Israeli scientists have identified a geometric principle that explains why expanding surfaces such as flower petals and biological tissue can abruptly develop dimples and folds rather than continuing to grow smoothly. The discovery could advance understanding of natural growth and inform engineering and materials science.
"This is a beautiful example, where seemingly abstract mathematical concepts, in this case, topological and geometrical constraints on surfaces, directly control a physical system," Said Professor Eran Sharon of the Hebrew University's Racah Institute of Physics. "Apparently, such principles are responsible for much of the morphological richness we find in nature."
The research, published in Physical Review Letters, describes a new form of "geometric frustration" That occurs even when existing mathematical models predict a surface should remain smooth. The finding contradicts previous assumptions about growing material behavior and could shape the design of self-shaping materials for applications including medicine and soft robotics.
The study was led by Yafei Zhang, Professor Michael Moshe and Sharon.
Using an analogy of an inflating balloon, researchers explained that while a conventional balloon expands smoothly, a growing surface expands normally until reaching an invisible threshold. Beyond that point, it cannot maintain a smooth surface and instead forms regular patterns of folds and dimples. The transition results not from material imperfections but from a previously unrecognized geometric constraint.
The team combined mathematical analysis, computer simulations and laboratory experiments with specially designed elastic shells. They found that as a surface accumulates curvature, it remains smooth until reaching a precise threshold, at which point geometry no longer permits a smooth shape.
"We usually look for frustration in a sheet by checking its geometry locally," Moshe said. "Here, every small patch passes those tests, yet once the surface accumulates enough curvature, the whole shape reaches a geometric horizon and cannot continue smoothly without stretching. The dimples are the sheet's way of accommodating this global, topological obstruction."
Unlike previous buckling caused by external forces or confinement, the growing sheet generates its own geometric constraint while remaining free-standing. The material spontaneously forms cone-shaped dimples that relieve accumulated stress. A single cut in the material eliminates the effect, allowing the surface to become smooth again, demonstrating that the phenomenon depends on overall topology and connectivity rather than local geometry alone.
The discovery could inform development of shape-changing materials for soft robotics, medical devices and space technology, potentially enabling flexible components that transform in controlled ways and minimally invasive devices that reshape inside the body.
"What excites us most," Zhang said, "is that this appears to be a completely new organizing principle. Nature has been using it all along - we're only just discovering it."