Roses Offer Mechanical Clues for Shape-Shifting Materials
The changing shape of a rose over its short and beautiful lifetime stems from a natural interplay between growth and mechanics that has long puzzled researchers. The young petals are almost entirely round, but as they grow, small and distinct points appear along their outer edges. Eran Sharon and colleagues at the Hebrew University of Jerusalem have now uncovered the mechanical mechanisms that sculpt the growing flower into this pointed form, offering new insights for designing materials and structures that can change shape on demand [1].
The elaborate shapes produced by many leaves and flowers typically arise from uneven growth across their surfaces. As an example, a ruffled lettuce leaf forms when its outer edge grows more quickly than its center. This difference in growth rate alters the preferred curvature of the leaf from the center to the outside, in turn generating stresses within the plant tissue. As these stresses accumulate, they eventually trigger a mechanical instability that causes the edge of the leaf to deform into a wavy shape.
Such ruffled patterns result from a geometric mismatch called Gauss incompatibility, which has also been studied as a way to generate curvature from flat sheets of elastic materials. However, the soft and wavy shapes produced through Gauss-type deformations cannot explain the highly localized points that form around the edges of rose petals. Sharon and colleagues suggest that these distinctive shapes are instead created by another type of geometric mismatch, called the Mainardi-Codazzi-Peterson incompatibility, which the team first identified in 2021 [2]. This Mainardi-Codazzi-Peterson incompatibility can be generated by a simple and highly symmetric growth profile that is similar to that of rose petals.
To test their theory, the researchers first examined around 100 petals from different species of rose. They observed that the younger petals have smooth sectors, with roughly uniform curvature along the edge, while the older petals appear more like polygons, with the rounded edges curled away. Sharp points, or cusps, form where the curled edges meet, with more cusps appearing as the petal matures. Cutting the petals from the center toward the outermost edge creates a downward curve, reflecting the change in preferred geometry across the petal, with the degree of curvature increasing with age.
Team members Yafei Zhang and Michael Moshe then modeled the petal as a simple disk. When the curvature is small, the disk curves symmetrically around its center into a bowl-like shape. As the curvature increases, the disk morphs first into a saddle shape and then into triangular and quadrilateral configurations. Thinner disks are found to produce cusps with sharper points, while thicker disks produce cusps that are curled less tightly.
To validate these simulated results, PhD student Omri Cohen fabricated a series of disks from two polymer layers. The lower layer was patterned with a regular matrix and the upper one consisted of thin lines radiating out from the center. When the disks were heated and then cooled again, the matrix layer remained the same, while the upper layer contracted by a varying amount along the radial direction. This difference induced a curvature in the disk, and the team was able to replicate the simulated series of shape transitions by varying the curvature and thickness of the disks.
Further analysis shows that the formation of each cusp acts as a focal point for the stresses that accumulate in the petal. In older petals this localized concentration of stress inhibits growth around the cusps, producing a concave distortion on the rounded edge of the petal. “This completes a nice feedback cycle,” explains Sharon. “Simple growth first generates Mainardi-Codazzi-Peterson incompatibility, leading to a mechanical instability that forms cusps. These cusps then focus the stress, which affects the further growth of the tissue.”
Understanding the mechanical mechanisms that alter the shape of rose petals as they grow could inform the design of self-shaping materials and structures for applications like soft robotics and deployable spacecraft. “The idea is to program internal forces to enable the material to shape itself, and this work offers a new strategy for creating more localized shaping,” explains Benoît Roman of ESPCI ParisTech, an expert in shape-changing materials. “But the real value of this study is that it provides a perfect example of using physics to uncover and describe a deep and general phenomenon.”
–Susan Curtis
Susan Curtis is a freelance science writer based in Bristol, UK.
References
- Y. Zhang et al., “Geometrically frustrated rose petals,” Science 368, 520 (2025), https://www.science.org/doi/10.1126/science.adt0672.
- E. Siéfert et al., “Euclidean frustrated ribbons,” Phys. Rev. X 11, 011062 (2021).