# Wrinkle predictions

(Nanowerk News) As a grape slowly dries and shrivels, its surface creases, ultimately taking on the wrinkled form of a raisin. Similar patterns can be found on the surfaces of other dried materials, as well as in human fingerprints. While these patterns have long been observed in nature, and more recently in experiments, scientists have not been able to come up with a way to predict how such patterns arise in curved systems, such as microlenses.
Now a team of MIT mathematicians and engineers has developed a mathematical theory, confirmed through experiments, that predicts how wrinkles on curved surfaces take shape. From their calculations, they determined that one main parameter — curvature — rules the type of pattern that forms: The more curved a surface is, the more its surface patterns resemble a crystal-like lattice.
The researchers say the theory, reported this week in the journal Nature Materials ("Curvature-induced symmetry breaking determines elastic surface patterns"), may help to generally explain how fingerprints and wrinkles form.
MIT researchers have developed a mathematical equation that predicts how surface patterns form on curved objects. Pictured is a sphere with a combination of hexagons and labyrinthine patterns, and a more complex, torus-shaped object with hexagonal dimples. (Image: Norbert Stoop)
“If you look at skin, there’s a harder layer of tissue, and underneath is a softer layer, and you see these wrinkling patterns that make fingerprints,” says Jörn Dunkel, an assistant professor of mathematics at MIT. “Could you, in principle, predict these patterns? It’s a complicated system, but there seems to be something generic going on, because you see very similar patterns over a huge range of scales.”
The group sought to develop a general theory to describe how wrinkles on curved objects form — a goal that was initially inspired by observations made by Dunkel’s collaborator, Pedro Reis, the Gilbert W. Winslow Career Development Associate Professor in Civil Engineering.
In past experiments, Reis manufactured ping pong-sized balls of polymer in order to investigate how their surface patterns may affect a sphere’s drag, or resistance to air. Reis observed a characteristic transition of surface patterns as air was slowly sucked out: As the sphere’s surface became compressed, it began to dimple, forming a pattern of regular hexagons before giving way to a more convoluted, labyrinthine configuration, similar to fingerprints.
“Existing theories could not explain why we were seeing these completely different patterns,” Reis says.
Denis Terwagne, a former postdoc in Reis’ group, mentioned this conundrum in a Department of Mathematics seminar attended by Dunkel and postdoc Norbert Stoop. The mathematicians took up the challenge, and soon contacted Reis to collaborate.