Group Theory in the Bedroom, and Other Mathematical Diversions
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Brian Hayes is one of the most accomplished essayists active today―a claim supported not only by his prolific and continuing high-quality output but also by such honors as the National Magazine Award for his commemorative Y2K essay titled "Clock of Ages," published in the November/December 1999 issue of The Sciences magazine. (The also-rans that year included Tom Wolfe, Verlyn Klinkenborg, and Oliver Sacks.) Hayes's work in this genre has also appeared in such anthologies as The Best American Magazine Writing, The Best American Science and Nature Writing, and The Norton Reader. Here he offers us a selection of his most memorable and accessible pieces―including "Clock of Ages"―embellishing them with an overall, scene-setting preface, reconfigured illustrations, and a refreshingly self-critical "Afterthoughts" section appended to each essay.
fairly arduous algebraic process, made worse by awkward and verbose notation. Furthermore, trial and error is still required, because there is no guarantee that a ratio generated by the method will be factorable. Camus reports seven failures before he hits on the ratio , which can be factored as . It was Brocot, a century later, who found a better way. An Eminent Maker I finally tracked down Brocot’s memoir in the Revue chronométrique at the Mariners’ Museum Library in Newport News,
differently. Suppose the sum S of a given set of integers is an even number; then a perfect partition must have a sum exactly equal to . If S is odd, on the other hand, subsets that add up to either or are considered perfect partitions. In effect, there are twice as many ways to construct a perfect partition of an odd-sum set. (If I had known where to look and what to look for, I would have found this fact already stated in a paper by Borgs, Chayes, and Pittel.) After the article was published,
empty slot. For each name added, a dot is marked on the graph at the horizontal position corresponding to the filling factor at that moment and at the vertical position indicating the number of slots checked. The spray of gray dots superimposes forty repetitions of the process; the black line averages 100,000 trials. The resemblance between name search and hashing is worth noting because the performance of various hashing algorithms has been carefully analyzed and documented. Much depends on the
showing the phase of the lunar month. In the next chamber is an armillary sphere tracking the equinoxes, the solstices, and the inclination of the Sun . . . The next chamber is the Lifetime room—a single blank, featureless disk of soft stone that rotates once a lifetime, onto which you can carve your own mark. The final chamber is much larger than the rest. This is the calendar room. It contains a ring that rotates once a century and the 10,000-year segment of a much larger ring that rotates
also, and continue in the same way. After infinitely many middle thirds have been erased, does anything remain? One way to answer this question is to label the points of the original line as ternary numbers between 0.0 and 0.222 . . . (Note that the repeating ternary fraction 0.222 . . . is exactly equal to 1.0, just as 0.999 . . . in decimal notation is also merely another way of writing 1.0.) Given this labeling, the first middle third to be erased consists of those points with coordinates