At its heart, this research uncovers the hidden geometric principles behind the unique shape of rose petals. While scientists extensively studied shape morphing in natural sheets such as leaves and petals, the team at Hebrew University discovered a new player: MCP incompatibility—a geometric principle that causes the petal’s signature cusps. It turns out that as the petal grows, stress builds at the edges, shaping the curves we recognize and love. The discovery not only uncovered the geometric origin of the shape of rose petals, but also introduces a new paradigm for understanding how complex forms emerge in nature—and how we might harness the same principles to design advanced materials that shape themselves with similar elegance and precision.

Link to pictures and video: Credit Yafei Zhamg https://drive.google.com/drive/folders/111p7Mjx8L7rlrFNFxf1Uxc3h-Fu7vkeo

[Hebrew University] – The soft, curving edges of rose petals have long enchanted poets, painters—and scientists. Now, a team of researchers from the Racah Institute of Physics at the Hebrew University of Jerusalem has discovered the mathematical secret behind this natural elegance.

The study, performed by Dr. Yafei Zhang (a postdoctoral fellow), Omri Y. Cohen (a PhD student), and led by Prof. Moshe Michael (Theory) and Prof. Eran Sharon (Experiments), published on the cover of Science, reveals that the signature cusp-like edges of rose petals are the result of a unique kind of geometric principle—not the kind previously recognized by scientists.

In the last two decades, scientists believed that shapes of slender structures such as leaves and petals emerged mainly due to what’s called “Gauss incompatibility”—a kind of geometric mismatch that causes surfaces to bend and twist as they grow. But when the Hebrew University team studied rose petals, they discovered something surprising: the petals don’t show signs of this kind of Gauss incompatibility.

Instead, the petal’s shape is governed by a new concept, also discovered at the Hebrew University, called Mainardi-Codazzi-Peterson (MCP) incompatibility. Unlike Gauss-based stress, MCP stress causes sharp points or cusps to form along the edge of the petal. The researchers tested this theory using computer models, lab experiments, and mathematical simulations—and the results were consistent across the board.

This discovery doesn’t just change how we understand flowers—it also opens new possibilities for designing self-shaping materials. These are materials that, like petals, change shape as they grow or are activated. The ability to form controlled cusps through MCP stress could lead to innovations in soft robotics, flexible electronics, and bio-inspired design.

One of the most fascinating aspects of the study is how growth and stress feed back into each other. The team found that as the petal grows, stress concentrates at the cusps, which then influences how and where the petal continues to grow. It’s a natural feedback loop—biology influencing geometry, and geometry shaping biology.

“This research brings together mathematics, physics, and biology in a beautiful and unexpected way,” said Prof. Eran Sharon. “It shows that even the most delicate features of a flower are the result of deep geometric principles.”

Prof. Moshe Michael added, “It’s astonishing that something as familiar as a rose petal hides such sophisticated geometry. What we discovered goes far beyond flowers—it's a window into how nature uses shape and stress to guide growth in everything from plants to synthetic materials.”

By uncovering the hidden rules behind rose petal formation, the team from Hebrew University has not only solved a botanical mystery—they’ve also added a powerful new concept to the toolbox of engineers and scientists seeking to mimic nature’s elegance in manmade systems.