogatz and Duncan Watts investigated this phenomenon and found that, when they worked with mathematical networks that consist simply of points and links between the points, throwing in just a few links between widely-spaced points greatly decreased the number of steps needed to get from a point to any other. Their work spurred a great deal of subsequent research into network theory, and this research could have substantial implications for designing networks like the Internet.

*--- Allyn Jackson*

** "Maths muddle." News story, ***New Scientist*, 17 January 2004, page 5.

This brief story reports on the case of the media stir caused by an announcement by a 22-year-old student who claimed to have proved a famous mathematics problem. The student, a graduate of Stockholm University, wrote a paper purporting to solve Hilbert's 16th problem, and the paper was accepted by the journal *Nonlinear Analysis.* Soon thereafter, mathematicians who read the paper found glaring errors; the journal has retracted its acceptance. The student is standing by her work, the article says, but "she will have to work hard to mend her relations with fellow mathematicians, who believe that she should not have talked to the media before she had talked to them."

*--- Allyn Jackson*

** "Infinite Confusion": Review of Everything and More: A Compact History of Infinity by David Foster Wallace. Reviewed by Rudy Rucker. **

Rucker, author of two books on infinity and a member of the Department of Computer Science at San Jose State University, did not like the book one bit. His opinion is best summed up in the first paragraph of the review: "I fully expected to enjoy *Everything and More*. But it's a train wreck of a book, a disaster. Nonmathematicians will find *Everything and More* unreadable, and mathematicians will view it with, at best, sardonic amusement. Crippling errors abound."

Additional reviews of this book

*--- Mike Breen*

** "Splitting Terrorist Cells," by Ivars Petersen. ***Science News*, 10 January 2004.

Mathematician Jonathan D. Farley (MIT) "uses order theory to quantify the degree to which a terrorist network is still able to function" after a cell member is captured or killed. He contends that using a graph model with nodes is insufficient to model terrorist cells, as graphs do not take into account hierarchies of leaders and followers. Although his method of analysis may be better than other current methods that law enforcers, policy- and decision-makers could use, he acknowledges that his "break the chains" model could be modified to take into account additional and more complicated factors. Farley's research was published in the November-December 2003 issue of *Studies in Conflict and Terrorism*.

*--- Annette Emerson*

** "Destined for destruction," by Kate Ravilious. ***New Scientist*, 10 January 2004, pages 42-45.

Why did the highly sophisticated civilization of the Maya disappear so suddenly? This article discusses recent research that attempts to use fractal analysis to unravel conundrum. Finding that the maps of Mayan cities and towns exhibit a fractal pattern, two researchers calculated the fractal dimension and "soon discovered that other phenomena, such as the devastation caused by forest fires and war, can be characterised by a very similar fractal dimension". Such phenomena often exhibit "self-organized criticality", meaning that a small change can precipitate a dramatic event. The standard example is a huge shift in a sand dune precipitated by the addition of just one additional grain of sand. In a similar way, the thinking goes, a relatively minor event, such as a single war or a change of government, might have caused the collapse of the Mayan civilization.

*--- Allyn Jackson*

** "Ocean devotion," by James Woodford. ***The Age*, 9 January 2004.

The author, a body surfing enthusiast, consulted mathematician professor Neville de Mestre (Bond University in Queensland) on the mathematics of body surfing. De Mestre's analysis begins with research on how waves carry inanimate objects to shore, and then describes the optimum shapes and acceleration techniques for body surfers, depending on wave shape and water depth.

*--- Annette Emerson*

** "Cake-cutting perfected," by Philip Ball. ***Nature Science Update*, 7 January 2004.

Political scientist Steven Brams (New York University) and his unnamed mathematician and economist colleagues have determined an "efficient, equitable and envy-free" (and honesty-reinforcing) method of cutting a cake for two or three parties. The story was also reported in *The Guardian*, by science editor Tim Radford, 14 January 2004.

*--- Annette Emerson*

** "Former IU and Olympic swim coach dies at 83," by Frank Litsky. ***The New York Times*, 5 January 2004.

This obituary of James "Doc" Counsilman, "perhaps the most innovative swim coach in American swimming history," quotes Counsilman's recollection that after observing and photographing a freestyle swimmer he finally realized he needed to apply the Bernoulli Principle to the swimmer's technique. "Bernoulli's Principle, named for the 18th-century Swiss mathematician who first defined it, states that the higher the speed of a flowing fluid or gas, the lower the pressure, and vice versa. That precept became the basis of Counsilman's 1968 book, *The Science of Swimming*, which was printed in more than 20 languages and reprinted numerous times."

*--- Annette Emerson*

** "Arithmetic and the Candidates," by John Allen Paulos. ****abcnews.com, 4 January 2004. **

In this piece, subtitled "Shouldn't Every President Know Math? A List of Proposed Quiz Questions," Paulos advocates that debate moderators apply "let no candidate be left behind" by asking a politically neutral question involving some thought and calculation. Paulos then puts forth 10 questions regarding population, percentages, the Electoral College, approximation, medians, fractions of a budget, pollsters, for instance. He provides the answers in the column.

*--- Annette Emerson*

** "The Curious History of the First Pocket Calculator," by Cliff Stoll. ***Scientific American*, January 2004, pages 92-99.

The first pocket calculator, a mechanical rather than electronic device, looked like a pepper grinder and calculated as the user spun a crank. Called the Curta, it was invented by Curt Herzstark in the 1940s. This article describes the device (with some nice graphics), its history, and the fascinating saga of Herzstark, who survived the Nazi concentration camp at Buchenwald. The Curta was manufactured in Liechtenstein and sold for two decades before being replaced by battery-powered calculators. Herzstark died in 1988 at the age of 86.

*--- Mike Breen*

** "Funny Songs About Math and Furniture," by Scott Mowbray. ***Popular Science*, January 2004.

The "FYI" column recommends a CD by singer-songwriter Jonathan Coulter, whose songs--deemed witty by the reviewer--are about science and technology and include "Mandelbrot Set."

*--- Annette Emerson*

** "2003: Mathematicians Face Uncertainty," by Keith Devlin. ***Discover*, January 2004, page 36.

The question "What is a proof?" can be answered in two ways, according to Keith Devlin. On the one hand, it is "a logically correct argument that establishes the truth of a given statement." On the other hand, it is "an argument that convinces a typical mathematician." Devlin argues that the former definition is generally an "unattainable ideal" and points to three events of the past year as evidence. The first was the apparent proof of a result closely related to the Twin Prime Conjecture; a few weeks after its announcement, the proof was found to be in error. The second was an outline of the proof of the Poincaré conjecture that has yet to be accepted as correct after many months. Finally, the *Annals of Mathematics* has agreed to publish a 1998 proof of a Kepler conjecture regarding the most efficient way to arrange spheres, but with a disclaimer regarding its veracity. These complicated proofs demonstrate to Devlin that, except for fairly simple cases, an assertion really becomes a proof when the mathematical community accepts it as such.

See also the Letter to the Editor in response to this article. The Letter, from Steven Goldberg, appeared in the March 2004 issue of *Discover.*

*--- Claudia Clark*

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