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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

Electronic topology of annulenes

hopf link

trefoil

The Hopf link (top) and the trefoil knot (bottom) are two of the "Möbius annulenes" studied by Rzepa et al. These images show the electron density function, contoured at a threshold of 0.025 a.u. Images courtesy of Henry Rzepa.

The Journal of Physical Chemistry published last July "The Geometry and Electronic Topology of Higher-Order Charged Möbius Annulenes" (A113, 11619-11629). By "Möbius annulenes" Henry Rzepa, Paul Schleyer and their coauthors mean circular molecules with one or more twists in their structure. They concentrate on the π molecular orbitals (characteristic of electrons shared between atoms) and show that the "π-torus" these orbitals form allows the characterization of a linking number for the annulene: it "subdivides into a torus knot for annulenes defined by an odd linking number... and a torus link for those with an even linking number...". They use it to report "that several new charged annulene systems with higher-order half-twists of 3 and 4 are also possible." Along the way there is a discussion of the exact meaning of "twist." If the central spine of the molecule is planar (as in the mathematician's ideal line-bundle picture of a Möbius strip) then "the twist will be distributed as local torsions between adjacent vectors on the surface of the strip, each rotating about the center line. These local torsions must by definition sum to π [in the simplest case]." But when "these center lines encroach into three dimensions, the term twist acquires a subtly different meaning." In that case the sum of all the torsions plus the writhe gives (π times) the linking number. In their arguments, "twist" is exactly that sum. More images, with cool Java interaction, here on the Imperial College Chemistry web-site.

"The Joy of x"

Stephen Strogatz's column has been appearing every Monday since January 31 in The New York Times online. He planned to revisit mathematics from an adult perspective starting with Kindergarten, and has now gone from 1 + 1 = 2 to differential geometry; "The Joy of x" on February 28 was about solving equations in high school. The columns are living up to their promise: "to give you a better feeling for what math is all about and why it's so enthralling to those who get it."

Issey Miyake meets William Thurston

fashion show

Models at the roll-out of the Issey Miyake Fall-Winter 2010 ready-to-wear collection (Paris, March 5). "The collection, in an eye-popping rainbow of mostly high-tech fabrics, riffed on the idea of oblong shapes and loops, circles and spirals--Thurston's model metaphors for the universe." Image courtesy Issey Miyake USA.

"Fashion and advanced mathematics collide at Japanese label Issey Miyake" said the press release, datelined Paris and picked up by abc News/Entertainment on March 5, 2010. "On hand at the display was Cornell University Professor William Thurston, whose Geometrization Conjecture inspired Miyake designer Dai Fujiwara. Thurston's theory was described in the collection notes as 'a comprehensive vision of eight geometries that are sufficient to form an ideal shape for all possible three-dimensional topologies.'" "Professor Thurston, sporting a Miyake blazer for the occasion, said his discussions with designer Fujiwara had uncovered real areas of overlap between their seemingly disparate professions." Hear them both on this YouTube clip.

Fermat's last stand

The Harvard Crimson's article about "Summer's Theory of Inequality" published January 25, 2010 (referring to Lawrence Summer's "infamous speech in which he implied that the reason few women seem to excel in mathematics is genetic") brought up another example of the confluence of fashion and higher mathematics. "Austrian artist Peren Linn has designed jeans with Fermat's Last Theorem imprinted on them, to merge elliptic curves with feminine ones." Article by Jonathan D. Farley and Autumn Stone. See "Math meets Fashion" on Peren's website.

fermat jeans

Fermat's Theorem on a pair of Peren Linn jeans. Image courtesy of Peren Linn.

New math symbols

When Byron Cook, a computer scientist and researcher at Microsoft's Cambridge UK research laboratory, needed new symbols to present his work on the halting problem, he "enlisted the help of a friend, New York City-based artist Tauba Auerbach, a former professional sign writer whose artwork has played with language and technology." This from a NewsFocus item by Angela Saini in the July 24 2009 Science. His research involves relations. As we are taught, a relation R on a set S is no more and no less than a subset of SXS. But you can do some pretty fancy things with relations, and some of them are tedious to spell out every time they are used. One of Prof. Cook's concepts is the restriction of a relation R on a set S to the Cartesian product [with itself] of the image of the reflexive, transitive closure of R. For Auerbach's symbol

symbol 1

Prof. Cook has graciously supplied us the standard notation: it is the restriction of R to

teds 1

where

ted 2

and

Rm is defined inductively by

ted 3

and

ted 4.

Tony Phillips
Stony Brook University
tony at math.sunysb.edu