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Imaginary numbers in The Observer. The May 18 2003 Observer ran a review of "Imagining Numbers (Particularly the Square Root of Minus Fifteen)" (Allen Lane) by Barry Mazur, the Harvard professor. The reviewer, Jonathan Heawood, leads us past the square root of two ("The geometry is definitive, but the maths goes on for ever") up to the foot of Mt. Imaginary. ("Divorced from the geometric world of shapes and their properties, maths gestures wildly towards a sphere of the unimaginable.") This may not be exactly what Mazur had in mind. Heawood nicely picks up Mazur's metaphor "rather like discovering that there is an efficacious way of getting from Brooklyn to Boston, but that somewhere in mid-journey one has to descend to the Underworld" (for the use of complex numbers in real calculations) as a (unwitting?) paraphrase of G.K. Chesterton's "A merry road, a mazy road, and such as we did tread / The night we went to Birmingham by way of Beachy Head," and ends with the judgment: "I found that I needed more than pencil and paper to make these calculations. I needed a bigger brain. Yet, even without following all his workings-out, the window which Mazur cuts into the world of imaginary numbers is just as exciting, and almost as provocative, as anything in Philip Pullman."
The Golden Mean in The New York Times. George Johnson contributed an essay to the May 20 2003 Science Times: "Deep in Universe's Software Lurk Beautiful, Mysterious Numbers." The piece begins with a reference to "The Da Vinci Code," a current thriller. "In one of the novel's typically awkward moments" the hero, in full flight from his pursuers, "pauses to reminisce. The subject: a lecture he recently gave at Harvard on the remarkable properties of the number phi." This number (one plus the square root of five, divided by two) is the Golden Mean. Johnson gives the value of φ as 1.6180339; more than Dan Brown, the "Da Vinci" author, who leaves it at 1.618; neither mentions the number's irrationality, which is certainly part of its charm. In the book, φ is connected to goddess-worship via the 5-pointed star "representing the sacred feminine," etc. (The first few Fibonacci numbers also have a role to play). Johnson: "In a world otherwise crippled by math anxiety, books about phi and other so-called constants of the universe are multiplying so quickly that 'The Da Vinci Code' threatens to become part of a genre." He mentions works on φ and Euler's constant γ, taking for granted our familiarity with the recent popularizations of 0, i, π and e, and then veers off into the consideration of numbers in physics. A quick tour of the fine-structure constant, creation theories, counting with stones and the Pythagoreans leads us back to Eugene Wigner and the "phenomenon which might always remain a mystery," The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
The Poincaré Conjecture (cont.) The recent developments were also covered by Science, in an April 18 2003 piece by Dana Mackenzie whose title, "Mathematics World Abuzz Over Possible Poincaré Proof," correctly suggests his Variety-style approach to the subject. "Furthermore, what was to keep the surgeries, like plastic surgeries on a Hollywood star, from going on endlessly?" Nevertheless Mackenzie gives the best layman's guide so far to the history of the problem and to Perelman's innovations. An excellent presentation, ending in a lovely quote from Bennett Chow (UCSD): "It's like climbing a mountain, except in the real world we know how high the mountain is. What Hamilton did was climb incredibly high, far beyond what anyone expected. Perelman started where Hamilton left off and got even higher yet - but we still don't know how high the mountain is."
Dog bites math. Lee Dye's May 29 2003 science/technology column for ABCNEWS.com is entitled "Good Dog - Mathematician Explains How His Dog Understands Calculus." Dye is picking up an article in the May 2003 College Mathematics Journal written by Tim Pennings (Hope College, Holland, Michigan). Pennings apparently had been teaching the standard calculus max-min problem that starts "So Tarzan is in the quicksand, and Jane is across the river and down the bank a ways, and she's got to get to him as quickly as possible" in its most modern incarnation. He was out on the shore of Lake Michigan tossing a tennis ball in the water for his Corgi ("Elvis") to retrieve, when he noticed that the dog would run part way down the beach and then cut diagonally out to the ball; just like Jane on her optimal way to rescue Tarzan. Apple : Newton :: Corgi : Pennings. Could it be calculus? Pennings, the story goes, clocked the dog on land and in the water, measured the xs and the ys for various tosses of the ball and plotted the points on a graph. As he told Dye, "it turns out that all the choices he made were right in line, or very close, to the optimal choice." Now Elvis is "on sort of a canine lecture tour, helping Pennings explain calculus to students of all ages." The moral of the story, as Pennings tells it: "Advanced math does have practical applications. If you end up as an industrialist ... and you manufacture a certain item, you will need to come up with a formula that will tell you how many you need to manufacture to maximize your profit, or minimize your cost. And as Elvis would say, if he could speak English, that's calculus." Column available online.
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