Physicists Conjure Curves From Flat Surfaces Using Japanese Folding Art

New research based on kirigami, the ancient Japanese art of cutting and folding paper, is providing a set of rules for imbuing flat surfaces with curvature. Elod Beregszaszi/Popupology | CC BY-NC-SA

Try gift-wrapping a soccer ball, and you will quickly encounter the geometric abyss between paper’s inherent flatness and a sphere’s natural curves.

“The very first bit seems to sort of match, but as you wrap the paper around, the crinkles get bigger and bigger,” observed Toen Castle, a physicist at the University of Pennsylvania.

With their concave curves, saddles are equally tough to wrap, but for the opposite reason. “There’s more saddle than there is paper,” Castle said.

The mismatch between soccer balls, saddles and sheets of paper lies in their “intrinsic” curvature, a property of surfaces known to mathematicians for centuries that no amount of folding can change. Scientists have sought a bridge across the divide — a systematic way of imbuing flat surfaces with curvature, which they say could revolutionize the design and assembly of three-dimensional structures and help extend a major theorem of geometry.

Now, Castle and several Penn colleagues have found just such a bridge in the same technique that tailors use to hug fabric around the curves of a body — namely, by making the right cuts. Reporting their work in December in Physical Review Letters , the physicists present a basic set of rules for cutting and reconnecting a piece of paper in order to add curvature to one point in its surface while subtracting it from another, maintaining the paper’s overall flatness while forcing it to bend into the third dimension.

“It’s a way of encoding three-dimensionality in a two-dimensional structure,” said Randall Kamien, a professor of physics at Penn who heads the research group behind the result. “The whole thing will just pop up all by itself.”