The New Math of Wrinkling
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 un-crumple). 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?