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What Comes Next? NEWSFOCUS For PRESIDENT What Comes Next? arrangement of toothpicks, in which a new batch is added at each stage, centered on and One of mathematicians’ most beloved at right angles to the exposed tips of the pre- e e e e e ic c ic o i ic ic Web sites is getting ready for a makeover. vious batch (see figure). The picture that (Rank candidates o o o o h h h h in order of choice) h C C C C C The Online Encyclopedia of Integer emerges displays surprising fractal growth. t d d h h s n r t t 1 2 3 4 5 Sequences, established by Neil Sloane at “It’s got beautiful structure,” Sloane says. He Candidate A 1 2 3 4 5 AT&T Labs Research in 1996 and run and his AT&T colleague David Applegate largely as a one-man shop, is poised to go have written a paper on the sequence’s math- Candidate B 1 2 3 4 5 “wiki,” with 50 associate editors taking over ematical properties, and Applegate has con- Candidate C 1 2 3 4 5 much of the workload. tributed a movie of its geometric growth, The OEIS, or simply “Sloane” as it’s linked to its entry in the OEIS. Candidate D 1 2 3 4 5 known to sequence fanatics, is a database of Sloane set up the OEIS Foundation last Candidate E 1 2 3 4 5 nearly 200,000 lists of numbers—a mathe- year and transferred intellectual-property matical equivalent to the FBI’s voluminous rights to the nonprofit organization. With *No more than one oval per column *No more than one oval per candidate fingerprint files. Much as fingerprints give Applegate’s help, he plans to move the data- police a quick way to link a new crime to Who’s on first? Optical illusion (left) mimics a par- earlier ones, sequences enable researchers adoxical “everybody wins” outcome that can actually to make connections between mathematical occur in ranked-choice voting. problems that might otherwise go unno- ticed. The innocuous-seeming sequence 1, “But it may be more than coincidence that 2, 5, 14, 42, 132, …, for example, arises in a one candidate almost always handily beats huge number of different contexts, from four opponents.” Saari is similarly unsure. counting the arrangements of nonintersect- “The only way such an empirical result can ing chords inside a circle to enumerating happen is if people have remarkably similar secondary structure possibilities of RNA. To preferences, much more so than we would sequence fanatics, such “Catalan numbers,” expect in general society,” he says. as they’re known, are even more famous Regenwetter agrees there’s a lot left to than the ubiquitous Fibonacci sequence 1, 1, be explained. But it’s becoming clear, he 2, 3, 5, 8, 13, … . says, that “the empirical world is highly Sloane began compiling sequences in different from the picture that has generally 1965 as a graduate student at Cornell Uni- emerged out of the mathematical theory of versity. By 1973 he had 2372 of them, social choice.” which he published as A Handbook of Inte- ger Sequences. An updated edition, with 5487 sequences, appeared in 1995, with the of these “perfect parallelograms” to serve as help of Simon Plouffe of the University of sides of candidate parallelepipeds and Quebec, Montreal. But by then, Sloane was devised a computer algorithm to pick out the already moving online. The OEIS made its perfect ones. debut a year later, with a database of “The biggest surprise for me was how 10,000 sequences. quickly we got results,” Sawyer says. Their The OEIS “is one of the most useful tools first example has edges of length 271, 106, available online for the working mathemati- and 103 (see figure, left). In all, the computer cian,” says Doron Zeilberger, a combinatorial- search found 30 perfect parallelepipeds, with ist at Rutgers University, New Brunswick. “It Chewy. Database managed by Neil Sloan (top) edge lengths up to 3920. Some are tantaliz- also is a great tool for determining the novelty includes the “toothpick” sequence (1, 3, 7, 11, 15, ingly close to being cuboids, with one or two of a new sequence. If it is not in Sloane, it is 23, 35,...), shown here in color-coded steps. rectangular sides. Such findings “may spark most likely to be new!” even more interest in the perfect cuboid The Web site invites users to submit new base to a wiki format, giving each sequence problem,” Sawyer says. sequences or comment on existing ones. its own Web page, with new submissions Ezra Brown, a number theorist at Vir- Such contributions have fueled the data- moderated by a board of editors. The transi- ginia Polytechnic Institute and State Univer- base’s steady growth. In 2009 alone, the tion has hit a snag, however: Search-engine sity in Blacksburg, agrees. “The size of total increased by 18,709 sequences, or software in the “wikiverse” can’t yet handle Reiter and Sawyer’s smallest solution is very more than 50 a day. “Sequences are still sequences of numbers. surprising,” he says. “Apparently, easing the pouring in,” Sloane says. Once that technicality is overcome, restrictions that the faces be rectangles is Even as he edits the constant stream of Sloane expects the wiki format to be an more crucial than anyone thought.” new sequences, Sloane takes time to admire improvement. “It’s the correct mechanism Nonetheless, he notes, perfect cuboids, if some of the contributions. His latest favorite for handling the database,” he says. As for they exist at all, are a long way off: Comput- is the “toothpick sequence,” added in 2008 his anticipated reduced workload, “it’ll ers have looked at all possible bricks with by Omar Pol, an OEIS contributor from mostly be a relief. On the other hand, I’ll edge lengths up to 10 billion without finding Buenos Aires. The toothpick sequence reg- miss all the e-mails.” CREDIT (PHOTO): SUSANNA CUYLER a single one that’s perfect. isters the increasing size of a geometric –BARRY CIPRA www.sciencemag.org SCIENCE VOL 327 19 FEBRUARY 2010 943 Published by AAAS.
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