The reason crumpling takes so long, says Witten, is because the ridges are strong and put up a fight. Virtually all the elastic energy in the sheet is concentrated in the ridges. Eighty percent of the energy, Witten’s team has calculated, comes from the bending of the sheet, and the other 20 percent comes from stretching. Even a piece of paper stretches a bit as you crumple it. If it didn’t, its surfaces would all be perfectly flat and the ridges all perfectly sharp—not the lowest-energy state. “To avoid that, it does a trade-off between extreme bending and a little bit of stretching,” Witten says.
That same trade-off is the essence of Mahadevan and Cerda’s theory of wrinkling—which Mahadevan, to distinguish it from brute crumpling, defines as “a roughly periodic arrangement of short-wavelength deformations of a thin sheet.” He and Cerda considered a simple model system: a thin sheet of polyethylene, longer than it is wide, clamped at both ends and stretched. What happens to the sheet, says Mahadevan, is “very bizarre: You’re pulling in one direction, and not only does the sheet get stretched in that direction, but in the other direction it actually gets compressed.” As the sheet narrows in the middle, the middle is deformed by a band of parallel wrinkles running lengthwise. You may have noticed this when you’ve tried to fold bedsheets with a partner.
Why not just one wrinkle? If you hold a sheet of paper, one side in each hand, and then compress it by bringing your hands together, it will form one large bend; this minimizes the curvature and thus the bending energy. But the total energy of a deformed sheet includes stretching as well as bending energy; and because Mahadevan’s plastic sheet, unlike the paper, is clamped at both ends, it would have to stretch enormously in the middle to accommodate a large wrinkle. To minimize stretching, the sheet should make lots of tiny wrinkles.
“Because you have these two competing effects, the optimal solution is a compromise,” Mahadevan says. That is, the sheet makes a medium amount of medium-size wrinkles. The wavelength and amplitude, Mahadevan and Cerda found, are proportional to the square root of the length of the sheet times its thickness.
A clamped plastic sheet may sound like an idealized example, but in nature there are lots of thin sheets that are clamped—not at the ends, but to a foundation. Our skin, for instance, is attached to the underlying flesh. As we age, our skin begins to sag as the fatty tissue beneath it loses its stiffness. Mahadevan can’t predict exactly what your face will look like as you age, nor can he design a better Botox. “Our theory is only physics; it doesn’t include biology,” he says. But it does explain why wrinkles are so prominent in bony places like the face, hands, and knees: The skin is thin there and thus easy to bend, and the underlying fatty tissue is thin and thus hard to stretch. As a result the energy compromise favors minimal stretching, which means lots of highly visible, relatively long-wavelength wrinkles.
Lately Mahadevan and Cerda have turned their attention to the physics of drapes, including the question of how cloth drapes a body. It’s a convoluted one. Whereas the polyethylene sheet in the model was merely bent and stretched, a drape is subject to gravity as well, so the energy trade-off becomes three-way. The sheet wants to hang flat to minimize its bending energy, unextended to minimize its stretching energy, and straight down to minimize its gravitational energy—except that it can’t penetrate the body it is draping, and it can’t do all that at once anyway. “The reason you see these fantastic patterns,” says Mahadevan, “is precisely because you’re trying to drape a curved surface with a flat sheet, which is impossible to do smoothly.”
Confirming our everyday experience with dresses and suits, he and Cerda have shown that there is no single energy-minimizing solution: A given fabric can drape a given body in a range of stable states. The researchers’ equations describing that range just might be of use to artists today—at least to artists trying to animate realistically clothed characters for video games or cartoons. The Renaissance masters had it easy by comparison; their feverish and gloriously draped saints and deities didn’t have to move. Modern computer animation, Mahadevan suggests, might profit from a judicious application of cutting-edge physics. “I’m sure you’ve noticed that most of the characters [in video games] wear armor,” he says. “They never wear real clothes.”