Scientists have discovered a previously unknown law of geometry that explains why some growing surfaces, whether in nature or engineered materials, suddenly stop being able to stay smooth and instead form dimples and folds. Until now, researchers believed they understood the geometric rules behind these shape changes, but this study reveals a new kind of "geometric frustration" that arises even when all the known rules are satisfied. The finding could reshape how scientists understand the formation of leaves, flowers, tissues and other natural structures, while helping engineers design smarter materials that can deliberately change shape for applications in soft robotics, medicine and advanced manufacturing.
[Hebrew University of Jerusalem] – Upon blowing up a balloon, it grows and expands smoothly. But now imagine a strange balloon that grows beautifully, until it reaches an invisible limit. The moment it crosses that line, it can no longer stay smooth. Instead, it suddenly erupts into a pattern of dimples and folds.
Scientists at the Hebrew University of Jerusalem have discovered that this invisible limit is not a flaw in the material, it is a previously unknown law of geometry that forms the basis for complex shapes in natural and synthetic growing matter.
The discovery, published in Physical Review Letters , by Dr. Yafei Zhang, Prof. Michael Moshe, and Prof. Eran Sharon of the Racah Institute of Physics at the Hebrew University of Jerusalem , identifies a new kind of geometric frustration: a point where a growing surface simply cannot remain smooth anymore, no matter how perfectly it is made.
Prof. Eran Sharon commented "This is a beautiful example, where seemingly abstract mathematical concepts, in this case, topological and geometrical constraints on surfaces, directly control a physical system. Apparently, such principles are responsible for much of the morphological richness we find in nature."
The finding began with a simple question: Why do some growing surfaces, like leaves, flower petals, biological tissues, or even engineered materials, suddenly develop complex patterns instead of continuing to curve smoothly?
To find out, the researchers combined mathematics, computer simulations, and laboratory experiments using specially designed elastic shells. They found that as a surface accumulates curvature, everything works normally, until it reaches a precise threshold. At that moment, the geometry itself becomes impossible.
"We usually look for frustration in a sheet by checking its geometry locally," explained Prof. Michael Moshe. "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." Remarkably, however, no external confinement is involved here. Instead, the growing sheet generates its own geometric constraint, causing frustration and buckling even while remaining completely free-standing.
Instead of remaining smooth, the material spontaneously creates a regular pattern of cone-like dimples that relieve the built-up stress. Even more surprisingly, the team showed that making a single cut in the material removes the problem entirely, allowing it to become smooth again. That revealed that the effect is fundamentally topological , it depends on the overall shape and connectivity of the surface, not just its local geometry.
Until now, scientists believed that two well-known geometric rules explained when growing sheets become frustrated. This study shows those rules were incomplete. The newly identified mechanism creates frustration even when the older conditions predict everything should work perfectly.
The discovery could help scientists better understand how shapes form in nature—from developing tissues and plant structures to microscopic biological systems—and may guide the design of future materials that deliberately change shape, such as soft robots, deployable spacecraft components, medical devices, and self-assembling materials.
"What excites us most," said Dr. Yafei Zhang , "is that this appears to be a completely new organizing principle. Nature has been using it all along—we're only just discovering it."
Physical Review Letters
Experimental study
Not applicable
Isometric Incompatibility in Growing Elastic Sheets
28-Jul-2026