Unlocking Puzzling Polygons

Proof settles a wickedly prickly question about unfurling crinkly polygons

Polygons come in all sorts of shapes: triangles, squares, hexagons, stars, and a host of other straight-edged forms.

Pulling apart this jaw-shaped polygon (blue) without allowing any segments to cross proved to be a tough exercise in computational geometry. Illustrations courtesy Demaine

Segment lengths determine whether it’s possible to unfurl this eight-petal tree configuration without allowing segments to cross.

Unlike a two-dimensional chain, this knotted, three-dimensional “knitting needle” chain in space can’t be untangled.

This polygon (left) can be unlocked, although the tree (right) on which its shape is based can’t be when its branches are close together.

The sequence of steps required to unlock the jaws configuration.

A complex folded paper structure based on the hyperbolic paraboloid presents new geometric challenges.

That’s not as easy to do as it may sound. Imagine, for example, the outline of a set of fearsome jaws with interlocking teeth.

Computational 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. In recent years, the geometric speculation turned into a sort of game. 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.

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.

No one ever came up with a stumper, however. Every tricky polygonal configuration anyone ever proposed was eventually cracked. “In a few cases, it took several months to find the answer,” says Erik D. Demaine, a 19-year-old computer science graduate student at the University of Waterloo in Ontario.

Now, the question is finally settled. Demaine, Robert Connelly of Cornell University, and Günter 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.

The researchers announced their proof last June in Minneapolis at a Society for Industrial and Applied Mathematics conference on discrete mathematics.

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. At the moment, however, the new result appears to have no obvious applications.

The puzzle’s real appeal has been aesthetic rather than practical. “It’s simply a natural question to ask and a beautiful problem,” insists computer scientist Joseph O’Rourke of Smith College in Northampton, Mass.

Mathematical questions

Over the years, studies of robotic arm movements have suggested purely mathematical questions about morphing one geometric shape into another. One important group of problems concerns chains made up of line segments. Such chains may be closed, like a polygon, or open-ended, like a segmented arc. Lines can also be linked to form a branched structure, termed a tree, where jointed segments sprout from a common vertex.

Suppose, for example, that a tree has eight chains emanating from a central point. Suppose further that each of these chains is made up of three segments folded so that the entire tree looks like a stylized flower with eight petals.

In 1998, Sue Whitesides of McGill University in Montreal and a large team of collaborators established that for certain segment lengths, the petals can’t be straightened out without letting segments cross. Opening up one petal necessarily impinges on others.

Researchers also found examples of three-dimensional chains, both open and closed, that are locked, or impossible to unfold. On the other hand, O’Rourke and Smith College colleague Roxana Cocan proved last year that in the roominess of four- or higher-dimensional space, one can straighten out any open chain and uncrinkle any polygon.

“That left the two-dimensional case as the major unsolved problem,” O’Rourke says.

Last year in July, Demaine, Rote, and Connelly all happened to be at a geometry conference in Ascona, Switzerland. In considering the polygon puzzle, Rote suggested that uncrinkling polygons had to require some sort of expansion—as if a balloon were inflating inside the polygon and forcing its sides outward.

“His suggestion was crucial, though we didn’t realize why it was so helpful until later,” Demaine remarks.

The trio did observe, however, that if they could somehow find a sequence of movements in which the distance between any pair of joints stayed the same or increased, then the segments could never cross. So, the problem could be converted from one about avoiding intersections into one about expanding movements.

Polygons as frameworks

By the time Demaine, Rote, and Connelly met again in November, this time in Budapest, Connelly had realized that the notion of expansion could be studied in the context of his own field of expertise: the rigidity of structures. With this concept, the team could look at polygons as frameworks of rods and invisible struts between nonadjacent joints, where rods have to stay the same size but struts can increase in length. The researchers could then consider stress patterns within that structure.

“This allowed us to apply some beautiful theorems in rigidity theory,” Demaine notes.

A proof that flat polygonal chains can’t lock followed from that insight. Along the way, Demaine, Rote, and Connelly also established that any open chain can always be straightened.

The big surprise is not the proof itself, O’Rourke comments, but the conceptual breakthrough that the opening move in any successful uncrinkling process has to be one in which each joint moves apart or stays the same distance away from every other joint.

Last summer, Demaine, Connelly, and O’Rourke added another element to the original argument. They showed that the area inside an uncrinkling polygon must increase. “This seems almost obvious,” Connelly notes, “but the proof that we have is not completely trivial.”

Now that the two-dimensional case is solved, Demaine is tangling with other fierce geometric beasts. An origami enthusiast, he’s tamed the hyperbolic paraboloid. Demaine developed instructions for folding and gluing this classic saddle shape into complex paper hats and starbursts.

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