# Synopsis: Folding on the Curve

The mystery of curved folds in origami artwork is partly unraveled by a mathematical modeling of the elastic forces in a folded ring structure.

Origami, the Japanese art of paper folding, attracts architects and engineers looking for inspiration for new structures. Curved folds can produce rigid, volume-enclosing structures (like the well-known french-fry box), but their potential has yet to be fully realized because scientists do not completely understand the underlying forces. A new scientific investigation of curved crease origami, described in Physical Review Letters, reveals how geometrical constraints and energy minimization combine to create a simple curving structure.

Paper cranes and other origami sculptures are a wonder to behold, but these art pieces also have a practical side in, for example, designing foldable solar panels, car airbags, and retractable stadium roofs. Most often the folds follow straight lines, but curved creases could open new possibilities if the induced mechanical stresses were better understood.

Marcelo Dias of the University of Massachusetts, Amherst, and his colleagues decided to explore the mechanics of the simplest curved fold. They started with a flat ring (or annulus) of paper and then folded it along a concentric circle midway between the ring’s inner and outer edges. The paper cannot stretch to accommodate this geometry, so the ring buckles up into a rigid saddle shape. The researchers characterized this geometric frustration by modeling the bending energy in the curved surfaces of the paper, plus an extra phenomenological energy in the crease, and derived the shape that would minimize this energy using analytical equations and numerical simulations. They imagine that this formalism could be used to design more complicated shapes with multiple creases. – Michael Schirber

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