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Mathematicians Solve Problem of Folding a Pop-up Tent

A series of experiments shed light on "overcurvature"



FROM “OVERCURVATURE DESCRIBES THE BUCKLING AND FOLDING OF RINGS FROM CURVED ORIGAMI TO FOLDABLE TENTS,” BY PIERRE-OLIVIER MOUTHUY ET AL., IN NATURE COMMUNICATIONS, VOL. 3, ARTICLE NO. 1290; DECEMBER 18, 2012

Camping enthusiasts and aspiring modern sculptors take heed: researchers have achieved a breakthrough in understanding and controlling overcurvature, which is found in such disparate settings as pop-up tents, DNA plasmids and curved origami. Overcurvature occurs when a ring is too curved to lie flat in a plane the way a normal circle does. For example, if you detached a segment of a Slinky and connected its ends to make a closed loop, you would have a hard time getting the whole thing to lie flat on the floor. The intrinsic curvature of the Slinky would cause the ring to buckle and assume a three-dimensional saddle shape.

In fact, the Slinky played a major role in this research project, the results of which were published in the journal Nature Communications last December. (Scientific American is part of Nature Publishing Group.) After observing overcurved rings of various sizes and materials, the researchers found a family of curves with fairly simple mathematical descriptions that they believed would model the shapes these overcurved rings take in space. They used loops made from portions of plastic Slinkys as the setting for precise measurements and found that their predicted curves were indeed what they observed in the Slinkys. “It was really surprising to us,” says Alain Jonas, a materials scientist at the Catholic University of Leuven in Belgium, who led the research. “It was this experience where you find something and it actually fits!”

The paper includes an efficient pathway for folding pop-up tents and other overcurved rings, as shown in the illustration above. To fold a ring into three loops, place your hands on opposite sides of the ring. As you lift up, bring your hands together and grab the opposite sides in one hand. Use your free hand to coax the two opposite sides down and toward each other to form a saddle shape. At both the top and the bottom, push one side over the other and collapse the loops together.

The proposal differs from the approach that people usually take. It requires more energy initially but uses less overall. “It's not very intuitive when you do it,” Jonas says, “but that's what the physics of the problem wants.” After performing the research, he borrowed a friend's tent to practice the technique he and his colleagues had developed. It was a success.

This article was originally published with the title "How the Slinky Buckles."

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