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This month's topics:

Summertime, when math losses come easy

An opinion piece by Ruth Peters in USA Today for June 18, 2006 bears the title: "Summer reading lists are OK, but math is where kids lag." Peters: "According to a study published in the Review of Educational Research, students lose about 2.6 months of grade-level equivalency in math skills over the summer." The study is presumably "The Effects of Summer Vacation on Achievement Test Scores: A Narrative and Meta-Analytic Review" by Cooper, Nye, Charlton, Lindsay and Greenhouse in that journal (66, 227-268, 1996), but the quoted statistic comes from the gloss on that study on the Johns Hopkins Cennter for Summer Learning webpage which reports the loss more precisely as one in "mathematical computation" -conceptual structures are more robust. Peters then reminds us of the economic consequences of these losses, referring to research by Richard Murnane and Frank Levy (presumably here) who "found that in the 1980s, high school seniors who did well in math earned more than those who did poorly by the time they were 24 years old. The difference was about $1.40 an hour on average for men and$1.80 for women." Peters ends by giving examples of the resources (interactive books and games, tutors, online programs) available today to help parents stem summer math losses. [The RER article mentions toward the end: "No study has examined the effect of summer break on students beyond the eighth grade," although the problem is known to persist through college. -TP]

Math at the World Cup

According to a news report in the June 15, 2006 Nature, it has been established mathematically that soccer goals are contagious, statistically speaking: scoring one goal increases the probability that your team will score more. Michael Hopkin, who write the piece, calls this "one of soccer's classic clichés," and attributes the result to Martin Weigel (Herriot-Watt University, Edinburgh) and his colleagues Elmar Bittner, Andreas Nussbaumer and Wolfhard Janke, all at Leipzig University. The four have posted a preprint on arXiv.org with the title "Football fever: goal distributions and non-Gaussian statistics." As they put it: "modifying the Bernoulli random process underlying the Poissonian model to include a simple component of self-affirmation seems to describe the data surprisingly well and allows to understand the observed deviation from Gaussian statistics." They analyzed "historical football score data from many leagues in Europe as well as from international tournaments, including data from all past tournaments of the 'FIFA World Cup' series" and concluded: "The best fits are found for models where each extra goal encourages a team even more than the previous one: a true sign of football fever." The group paid special attention to three German soccer leagues: the East German Oberliga, the West German Bundesliga and the women's league, the Frauen-Bundesliga. They found that their self-affirmation factor κ was higher for the East German league and highest of all for the women.

Gravitational waves in the news

The intense burst of gravitational waves at the moment of merger of two black holes. High resolution image. Image credit: Eric Henze, NASA

This image, minus the spurious starry background, appears about 14 inches tall on page 1 of the Science section of the May 2, 2006 New York Times. "Black Holes Collide, and Gravity Quivers" is the headline below. The story, written by Kenneth Chang, tells of the recent successful numerical simulation of the collision of two orbiting black holes and of the ensuing gravitational radiation. It is embedded in an account of the first six months of operation of the Laser Interferometer Gravitational-Wave Observatory (LIGO); black hole collisions are considered to be the only cosmic candidates for producing gravitational waves strong enough for LIGO detection. Chang gives a 2-dimensional analogy for the modern, geometric view of of gravity: "Imagine a rubber sheet pulled taut horizontally and then tossing a bowling ball and a tennis ball onto it. The heavier bowling ball sinks deeper, and the tennis ball will move toward the bowling ball not because of a direct attraction between the two, but because the tennis ball rolls into the depression around the bowling ball. In this two-dimensional analogy of space-time, one can also imagine a sudden collision of objects creating ripples that skitter across the sheet. Those are the gravitational waves LIGO hopes to detect." The simulation (report published in Physical Review D 73 104002) was carried out by Joan Centrella and her team (John Baker, Dae-Il Choi, Michael Koppitz, James van Meter) at the NASA Goddard Space Flight Center. It is a tour de force of applied mathematics. "The wave forms from the final merger ... demand full 3-D numerical relativity simulations of the full Einstein equations." All previous modeling attempts had crashed because of "pernicious numerical instabilities that prevented the simulation codes from running long enough to evolve any significant fraction of a binary orbit." (These quotations from the Phys Rev D article). What worked this time was an appropriate choice of corotational gauge, a sophisticated adaptive computation scheme triggered by curvature, and a gigantic parallelization effort involving 3 days of computation on 2032 processors.

The image above is just one frame from an animation posted on the NASA website. With each black hole measuring about 4 million solar masses, the animation represents about 5 minutes in real time. The red contour lines represent gravitational waves as manifested by fluctuations in the Ψ4 component of the Weyl tensor.

Nanoscale Minimal Surface?

"Mesostructured germanium with cubic pore symmetry," by the MSU chemists Gerasimos Armatas and Mercouri Kanatzidis, appeared in the June 29, 2006 Nature. The article describes a preparation of germanium resulting in "two three-dimensional labyrinthine tunnels obeying space group symmetry and separated by a continuous germanium minimal surface." The thickness of the walls of this germanium structure is given as one nanometer. The "minimal surface" separating the labyrinths is identified as the gyroid, a triply periodic surface first described by Alan Schoen in an NASA Technical Note dated May 1970. Schoen gives the (x, y, z) coordinates of a point on the surface in terms of complex integrals:

where θG = 38.0147740o approximately is calculated using elliptic integrals. (The 3-integral format goes back to Weierstrass; the specific was used, with 0o and 90o instead of θG, in H. A. Schwarz's 1865 construction of the first known triply periodic minimal surfaces). Armatas and Kanatzidis, on the other hand, use the much more simply defined level surface cosx siny + cosy sinz + cosz sinx = 0. What is going on? As David Hoffman explained to me, these two surfaces, although extremely close, are not the same.

The Gyroid (red) and the surface cosx siny + cosy sinz + cosz sinx = 0 (green) plotted together. Image: James T. Hoffman and David Hoffman, Scientific Graphics Project, used with permission.

The coincidence is mysterious. As I understand it, chemists start with the symmetry group, which they determine by Fourier analysis of transmission electron micrographs of their sample. From the symmetry group they calculate the equation of a periodic nodal surface as a Fourier series. Our level surface equation comes from setting the sum of the lowest order terms to zero. Taking more terms gives better approximations to the gyroid, but why this procedure leads to a minimal surface is, as far as I can tell, unknown.

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