Math in a fly's eye Retina from a *Rough-eye* mutant fly at 35% of pupal life. In this mutation, ommatidia contain variable numbers of cone cells. Image from *Nature* **431** 648, used with permission. | This image illustrates the article "Surface mechanics mediate pattern formation in the developing retina" by Takashi Hayashi and Richard W. Carthew (*Nature*,October 7, 2004). The article addresses the general problem of how, during the growth of an organism, cell aggregates acquire their characteristic shapes. In certain organs, cell aggregates seem to organize themselves into structures that minimize their total surface area; since this is the same principle that governs aggregates of soap bubbles, patterns familiar from the behavior of soap films will turn up in the structure of cell aggregates. This observation goes back to D'Arcy Thompson (*On Growth and Form*, 1917). The main result of the article is a characterization of the bio-molecular mechanism that plays the part, for living cells, of the physical mechanism which allows two soap-bubble surfaces to fuse when the bubbles are juxtaposed. But on the way the authors give convincing evidence for the presence of a total-surface-area minimizing action in their experimental arena, the retina of the fruit fly *Drosophila*. The eye of a fruit fly is compound, made up of a regular hexagonal array of some 800 unit eyes, or *ommatidia*. The retina is correspondingly tiled: each ommatidium contributes a 20-cell structure, which includes four cone cells. The four cone cells are squeezed together at the center of the ommatidium in an array (see image at left) replicating the unique stable planar configuration of four juxtaposed equal-volume soap film cells. These arrays were first studied by Plateau, in a work cited by Hayashi and Carthew. For more evidence, the authors turned to the *Rough-eye* mutant strain of fruit flies. In *Rough-eye* retinas, an ommatidium may have from one to six cone cells (the image above shows ommatidia with 2, 3, 4 and 5), and the hexagonal tiling gets perturbed (hence *Rough-eye*). In every case, the cone cells replicate the stable soap-film array. The single cone has a circular cross-section, and arrays of six cones manifest all three of the possible stable six-cell configurations enumerated by Plateau | Ommatidium in a normal fruit-fly retina. Image from *Nature*, used with permission. | | Tradition vs. Modernity Samuel G. Freedman contributed the Wednesday "On Education" column to the October 20 2004 *New York Times*. His title: "Math's Tradition vs. Modernity Forms a Debatable Equation." The column gives a report from the trenches in the "math wars in America," specifically from a fourth grade classroom in Ossining, New York. There a visitor observed that one student (just transferred in from Catholic school) was able effortlessly to multiply 23 times 16 while the rest of his class were busy with yellow markers, coloring in multiples of two. "Jimmy had learned multiplication the old-fashioned way, with drills, algorithms and concepts like place-value. The rest of the students were using a curriculum called Investigations, one of the new constructivist models, which teaches reasoning out a solution." Freedman briefly characterizes constructivism ("so named because proponents say students learn better when they construct their own knowledge"), its supporters (the NCTM, the NSF and "the colleges and graduate schools of education") and its detractors ("College and university professors of mathematics and various sciences have stood against this new orthodoxy"). Ossining presents "a case history of how the constructivists are winning." Freedman describes the problems the district faced, how they sought help and how they evaluated competing curricula. The vast majority of the town's teachers were "more confident in their judgment, and more able to resist cant and dogma, in the humanities rather than in math." So they chose Investigations among the programs "approved by the national bodies." Freedman ends by remarking how much the teachers and the students seem to be enjoying the new program. "Yet it is impossible not to be haunted by the image of Jimmy doing 23 times 16 while everyone else was charting multiples of two, and not to wonder if he knew something nobody else in the room did." [As Freedman reports, when the visitor asked Jimmy how he had gotten his answer, "Jimmy offered her a shy, yearning face and said nothing." Readers curious about what is actually going on in fourth grade classrooms can take the G4 Mathematics Online Test prepared by the Texas Education Agency. TP] Topology and the Aharonov-Bohm effect The Aharonov-Bohm effect is part of the differential geometry of the physical world: the electromagnetic vector potential is a connection in the bundle of phases; as a charged particle moves through the field, its phase advances by parallel transport. If an electron beam is led around an enclosed magnetic flux, the resulting phase difference can be detected by an interference pattern. This is the "effect." Philip Ball, in the "Research highlights" section of the September 9 2004 *Nature,* picked up an article in the August 2004 *Physics Review Letters* which shows an interplay between the Aharonov-Bohm effect and the topology of knots and links. The article, "Aharonov-Bohm Effects in Entangled Molecules," by J. C. Kimball and H. L. Frisch, explains how molecules which are magnetic and conducting can show a change in quantum energy levels if they are non-trivially linked or knotted. If a conducting molecule links a magnetic one, then "this is a molecular version of the AB experiment: an electron traversing the first link circumnavigates the magnetic flux of the second link," in Ball's words. The energy shift depends on the linking number. If a molecule which is both conducting and magnetic is tied in a knot, "the energy shift then depends on the 'writhe,' a measure of the number of self-crossing points." Solve the equation, get the job National Public Radio's "Morning Edition" for September 14, 2004 reported that Google was running a mysterious ad campaign at the Harvard Square subway stop: three banners, all with the same incrutable message: "{first 10-digit prime found in consecutive digits of *e*}.com" (The same message appeared on a billboard along Highway 101 in Silicon Valley, shown below). Photo credit: Benjamin Tegarden and Kristina Chu. Image used with permission. | NPR reporter Andrea Shea was on the scene. She pulled a blank with a passing web designer ("Is *e* some sort of constant I should know?") but hit the jackpot with Josh Nichols-Barrer, a grad student in math at MIT. "You list out all the digits of *e*, and they're infinitely many of them; you look at the consecutive ten-digit strings and you see which ones are prime. A prime is ..." NPR inexplicably fades him out. Shea describes Nichols-Barrer as the perfect fit for Google: one of those "geeky enough to be annoyed at the very existence of a math problem they haven't solved, and smart enough to rectify the situation." Nichols-Barrer starts to tell us the number, but NPR fades him out again. Anyone else astute enough to figure it out could go to the website www.xxxxxxxxxx.com, where another puzzle was posed. That solution got you to a web page that asked for your résumé. Recording, and written paraphrase, available online. The ads came up again on the NPR Weekend Edition, Saturday, October 9. The number is 7427466391, according to Keith Devlin, NPR's Math Guy (he found it on Google). He explains: "High-tech companies in Silicon Valley and elsewhere have been using mathematical and logical puzzles since at least the 1950s in order to try and help recruit people and weed out personnel applicants. ... It's not clear that this really is an effective way of screening applicants, but it certainly does create an image of a company as being very demanding of intellectual talent." When pressed by Scott Simon for an example of a problem, Devlin comes up with one supposedly used by Microsoft: "You have to imagine an old-fashioned clock which has two hands on it. Now when it is at 12 o'clock the minute hand is directly on top of the hour hand. The question is, how many times a day does it happen that the minute hand is right over the hour hand, and how would you in fact determine the exact times of the day that this occurs." The solution is given on the NPR web version of the broadcast, along with three additional brain twisters. The original broadcast can also be heard there. Note: Devlin tells us that the Google site has been closed. [Here is where the mystery number occurs in the decimal expansion of *e*:
| *e* = 2.718281828459045235360287471352662497757247 0936999595749669676277240766303535475945713821 78525166427427466391932003059 ... ] | Tony Phillips Stony Brook University tony at math.sunysb.edu |