Space and Physics

# Mathematical Theory Explains Wrinkle Patterning

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Fingerprints are often used as forensic evidence when solving crimes, because individuals have wrinkle patterns that are unique. Though they are unique, they might not be random. A new paper published in Nature Materials by lead author Norbert Stoop of MIT describes experiments that support their mathematical theory which predicts wrinkling patterns based on the curvature of the surface. This theory could help predict wrinkling patterns on a number of surfaces, including those used to create microlenses.

“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,” senior author Jörn Dunkel said in a press release. “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 current paper builds off of previous work from co-author Pedro Reis, who explored the relationship between drag and surface patterns on polymer balls. Reis removed air from the balls, which initially formed a simple hexagonal pattern that significantly reduced air resistance, but a more complex pattern eventually emerged.

Though Reis was unable to explain the wrinkling pattern of his material, Stoop and Dunkel attempted to make mathematical sense of the data through elastic theory. The elastic theory equations when first applied were not perfect, as they did not precisely predict when the pattern would emerge or what made the material develop those patterns. However, applying principles of fluid mechanics to the elastic theory equations yielded a formula that was able to predict the wrinkling patterns that held up through testing with computer simulations.

“What type of stretching and bending is going on, and how the substrate underneath influences the pattern — all these different effects are combined in coefficients so you now have an analytically tractable equation that predicts how the patterns evolve, depending on the forces that act on that surface,” Dunkel continued.

The team was ultimately able to determine that the curvature of a material’s surface was the main indicator of wrinkle pattern. Though the current paper has been applied to polymer spheres about the size of golf balls, the researchers state that future research could allow them to interpret the wrinkling pattern of more donut-shaped objects or other irregular shapes.

“This theory allows us to go and look at shapes other than spheres,” Reis added. “If you want to make a more complicated object wrinkle — say, a Pringle-shaped area with multiple curvatures — would the same equation still apply? Now we’re developing experiments to check their theory.”

### ARTICLE POSTED IN

Space and Physics
• wrinkling,

• patterns,

• fingerprints,

• forensic evidence,

• elastic theory