Writing in the Feb. 13 Science, Barry Cipra highlighted presentations on fractal billiards, geometric gerrymandering, and overhanging bricks in his reports from the recently concluded 2009 Joint Mathematics Meetings in Washington, D.C.
Cipra's article "Taking a Cue from Infinite Kinkiness" was about research on the motion of idealized billiard balls on tables with a fractal geometry. Fractal billiards, Cipra noted, fits well with an experimental approach to mathematics in which computers explore complex phenomena.
Robert G. Niemeyer of the University of California, Riverside and advisor Michel L. Lapidus investigated how a point-mass cue ball rattles about within a shape whose boundary seems to be all corners. They sought examples of periodic orbits—finite-length trajectories that retrace themselves—on fractal shapes, such as a Koch snowflake.
In a paper titled "Periodic Orbits for Billiards on an Equilateral Triangle," published last year in the American Mathematical Monthly, Andrew Baxter of Rutgers University and Ronald Umble of Millersville University had classified the periodic orbits on an equilateral triangle. Niemeyer and Lapidus described their efforts to find periodic orbits in successive approximations of the Koch snowflake, which are formed from equilateral triangles.
"It's very cool," Victor Hugo Moll of Tulane University said. "It's a new angle on a very classical, well-studied problem."
"There's a lot of interesting behavior when we look at these things analytically," Niemeyer claimed. Eventually, studies of fractal billiards could, for example, provide insights into the ways sound bounces off ceilings and walls in a concert hall or off the seafloor.
Cipra's report "Can Mathematics Map the Way toward Less-Bizarre Elections?" focused on a special session devoted to geometric gerrymandering and the use of political redistricting algorithms. Discussions ranged from the "pie-in-the-sky theoretical to crust-on-the-ground practical" ideas and solutions, Cipra said.
Several presentations addressed the problem of gerrymandering—the shaping of districts to the advantage of parties in power. The problem "is much worse than it used to be," argued Richard Pildes of New York University. Gerrymandering "gives people the sense that they're not really in control of their democracy."
Mathematics, Pildes suggested, "can give you tools for creating processes that are likely to lead people to feel that the process is fair and that the outcome is therefore something to be respected."
"The mathematics of redistricting starts with arithmetic and geometry," Cipra explained. "Ideally, every district in a state would have an equal population and would be, in some sense, both 'contiguous' and 'compact.'"
Alan D. Miller of the California Institute of Technology proposed a method for quantifying the "bizarreness" (or compactness) of geometric shapes, which could be used as one criterion in producing redistricting maps and comparing alternatives. "You can use it to reject districts that are badly shaped," Miller said. He ranked the shapes of current Congressional districts according to shape, ranging from highly compact to convoluted.
Lawyer Sam Hirsch of Washington, D.C., a specialist in election law and voting rights, and Charles R. Hampton of the College of Wooster, in Ohio, offered other remedies. However, Hirsch said, redistricting is "ultimately a political problem."
The complex situation presents "contradictions out the wazoo," commented Kimball Brace, head of Election Data Services in Manassas, Va., and a member of the 2010 Census Advisory Committee.
Cipra's "The Joys of Longer Hangovers" focused on new results in a classical brick-stacking problem. In an invited address, Peter M. Winkler of Dartmouth College described how he and his coworkers found better ways to stack a large number of bricks to create an overhang.
Winkler, Mike Paterson (University of Warwick), Uri Zwick (Tel Aviv University), Yuval Peres (Microsoft Research), and Mikkel Thorup (AT&T Labs-Research) showed that a better way to get an overhang with a large set of bricks is to build what looks like a parabolic brick wall with jagged edges. Unexpectedly, such a stacking boosts the overhang from a multiple of the logarithm of n to a multiple of the cube root of n, where n is the number of bricks.
That "hadn't occurred to people," American Mathematical Monthly (AMM) editor Daniel Velleman of Amherst College said. "It's not obvious it's going to help. And the fact that it helps so much is surprising." Velleman has accepted their findings for publication in the AMM.
Moreover, the mathematicians then translated the problem of stacking bricks into a problem about random walks. "Adding a brick," Cipra reported, "spreads force in essentially the same way that taking a random step spreads the walker's probability of being at a given location equally in each direction."
Still, friction makes a real difference, Winkler said. "And real bricks have a lot of that—not to mention mortar!"