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

This month's topics:

Bees do it

They subitize. To subitize is to "estimate small quantities at a glance" --just as we can tell at a glance if there are four people or five seated at a table. But how about eight or nine? Jürgen Tautz and a team at the University of Würtzburg tested the same ability in honeybees. Their report, in PLoS One for January 28, 2008, was picked up in the January 30 Science, under the "Random Samples" rubric edited by Constance Holden. "Using a y-maze we found that bees can not only differentiate between patterns containing two and three elements, but can also use this prior knowledge to differentiate three from four, without any additional training. However, bees trained on the two versus three task could not distinguish between higher numbers, such as four versus five, four versus six, or five versus six." And they add: "To our knowledge, this is the first report of number-based visual generalisation by an invertebrate." In their concluding section, the authors cite recent research in (human) psychology which gives evidence that " due to attentional limitations, the number of items that humans can hold in their short-term memory and subsequently recall is four, or very close to it."

Congresswoman Waters disses mathematicians

It happened on the Bill Maher show (HBO, Fridays at 11) on February 20, 2009 (Episode 3, at 2:20). The discussion is about the future of the economy; Tina Brown has just suggested that the catastrophe might lead to some moral progress. Representative Maxine Waters (D-California) breaks in with "But we're not going to change that until we put some people in jail. [Applause. Cheers.] We have got to take the schemers who have conspired by hiring these mathematicians and others to come up with these exotic products that rip people off and put them in homes that they could not afford ..." The discussion then veers away from mathematics.

[A cursory Google search turns up the name of one David X. Li, who came up in 2000 with a Gaussian copula function that allowed "hugely complex risks to be modeled with more ease and accuracy than ever before" (this quote from "Recipe for Disaster, the Formula that Killed Wall Street" by Felix Salmon in Wired, February 23, 2009). And here is where mathematics meets the mortgage meltdown: "Li's copula function was used to price hundreds of billions of dollars' worth of CDOs [Collateralized Debt Obligations] filled with mortgages." As Salmon tells us, the problem is that financiers did not really understand the formula and underestimated its vulnerability to rare events. "And even then, the real danger was created not because any given trader adopted it but because every trader did. In financial markets, everybody doing the same thing is the classic recipe for a bubble and inevitable bust." But please, folks, it was not the copula function or even the CDOs who put people in homes they could not afford. They just allowed a lot of bankers and traders to think, for a time, that they had found the ultimate risk-free investment. The rest was competition, and greed.-TP]

Thanks to Jonathan Farley for bringing Rep. Waters' outburst to my attention.

Barry Cipra at the Joint Meetings

 

It's February again, and Science has a two-page spread of general-interest highlights from the AMS-MAA-SIAM annual meeting (issue of February 13, 2009).

  • The mathematics of redistricting. Cipra reports on research presented by Alan Miller (Caltech), who has developed a method to quantify the "bizarreness" of geometric shapes, as a way to evaluate the extent to which electoral districts have been gerrymandered. Can math help? Cipra quotes Richard Pildes, an expert on election law at NYU: "Math can give you tools for creating processes that are likely to lead people to feel that the process is fair and that the outcome is therefore something to be respected."
  • "Taking a Cue From Infinite Kinkiness." (Cipra's title) We look into research into billiards on a fractal-shaped table. Robert Niemeyer and Michel Lapidus (UC Riverside) are looking what happens when "a point-mass cue ball rattles around inside a shape whose boundary seemingly consists of nothing but corners." They're concentrating on the Koch snowflake.

     

    Casselman's Koch Snowflake

    The Koch Snowflake is this fractal set, produced by the infinite iteration of a process whose first two steps are an equilateral triangle and a six-pointed star. Image generated by UBC student Beau Skinner in Bill Casselman's 2003 graphics course, used with permission.

    As Cipra tells us, periodic orbits are easy to find inside the finite approximations to the Snowflake, but "are much more challenging in the infinitely wiggly real thing."
  • Longer hangovers. Here the topic is a big improvement on an old trick. It used to be that you could stack bricks at the edge of a table to get an arbitrarily large overhang: if you want overhang ln(n) you let your first brick hang (1/n) of the way over the base, and the second brick 1/(n-1) over the first, etc., all the way to the last brick which extends (1/2) of its length over the next-to-last. Looked at the other way, it takes about ex bricks to get extension x. Peter Winkler (Dartmouth) told the guests at the joint meetings how he and four collaborators have devised a completely new method (it uses vertical towers of bricks as counterweights) where extension x costs a constant times x3 bricks, a huge improvement.

 

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