Mathematicians have broken through a long-standing barrier in the study of “minimizing surfaces,” which play an important role in both math and physics.

In the mid-19th century, the Belgian physicist Joseph Plateau — who had been designing and conducting scientific experiments since he was a child — submerged loops of wire in a soapy solution and studied the films that formed. When he bent his wire into a circular ring, a soap film stretched across it, creating a flat disk. But when he dipped two parallel wire rings into the solution, the soap stretched between them to form an hourglass shape instead — what mathematicians call a catenoid. Different wire frames produced all sorts of different films, some shaped like saddles or spiraling ramps, others so complicated they defied description.

These soap films, Plateau posited, should always take up the smallest area possible. They’re what mathematicians call area-minimizing surfaces.

It would take nearly a century for mathematicians to prove him right. In the early 1930s, Jesse Douglas and Tibor Radó independently showed that the answer to the “Plateau problem” is yes: For any closed curve (your wire frame) in three-dimensional space, you can always find a minimizing two-dimensional surface (your soap film) that has the same boundary. The proof later earned Douglas the first-ever Fields Medal.

Since then, mathematicians have expanded on the Plateau problem in hopes of learning more about minimizing surfaces. These surfaces appear throughout math and science — in proofs of important conjectures in geometry and topology, in the study of cells and black holes, and even in the design of biomolecules. “They’re very beautiful objects to study,” said Otis Chodosh of Stanford University. “Very natural, appealing and intriguing.”

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