Hebrew University physicists discover why growing surfaces suddenly dimple

Physicists at the Hebrew University of Jerusalem's Racah Institute of Physics reported in Physical Review Letters, announced on 3 August 2026, a previously unknown geometric principle that explains why growing surfaces such as leaves, petals and developing tissues can suddenly buckle into dimples instead of expanding smoothly. The study, titled Isometric Incompatibility in Growing Elastic Sheets, was carried out by Dr. Yafei Zhang, Prof. Michael Moshe and Prof. Eran Sharon.

When a thin sheet grows unevenly, it often cannot stay flat and must wrinkle or fold. Existing mathematical rules predict when this happens by checking the geometry at each small patch of the surface. The Jerusalem team found a new kind of 'geometric frustration' that those local checks miss. A growing sheet stays smooth until its accumulated curvature reaches a precise threshold. Beyond that point, its overall shape can no longer continue smoothly without stretching, and it spontaneously forms a regular array of cone-shaped dimples.

Surprisingly, the constraint is created by the sheet itself, without any outside pressure or confinement, and it depends on the surface's overall connectivity rather than its local shape. Making a single cut in the material makes the effect disappear. The team combined mathematical analysis, computer simulations and laboratory experiments with elastic shells to establish the result. Moshe described the dimples as the sheet's way of accommodating a global, topological obstruction, and Zhang called the effect an apparently new organising principle.

Sharon, who has long studied how growth shapes leaves and flowers, said such principles appear to be responsible for much of the variety of forms found in nature. The researchers say the insight could help explain shape formation in plants, tissues and microscopic organisms, and guide engineers designing materials that change shape on their own, including soft robots, deployable spacecraft structures, minimally invasive medical devices and self-assembling products.