September 15, 2008
MathFest 2008, which was held in Madison, Wis., in early August, was highlighted in the "News of the Week" section of the Sept. 5 Science. Barry Cipra wrote four mathematical news briefs about this summer's MAA gathering of more than 1,400 participants. He focused on hinged dissections, efficient wrapping of spherical chocolates, a queens-and-pawns chessboard problem, and packing circles on a torus.
In "Shapeshifting Made Easy," Cipra highlighted the work of computational geometers who proved that any polygon can be transformed into any other polygon with the same area by cutting it into pieces connected by hinges. They proved that "it's possible to do mathematical dissections without falling to pieces," Cipra wrote.
"It was a surprising result to me, because I thought it was false," said Erik Demaine of the Massachusetts Institute of Technology, who described the proof at MathFest.
In the key insight, Demaine and his coworkers found a way to turn an arbitrary unhinged dissection into a hinged dissection. Their proof involved taking an unhinged dissection of one polygon, adding hinges, then subdividing the pieces and adding more hinges until the polygon could be contorted into an equal-area equivalent.
Although "the movement is magical," Greg Frederickson of Purdue University said, "you don't get very pretty dissections this way." Still, the results suggested a first step toward applications, particularly in designing robots that can reconfigure themselves. "Now the optimization begins," Demaine observed.
Optimization has already taken hold in the wrapping of Mozartkugel, the subject of Cipra's second piece, called "Sweet Inspiration." Nonetheless, Demaine and his colleagues set out to see if they could do a more efficient job of enclosing a piece of the famous Austrian chocolate, which is spherical, than with traditional square or rectangular foil wrappers.
Demaine and his "computational chocolatiers found that they could achieve a 0.1% savings over current practice with an equilateral triangle," Cipra wrote. The savings in area and perimeter are even greater if the wrapping has a three-petal configuration, embedded within the triangle. With its smaller area and shorter perimeter, such a trefoil wrapping would save foil and be cheaper to cut. See "Wrapping a Perfect Sphere."
In another talk at MathFest, Doug Chatham of Morehead State University described a classic chessboard problem: placing eight queens on a chessboard so that no two queens attack one another. Writing about this area of recreational mathematics in "A Royal Squeeze," Cipra described a variation of the problem in which Chatham and his collaborators introduced pawns that block the queens' lines of sight. How many more queens, they asked, do such pawns make possible?
Chatham and his crew demonstrated that each additional pawn permits an extra queen, provided the chessboard is large enough. "There's a lot of interesting theory behind these questions," Loren Larson of Northfield, Minn., told Cipra. "They're also nice programming exercises. They're good examples of backtracking algorithms."
In "Taking the Edge Off," Cipra wrote about packing circles into squares and other shapes. "Math has a lot to say about packing things together," he noted. "The abstract problem of cramming, for example, equal-sized circles into a larger square has applications as far-flung as error-correcting codes for digital communications and in the physics of granular materials such as sand."
But what if the square in question lacks edges? Cipra asked.
William Dickinson of Grand Valley State University, in Allendale, Mich., and several undergraduates have studied how packing works in such a borderless space. The space in question is a torus. Dickinson and the students classified the graphs that can result when lines are drawn connecting centers of tangent circles, then analyzed the ones that lead to the densest packings. For five circles, they came up with 20 ways to arrange the five circles on a torus.
"In general, it is very difficult to prove that a particular packing is optimal," said Ronald Graham of the University of California, San Diego. Working without boundaries may make proofs easier to come by, he said, "but that is just an impression."
Unfortunately, the two-page spread in Science magazine failed to mention the entity that brought so many mathematicians together for MathFest: the Mathematical Association of America.
Source: Science, Sept. 5, 2008.