Proof Positive ﬁrst Written Formulation

RIGHT NOW induced learning disability.” study showing how sensory overload can iors themselves. (“I have yet to meet a Similarly, even though there is evidence hinder undergraduates’ ability to recall pregnant teenager who didn’t know bio- that sleep is important for learning and words. “It’s truly a brave new world. Our logically how this transpired,” he says.) memory, teenagers are notoriously sleep- brains, evolutionarily, have never been By raising awareness of this paradoxi- deprived. Studying right before bedtime subjected to the amount of cognitive cal period in brain development, the neu- can help cement the information under input that’s coming at us,” she says. “You rologists hope to help young people cope review, Jensen notes. So can aerobic exer- can’t close down the world. All you can with their challenges, as well as recognize cise, says Urion, bemoaning the current do is educate kids to help them manage their considerable strengths. lack of physical-education opportunities this.” For his part, Urion believes pro- debra bradley ruder for many American youths. grams aimed at preventing risky adoles- Teens are also bombarded by informa- cent behaviors would be more e≠ective if frances jensen e-mail address: tion in this electronic age, and multitask- they o≠ered practical strategies for mak- [email protected] ing is as routine as chatting with friends ing in-the-moment decisions, rather than david urion e-mail address: on line. But Jensen highlights a recent merely lecturing teens about the behav- [email protected] MYSTERIES OF MATH mately 1.6 to 1). Some analyses find the ratio in structures—most famously the Parthenon—built centuries before its Proof Positive first written formulation. More recently, scientists have found that the faces people find most beautiful are those in s academics work to under- of mathematics: curved spaces, from geom- which the proportions conform most stand the architecture of the etry, and modular arithmetic, which has to closely to the ratio. A universe, they sometimes un- do with counting. Taylor has spent his ca- The geometry-arithmetic connection cover connections in mysteri- reer studying this nexus, and recently explored by Taylor solves another puzzle ous places. So it is with Smith professor proved it is possible to use one domain to that has enticed mathematicians across of mathematics Richard L. Taylor, whose solve complex problems in the other. “It centuries. In 1637, French mathematician work connects two dis- just astounded me,” he says, “that there Pierre de Fermat scrawled in a book’s crete domains should be a connection between these margin a theorem involving equations like two things, when nobody could the one in the Pythagorean theorem (a2 + see any real reason why there b2 = c2), but with powers higher than two. should be.” Fermat’s theorem said such equations This is not the first have no solutions that are whole num- instance of finding bers, either positive or negative. Go ahead, in geometry an el- try—it is impossible to find three inte- egant explana- gers, other than zero, that work in the tion for a seem- equation a3 + b3 = c3. ingly unrelat- The French mathematician also wrote ed phenom- that he had discovered a way to prove enon. Schol- this—but he never wrote the proof down, ars during or if he did, it was lost. For more than 350 the Renais- years, mathematicians tried in vain to sance, seek- prove what became known as Fermat’s ing a math- Last Theorem. They could find lots of ex- ematical ba- amples that fit the pattern, and no coun- sis for our terexamples, but could not erase all conceptions doubt until Princeton University mathe- of beauty, fin- matician Andrew Wiles presented a gered the so- proof in 1993. called Golden His discovery made the front page of Ratio (approxi- the New York Times, but six months later, M.C. Escher’s Circle Limit III illustrates the concept of hyperbolic space. Although the fish appear to get smaller toward the edge of the image, in the non-Euclidean world of hyperbolic geometry, the white lines along all the fishes’ spines are actually the exact same length. Each fish is the same size as all the others, and an inhabitant of this world would have to walk an infinite distance to reach the circle’s edge. ©2008 THE M.C. ESCHER COMPANY-HOLLAND/ALL RIGHTS RESERVED 10 September - October 2008 This type of unconventional clock is commonly used to illustrate problems in modular arithmetic. This particular clock has 7 “hours”; each “hour” position holds a set of numbers that are congruent modulo 7, or separated by multiples of 7. For one particularly elusive set of equations, earlier mathematicians located an ingenious but labor-intensive solution method in modular arithmetic. Richard Taylor’s work finds a shortcut in geometry. the Times reported that another mathe- equation required counting one’s matician had found a mistake in the new way around the “clock” (see figure proof. In the spotlight and under pres- at right). In proving the conjecture, sure, Wiles called on Taylor, his former Taylor essentially mapped the direc- doctoral student, for help. Together, they tions for a massive shortcut. But the published the corrected proof in 1995, and proof was, as Taylor puts it, “a nice Wiles won the Fermat Prize for mathe- by-product” of his main work on the matics that year. Taylor won the same connection between arithmetic and prize in 2001, and last year shared the geometry. prestigious, million-dollar Shaw Prize for Taylor’s Shaw Prize co-recipient, mathematician Robert Langlands, had “It just astounded me been the first to realize, back in the 1970s, that instead of counting the number of that there should be a solutions for these equations, it would be from undergraduate calculus to advanced possible to get the same answer using graduate courses, his primary research is connection between geometry. Specifically, the shortcut lay in work on a kind of foreign-language dic- the symmetry of hyperbolic space, one of tionary that translates between these two these two things.” the mathematical concepts illustrated by mathematical domains—between the the artist M.C. Escher. In this weird, language of clocks and that of saddles. proving the 1963 Sato-Tate Conjecture, warped world, pairs of parallel lines Taylor describes the dictionary this way: which said that one could draw a curve break all the laws of Euclidean geometry “It says, ‘Here’s an equation. I’m going to that would predict the number of solu- and bend away from one another, and give you a problem about symmetry that tions for any equation describing an ellip- lines that, to the human eye, look wildly has the same behavior.’ ” He and other tic curve. di≠erent are actually the same length. number theorists are busily filling in more The conjecture itself was a brilliant in- The analogy commonly made is to a entries to illustrate patterns and perhaps sight, but applying it is labor-inten- horse’s saddle: viewed from above, a sad- solve still-unidentified mathematical sive—it relies on modular arithmetic, a dle appears as an oval, but its circumfer- mysteries. “We’ve done A and B so far,” he domain of mathematics that studies the ence is unexpectedly long because it says. “All the way up to Z is left.” relationships between numbers, often il- curves downward along the sides of the elizabeth gudrais lustrated using clock-like figures with horse’s body and upward toward the unusual numbers of hours (i.e., not 12). horse’s head and tail. richard taylor e-mail address: Counting the number of solutions for an Although Taylor teaches everything [email protected] ANNALS OF DE-MINING Man, Mongoose, and Machine tanding outside a Sri Lankan machines are too expensive. Dogs weigh harness to keep the mongoose under con- army base in the spring of 2007, enough to trigger mines, injuring or kill- trol and a video camera to record its find- SThrishantha Nanayakkara mapped ing themselves and their handlers. ings. Although the mongoose walks a few an entire minefield without once Nanayakkara, a visiting scholar at the feet ahead, the robot with its eight metal setting foot in it. Nanayakkara held a re- School of Engineering and Applied Sci- legs sets the pace. During the test run, the mote control and periodically made a note ences and a 2008-09 Radcli≠e Institute pair went back and forth across a 10-by- on his computer. A mongoose hitched to a fellow, picked an indigenous mongoose 10-meter plot, stopping whenever the robot did most of the work. for its temperament, size (roughly 2.5 mongoose detected a mine, which it indi- This unorthodox de-mining team kilograms, light enough to step on a mine cated by sitting up (as it was trained to avoids most of the traditional pitfalls. without detonating it), and sense of smell do). In a morning’s work, the mongoose Metal detectors give too many false (able to detect explosives three meters found every mine. alarms, fooled by bullets or other debris. away). He equipped his robot (roughly a The land mines in Sri Lanka—and Hand-held ground-penetrating radar meter long and half a meter wide) with a other war-torn nations—are both physi- Harvard Magazine 11.

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