The New Math of Wrinkling Patterns

Ian Tobasco crumpled a large piece of paper into a ball of chaos while giving a speech at the University of Michigan. He squeezed it for good measure and then spread it out again.

He said that he got a mass of folds that emerged. Why does this pattern come from another pattern?

He pressed the second piece of paper flat after holding it up and folding it into a Miura-ori pattern. He said the force he used on each sheet of paper was the same, but the outcome was different. The Miura-ori was neatly divided into geometric regions and the ball was a mess.

He pointed to the scattered arrangement of creases on the crumpled sheet and said, "You get the feeling that this is just a random disorder of this." The orderly Miura-ori was indicated by him. We don't know if that's true or not.

Establishing mathematical rules of elastic patterns is what it would take to make that connection. The equations that describe thin elastic materials have been studied by Tobasco for years. If you poke a balloon hard enough and remove your finger, there will be a pattern of radialwrinkles. You can squeeze the paper and it will expand when you release it. Engineers and physicists have studied how these patterns come about, 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 the question in the affirmative in the case of a smooth, curved, elastic sheet pressed into flatness. His equations show how seemingly randomwrinkles have a pattern. He co-authored a paper in August that showed a new physical theory that could predict scenarios.

wrinkling can be seen as the solution to a geometric problem according to Tobasco. Mller is the director of the Hausdorff Center for Mathematics at the University of Bonn.

The mathematical rules are laid out for the first time in this case. Robert Kohn, a mathematician at New York University's Courant Institute and Tobasco's graduate school adviser, said that the role of the math was not to prove a hypothesis but to provide a theory.

The person is stretching out.

The goal is to develop a theory of elastic patterns. The mathematician George Greenhill wrote a review in 1894 about the difference between theorists and useful applications.

Scientists mostly made progress on the latter in the 19th and 20th centuries. Forging smooth, curved metal plates for seafaring ships is one of the early examples.