Reporting in today's issue of Proceedings of the National Academy of Sciences, the team describes a method of mathematically defining not just the surface dimensions of a soap bubble within a wire ring, but also the bubble's surface weight and thickness.
"We think there's a lot of power in the idea," said Andrew Kahng of the University of California Los Angeles.
T.C. Hu, a professor of computer science at the University of California San Diego, and Gabriel Robins of the University of Virginia collaborated with Kahng on the research.
The genesis of their work goes back roughly 150 years to Joseph Plateau, a Belgian physicist who, after dipping a wire ring in soapy water, asked the original question:
What is the shape of a soap bubble which has wire as its boundary?
For decades, no one could mathematically solve what became known as the Plateau Problem. Then in 1930, American scientists Jesse Douglas and Tibor Rado proposed a general analytical solution that ultimately earned Douglas the first Fields Medal, the equivalent of a Nobel Prize in math.
The problem with the Douglas-Tibor solution was that while it looked good in theory, it lacked a certain weight. In particular, it presumed that objects like wire-ring soap bubbles lack heft and thickness.
"In pure mathematics, the line between point A and B has no thickness and a point has no weight," said Hu, who has been intrigued by the Plateau Problem for more than 20 years.
The real world, of course, is different. A line, no matter how thin, separates space and a point, no matter how small, displaces something. Such realities are important when developing mathematical directions for, say, a robot.
"Suppose you want a robot to go from point A to point B in the quickest, safest possible way," said Hu. "Before, we had solution methods that could only come up with the ideal paths that 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, so that it is safer and more realistic."
In other words, it would be helpful if mathematical equations could take into account that 1-foot-wide robots can't pass through 11-inch-wide gaps.
The three scientists conducted their research using a computer image of a curved slab defined by a wire shape embedded along the inside wall of a cylinder. Initially, the image had a thick, heavy surface like an orange peel. Over time, they shaved the surface thinner and thinner by changing the numbers that described it. Eventually, the surface of the computer object neared zero thickness and the old solution to Plateau's problem.
Hu said the scientists have not completely solved the problem. "We get a very nice, very deep result (using computer graphics to confirm their methodology) but it isn't a precise answer. Pure mathematics requires precise answers and this is just an approximate answer. We don't actually describe (the solution), we simply get it."
Hu and his colleagues hope others will take the findings further, particularly in pursuing practical applications. Besides robotics, they say there may be practical benefits in computer chip design and architecture, where figuring in the theoretical weight and displacement of, say, a domed roof is an obvious advantage.
Adding factors like weight and thickness may also compel mathematicians to rethink some old notions and perhaps lead to a field Hu calls "dimensional geometry."
"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 of Egypt, so the question of minimum surfaces with thickness and weight may lead to a new branch of mathematics."
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