River Gazette • Page 11
Math in Our Everyday Lives
“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?” asked Albert Einstein. It’s a remarkable question that captures the dual wonders of that discipline we call mathematics: its eternal mystery and its supreme practicality.
Photo by Barbara Woodel
Soup Cans and Soap Film
by Alex Meadows, Assistant Professor of Mathematics because the solution is somehow the best. Sometimes optimization problems do not have solutions. For example, if we eliminate the above requirement that the can hold 20 cubic inches of soup, then there is no cheapest can because smaller cans are cheaper and there is no smallest can. One of the great mathematical questions that existed at the turn of the twentieth century was a question about geometric optimization, called the Plateau Problem. Consider this physical experiment: “Start with a circle of wire that has been twisted, bent, and stretched into some new shape. If we dip it into soapy water and pull it out again, there will be a soap ﬁlm stretching across it. Physical tensions make the ﬁlm want to have the least possible area while still spanning the wire frame.” Here is the mathematical Plateau Problem: “You are given a bent circular curve in three-dimensional space, like the wire. There are many different possible two-dimensional surfaces touching the entire given curve, like attached sheets of plastic wrap. Show that there must be one that has the smallest total area.” Solving the Plateau Problem requires proving that there exists an optimal least area surface for every set of parameters, i.e. the given curve. It is similar to showing the existence of a solution to the soup can problem. The trouble is that instead of the solution being two numbers (the radius and height of the best soup can), the solution comes from an inﬁnite dimensional space of surfaces. Various precise versions of this question have been answered, starting in the 1930s with the awarding of the ﬁrst ever Fields Medal, and it has inspired vast generalizations and beautiful mathematics. Currently there are hundreds of mathematicians around the world (including undergraduates) working on problems in minimal surfaces (soap ﬁlms) and bubble clusters, with many questions both big and small still unresolved. To learn more about soap ﬁlms, visit my Web site at http://www.smcm.edu/users/ ammeadows/create/.
by David Kung, Associate Professor of Mathematics and Simon Read, Assistant Professor of Computer Science “Now boarding rows 25 to 30 … Now boarding rows 20 and up.” We’ve all heard this monotonous call as we patiently wait to board a plane. When your row ﬁnally wins this airplane boarding lottery, you shufﬂe down the gangway only to wait to get to the plane, wait while your fellow passengers stow their luggage, and wait some more for those in aisle seats to get up to allow the window and middle-seat passengers to slide into their cramped, coach-class quarters. The whole process seems to go on forever—couldn’t it be faster? Actually, it could! Using a mathematical model and a computer simulation, we compared a variety of boarding strategies. By putting passengers in a random order within their boarding group and running a large number of simulations (3,000 for each strategy), we determined the mean, best-, and worst-case boarding times. The results were surprising, given our experiences with the supposedly efﬁciency-conscious airline industry. The fastest way to get passengers onto a plane is to board them from the outside in, starting with the window passengers, then the unlucky middle-seat holders, and ﬁnally the aisle passengers. Such strategies save about ﬁve minutes on average, compared to the back-to-front strategies used by most airlines. Surprisingly, those back-to-front strategies actually get worse as you split the...