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Such chains may be closed, like a polygon, or open-ended, like a segmented arc. One important group of problems concerns chains made up of line segments. Over the years, studies of robotic arm movements have suggested purely mathematical questions about morphing one geometric shape into another. 'It's simply a natural question to ask and a beautiful problem,' insists computer scientist Joseph O'Rourke of Smith College in Northampton, Mass. The puzzle's real appeal has been aesthetic rather than practical. At the moment, however, the new result appears to have no obvious applications. Related geometric problems have practical applications, such as checking the range of movements of a jointed, robotic arm, designing a complicated antenna that opens up properly in space, or studying how a protein strand folds into a compact blob. The researchers announced their proof last June in Minneapolis at a Society for Industrial and Applied Mathematics conference on discrete mathematics. Pulling apart this jaw-shaped polygon (blue) without allowing any segments to cross proved to be a tough exercise in computational geometry.

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Demaine, Robert Connelly of Cornell University, and GUnter Rote of the Free University of Berlin have proved that any polygon can be uncrinkled in two dimensions without any sides crossing each other during the unfolding. Demaine, a 19-year-old computer science graduate student at the University of Waterloo in Ontario. 'In a few cases, it took several months to find the answer,' says Erik D. Every tricky polygonal configuration anyone ever proposed was eventually cracked. No one ever came up with a stumper, however. Most of those who tackled the polygon problem believed that someone ultimately would come up with a polygon that could not be unfurled, at least not in two dimensions.

Someone would propose a complicated configuration that appears to stay locked, and other enthusiasts would spend hours, even weeks, looking for the key to opening it up. In recent years, the geometric speculation turned into a sort of game.

Imagine, for example, the outline of a set of fearsome jaws with interlocking teeth.Д¬omputational geometers and assorted others puzzled over this problem for more than a decade, ever since it came to the attention of robotics engineers who were trying to make a robot arm move from place to place. That's not as easy to do as it may sound. Can you always find a sequence of moves that removes the indentations-unfurling the polygon into what mathematicians describe as a convex shape, like a triangle-without ever letting the rods cross each other?

Start with any configuration, no matter how complex and intricately indented, or crinkly. Think of a polygon as a chain of rigid rods connected to each other in two dimensions with flexible joints. Polygons come in all sorts of shapes: triangles, squares, hexagons, stars, and a host of other straight-edged forms. Proof settles a wickedly prickly question about unfurling crinkly shapes