Take a piece of paper. Crumple it. Before you sink a three-pointer in the corner wastebasket, consider that you’ve just created an object of extraordinary mathematical and structural complexity, filled with mysteries that physicists are just starting to unfold.

“Crush a piece of typing paper into the size of a golf ball, and suddenly it becomes a very stiff object. The thing to realize is that it’s 90 percent air, and it’s not that you designed architectural motifs to make it stiff. It did it itself,” said physicist Narayan Menon of the University of Massachusetts Amherst. “It became a rigid object. This is what we are trying to figure out: What is the architecture inside that creates this stiffness?”

Menon’s expedition into the shadowy heart of a crumpled sheet — of aluminum foil, to be precise — was undertaken with fellow Amherst physicist Anne Dominique Cambou and published in an August 23 Proceedings of the National Academy of Sciences article. The pair think they’ve mapped the mathematical underpinnings of its rigidity.

Of course, it may seem surprising that a balled-up sheet of paper or foil should contort itself beyond knowledge. But Menon noted that when physicists finally described the precise dynamics of conical crumpling, which you can achieve by laying a sheet of paper over a coffee cup and poking down with one finger, it was regarded as a mathematical tour-de-force.

A crumpled cone is a far simpler example of the tendencies that produce a crumpled ball: when a flat plane is subjected to distortional stress but only permitted to bend, not stretch, it transforms suddenly and unpredictably into a landscape of folds and facets, each representing an entirely new surface. It’s what researchers call a “far from equilibrium” process, guided by strange rules and non-linear effects. The mechanics of an individual crease are understood, but when physicists try to predict where that crease will appear or how it will influence the next, understanding goes dim.

Trying to peer inside a crumpled ball by simulating the process in three dimensions is “mathematically nasty,” a problem that quickly pushes lab-grade computers to their limits, said Menon. And trying to reverse-engineer structure from patterns revealed upon unfolding just isn’t possible. What happens in a crumpled ball stays in a crumpled ball.

‘I love it that these simple-looking problems are so nasty sometimes.’

“If you’re not talking about simulation, but mathematical understanding of these things, that’s one step harder,” said Menon. “We understand the underlying equations of the mechanics of a thin sheet very well. Those have been around for a century. But solving those equations, to produce a physical understanding, is difficult even in simple cases. If you’re talking about a structure that owes its properties to 1,000 or more of these structures, interacting in complicated ways, that’s asking more than we can do now.”

To look into crumpled balls, Menon and Cambou used X-ray microtomography, an imaging technique that, like a medical CT scan, assembles three-dimensional images from thousands of two-dimensional, cross-section snapshots. They imaged dozens of balls of different sizes, searching for statistical patterns in their internal geometries.

They found that a crumpled ball is most dense in its outer regions, and least dense in its core. Once inside its folds, there’s no way of knowing from their shape which direction is out and which is in (as, for example, one can determine from an onion, which has layers of skin arranged in curves parallel to its outer surface.) “If I was a creature that lived inside this ball, could I make my way out by looking at the way things are arranged? The answer is no,” said Menon.

When he and Cambou studied arrangements of creases and folds, they found a distinctive pattern. Planes often lie flat against other planes. “It’s a fairly uniform object, though you’ve created it by a random, not-so-uniform process,” said Menon. “That’s the most surprising thing. There is no real geometrical reason why things should stack and layer in that way.” But if the researchers don’t know why this happens, they can speculate as to its effect: strength.

Multiple layers of a thin sheet soon become walls. Per the lack-of-orientation observation, these walls are aligned in thousands of random directions. Press down and, from any angle, you’re pressing against down columns. “It can resist being crushed in all different directions,” said Menon.

To explore why this happens, he and Cambou are now using transparent plastic sheets to make three-dimensional movies of crumpling. The implications extend far beyond Menon’s lab. “You’ve heard of crumple zones,” he said. “I’m just as interested in understanding leaves, or thin membranes of animal tissue, or the conformation of the Earth’s crust when it’s folded into mountains. I love it that these simple-looking problems are so nasty sometimes.”

Images: 1) Reconstructed cross-section image of a foil ball approximately 4 inches in diameter. (Menon & Cambou/PNAS) 2) Turinboy/Flickr

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Citation: “Three-dimensional structure of a sheet crumpled into a ball.” By Anne Dominique Cambou and Narayanan Menon. Proceedings of the National Academy of Science, Vol. 108 No. 33, August 23, 2011.