|Mathematician's Solution to "$1 Problem" Wins National Award for Top Journal Articles
August 07, 2007
Contact: Bill Giduz
A $1 problem might not sound very daunting. But it doesn’t take much money to get a mathematician excited about a thorny problem!
It’s not uncommon in the world of mathematics for a leading expert to offer a “bounty” for the solution of a conundrum. The prize can be enormous. The Clay Mathematics Institute offers a million dollars for the solution of each of six famous problems. More commonly, though, the bounty offered is more modest.
Davidson College mathematician Michael Mossinghoff employed that insider notion to weave a clever double entendre and win an award for publishing one of the best expository articles of the year in The American Mathematical Monthly.
Mossinghoff’s Lester R. Ford Award recognizes his article in the May edition about maximizing the area of polygons. Rather than approaching the subject from a strictly theoretical angle, Mossinghoff explained it in a clever popular context.
He received the award August 4 at the annual “Summer Mathfest” meeting of the Mathematical Association of America in San Jose, Calif. Davidson’s math department actually brought home two of the total fifteen prizes given at the meeting. Mossinghoff’s departmental colleague, Assistant Professor Tim Chartier, was also honored at the meeting with the Henry L. Alder Award as one of three top new young mathematics teachers nationwide. An earlier article about Chartier’s award can be found here.
The problem that Mossinghoff addressed is not new—if you fix the diameter of a hexagon (six sides), how large can you make the area? Here, the diameter of a polygon refers to the maximum distance between two of its vertices. This problem was originally posed in the 1950s, and Mossinghoff learned about it at a math meeting in 2000.
Most people assume that the maximum area would be achieved with a figure shaped like a beehive cell—with all angles and side lengths equal. But that’s not true. Whereas equal angles and lengths provide the correct answer for maximizing polygonal figures with an odd number of sides, the problem is much more complex in shapes with even numbers of sides. “It turns out that no one knew the optimal shape when the number of sides was even, starting with ten sides,” Mossinghoff said. “It’s a problem that had to wait on more modern computational tools.”
Mossinghoff began by creating improved polygons in the even case with up to twenty sides, employing computers to optimize some complicated expressions for the area in these cases. “After that, I had a good amount of data to see some semblance of a pattern, and could design from it a method for building better polygons for an arbitrary even number,” he explained.
|Mossinghoff working in his office with a Davidson student.|
The article also treats a related problem in a similar way: how can you maximize the perimeter of a polygon if the diameter is fixed? Here again, the regular polygon is not optimal when the number of sides is even (not even for four sides!). It turns out that the most interesting case here is when the number of sides is a power of 2, so 4, 8, 16, 32, etc. The article shows how to construct a better polygon for each of these numbers.
As important to the magazine as his results, however, was the interesting way he presented it. An avid former coin collector, Mossinghoff wrote up the polygon problem as a real-world exercise in numismatics.
How, he asked, might designers of the country’s new Presidential one dollar coin create it to have a polygonal shape, fixed diameter, maximal area, and large perimeter? Mossinghoff noted that attention to those parameters are necessary to satisfy the needs of parties interested in the new coin – the Secretary of the Treasury, the vending machine industry, sculptors of the coin’s artwork, and federal law.
His introduction to the problem, and the conclusion of the article, explain the challenges clearly in lay language. The mathematics involved in determining the ideal shape get pretty deep pretty quickly, however, and involve geometry, calculus, and linear algebra.
|Mossinghoff received the award at the MAA's recent meeting in San Jose, California. (photo courtesy of Tanya Chartier)|
Mossinghoff earned his B.S. in mathematics summa cum laude at Texas A&M University in 1986, coompleted his M.S. in computer science at Stanford University in 1988, and earned his Ph.D. in mathematics at the University of Texas at Austin in 1995. He has taught mathematics at Appalachian State University and computer science at UCLA, and has taught both subjects at Davidson since joining the faculty here in 2002. He received teaching awards at both UCLA and the University of Texas. He teaches a wide variety of courses at Davidson, including cryptology, the theory of computation, calculus, discrete math, number theory, object-oriented programming, and linear algebra.
Davidson is a highly selective independent liberal arts college for 1,700 students. Since its establishment in 1837 by Presbyterians, the college has graduated 23 Rhodes Scholars and is consistently ranked among the top liberal arts colleges in the country by “U.S. News and World Report” magazine.
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Posted By: Bill Giduz