The New Math of Wrinkling Patterns

A few minutes into a 2018 talk at the University of Michigan, Ian Tobasco picked up a large piece of paper and crumpled it into a seemingly disordered ball of chaos. He held it up for the audience to see, squeezed it for good measure, then spread it out again.

“I get a wild mass of folds that emerge, and that’s the puzzle,” he said. “What selects this pattern from another, more orderly pattern?”

He then held up a second large piece of paper—this one pre-folded into a famous origami pattern of parallelograms known as the Miura-ori—and pressed it flat. The force he used on each sheet of paper was about the same, he said, but the outcomes couldn’t have been more different. The Miura-ori was divided neatly into geometric regions; the crumpled ball was a mess of jagged lines.

“You get the feeling that this,” he said, pointing to the scattered arrangement of creases on the crumpled sheet, “is just a random disordered version of this.” He indicated the neat, orderly Miura-ori. “But we haven’t put our finger on whether or not that’s true.”

Making that connection would require nothing less than establishing universal mathematical rules of elastic patterns. Tobasco has been working on this for years, studying equations that describe thin elastic materials—stuff that responds to a deformation by trying to spring back to its original shape. Poke a balloon hard enough and a starburst pattern of radial wrinkles will form; remove your finger and they will smooth out again. Squeeze a crumpled ball of paper and it will expand when you release it (though it won’t completely uncrumple). Engineers and physicists have studied how these patterns emerge under certain circumstances, but to a mathematician those practical results suggest a more fundamental question: Is it possible to understand, in general, what selects one pattern rather than another?

In January 2021, Tobasco published a paper that answered that question in the affirmative—at least in the case of a smooth, curved, elastic sheet pressed into flatness (a situation that offers a clear way to explore the question). His equations predict how seemingly random wrinkles contain “orderly” domains, which have a repeating, identifiable pattern. And he cowrote a paper, published in August, that shows a new physical theory, grounded in rigorous mathematics, that could predict patterns in realistic scenarios.

Notably, Tobasco’s work suggests that wrinkling, in its many guises, can be seen as the solution to a geometric problem. “It is a beautiful piece of mathematical analysis,” said Stefan Müller of the University of Bonn’s Hausdorff Center for Mathematics in Germany.

It elegantly lays out, for the first time, the mathematical rules—and a new understanding—behind this common phenomenon. “The role of the math here was not to prove a conjecture that physicists had already made,” said Robert Kohn, a mathematician at New York University’s Courant Institute, and Tobasco’s graduate school adviser, “but rather to provide a theory where there was previously no systematic understanding.”

Stretching Out

The goal of developing a theory of wrinkles and elastic patterns is an old one. In 1894, in a review in Nature, the mathematician George Greenhill pointed out the difference between theorists (“What are we to think?”) and the useful applications they could figure out (“What are we to do?”).

In the 19th and 20th centuries, scientists largely made progress on the latter, studying problems involving wrinkles in specific objects that are being deformed. Early examples include the problem of forging smooth, curved metal plates for seafaring ships, and trying to connect the formation of mountains to the heating of the Earth’s crust.