This month's topics:
Leonardo, Michelangelo and the elevated dodecahedronThe lantern atop the dome of the Sagrestia Nuova (San Lorenzo, Florence; in March 2010, before the restoration of the coronamento; the polyhedral structure of the "sphere," legible in this enlargement, is almost imperceptible from the street). The Sagrestia Nuova, essentially a mausoleum for Lorenzo and Giuliano de' Medici and two of their kinsmen, was built by Michelangelo between 1520 and 1534. The architecture, the interior decoration and the sculpture are considered among his greatest masterpieces. Photo: Tony Phillips. "When Michelangelo 'copied' Leonardo" was the headline in La Repubblica, January 17, 2013. The gilded bronze coronamento of the lantern above the dome of the Sagrestia Nuova had been removed, restored and replaced by a copy; the original was now on display in the crypt of the Medici chapels. As Laura Larcan tells the story (article, slide show) Florentines for the first time realized that the spherical element in the spire was actually an elevated dodecahedron. To make one you replace each face of a dodecahedron with a pyramid of equilateral triangles. It is combinatorially the same as the stellated dodecahedron (there was one on the pavement of Saint Mark's in Venice, perhaps due to Paolo Uccello when he worked there in 1425) except for the shape of the triangles; and in fact for San Lorenzo the triangles have been additionally flattened to make the solid more spherical (see below). Leonardo comes into the story because he had drawn an elevated dodecahedron as one of his illustrations for Luca Pacioli's De Divina Proportione (14961498, published in Venice, 1509); he may well have invented this object.
Hollow Elevated Dedocahedron, woodcut after a design by Leonardo da Vinci, from a copy of De Divina Proportione sold at auction for £110,000 on May 27, 2010. Image courtesy of liveauctioneers.com and Bloomsbury Auctions. Larcan quotes Vincenzo Vaccaro of the Soprintendenza per i Beni Architettonici di Firenze on the symbolism of this polyhedron. Leonardo's picture represents "the force inherent in the ether, in that space of which the dodecahedron is the symbol, that tries to expand, to emerge in every direction. But this image is too far from the familiar and reassuring sphere, allcomprehensive and allinclusive. Michelangelo borrows the image of force and expansion from the original design of Leonardo, but hides it using isosceles triangles which give the pentagonal pyramids a lesser height and make the polyhedron resemble a crystal that amplifies and decomposes light." [My translations TP] New prime in the newsOn February 5, 2013 Jacob Aron, writing on New Scientist, picked up a press release from the Great International Mersenne Prime Search (GIMPS): a new Mersenne prime had been identified by GIMPS volunteer Curtis Cooper (University of Central Missouri). As Aron put it: "The largest known prime number has just shot up to $2^{57,885,161}1$, breaking a fouryear dry spell in the search for new, everlarger primes." Aron describes GIMPS as a "distributed computing project designed to hunt for a particular kind of prime number first identified in the 17th century." These are the Mersenne primes, of the form $2^p1$, where $p$ is itself a prime. "The new prime, which has over 17 million digits, is only the 48th Mersenne prime ever found and the 14th discovered by GIMPS." What is the point? Aron quotes Chris Caldwell (Tennessee): "It's sort of like finding a diamond. For some reason people decide they like diamonds and so they have a value. People like these large primes and so they also have a value." But Aron adds: "Primehunting isn't a completely esoteric pastime though, as these numbers underpin the cryptographic techniques used to make online transactions secure." The discovery was also reported on Fox News, CNN ("You know you're a geek if you felt all warm and fuzzy inside when you read that headline"), the BBC ("'No practical use' for 17million digit prime number") and CBS News, who quote George Woltman, the creator of GIMPS: "It's analogous to climbing Mt. Everest. People enjoy it for the challenge of the discovery of finding something that's never been known before." "Chaos" in Le MondeA still from the trailer for "Chaos." The work is distributed under a Creative Commons license. Posted on Le Monde's Science blogs January 17, 2013 was "Discovering Chaos in animation" (Découvrir le chaos en dessin animé) by David Larousserie. Larousserie is pointing us to "Chaos," a recently released 9part video (the sections are independent) made by Aurélien Alvarez, Etienne Ghys and Jos Leys. It is indeed, as he tells us, "free, fascinating and spectacular." "And original, since it is made by a trio of mathematicians careful of their pedagogy and motivated by their desire to share their scientific knowledge with the greatest possible number of people." Larousserie remarks that "[the authors] themselves had to deal with the unpredictability of nature, notably the sensitivity to initial conditions, because of which two trajectories which start off very close can end very far apart, " and he quotes Alvarez: "We didn't want to cheat in our simulations. So we had to find the right parameters to get the trajectories we wanted." The series took two years to make (supported by the Centre National de la Recherche Scientifique and by the Ecole Normale SupérieureLyon). It can be viewed in English. [My translations TP] Is Algebra Necessary (Cont.)Last year Andrew Hacker, an emeritus professor of political science at Queens College, CUNY, contributed "Is Algebra Necessary?" to the New York Times OpEd page. (See this column for September, 2012). Hacker suggested in particular that instead of focusing on "equations used by scholars when they write for one another" the highschool algebra curriculum "could, for example, teach students how the Consumer Price Index is computed." This point is taken up vigorously by Berkeley professor Edward Frenkel, in "Don't Let Economists and Politicians Hack Your Math: Of course kids need to learn algebra," an article posted on Slate, February 8, 2013. Frenkel recounts some recent actual and proposed changes in the way the CPI is calculated and remarks: "What seems to be completely lost on Hacker and authors of similar proposals is that the calculation of the CPI ... is in fact a difficult mathematical problem, which requires deep knowledge of all major branches of mathematics including ... advanced algebra." In more detail, he explains how complicated it is to compute a single, accurate index measuring the cost of living from year to year. Since the prices of many different items are concerned, along with the prices "we also need to know the levels of consumption today and a year ago; economists call these 'baskets.' ... But consumption tends to change in part because our tastes change, but also in response to price variations. ... [E]ven to begin talking about this problem, we need a language that would enable us to operate with symbolic quantities representing baskets and pricesand that's the language of algebra!"
Tony Phillips 
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