Cartographers have long contended with the distortions to a map that occur when flattening the curved two-dimensional surface of Earth onto a plane. In the Mercator projection, for example, landmasses and other features farther away from the equator appear larger than they really are. Equal-area projections, on the other hand, distort the shapes of various features on a map.
The mismatch in the length of a curved 2D surface and that of a flat plane causes wrinkle patterns to form. Nonlinear mechanics and tension-field theory do a good job of describing wrinkles on a confined surface. But that approach fails to explain why wrinkle patterns appear on surfaces subject to weak tension forces.
Ian Tobasco (University of Illinois at Chicago), Joseph Paulsen (Syracuse University), Eleni Katifori (University of Pennsylvania), and their colleagues considered the issue from a mathematical angle rather than as a physics problem. The simple set of geometric rules they derived are an exact solution for the wrinkle patterns that arise when thin, curved sheets are flattened.
The researchers found that the patterns fall into two categories. The first rule predicts that for negative curvatures, as in panels a and d of the figure below—think of a horse saddle, for example—wrinkles form along line segments (blue) that are perpendicular to the surface boundary. The wrinkles meet each other along the medial axis, defined as the set of points that can most quickly reach the edge from two or more paths. Notably, those segments make up the equal legs of a special family of isosceles triangles (panels b and e).
The second rule predicts that for globes and other positive-curvature surfaces (panels c and f), wrinkles form along the opposite legs (green) of the same isosceles triangles. To the researchers’ surprise, the wrinkle patterns in positively and negatively curved sheets are related: Their reciprocal relationship allows for the pattern in one to be deduced from the other.
The areas covered by the isosceles triangles harbor ordered wrinkle patterns, and they hold even on nonuniformly curved surfaces. The remaining areas support disordered patterns—geometrically, the areas correspond to points on the medial axis with three or more shortest paths to the boundary (green polygon in panels e and f).
To test the new geometry-based predictions, Paulsen and some of the coauthors observed the wrinkles that developed when polystyrene films of various shapes were formed on curved glass surfaces and subsequently placed over a flat liquid surface. The results of those experiments, and simulations of the phenomena led by Katifori, reassuringly agreed with the geometric interpretation.
The researchers, led by Tobasco, found that the wrinkle patterns could be derived as a consequence of the curved sheet trying to cover as much of the liquid bath as possible, in a limit where the wrinkles are infinitely fine. That coverage maximization is driven in part by the surface tension of the water pulling the sheet’s edges as far apart as possible, which is the dominant role of the liquid in the experiment. But gravity is also at play on the system, and simulations driven entirely by gravity with zero surface tension produced the same coverage maximization and the same wrinkle patterns as in the experiment. (I. Tobasco et al., Nat. Phys. 18, 1099, 2022.)
