Math Digest

July 2006

"The Topology and Combinatorics of Soccer Balls," by Dieter Kotschick. American Scientist, July-August 2006, pages 350-357.

This article, which appeared in print at the time of the opening of the 2006 World Cup, takes a mathematical look at the design of soccer balls. The standard black-and-white soccer ball, consisting of 12 black pentagons and 20 white hexagons, has become an iconic symbol of the game recognized all over the world. From a mathematical point of view, the iconic image has three distinctive features: it consists only of pentagons and hexagons, the sides of every pentagon meet only sides of hexagons, and the sides of every hexagon alternately meet pentagons and hexagons. Taking these features as the definition of a mathematical soccer ball, the article analyzes the possible balls using tools from topology and combinatorics. If one specifies that only three polygons can meet at any vertex, then the only possible design is that of the standard soccer ball made up of 32 polygons. But if there is no restriction on the number of polygons meeting at a vertex, then there are infinitely many soccer ball designs that fulfill the three criteria listed above. It is a mathematical theorem that all such soccer balls can be generated from the standard one by a topological construction called a "branched covering".

Other articles on mathematics and soccer balls:
"Bending a soccer ball," by Ivars Peterson, Science News Online, 8 July 2006;
"Soccer 'Sphere' Kicks Off a Circular Argument," an interview with Keith Devlin on Weekend Edition, National Public Radio, 8 July 2006.

--- Allyn Jackson

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"Irving Kaplansky, Scholar: 1917-2006," by F. F. Langan. The Globe and Mail, 3 August 2006.
"Irving Kaplansky, 89, a Pioneer in Mathematical Exploration, Is Dead," by Jeremy Pearce. The New York Times, 13 July 2006.
"Irving Kaplansky--mathematician and author," by Keay Davidson. The San Francisco Chronicle, 2 July 2006.

Irving Kaplansky.

These obituaries are tributes to Irving Kaplansky's significant contributions to research in algebra and other fields. The articles also note that he authored many important texts, was elected a member of the U.S. National Academy of Sciences and the American Academy of Arts and Sciences, and served as President of the AMS. Family and colleagues are quoted on Kaplansky's career, his influence as a teacher, and his musical talent (he was an accomplished pianist). --- Annette Emerson

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"Benjamin Franklin hat's erfunden (Benjamin Franklin's got it)," by George Szpiro. Neue Zürcher Zeitung, 30 July 2006.

This article discusses Benjamin Franklin's version of Sudoku. Apparently the founding father whiled away his time at the Pennsylvania Assembly filling in 8x8 squares with numbers. A recent article in the Proceedings of the Royal Society analyzes Franklin's squares.

--- Allyn Jackson

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"Mathematical Theorist Frederick Mosteller," by Adam Bernstein. The Washington Post, 25 July 2006, page B6.
"Dr. Frederick Mosteller, 89; put statistics to a world of uses," by Bryan Marquard. The Boston Globe, 26 July 2006.
"C. Frederick Mosteller, a Pioneer in Statistics, Dies at 89," by Kenneth Chang. The New York Times, 27 July 2006.

Frederick Mosteller, a premier statistician and founder of Harvard University's statistics department, died on July 23, 2006, in Falls Church, VA at the age of 89. He wrote hundreds of papers and books, was president of several professional associations, served on scientific research boards, and was active in research until 2004. Mosteller received his Ph.D. in mathematics from Princeton in 1946. He began as a faculty member at Harvard in the social relations department, was chair of the statistics department from 1957 to 1971 and retired as chair of the department of health policy and management in 1987. One of his earliest papers was the first known academic analysis of baseball, which he wrote in 1946 after his favorite baseball team, the Boston Red Sox, lost the World Series to the St. Louis Cardinals. In 1961 Mosteller taught probability and statistics in Continental Classroom on NBC. In Inference and Disputed Authorship (1962), he and David Wallace applied Bayes' Theorem to identify James Madison as the author of some disupted Federalist Papers. He said that he first became interested in mathematics at Carnegie Mellon University when a professor showed him some sophisticated mathematical techniques. The Washington Post article quotes him as saying: "It was the most marvelous thing I had ever seen in mathematics.... It used mathematics that, up to that time, in my heart of hearts, I had thought was something that mathematicians just did to create homework problems for innocent students in high school and college."

--- Mike Breen

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"The Joy of Symmetry": Review of The Equation that Couldn't Be Solved, by Mario Livio, and Fearless Symmetry, by Avner Ash and Robert Gross. Reviewed by Justin Mullins. New Scientist, 29 July 2006, page 48.

These two books provide "a tour of the beautiful but mysterious world of symmetry", the reviewer writes. He finds that "Livio is a knowledgeable guide who reveals a part of the mathematical world of symmetry that most people would be unlikely to find on their own." The book by Ash and Gross is more challenging. The two authors "make one of the bravest attempts yet to guide the mathematical novitiate up the highest mountains in the world of symmetry."

--- Allyn Jackson

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"Dear Mom and Dad: Please Send Cookies And a New Calculator: Math Camps Spread for Kids Who Can't Get Enough," by John Hechinger. Wall Street Journal, 28 July 2006, page A1.

In this article, John Hechinger writes about the experiences of a few of the blossoming mathematicians at MathPath, a 4-week summer program in Santa Cruz, California where highly gifted 10–14 year olds work with renowned mathematicians. MathPath is unusual for the age of its campers: Most summer math programs are geared toward high-school-age students. It is also a highly selective program: some 71 out of 400 students who took the 8-question qualifying quiz were chosen for this July’s program.

Hechinger interviews both campers and their parents. Parents speak of their reasons for sending their children, including their own summer academic experiences as well as a desire to help their children "shine" doing something they enjoy. Many of these parents, Hechinger notes, are "highly-educated immigrants from China, South Korea, and other countries who don’t share native U.S. notions of carefree summer days."

The campers speak of their love of mathematics and feeling like they "belong" at MathPath. While they have the opportunities to play sports and participate in other activities, the students Hechinger interviewed were quite serious about studying mathematics. Referring to an opportunity to "goof off" with other kids who had finished their homework one evening, 14-year-old Alice Xie states, "I could go outside and play soccer instead of finishing my homework, but I'd be wasting my time. I came here to do math."

If you want to try it, a printable copy of the quiz taken by this summers’ applicants is available on the MathPath website.

--- Claudia Clark

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"New Life for Neural Networks," by Garrison W. Cottrell. Science, 28 July 2006, pages 454-455.

Cottrell summarizes research by G.E. Hinton and R.R. Salakhutdinov ("Reducing the Dimensionality of Data with Neural Networks," pages 504-507 of the same issue) that uses neural networks to convert data in a high number of dimensions to a more compact description. He gives the example of a helix whose one-dimensionality is hard to see simply by looking at its points' coordinates. Hinton and Salakhutdinov's method "learns" to encode data and to decode it. Their mapping is invertible, so that new data can be mapped into the low-dimensional space.

--- Mike Breen

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"All aboard, with a little help from Einstein," by James Randerson. The Guardian (UK), 27 July 2006;
"Getting passengers in their seats is as easy as E=mc²," by Ian Johnston. The Scotsman, 27 July 2006.

Both of these articles cite "Let Einstein help you board your plane," by Ben Longstaff (New Scientist, 29 July 2006), which reports that this is "the first practical application of Einstein's work outside physics." The research team used Lorentzian geometry ("usually used to describe the path of an object through the space-time continuum, which is governed by Einstein's theory of relativity") to illustrate passenger congestion. Johnston sums it up with "the key factor is to reduce the amount of contact between passengers, which cuts delays to a minimum." The researchers found it makes little difference whether airlines board passengers back to front or randomly (first-come, first-serve). Perhaps, Randerson notes, the best scenario is to board window seats first, followed by middle seat then aisle seat. A recent AMS Mathematical Moment discusses the mathematics of boarding. --- Annette Emerson

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"Celebrating Puzzles, in 18,446,744,073,709,551,616 Moves (or So)," by Margaret Wertheim. The New York Times, 25 July 2006.

Photo: Michael Taylor/Lilly Library, Indiana University

If you love puzzles, you might want to take a trip to the Lilly Library at Indiana University. That's where a portion of a collection of more than 30,000 mechanical puzzles has recently been placed on display. This collection has been donated to the library by former Hughes Aircraft vice president Jerry Slocum, who has been collecting and studying puzzles for more than 50 years. Writer Margaret Wertheim interviewed Slocum for this article in his private puzzle museum at his home. Slocum speaks about a few different puzzles in his collection. These include a Chinese rings puzzle, a recursive puzzle related to the Towers of Hanoi problem: the typical 9-ring version in his collection requires 341 moves to solve while the 65-ring version could theoretically be solved in the number of moves alluded to in the title of the article. Slocum describes this as a type of "disentanglement puzzle," one of 10 categories into which he has divided his puzzles. An original Rubik's cube, which falls into the "sequential movement" category, is now on display at the Lilly Library exhibit. Another puzzle in the exhibit is a compartment puzzle made by Japanese puzzle master Akio Kamei. This puzzle belongs to Slocum's "take-apart puzzles" category. In case you need further persuasion, Wertheim reports that, "In keeping with the spirit of the show, the drawers and cupboards that hold the puzzles will themselves be puzzles." For, as Slocum asserts, "With puzzles, there is really no substitute for trying them out yourself." You can read more about the exhibit online (and find out more about the museum location and hours of operation). --- Claudia Clark

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"Genialer Einsiedler (Brilliant hermit)," by George Szpiro. Neue Zürcher Zeitung, 23 July 2006.

This article presents a profile of the enigmatic Grigory Perelman, whose breakthrough work on the Ricci flow seems to have provided a way to prove two important questions in mathematics, the Poincaré Conjecture and Geometrization Conjecture.

--- Allyn Jackson

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"Grim Statistics," by Robin Mejia. Science, 21 July 2006, pages 288-290.

"In a soon-to-be-released statistical analysis funded by the U.S. State Department, scientists have put together the most credible figures yet on the tragedies that unfolded [during the 1991-2000 civil war in Sierra Leone]," writes Robin Mejia for this issue of Science. The statistics are indeed grim: More than 25% of the country's 5 million people were forcibly displaced, about one-quarter million had property destroyed, and approximately 140,000 citizens were assaulted or beaten.

The project was a joint effort of Benetech, a California company that "produces technology for social good," and the American Bar Association (ABA). Statistician Jana Asher, at the time a Ph.D. candidate at Carnegie Mellon University in Pittsburgh, PA, was hired to lead the survey teams. All in all, the teams conducted "face-to-face interviews with 3633 randomly selected households from across all of Sierra Leone's 150 chiefdoms," Mejia writes.

The obstacles were manifold. For one, the country, although not quite as large as South Carolina, contains many impassable regions: Mejia reports that "one survey team hiked for 16 kilometers to get to a site-after a 16-hour boat ride." In order to persuade individuals to respond, local staff suggested initial meetings with Sierra Leone chiefs prior to interviews with individuals living within their territories. These staff also suggested ways to deal with difficult subjects, including rape, and worked with Asher to find a way to determine the dates when crimes occurred to within a year.

Mejia notes that, while the confidence intervals are good-sized, "the numbers are more precise than what is normally available after a decade of civil fighting in a developing country." Raul Suarez de Miguel, who works with the Organization for Economic Co-operation and Development, describes the result as "the difference between having an idea that something occurred and having data structured" so that it can be used to identify the magnitude and location of abuses, as well as whether conditions are getting better or worse.

The survey results are available for the Special Court for Sierra Leone to use in legal proceedings against the persons indicted for war crimes. But some defendants might benefit from survey results, which apportioned responsibility for the crimes to various groups. Benetech Human Rights Program director Patrick Ball, who is also working on the project, is concerned that the numbers of defendants representing various factions do not match the percentages found by the survey. For example, the survey found that the Civil Defense Force (CDF) perpetrated 4% of the crimes, but 3 out of 13 defendants are CDF leaders. "Even the most casual observer of the statistics can see that the CDF is not responsible for the majority of crimes... That's false moral equivalence," Ball maintains.

Surveys of this type might be used to help future human-rights tribunals determine how to allocate resources by determining "proportional responsibility." Meanwhile Asher is working with the Science and Human Rights Program at AAAS where she is "reanalyzing data she gathered to document humanitarian needs." And she and the ABA will make a "needs map" for Sierra Leone available to nongovernmental organizations.

--- Claudia Clark

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"Major Math Problem Is Believed Solved By Reclusive Russian," by Sharon Begley. The Wall Street Journal, 21 July 2006, page A9.

Begley writes about the Poincaré Conjecture and Grigori Perelman's purported proof. A proof of the conjecture in a refereed journal earns the solver US$1,000,000 from the Clay Mathematics Institute. Perelman posted his proof in two papers on the non-refereed mathematics preprint server called the arXiv. The papers have not appeared in any journal and may never appear in a journal. Bruce Kleiner (Yale University) and John Lott (University of Michigan) have been studying Perelman's papers and their "Notes on Perelman's Papers" like the papers, also on the arXiv explains the proof in detail. Also a "complete proof" of the conjecture, based on Perelman's work, by Huai-Dong Cao and Xi-Ping Zhu was published in June 2006 in the Asian Journal of Mathematics. On July 25, 2006, John Morgan (Columbia University) and Gang Tian (Princeton University) posted on the arXiv their book manuscript that they say contains a complete proof of the Poincaré conjecture, based on Perelman's ideas.

A radio program on the same subject: "Mathematician May Have Solved 100-Year-Old Problem," an interview with Keith Devlin, ran on NPR's Weekend Edition. National Public Radio, 29 July 2006.

--- Mike Breen

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"Revisiting the Relevance of the Queen of the Sciences," by M. A. Sofi. Greater Kashmir (Srinagar, Inda), 18 July 2006.

Sofi, a professor of mathematics and dean of academics at Kashmir University, starts off with a quote about mathematics by Bertrand Russell on the beauty of mathematics, and proceeds to dispel some myths about the subject. He notes that the definition of doing mathematics as "working with numbers" has been out of date for nearly 2500 years. He offers that mathematics is "an important human endeavor, essentially a science of patterns---real or imagined---visual or mental, arising from the natural world or from within the human mind. Far from being too abstract to matter and besides being a uniquely human endeavor, mathematics helps us understand the universe and ourselves." Beyond that, Sofi suggests that "mathematics surpasses every other human endeavor in terms of the value it has brought to human life through science and technology, where the role of mathematics has been immense." He concludes by sayng that one must do mathematics to truly appreciate its fun, value, and beauty.

--- Annette Emerson

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"Integrating Mathematics." NetWatch, Science, 14 July 2006, page 151.

Larry Riddle of Agnes Scott College maintains a site with biographies of approximately 200 women mathematicians. One woman mentioned in this short article is Christine Ladd-Franklin who in 1876 applied for a mathematics fellowship at Johns Hopkins University under the name of C. Ladd. She received the fellowship even though the school was closed to women at the time.

--- Mike Breen

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"Small Movements: New Devices Help the Paralyzed," with guests John Donoghue and Krishna Shenoy. Talk of the Nation, on National Public Radio, 14 July 2006.

New research involves planting electrodes in monkey brains to record impulses of neurons. One paralyzed man with a similar tiny sensor implanted in his brain was able to move a robotic hand just by thinking about doing it (the brain signals drive decoding computer programs), and reportedly said that it only took a few days to learn how to control the devices using the sensor. The researchers said it is striking "how well the mathematical algorithms and lessons learned in the intact monkeys translate to paralyzed humans."

--- Annette Emerson

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"AI research dealing with complicated matter," by Dave Brooks. The Telegraph (Nashua, NH), 12 July 2006.

The article covers a conference at Dartmouth College celebrating the 50th anniversary of Artificial Intelligence (AI). In 1956 Dartmouth hosted the first Summer Research Project on Artificial Intelligence, raising the hopes of computer scientists and the general pubilc that one day computers and robots would be able to mimic human thought processes. While the discipline has contributed to the creation of robotic vacuum cleaners, the Big Blue chess champ, and the DARPA Grand Challenge self-driving car, the field hasn't really shed any light on "how people think, or what intelligence is." Mathematician Dan Rockmore notes that perhaps that is something to celebrate, knowing that "we're much harder to duplicate then we thought." Rockmore is currently working on a documentary about the 1956 conference.

--- Annette Emerson

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"Narratively Imaginative," by Xu Wei. Shanghai Daily, 12 July 2006.

Batman drives his Batmobile; Superman flies with the help of his superpowers, but Jac Weir, the hero in an about-to-be-released science fiction film, travels through space—and time—by way of a Möbius strip. The film, aptly titled Through the Möbius Strip, marks the first full-length 3D animated film to be made in China, reports Xu Wei, writer for the Shanghai Daily. The film, "based on a concept by Jean Giraud, a noted French comic artist, looks set to not only reap big profits at the box office," Wei writes, "but also ring a clarion call for the renaissance of [the] Chinese animation industry." Through the Möbius Strip was over five years in the making and had a budget of around US$18.8 million. Over 400 domestic and international cartoon professionals worked on the 3D CGI (computer-generated imagery). Some notable Hollywood professionals, including Star Wars actor Mark Hamill, took part in the production. Currently, the producers are searching for distributors to organize the film’s international release.

--- Claudia Clark

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"Geometric maps reveal hidden beauty of music," by Roxanne Khamsi. New Scientist, 7 July 2006;
"Exploring Musical Space," by Kulian Hook. Science, 7 July 2006;
"Calculated tones," by Richard Webb. Nature, 13 July 2006;
"Music's inner map revealed, with some help from geometry," by Gareth Cook. The Boston Globe, 31 July 2006.

Dmitri Tymoczko of Princeton University has devised a new mapping method using non-Euclidean geometry to visualize chord progressions to "shed light on why conventional and experimental music compositions both sound pleasant to the ear." All three articles summarize his report, "The Geometry of Muscial Chords," published in the 7 July 2006 issue of Science (page 72) and provide examples. Nature states that Tymoczko's system is the latest attempt to use mathematics to systemize the relationships of notes to answer questions of "how simultaneous notes combine to form aesthetically pleasing (or displeasing) harmonies, and how each harmony best evolves to the next one." --- Annette Emerson

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"Van Gogh painted perfect turbulence," by Philip Ball. news@nature.com, 7 July 2006.

Physicist Jose Luis Aragon has found that many of Vincent van Gogh's paintings have a pattern of brightness that closely resembles turbulence. Aragon and co-workers took digital images of some of van Gogh's paintings and determined the probability that "two pixels a certain distance apart would have the same brightness, or luminance." The team found that several of the paintings, for example The Starry Night, showed Kolmogorov scaling in their luminance probability distributions. (Kolmogorov did foundational work in turbulence; the scaling that bears his name refers to the probabilities of certain velocity differences between points in a fluid.) Aragon's team did not find turbulence in some of van Gogh's other, calmer, works, or in other artists' works.

--- Mike Breen

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"'No Child' Law Leads States to Weaken Student Tests, Study Says," by Paul Basken. Bloomberg.com, 5 July 2006.

Policy Analysis for California Education studied schools in 12 states, chosen for geographical and educational diversity, and found that fourth graders' math and reading proficiency rates were reported to be about twice as high as they actually were. In math, 65 per cent of fourth grade students were reported to be proficient whereas the National Assessment of Educational Progress test put the proficiency rate at 30 per cent. The study, which looked at data from 1992 to 2005, claims that states are "dumbing down" their tests to avoid high failure rates, which trigger penalties under the No Child Left Behind law.

--- Mike Breen

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"The Math Behind Pellicano's Code": Interview with Keith Devlin. Weekend Edition, National Public Radio, 1 July 2006.

NPR host Scott Simon leads the segment with "encryption is the sport of mathematicians," and with that introduces "Math Guy" Keith Devlin to give a mathematical perspective on the government's case against Hollywood private eye, Anthony Pellicano. Pellicano is charged with illegally wiretapping for his clients, and the government can't make its case yet because it hasn't been able to break the code on his computers. Devlin explains that Pellicano has used a widely available encryption method called Pretty Good Privacy (or "PGP"), developed by Phil Zimmerman in 1991. The email software package program is a very secure system, using an encryption algorithm and requiring a key or password set by the user. Pellicano has apparently added another layer of security on his hard drive. Without going into detail, Devlin explains that to decrypt the code there are several mathematical tools needed: prime numbers, number theory, and advanced, sophisticated analytic theory. --- Annette Emerson

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"The net reloaded", by Kim Krieger. New Scientist, 1 July 2006, pages 40-43.

This article discusses a rethinking of the mathematical models used to study and understand networks, in particular the Internet. Power-law models, widely used to describe the Internet, made news some years back through the startling prediction that the Internet could be crippled if hackers took down just a few key hubs. Power-law models spawned "scale-free" network theory, which became a fashionable area of study. "Now a growing number of biologists, mathematicians and computer scientists are complaining that the idea has been overhyped, and that the power-law pattern does not reveal anything fundamental about what makes networks tick," the article says. New models, such as "highly optimized tolerance", or HOT, have now been proposed for modeling networks like the Internet.

--- Allyn Jackson

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"The Extreme Sport of Origami," by Jennifer Kahn. Discover, July 2006, pages 60-63.

Since quitting his job as a laser physicist in 2002, Robert Lang now gets paid to simulate the folds of a car airbag and design medical implants; his triumphs include a six-legged five-horned beetle, a full-scale human being, and a Maine lobster---each crafted from a single, un-cut sheet of paper. He is a professional origami folder, and the key to his success is a self-written computer program that translates a shape into a series of fold lines which, when executed on a sheet of paper in the correct order, will result in the desired form. For Lang, the tricky part lies in the years of trial-and-error that can be needed to determine the correct folding sequence. While other folders appreciate the abilities of his program, his most formidable competitor prefers human ingenuity to computer-generated solutions: Satoshi Kamiya, a 23-year-old Japanese folding genius, produces complex creations by visualizing an object and "unfolding it in his mind," a process that can require over 200 steps.

--- Lisa DeKeukelaere

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