Cartographers have long struggled with map distortions that occur when the Earth’s curved, two-dimensional surface is flattened into a flat surface. For example, the Mercator projection makes landmasses and other features far from the equator appear larger than they actually are. Equal area projections, on the other hand, distort the shape of various features on the map.
A wrinkle pattern is formed by the length mismatch between the curved 2D surface and the plane. Nonlinear mechanics and tension field theory successfully explain limited surface wrinkling. However, that approach cannot explain why wrinkle patterns appear on surfaces with low tension.
Ian Tobasco (University of Illinois at Chicago), Joseph Paulsen (Syracuse University), Eleni Katifori (University of Pennsylvania) and their colleagues looked at the problem from a mathematical angle rather than a physical one. A simple set of geometric rules they have derived is a precise solution to the wrinkle patterns that occur when thin, curved sheets are flattened.
Researchers have found that the patterns fall into two categories. The first rule is that for negative curvature (think of a horse saddle, for example), wrinkles occur along lines (blue) perpendicular to the surface boundary, as in panels a and d of the figure below. predict it will form. The wrinkles intersect each other along the central axis, defined as the set of points where the edge can be reached fastest from two or more paths. In particular, these segments constitute equal legs of a special family of isosceles triangles (panels b and e).
The second rule predicts that for spheres and other positive curvature surfaces (panels c and f), wrinkles will form along opposite legs (green) of the same isosceles triangle. To the researchers’ surprise, the wrinkle patterns of the positively and negatively curved sheets are related: their interrelationship allows one pattern to be inferred from the other.
The area covered by the isosceles triangles has a regular wrinkle pattern that is preserved even on uneven curved surfaces. The remaining regions support chaotic patterns. Geometrically, regions correspond to points on the central axis that have 3 or more shortest paths to the boundary (green polygons in panels e and f).
To test the new shape-based prediction, Paulsen and some co-authors observed the wrinkling that occurs when polystyrene films of various shapes are formed on a curved glass surface and then placed on a flat liquid surface. Did. The results of these experiments, and the simulations of the phenomenon led by Catifoli, showed encouraging agreement with the geometric interpretation.
Researchers led by Tobasco found that at the limit of infinitely fine wrinkles, the curved sheet tried to cover as much of the liquid bath as possible, resulting in a wrinkled pattern. That coverage maximization is driven in part by the surface tension of the water pulling the edges of the sheet apart as much as possible. This is the dominant role of liquids in experiments. However, gravity is also acting on the system, and simulations driven entirely by gravity with zero surface tension produced the same coverage maximization and the same wrinkle pattern as the experiment. (I. Tobasco et al., nut. Physics. 181099, 2022.)