Brie Finegold summarizes blogs on a trick with coins and the Menger Sponge:
"Flipping Miracles (or Bar Bets to Amaze Your Friends)," by Colm Mulcahy. Huffington Post, 14 December 2012.
Start with one jar each of orange juice and lemonade. Ladle one cup lemonade into the orange juice. Then ladle one cup of the new mixture back into the lemonade. Is there more orange juice in the lemonade or more lemonade in the orange juice? Colm Mulcahy writes about variations on this theme. Here's another one of his challenges. Take ten seconds to observe about a dozen pennies spread out on a table. Ask a friend to mix them up without flipping any over while you cover your eyes. With eyes still closed, can you then separate the pennies into two piles both showing the same number of heads? The hint given (only via the title of the post) is that you are allowed to flip coins.
The great thing about both of these questions is that adults and children are equally able to answer them, and there is no real advantage for those knowing a lot of advanced math. Recently, when I led a professional development course for teachers , it was the kindergarten and first-grade teachers who solved these types of problems first. They were not afraid of using manipulatives to experiment with their conjectures. So get out your pennies, your marbles, and/or your index cards, before reading the post.
"The Surprising Menger Sponge Slice," by George Hart. Simons Foundation: Mathematical Impressions, 10 December 2012.
The Menger Sponge is a three-dimensional fractal formed by removing portions of a cube ad infinitum. George Hart quickly explains via animation the construction of the sponge and then discusses a particular cross-section whose appearance will most likely surprise you. There are many ways to slice a cube in half, the least well-known of which results in a hexagonal cross-section. So what happens when you slice the Menger sponge in half along its diagonal? This delightful six-minute video really exploits the power of 3-D printing, animations, and simple, well-thought-out explanation. Without even knowing what a fractal is, anyone can understand the punchline. Hart even explains with finesse why the cross-section appears as it does. This is one of a series of videos on the Simon's Foundation site called Mathematical Impressions. The videos are aimed at both the general public and mathematicians alike. So far, Hart has produced one per month starting this past August. November's post featured 3D printing. And yes, George Hart is Vi Hart's father. For a quick view of the new Museum of Mathematics that just opened in New York, MoMath, see Mr. Hart's preview, which has been viewed almost 40,000 times, posted the day after the opening. He helped design many of the exhibits. [For more on MoMath, see Ben Polletta's Digest below.]
--- Brie Finegold
"The Life of Pi, and Other Infinities," by Natalie Angier. New York Times, 31 December 2012.
To put an end to the year 2012 the Time's science section ran a piece on the concept of infinity, or rather infinities. The piece nicely illustrates that there are many different types of infinities, and these infinities arise in diverse fields of research such as mathematics, cosmology and even theology. The most common infinity is exemplified by the number pi, which has a never-ending tail of non-repeating digits to the right of the decimal point. A little more mind boggling is that recent cosmology studies suggest that our known universe is just a tiny fraction of an infinite universal fabric, which has some strange consequences. “If you take a finite physical system and a finite set of states, and you have an infinite universe in which to sample them, to randomly explore all the possibilities, you will get duplicates,” said Anthony Aguirre, an associate professor of physics at the University of California, Santa Cruz. "If I ask, will there be a planet like Earth with a person in Santa Cruz sitting at this colored desk, with every atom, every wave function exactly the same, if the universe is infinite the answer has to be yes." There could even be a universe, Dr. Aguirre said, "where the Nazis won the war."
See also: "An eternity of infinities: the power and beauty of mathematics," by Ashutosh Jogalekar. Scientific American, 22 January 2013.
--- Baldur Hedinsson
"Tim Gowers: Seed of Discontent," by Richard van Noorden. Nature, 20/27 December 2012, page 341.
Nature selected British mathematician and Fields Medal winner Tim Gowers as one of the "Ten People Who Mattered in 2012" based on his outspoken criticism of scientific publishing giant Elsevier. In a strongly worded blog post, Gowers took aim at Elsevier's high prices, coercion of libraries into buying unwanted journals, and opposition to open-access publishing, and announced that he would boycott Elsevier in the future. More than 13,000 scientists worldwide subsequently pledged a similar boycott, leading Elsevier to drop its support for a U.S. act that would prohibit requirements for open-access publication of U.S. government-funded research. Though Elsevier publicly disagreed with Gowers' charges, a senior Elsevier official noted that Gowers activism helped the company better understand the concerns of the mathematical community.
--- Lisa DeKeukelaere
"Juggling by numbers: How notation revealed new tricks," by Laura Gray. BBC News Magazine, 19 December 2012.
Mathematician Colin Wright's system for notating juggling maneuvers, known as Siteswap, not only provides jugglers with a code for describing and discussing their tricks, but also provides opportunities to try out new moves. Siteswap translates each ball toss into a number corresponding to how many beats the ball spends in the air, which is directly related to how high the ball must be tossed. Even numbers are tosses caught with the same hand with which they are thrown, odd number tosses change hands, and numbers are strung together in sequences to describe tossing multiple balls simultaneously. Wright developed Siteswap in the 1980s after becoming frustrated with the difficulty in recording a single juggling move on paper, and since then jugglers around the world have adopted his method. With a system like Siteswap for cataloguing tricks, jugglers were able to identify sequences that produced fun and novel tricks.
--- Lisa DeKeukeleare
"The maths of the pop-up tent," by Philip Ball. Nature, 18 December 2012;
"How the Slinky Buckles, by Evelyn Lamb. Scientific American, March 2013, page 28.
Pitching up a pop-up camping tent is pretty easy, but wrestling it back in the bag is usually much harder. Nature's News section and Scientific American have pieces on how a mathematical theory is used to describe the shapes that flexible rings found in pop-up tents can adopt. "We have found the best way to fold rings," says Alain Jonas, a materials scientist at the Catholic University of Leuven in Belgium, who led the research. Jonas and his team showed that buckled rings can be predicted accurately using a theory that invokes a single key mathematical concept. "There is a lot of interest currently in this kind of fundamental mechanical problem," says Basile Audoly, a mathematician at Marie Curie University in Paris. Jonas thinks that the results might also apply on the molecular scale for understanding the shapes of circles of DNA found inside organisms such as bacteria and other ring-shaped polymers. |
Images: Pierre-Olivier Mouthuy and Alain Jonas.
--- Baldur Hedinsson
"Opening the Doors to the Life of Pi," by Edward Rothstein. The New York Times, 13 December 2012.
At long last, the Museum of Mathematics (MoMath) has opened in Manhattan, where it lies down the street from The Museum of Sex, and squarely in the jurisdiction of the critical minds at The New York Times. This Museum Review by author Ed Rothstein--for whom Museum Reviews are a regular thing, although I can't recall ever reading one before--enthusiastically welcomes MoMath (not to be confused with MoMA, or, God forbid, MoSex) to the neighborhood, even as it unflinchingly dissects the good and bad points of the new museum (not to be confused with the New Museum). Rothstein's attitude towards mathematics is both sympathetic and understanding. "So profound are the effects of math for those who have felt them," he writes, "that you never really become a former mathematician ... but one who has 'lapsed,' as if it were an apostasy." So Rothstein is deeply supportive of MoMath's mission as a "proselytizing museum"--an aim it pursues with an abundance of spectacle. "This is not a museum, you might think," writes Rothstein, "it is a high-tech playground," full of screens, rides, interactive sculptures, and captivating sensory experiences. Many of the exhibits sounds incomparably cool--like the "Hyper Hyperboloid" (pictured), the "Square-wheeled Trike," and mathematician and music theorist Dmitri Tymoczko's "Harmony of the Spheres" (which I admit this article does not describe in detail).
Others are innovative and have serious growth potential--such as the "Enigma Cafe" and its expanding library of problems, and the programmable pixellated, illuminated floor of the "Math Square." The tricky part, observes Rothstein, is connecting these vivid experiences to illuminating expositions of the underlying math, and ushering museum-goers at all levels of mathematical maturity into the sometimes recalcitrant heart of mathematics: "the enthusiasm the subject inspires is not easily communicated and not readily discovered," he writes. "To attract the uninitiated, a display must be sensuous, readily grasped and memorable. Yet the concepts invoked are often abstract, requiring reflection and explanation." According to Rothstein, some of MoMath's exhibits succeed admirably at these tasks--such as the digital screen that amplifies and reflects visitors' images to turn them into fractals ("The Human Tree")--while others fall short. MoMath's innovative technique of providing multiple tiers of explanations--cued by cards visitors receive upon entering--seems to be only partially successful. But while addressing these challenges will require thoughtful tinkering, it does not require new construction. In the end, Rothstein seems to share founder Glen Whitney's optimistic hope that MoMath's sensual approach--one which offers an interesting counterpoint to the museum down the street--will succeed in spreading the word about the wonders of mathematics. Image of "Hyper Hyperboloid" courtesy of MoMath.
[Ed. Note: The Times's puzzle blog Wordplay posed some problems based on MoMath exhibits. WABC-TV (New York) did a video segment ,"Creativity at the Museum of Mathematics," about the museum. See also the video and online piece published by USA Today: "Can math be fun and cool?" by Bob Minzesheimer. (14 December 2012). The AMS's next Arnold Ross Lecture will take place at MoMath on April 15.]
--- Ben Polletta
"Know when your numbers are significant," by David L. Vaux. Nature, 13 December 2012, pages 180-181.
Nature has commentary by David L. Vaux, professor of cell biology at the University of Melbourne in Australia, on the use and misuse of elementary statistics in experimental biology. Vaux does not shy away from criticizing biology journals for the prevalence of basic statistical errors found in many published articles. Vaux is clearly concerned with improving the quality of elementary statistics in biology journals and illustrating the advantages of statistical analysis. He addresses this by laying out some basic guidelines for researchers and reviewers. He also summarizes the most important concepts in a very helpful statistics glossary.
--- Baldur Hedinsson
"Symmetry Wars": Review of I Died for Beauty: Dorothy Wrinch and the Cultures of Science by Marjorie Senechal. Reviewed by Philip Ball. Nature, 6 December 2012, pages 37-38.
In this article, science writer Philip Ball reviews I Died for Beauty, a book about brilliant, complicated, and controversial scientist Dorothy Wrinch, "a name that few now recognize and that is often derided by those who do." Mathematician Marjorie Senechal, for whom Wrinch was a mentor, "has written a sympathetic apologia" that asks why this woman, a member of the Theoretical Biology Club, and whose work was so highly thought of by some of her scientific contemporaries--D'Arcy Thompson, Irving Langmuir, and Harold Jeffreys among them--ended up "relegated to obscurity?"
"The too-easy answer is: Linus Pauling," who published a paper in 1939 that destroyed Wrinch's best-known work--the "cyclol theory" of protein structure--and with it, her career. However, Senechal's book reveals a series of other issues that may have contributed. There was "her magpie mind: seemingly unable to decide how to use her substantial abilities, Wrinch never really made important contributions to one area before flitting to another." There is some evidence that suggests Wrinch had a "problematic personality." Then there was her "talent for making enemies." But "Wrinch's central problem, it seems, was that, working at a time when most male scientists assumed that women thought differently from them, she seemed to conform to their stereotype: headstrong, strident and reliant on intuition rather than facts." And while the same could be said of Wrinch's arch-enemy Pauling, Senechal "sees injustice in the way Pauling's blunders…were forgiven, whereas Wrinch's were not." All in all, Ball finds the book to be "a gripping portrait of an era and of a scientist whose complications acquire a tragic glamour. It is a cautionary tale for which we must supply the moral ourselves."
--- Claudia Clark
"Poet of the infinite": Review of Henri Poincaré: A Scientific Biography by Jeremy Gray. Reviewed by George Szpiro. Nature, 6 December 2012, pages 38-39.
Columnist George Szipro gives rave reviews to Henri Poincaré: A Scientific Biography, a new book by John Gray. The book details not only Poincaré's famous conjecture from 1904, solved only in 2003 by reclusive mathematician Gregory Perelman, but also Poincaré's efforts in philosophy and physics. Szipro praises the book as a "triumph of readability," and notes that Gray delivers both the good and the bad about Poincaré. For example, the book details Poincaré's earlier conjecture, for which he subsequently found counterexamples, as well as the fact that the first version of the famous conjecture contained a serious flaw. Szipro notes in particular that portraying Poincaré "warts and all" is useful because it allows the reader to observe how science progresses as mistakes are corrected, and he posits that presenting only the finished product of scientific work would be "an injustice to the scientific process."
--- Lisa DeKeukeleare
"The Cooperation Instinct," by Kristin Ohlson. Discover, December 2012, pages 34-41, 77.
For the past 50 years or so, most evolutionary biologists have explained the existence of cooperation in light of Darwin's theory of natural selection with the theory of "inclusive fitness": the idea that "cooperative behavior arose from the individual's need to ensure the survival of the genes of close family members." But in 2010, biologist and mathematician Martin Nowak, along with legendary sociobiologist E. O. Wilson and mathematician colleague Corina Tarnita, published a paper in Nature in which they challenged this theory. Nowak had found the mathematics behind inclusive fitness to be "so unwieldy as to be useless." Instead, they promoted the idea that "cooperation was not merely the product of evolution but an engine, driving the process along with mutation and natural selection itself." In this article, Ohlson tells the story of how Nowak, currently the director of Harvard University's Program for Evolutionary Dynamics, and author of the recently published book SuperCooperators, arrived at this conclusion. In the process, she provides a primer on the prisoner's dilemma and its impact on Nowak's thinking about cooperation and the mechanisms that drive it.
--- Claudia Clark
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