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

This month's topics:

Random molecular tilings

A team of physicists and chemists at Nottingham reports, in the November 14 2008 Science, "a random tiling formed in a two-dimensional molecular network of p- terphenyl-3,5,3'5'-tetracarboxylic acid [TPTC] adsorbed on graphite." In the TPTC monolayer constructed by the team (led by the physicist Peter Beton and the chemist Neil Champness), the molecules only link in two ways, in a parallel or in an "arrowhead" configuration where the two backbones form a 60-degree angle. As a consequence each of the backbones is lined up along one of three global axes. Substituting a 60-120-degree rhombus for each of the molecules gives a planar tiling.

SEMicrograph morphs to tiling

The scanning electron micrograph of a TCPM molecular network and its interpretation as a rhombic tiling. Size of molecules about 12 Å = 1.2 nm. Image courtesy of Peter Beton.

Such a tiling presents itself (in 2 ways!) as the surface in perspective of a 3-dimensional array of cubes. (In their illustrations the authors enhance the optical effect by coloring each rhombus according to its orientation.) But there are exceptions: "topological defects" the authors observed where three neighboring rhombi enclose a triangle.

topological defect

A "topological defect" in the molecular layer: three rhombi border a triangle. Image courtesy of Peter Beton.

Besides creating "impossible" configurations of cubes these defects are energetically unstable; the authors recorded a defect migrating across the surface by swapping places with adjacent rhombi.

migrating defect-1 migrating defect-2

Two steps in the migration of that defect across the surface. Image adapted from Science.

Roach trigonometry

"Antipredator behavior is vital for most animals and calls for accurate timing and swift motion." This is the beginning of "Cockroaches Keep Predators Guessing by Using Preferred Escape Trajectories" in Current Biology for November 25, 2008. The authors (Paolo Domenici, David Booth, Jonathan Blagburn and Jonathan Bacon) "demonstrate that individual cockroaches (Periplaneta americana, a much-studied model prey species), keep each escape unpredictable by running along one of a set of preferred trajectories at fixed angles from the direction of the threatening stimulus."

experimental setup

The roach reacts to a puff of air (grey arrow) by running away (black arrow: escape trajectory), but at an unpredictable angle from the direction of the puff. Images from Current Biology, used with permission.

The authors report that statistically, each individual roach has a favored set of escape angles, roughly at 0, 30, and 60 degrees from the line of the puff.

escape
trajectories

93 escape trajectories for each of two different specimens; data from left-handed and right-handed experiments are aggregated as if they were all on the right.

As the authors remark, "The neural mechanism for generating these multiple ETs is completely unknown." This work was picked up in the New York Times science section on November 18 2008.

Math on the Fringe

 

When all plausible plot lines have been exhausted, whatever remains, however absurd, will be shown on television. Take the Fringe. A recent episode ("The Equation") involves using "an intricate pattern of flashing lights intended to create a suggestible state of hypnosis" to further the evil quest for a mysterious and powerful formula (seems to involve 8πε/c2); the bad guys drive an astrophysicist crazy trying to extract it from his brain; then they kidnap a child obsessed with composing a tune he can't finish. The good guys finally catch on: "Ben's piece is the equivalent of Dashell's mathematical formula! Curious minds often converge on the same idea. Newton and Leibniz without knowing each other independently invented calculus." And "Music is a mathematical language; chords have numerical values; the notes, quarter-notes, eighths, sixteenths, they're all just fractions, variables." Final dialogue between bad guys: "Seems crazy that some numbers can make a machine like this work." "Look around your home, your office, your kitchen. Numbers make everything work."

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