Math Digest

Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Contributors: Mike Breen (AMS), Claudia Clark (writer and editor), Lisa DeKeukelaere (2004 AMS Media Fellow), Annette Emerson (AMS), Brie Finegold (University of Arizona), Baldur Hedinsson (2009 AMS Media Fellow), Allyn Jackson (Deputy Editor, Notices of the AMS), and Ben Polletta (Harvard Medical School)

A diagram showing the brief obstruction of the officer's view. Dmitri Krioukov provided the mathematical explanation to justify not being given a traffic fine.

This Month's Math Digest Summaries Posted here May 2012: , Mobile app of IBM's Math History Timeline, Turing and the Fibonacci sequence in nature, the Liar Paradox ...

Brie Finegold summarizes blogs on the best geometric shape for a doughnut, and a "toroflux".

After almost 50 years, Fernando C. Marques and André Neves have identified the doughnut whose shape minimizes the integral of the mean curvature squared. This quantity, called the Willmore Energy, is invariant under conformal transformations and is relevant to biologists who study cell membranes. Mathematician Thomas Willmore conjectured that the doughnut in question would have a small hole, whose radius was about 17% of its width. But it was only this February that Marques and Neves published their preprint detailing the proof on arxiv. In May, the authors will give a workshop on the details of their proof. For a summary of the results as well as a discussion of their context, see the latest entry at Disquisitions Mathematicae listed above, which references the Huffington Post Blog--the fourth post of Dr. Morgan's published on the Huffington website. Dr. Morgan is a professor at Williams College who studies minimal surfaces. Read his personal blog.

Trying to make the Hopf fibration come to life for your topology students? (The Hopf fibration allows one to view the three dimensional Euclidean space as a union of nested tori.) Maybe you should check out this blog post and buy a "toroflux". The toroflux is made of a string of thin metal, which is bent into a torus knot that can be completely flattened and put in your pocket. The torus knot wraps once around the "hole" and several times around the thickness of the torus. While playing with the toroflux may not give instant insight into mathematics, it would provide a kinesthetic experience of a typically intellectual concept. The toroflux seeks to minimize its energy (see the post above). But because it can pass through itself, the inner radius of the torus it would encompass is zero-- unless the manipulator of this toy were to thread a pole through the center of the toy. Then the toy grabs the pole and "rolls" down it, creating the illusion of a beautiful spherical halo. Walsh explains the physics behind kinetic properties of this toy, and some of its connections to topology.

The Goldbach conjecture states that every even prime number bigger than 2 can be written as the sum of two prime numbers (ex. 30 = 13 + 17). The statement is still a conjecture even after almost 300 years after its first statement. The weak Goldbach conjecture states that every odd number greater than 7 can be written as the sum of three primes (ex. 15 = 3 + 5 + 7). This statement follows from the Goldbach conjecture. Each conjecture has been verified for numbers up to 19 digits. Now Terry Tao (UCLA) has shown that you can write any odd number as the sum of at most 5 primes. He hopes to extend his proof technique to prove the weak Goldbach conjecture, but does not think that the approach will extend to the original conjecture.

In the same issue of the magazine (but at the other end of the mathematical spectrum) is "Math rules," in which Steve Mirksy (author of the monthly Anti Gravity column), in response to Ian Stewart's book In Pursuit of the Unknown: 17 Equations That Changed the World, looks at some additional equations not in the book, such as 20x + 10y + 5z = 0: "when attempting to use a vending machine that takes singles, you will have in your possession some integer numbers of 20s, 10s and fives but no ones--and, therefore, no candy."

This article offers a perspective on the low representation of women in the so-called STEM fields (science, technology, engineering, and mathematics). The authors argue that women are less creative than men and are therefore less successful in mathematically intensive fields. "[T]here is a broad consensus among experts and lay observers alike that, with the exception of creative writing and acting, men exhibit substantially more creative achievement than women," the authors write. To support this assertion, they offer a quotation from a "creativity expert" at Ashland University: "Where are the publicly and professionally successful women visual artists, musicians, mathematicians, scientists, composers, film directors, playwrights, and architects?" They also state that studies have found that males are more playful than females, more curious, and more willing to take risks, and that these characteristics are important in creative achievement. If colleges offered more opportunities to be playful, curious, and engage in risk-taking, the article suggests, more women might enter STEM fields.

Did the traffic cop really see Dmitri Krioukov run a stop sign? Although it may have appeared to be so, Krioukov convinced a California judge that a passing vehicle and the difference between linear and angular velocity had obscured the police officer’s view of reality. Using graphs and a four-page paper, Krioukov showed that from the vantage point of a patrol car parked on a perpendicular side street, the movement of a vehicle that quickly stopped and then accelerated would appear nearly identical to that of a vehicle that maintained a constant speed (and skipped the stop sign) if the officer’s view was briefly blocked at the exact time of the stop—which Krioukov's car was. End result of Krioukov’s mathematical handiwork: the judge bought the argument, and Krioukov avoided a $400 ticket.

The AP news release was picked up widely in the media (by San Francisco Chronicle, Washington Post, Boston Globe, San Jose Mercury News, among others), and covered by this blog: "Research paper saves UCSD scientist from $400 traffic fine," by Matt Stevens, Los Angeles Times Blog, 18 April 2012.

Krioukov's paper, "The Proof of Innocence," from which the captions were adapted, is posted on the ArXiv. (Images from the paper, courtesy of Dmitri Krioukov.)

A week after this story broke, another news item was posted in which the traffic court judge offered a different, and more mundane, explanation of her decision. Superior Court Commissioner Karen Riley told U-T San Diego that she listened to the physics argument but much of it went over her head. Riley says she found Krioukov not guilty because the officer who cited him wasn't close enough to the intersection to have a good view.

Two math projects won spots in the top ten of the 2012 Intel Science Talent Search. David Ding, of Albany, CA, won fourth place and a US$40,000 award for his project on Cherednik algebras. Anirudh Prabhu, of West Lafayette, IN, won seventh place and $25,000 for finding a lower bound for odd perfect numbers. The first-place prize of $100,000 went to Nithin Tumma for his project on how a particular protein aids in the development of cancer.

The world record holder for reciting digits of pi is Chao Lu of China, who recited 67,890 digits of pi in 2005. This short article is a summary of an article in the June issue of Cognitive Psychology, which explains his memorization technique. Lu set the world record by first assigning people and objects with pairs of digits and then constructing stories based on the associations. Five years later he could remember only 39 digits.

Earlier this month, IBM released a new iPad app, Minds of Modern Mathematics, "an interactive timeline of the history of mathematics and its impact on society from 1000 to 1960." The app is based on the IBM-sponsored exhibition, Mathematica: A World of Numbers…and Beyond, which first opened in 1961 at the California Museum of Science and Industry. Legendary husband-and-wife design team, Charles and Ray Eames, produced the exhibit. The app includes a high-resolution version of the widely distributed 2' x 12' Men of Modern Mathematics timeline that was published some five years later, as well as "an interactive timeline with more than 500 biographies, math milestones and images of relevant artifacts." (Note that, as was true in the original exhibition, Emily Noether remains the sole female mathematician in the bunch.) The app also includes nine 2-minute animated clips that were also created by the Eames team. Learn more about and download the app.

The IBM news release (picked up by some media) includes more details and a video demo.

The news section of Science reports on a large crowd sourcing project aimed at better understanding mathematical patterns that are found in nature. One pattern commonly found is the Fibonacci sequence, 1,1,2,3,5,8,…, where each number is the sum of its two preceding numbers. English mathematician Alan Turing, famous for the Turing machine, was fascinated with why natural features such as the number of spirals in which seeds grow on a sunflower follow the Fibonacci sequence. Late in his life he put forward a hypothesis that simple geometry constrains new organic growth so that over time an organism's features take on higher Fibonacci numbers. Now a project based at the University of Manchester is asking gardeners all over the world to grow sunflowers and count the number of seed spirals to test Turing's hypothesis.

See the website for more information about the project. (Image: Fibonacci poster, click on the image to see the poster which can be ordered free of charge from the AMS Public Awareness Office: paoffice at ams dot org, subject line: Fibonacci poster).

In this opinion piece, Paulos explores "a few classic puzzles related to lying and self-reference" using contemporary settings. He starts with the liar paradox. This results if a person, in effect, says, "This statement I’m making is false." If the statement is true, then it's false, and if the statement is false, then it's true. This paradox may also arise if two non-paradoxical statements are combined: for example, combine the statement that "Senator S says that Senator T's comment about the health care bill is false" with the statement that "Senator T says that Senator S's remark about the issue is true."

Paulos then considers a different puzzle using the example of a reporter with two sources, one of whom always lies while the other always tells the truth. If the reporter does not remember which one is which, and can only ask a single question, what question should the reporter ask to determine "if Senator S is involved in a certain scandal"? A more difficult problem arises if the reporter has three sources, one of whom always lies, one of whom always tells the truth, and one of whom sometimes lies and sometimes tells the truth. The article provides the solutions, why "complete liars can be as informative as truth-tellers."

Goofy laughs and spooky music abound in this lavishly propped and locationed hour-long documentary on the subject of string length, an episode of the BBC science series Horizon. We follow British comedian Alan Davies--a shaggy-haired, pot-bellied everyman--on his quest to find an accurate and definitive length for a piece of string purchased at his local hardware store. Davies' first answer--32 cm, obtained using a ruler at the very same store--is rebutted by mathematician Marcus du Sautoy, Simonyi Professor for the Public Understanding at Oxford. Du Sautoy insists that different rulers may give different measurements. To get the last word in standards measurements, Davies and du Sautoy visit the National Physical Laboratory, and Britain's measurement standards old (a meter bar made of 90% platinum and 10% iridium) and new (the distance light travels in one 299,792,458th of a second). Next, du Sautoy takes Davies to the coast of England to explain how the question of its length gave birth to the concept of fractal curves like Koch's snowflake, which they draw in the sand. "Because your string is a bit crinkly, a bit fractal-like," suggests du Sautoy, "that piece of string could actually be infinite in length." Of course, they agree, no real string could be truly fractal or infinitely long--eventually, zooming in will reveal not increasing complexity but the smallest measurable particles. So, Davies decides to measure his string "in the smallest units possible--and surely that means measuring in atoms". With that, Davies begins a journey into quantum mechanics. At Simon Langton School for boys, the amazingly animated physics teacher Becky Parker tries using a bag full of soccer balls to explain that objects on the quantum level can be in two places at once. Parker takes Davies to Johnny Hudson's lab at Imperial College London, where the ominously revealed interference patterns of a double-slit experiment convince Davies that in the quantum world, "position is not a good idea anymore". In a downward spiral of uncertainty about his own "blurriness at the edges", Davies seeks out MIT Professor of Engineering Seth Lloyd. In what appears to be a taxidermist's shop or a natural history museum, the supremely entertaining Lloyd introduces Schrodinger's cat, and does a nice job explaining how a broad interpretation of the term "observing" makes that thought experiment impossible to carry out. Lloyd takes Davies to yet more locations while explicating the fundamental role quantum theory plays in biology: it turns out that photons traveling all possible paths are essential to photosynthesis, and quantum tunneling may be key to the function of olfactory receptors. The documentary is larded with lots of neat facts--for example, Ms. Parker reveals the fact that the massive particles (protons and neutrons) making up the entire human race could be collapsed into a volume smaller than a sugar cube (surely at tremendous energetic costs)--as well as some entertaining visuals and good insights into math and physics. If this hour leaves you wanting more, check out the earlier episode of Horizon entitled "Alan and Marcus Go Forth and Multiply".

"Quantum Gravity in Flatland," by Steven Carlip. Scientific American, April 2012, page 40.

This complex physics article explains how physicists have attacked a complex problem in three dimensions by looking for clues from how things work in a two-dimensional world. While the concepts of quantum mechanics, which describes the behavior of the smallest building blocks in the universe, and gravity, which governs on a much larger scale, are well accepted within the physics community, physicists are still working to understand how the two concepts come together in spaces like black holes. Inspired by the tales of "A Square" in the two-dimensional world of Edwin Abbott's literary satire "Flatland," physicists pondered the representation of gravity in 2-D and discovered that a torus (a doughnut-shaped topological object with a flat internal geometry) was a powerful tool. In “Doughnutland,” as the author describes it, physicists are able to construct the concept of time at the nexus of quantum mechanics and physics, one of the key hurdles for understanding black holes and even possible wormholes in 3-D.

Mathematician Donal B. O'Shea (left), whose research area is singularities of high-dimensional manifolds, has been hired as the president of New College of Florida, a liberal arts and sciences honor college. O'Shea had been chair of the math department at Mount Holyoke College as well as the college's dean of the faculty and vice president for academic affairs since 1998. Monaghan writes that Mount Holyoke's former president said that O'Shea "has boosted science education at Mount Holyoke as well as the number of minority students and the representation of women on the faculty." (Photo: Courtesy of New College of Florida.)

Nature's 'Books & Arts' segment talks to Harvard mathematician Barry Mazur about the overlapping realms of mathematics and the imagination. Mazur recently edited a collection of essays on mathematics and narrative with author Apostolos Doxiadis. Mazur and Doxiadis asked mathematicians, historians and literary scholars to write about mathematics and narrative and then gathered all the authors to a conference in Greece. To make their essays more accessible the authors interviewed each other. The interviews are available online courtesy of the non-profit Thales + Friends, which works to bridge the chasm between mathematics and other forms of cultural activity.

Summaries of Media Coverage of Math

This Month's Math Digest Summaries Posted here May 2012: , Mobile app of IBM's Math History Timeline, Turing and the Fibonacci sequence in nature, the Liar Paradox ...

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|