This month's topics:
The "Fib" is a new poetical genre: a six-line 20-syllable poem where the line lengths (in syllables) are given by the first six Fibonacci numbers. This information from an article in the April 24 2006 New York Times, "Fibonacci Poems Multiply on the Web After Blog's Invitation," by Motoko Rich. The invitation initially came from Gregory K. Pincus, "a screenwriter and aspiring children's book author in Los Angeles," who posted the rules on his blog. The Times ran a photograph of Pincus next to one of his poems:
Rich tested poets and mathematicians for their reactions to the new genre. Annie Finch, a poetry professor at the University of Southern Maine, was enthusiastic: "Poets ... love constraints that allow the self to step out of the picture a little bit. The form gives you something to dance with so it's not just you alone on the page." Judith Roitman, a math professor at Kansas, reported that she "found the phenomenon pretty uninteresting." But then, we are told, she went on to write:
That's the title of Fenella Saunders' piece in the May-June 2006 American Scientist. The subject is what its authors --Mathias Weber (Indiana), David Hoffman (Stanford) and Michael Wolf (Rice)-- describe as "the first properly embedded minimal surface with infinite total curvature and finite topology to be found since 1776, when Meusnier showed that the helicoid was a minimal surface." (Their paper, which appeared in the November 15, 2005 PNAS, is available online). In fact the new surface is closely related to Meusnier's; it is properly described as a helicoid with a handle, or a "genus-one helicoid", and is asymptotic to the helicoid at infinity.
The helicoid and the genus-one helicoid. This picture shows a segment of a cylindrical core through each of the surfaces, which actually extend to infinity in every direction. Image courtesy Mathias Weber.
Saunders tries to start her readers off gently: "Dip a loop of wire into a soapy solution, and the film that covers the loop will be what mathematicians call a minimal surface." But soon we hear: "At any point, a minimal surface is maximally curved in one direction and minimally curved in the opposite direction, but the amount of curvature in each direction is exactly the same." The readers may have better luck with the project's interesting history. "Over a decade ago, Hoffman, with Fusheng Wei ... and Hermann Karcher ... , had created computer simulations of such handled helicoids, but an airtight demonstration of minimal surfacehood eluded them." They knew what it looked like, but they could not prove that it really was an embedded minimal surface. Saunders quotes Hoffman: "I think the information about how to solve this problem was lurking in the pictures all the time, but we just had to think about it for a long time and have the theory catch up with the evidence we had."Dynamics of Roach Congregation
"Group-living animals are often faced with choosing between one or more alternative resource sites." Thus begins the abstract of a paper published April 11, 2006 in the Proceedings of the National Academy of Sciences (105 5835-5840). The authors, a French-Belgian team led by Jean-Marc Amé and José Halloy, report on "an experimental and theoretical study of groups of cockroaches (Blattella germanica) tested in a circular arena ... with identical shelters." When the number of shelters is two, the phenomenon can be described by the graph below, giving the occupancy of shelter 1 as a function of shelter size. Until the size of a shelter is enough for the whole population, the roaches split between the two shelters. But as soon as there is room for everyone in each of the shelters, then the roaches all occupy one and not the other.
Occupancy of two shelters as a function of shelter size S for a fixed number N of individuals. When S/N is less than .5, both shelters are filled; for S/N between .5 and 1, the animals split evenly between the shelters; if S/N is 1 or more, all the animals congregate in one of the shelters. Adapted from PNAS 105 5835-5840, from which the equations below are taken.
This behavior is predicted by a mathematical model. First, the researchers determined from experiment that the probability Qi of an individual leaving shelter i varies inversely with the crowdedness (the ratio of the number xi of animals in the shelter to the shelter size S):
where θ, ρ and n are experimentally derived parameters. On the other hand, the probability Ri for an exploring cockroach to join shelter i decreases linearly with the crowdedness:
where μ is experimentally derived. These two laws can be combined into a system of differential equations:
(here xe is the number of unattached individuals) subject to the constraint xe + x1 + x2 + ... + xp = N if there are p shelters and a total of N individuals. The authors solved numerically for the steady states, which for p = 2 appear schematically in the graph above. Larger numbers of shelters give more complex bifurcation. The authors explain why this collective behavior gives the optimal outcome for each individual roach, and speculate that "the collective decision-making process studied here should have its equivalent in many gregarious animals ... . " This work was picked up in the April 6 2006 Nature Research Highlights.
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