Photo by
Rivka Kahana/TPS-IL on 21 December, 2025
Israeli Study Reveals the Geometry That Could Drive Future Smart Materials
Jerusalem, 2 August, 2026 (TPS-IL) -- Israeli scientists have identified a previously unknown geometric principle that explains why growing surfaces like flower petals or biological tissue can suddenly stop expanding smoothly and instead develop dimples and folds, a discovery that could deepen understanding of natural growth and lead to advances in 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 Prof. 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.”
A bee hovers by a flower near Moshav Mesilat Zion in central Israel on Dec. 21, 2025. Photo by Rivka Kahana/TPS-IL
The research, published in the peer-reviewed Physical Review Letters, describes what the team calls a new form of “geometric frustration” that emerges even when existing mathematical rules predict a surface should remain smooth. The finding challenges long-held assumptions about how growing materials behave and could influence the design of self-shaping materials for applications ranging from medicine to soft robotics.
The study was led by Dr. Yafei Zhang, Prof. Michael Moshe and Sharon.
Researchers explained the phenomenon using an analogy of an inflating balloon. A conventional balloon expands smoothly as it fills with air, but the team described a hypothetical balloon that grows normally until it reaches an invisible threshold. Beyond that point, it can no longer maintain a smooth surface and instead forms a regular pattern of folds and dimples.
According to the researchers, the transition is not caused by imperfections in the material but by a previously unrecognized geometric constraint.
A Hidden Geometric Rule
The discovery began with a question that has puzzled scientists studying biological and engineered materials: Why do some growing surfaces, including leaves, flower petals, biological tissues and synthetic materials, abruptly develop complex patterns instead of continuing to bend smoothly in nature and engineered systems?
To investigate, the team combined mathematical analysis, computer simulations and laboratory experiments using specially designed elastic shells. They found that as a surface accumulates curvature, it remains smooth until it reaches a precise threshold. At that point, the geometry itself no longer permits a smooth shape.
Moshe said the results reveal a form of geometric limitation that cannot be detected by examining small sections of a surface in isolation.
“We usually look for frustration in a sheet by checking its geometry locally,” he 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 examples of buckling caused by external forces or confinement, the researchers found that the growing sheet generates its own geometric constraint while remaining completely free-standing.
Rather than remaining smooth, the material spontaneously forms a regular array of cone-shaped dimples that relieve the accumulated stress. The team also discovered that making a single cut in the material eliminates the effect, allowing the surface to become smooth again. This demonstrated that the phenomenon depends on the overall topology and connectivity of the surface rather than on its local geometry alone.
The discovery could eventually help engineers develop shape-changing materials for applications in soft robotics, medical devices and space technology. In soft robotics, the new geometric principles could guide the creation of flexible components that fold or transform in controlled ways without relying on complex mechanical systems. In medicine, similar materials could contribute to future minimally invasive devices designed to expand or reshape inside the body.
By understanding how geometric constraints drive folding and buckling, engineers may be able to create materials that assemble themselves for use in manufacturing, packaging and other technologies where traditional construction methods are difficult or impractical.
“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.”
