f you take a wire, bend it to create a loop, and stick it in soapy water, what shape bubbles will you create?
This question has tormented mathematicians for 150 years.
A team of three researchers has come up with a solution, one it believes may even have some practical applications for solving problems with robots and computer chip design. In fact, the scientists believe, their work may pave the way to a new branch of mathematics: dimensional geometry.
In devising a solution, the three added thickness and weight to the equation, removing the problem from the ideal realm of mathematics and putting it into everyday world, in which objects have dimensions.
"Just as the subject of probability started from a gambler's question and the subject of geometry started from partitioning flooded land in the Nile River valley in Egypt, so the question of minimum surfaces with thickness and weight may lead to a new branch of mathematics." said T. C. Hu, a professor of computer science and engineering at UC San Diego.
Hu's quest for the soap bubble solution, published in today's issue of the journal Proceedings of the National Academy of Science, began in 1963. If you ask Hu why he has battled this problem, he'll give you a because-it-was-there answer.
"These kinds of question should be answered," Hu said. "The Plateau problem was dealing with the ideal, but everything has thickness and weight. So we solved that kind of problem."
Other mathematicians applauded Hu's efforts, saying the findings will help solve similar as well as more complicated problems down the road.
"This is quite significant. It provides a practical solution of a fundamental problem that's typical of many other problems," said Russel Caflisch, a professor of mathematics at UCLA. "It will serve as a method for constructing these solutions, and their solution will be a guide to solutions of more complicated problems."
"Nobody cares too much about soap films," Caflisch said. "Oh, I don't know, some people do. But it's a class of problems; minimizing the area of a surface."
Belgian physicist Joseph Antione Ferdinand Plateau first posed the soap bubble problem 150 years ago. It is not known why this question irked him so. In the years since, the brain teaser became known as the "Plateau problem," or the "minimal surface problem," a conundrum that mathematicians attempted to crack.
Andrew Kahng, an assistant professor of computer science at UCLA and a coauthor of the paper, said the problem boiled down to this:
"Given any curve, you want to know the minimum surface that spans the curve. Spanning the curve means that the surface covers all points of the curve and doesn't have any holes in it."
In 1930, mathematicians Jesse Douglas and Tibor Rado came up with a purely analytical solution: For any curve, they could provide a mathematical description of the surface of the minimum area defined by the borders of the curve. With their solution, however, you couldn't actually construct the surface. And it made sense only in the surreal world of pure mathematics because it was based on the premise that the minimum area - or the bubble within the wire - had no depth and no weight.
Thirty years later, Hu decided to tackle the same problem, but in real-world terms. He and a colleague devised the idea of constructing minimum surfaces that had dimension - an idea that was ignored until last year, when Hu, Kahng, and a UCLA graduate student, Gabriel Robins, now of the University of Virginia, dusted it off to solve the Plateau problem.
In their study, the three describe their method for constructing a solution for determining a given wire shape and a specified thickness of surface. Using a computer they showed how the method worked even as the surface's thickness diminished to almost nothing.
"We solved for the optimum thick surface [like an orange peel] and then progressively changed this dimension so the surface became thinner and thinner and smoother and smoother." Hu said, acknowledging that his approach offers an approximate rather than a precise answer.
While all this sounds fairly abstract, Hu believes it could one day be used to construct smaller computer chips and plan paths for robots to traverse.
"Suppose you want a robot to go from point A to point B in the quickest, safest possible way." he said. "Before, we had solution methods that could only come up with the ideal paths, which had zero width.
"Since robots have physical dimension and can stray slightly from the paths we plan for them, it is nice that we can now find the best possible path, which has a prescribed width," Hu said. "It's safer and more realistic."
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