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This month's topics:

The math of meniscus mountaineering

Mesovelia approaches a meniscus. Image from Nature 437 733-736, courtesy John W. M. Bush and David L. Hu.

Walking on water is a way of life for many species of insects and spiders. But when they need to get onto dry land, they face a problem: surface tension, the same phenomenon that allows these creatures to exist, makes water curve upwards at the shore; the inclined surface marking the edge between wet and dry is called the meniscus. For small insects (say, millimeter-sized) the meniscus appears as a perfectly slippery slope. If they try to walk up, they slide back down. But some species have developed a method that seems like magic: they adopt a special posture and slide up the meniscus. David Hu and John Bush of the MIT Mathematics Department have recently worked out the math that makes this possible. Their article, "Meniscus-climbing insects," appears as a Letter in the September 29 2005 Nature.

A diagrammatic version of the photograph above, showing the positive and negative meniscus pockets created by Mesovelia's three pairs of legs. Image from Nature 437 733-736, courtesy John W. M. Bush and David L. Hu.

Their analysis is based on the well known observation that "lateral capillary forces exist between small floating objects, an effect responsible for the formation of bubble rafts in champagne and the clumping of breakfast cereal in a bowl of milk." More precisely, they calculate that a body of buoyancy T at distance x from the wall is attracted to the wall by a force

F = ATe-Bx.

where A and B depend on properties of water and on the contact angle θ shown in the diagram. An insect like the water-walker Mesovelia faces the meniscus and exploits its three pairs of legs: it pulls up on the surface with the front and rear pairs as it pushes down with the middle pair. Even though the three sets of Ts must add up to something negative (the weight of the insect) the exponential advantage gained by the front legs being closer to the wall will propel the insect forward and up the hill. Where does the work come from? As the authors explain at the end, "by deforming the free surface, the insect converts muscular strain to the surface energy that powers its ascent." Many images and movies (recommended!) available at the project website.

Hermann Weyl, remembered in Nature

"Every great mathematician is great in their own way, but Weyl's way was particularly special." This sentence occurs towards the beginning of an "Essay" on Hermann Weyl in the October 20 2005 Nature, perhaps commemorating the 50th anniversary of his death. The author, Frank Wilczek (Center for Theoretical Physics, MIT), continues: "Unlike most modern scientists, ... Weyl surveyed the whole world. He sought truth and beauty with a discriminating and far-seeing eye." Wilczek gives us Weyl's mathematical genealogy (Gauss, Riemann, Dirichlet, Hilbert) and remarks how, more than either of his Institute for Advanced Study contemporaries Einstein and Von Neumann, Weyl embodied "the grand German literary and pan-European cultural tradition" of the 19th and early 20th centuries. He points to Weyl's Philosophy of Mathematics and Natural Science (1926, revised in 1949) as reflecting that tradition, and quotes for us one particularly striking passage: "The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time." The essay touches, without naming them, on Weyl's contributions to mathematics. But their scale is suggested by a quote from Sir Michael Atiyah's biographical memoir: "The last 50 years have seen a remarkable blooming of just those areas that Weyl initiated. In retrospect, one might almost say that Weyl defined the agenda and provided the proper framework for what followed."

A new topology for the internet

Science News for October 8, 2005 ran a short report by Katie Greene with the title "Untangling a Web. The Internet gets a new look." Greene is describing work to be published in the PNAS, in which John Doyle (Caltech) and his colleagues "offer a new mathematical model of the Internet." The conventional ("scale-free") model "indicates that a few well-connected master routers direct Internet traffic to numerous, less essential routers in the network's periphery." Doyle et al. prefer HOT models (the letters stand for Highly Optimized/Organized Tolerance/Tradeoffs), based on insights from biology and engineering. For the Internet, HOT modeling would predict "no ... central hubs and any highly-connected routers lie at the periphery." For security, a HOT model is clearly preferable to a scale-free one, since, as Greene puts it, "if one of those well-connected, outlying routers were taken out, Internet traffic would simply divert to another well-connected router." Whereas in the scale-free Internet, "a targeted attack on a central router could halt virtually all data flow." This is not completely hypothetical: as Greene reports, Doyle and his team have tested their model on Internet2 (an academic subnetwork whose map is known, and which according to Doyle is a "good representation" of the structure of the entire Internet). "The researchers report that their proposed model corresponds well to the structure of Internet2."

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