### Tony Phillips' Take Blog on Math Blogs

 Mail to a friend · Print this article · Previous Columns Tony Phillips' Take on Math in the Media A monthly survey of math news

# This month's topics:

## Leonardo, Michelangelo and the elevated dodecahedron

The 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 (1496-1498, 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, all-comprehensive and all-inclusive. 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 news

On 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 four-year dry spell in the search for new, ever-larger 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 $2^p-1$, 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: "Prime-hunting 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 17-million 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."