A few minutes at a 2018 talk at the University of Michigan, Ian Tobasco picked up a large sheet of paper and crumpled it into a chaotic ball that seemed sprawling. He held it up for the audience to see, squeezed it beautifully, then spread it out again.
“I get a large volume of raised folds, and that’s the puzzle,” he said. “What chooses this pattern from another, more orderly pattern?”
Then he held up a second large sheet of paper – which was folded into a famous origami parallelogram known as Miura-ori – and pressed it flat. He said that the force he applied on each sheet of paper was the same, but the result could not have been more different. Miura-ori is divided into geometric regions; The crumpled ball is a mess of jagged lines.
“You get the feeling that this,” he said, pointing to the scattered arrangement of folds on the crumpled paper, “is just a random version of this thing.” He pointed out that Miura-ori was neat and orderly. “But we haven’t put a finger on whether that’s true or not.”
Making that connection would require nothing more than the establishment of universal mathematical rules of elastic patterns. Tobasco has been working on this problem for years, working on equations that describe thin elastic materials — which respond to deformation by trying to bounce back to its original shape. Poke a ball hard enough and a star-shaped radial crease will form; remove your fingers and they will be smooth again. Squeeze a crumpled piece of paper and it will expand when you release it (though it won’t be completely unbulged). Engineers and physicists have studied how these patterns appear under certain circumstances, but for a mathematician, those actual results suggest a more fundamental question: Is it possible understand, in general, what chooses one sample over another?
In January 2021, Tobasco publishes a piece of paper answered that question in the affirmative—at least in the case of a flat, curved, elastic sheet pressed into flat (a situation that provides an unambiguous way to explore the question). His equations predict seemingly random wrinkles containing “ordered” domains, having a repeatable, identifiable pattern. And he wrote a paper, published in August, showing how a new physical theory, based on rigorous math, can predict patterns in real-life situations.
Notably, Tobasco’s work shows that wrinkles, in its many forms, can be thought of as the solution to a geometric problem. “It’s a great part of mathematical analysis,” says Stefan Muller of the Hausdorff Mathematical Center of the University of Bonn in Germany.
For the first time, it elegantly lays out the mathematical rules – and a new understanding – behind this ubiquitous phenomenon. “The role of mathematics here is not to prove a conjecture that physicists have made,” says Robert Kohna mathematician at New York University’s Courant Institute, and Tobasco’s graduate school advisor, “which was to provide a theory for which there was previously no systematic understanding”.
Extend
The goal of developing the theory of wrinkles and elastic patterns is an old theory. In 1894, in a review in NatureMathematician George Greenhill pointed out the difference between theorists (“What should we think?”) and the useful applications they could find (“What shall we do?”).
During the 19th and 20th centuries, scientists largely made progress in this area, studying the problems associated with wrinkles on specific objects that are being deformed. Early examples included the problem of forging smooth, curved sheets of metal for seafaring ships, and the attempt to connect mountain formation with warming of the Earth’s crust.
