Brie Finegold summarizes blogs on the best geometric shape for a doughnut, and a "toroflux".
"Math Finds the Best Donut," by Frank Morgan. The Huffington Post, 2 April, and "The Willmore conjecture after Fernando Coda Marques and André Neves," by "matheuscmss". Disquisitiones Mathematicae, 23 April 2012.
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.
"Playing with a Flowing Torus," by Dan Walsh. Dan's Geometrical Curiosities, 7 March 2012.
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.
--- Brie Finegold
"Wrinkled doughnut solves geometrical mystery," by Jacob Aron. New Scientist, 30 April 2012.
From the perspective of Riemannian geometry, there is no isometric embedding of the flat (square) torus into three-space: the curvature of the flat torus is zero, and the curvature of the curved torus - that old familiar donut - is, well, not. But "isometric" really only means "length-preserving", and Nash and Kuiper's embedding theorem says that we can obtain a length-preserving embedding of the flat torus into three-space if we are willing to compromise on its regularity, accepting an embedding smooth enough to admit tangent planes but not curvature tensors. Furthermore, this embedding can be as close as we like to any smooth embedding that is "short," or shrinks distances; in other words, it can be compressed into an arbitrarily small volume in three-space. Nash and Kuiper prove the existence of such C^{1} embeddings locally - chart by chart - making their visualization, and the exploration of their global properties, difficult.
Now, a team of mathematicians (V. Borrelli, S. Jabrane, F. Lazarus, and B. Thibert) from three laboratories in Grenoble and Lyon have visualized such an embedding, by converting the quasi-constructive technique of convex integration, pioneered by Mikhail Gromov, into an explicit algorithm. This algorithm begins with a short embedding of the torus into three space and introduces periodic wrinkles or corrugations along different directions. These wrinkles change the lengths of curves over the surface. For example, initial corrugations in the vertical direction, making the torus look like a cruller or perhaps a churro eating its own tail, increase the lengths of meridians without changing the lengths of latitudes too much. By carefully choosing the directions, amplitudes, and frequencies of successive corrugations, a limiting surface which is isometric to the flat torus and C^{1} is obtained. The chosen corrugations have successively decreasing amplitudes and increasing frequencies, and if wrinkles upon wrinkles, decreasing in size, ad infinitum, makes you think of fractals, you're partially right. The resulting embedding does have a fractal-like structure, but the fact that it's C^{1} prevents it from being irregular at small scales; to provide control of the first derivative, the amplitudes of the corrugations must decrease very fast - too fast for the embedding to have real fractal structure. This is all to the good - to admit tangent planes, the dimension of the embedded surface must be a whole number, namely two. But Lazarus and his colleagues believe that the normal vectors to the embedding, when rooted on the square torus in the plane, may trace out a fractal graph. Check out their page for more interesting explanations and pictures. (Image from "Flat tori in three dimensional space and convex integration," V. Borrelli, S. Jabrane, F. Lazarus and B. Thibert, Proceedings of the National Academy of Sciences.)
--- Ben Polletta
"Interview: Matt Parker on being a stand up mathematician," by Alasdair Morton. TNT Magazine, 20 April 2012.
Matt Parker brought his show, Your Days Are Numbered, to the Leicester Square Theater at the end of April, and this short interview makes me wish I'd seen it. Parker works part-time as a stand up comedian and part-time as a "Maths Communicator" (I'm not sure what it is, but it's definitely British), and he manages to fold some math, as we call it in the states, into his comedy. Your Days Are Numbered, he kills off the audience according to statistics on life expectancy and cause of death, as if they were a "normal population" aging at the rate of one year every minute of the show. Over the course of the show, one lucky audience member gets to "try their odds" with the Grim Reaper - "you never know when he will appear or how he will finish you," says Parker. The comedian was pleasantly surprised by how positive many statistics are - for example, the British lifespan is growing so fast that during the two hours spent watching his show, an audience member's life expectancy increases by 24 minutes. Parker avoids math aversion by making sure everything in his show is entertaining on its own. But why is it so hard to get people interested in math? "I find that most people base their impression of maths on what the 14-year-old version of themselves thought when they were forced to learn about trigonometry in Year 9," he says. "Very few of my opinions have remained unchanged since I was 14. When people take a second-look at maths now that they're adults, they find it much bigger, more interesting and easier than when they were an adolescent."
--- Ben Polletta
"For Women to Think Mathematically, Colleges Should Think Creatively," by Theodore P. Hill and Erika Rogers. The Chronicle of Higher Education, 20 April 2012, page A25.
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.
--- Allyn Jackson
"Now that's calculated! Physicist writes four-page math paper to beat $400 traffic ticket," Daily Mail, 16 April 2012, and
"Victory of science over traffic ticket debunked," by Pat Flynn. U-T San Diego, 23 April 2012.
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.
--- Lisa DeKeukelaere
"Cancer research wins top prize in science contest," by Devin Powell. Science News, 7 April 2012, page 10.
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.
--- Mike Breen
"Pi master relates secrets of recall," by Bruce Bower. Science News, 7 April 2012, page 12.
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.
--- Mike Breen
"New IBM App Presents Nearly 1,000 Years of Math History," by Alexandra Chang. Wired, 6 April 2012.
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.
--- Claudia Clark
"Turing's Ideas Blossom," Random Sample, Science, 6 April 2012, page 18.
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).
--- Baldur Hedinsson
"Logical Liars, Paradoxical Politicians," by John Allen Paulos. ABC News, 3 April 2012.
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."
--- Claudia Clark
"How Long Is A Piece Of String?," a documentary produced and directed by Rob Liddell for BBC Horizon, WGBH-TV, 2 April 2012.
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".
--- Ben Polletta
"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.
--- Lisa DeKeukelaere
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