Editor’s summary
Stressed thin sheets can form complex shapes that are caused by mechanical instabilities. An example of this in nature is the growth of leaves. Many of these shapes can be described by Gauss incompatibility. An exception to this behavior is the shape of rose petals, which do satisfy Gauss conditions. However, Zhang et al. have shown that the formation of localized cusps along the edges of rose petals during their growth is driven by Mainardi-Codazzi-Peterson incompatibility instead (see the Perspective by Cui and Jin). Although this theory is known, it has not been connected to natural growth. By combining theory, simulations, and experimental results, including the fabrication of synthetic petals, the authors provide an expanded perspective on the geometric frustrations that affect morphogenesis in biological systems and qualitatively demonstrate the role of stress concentration in petal growth. —Marc S. Lavine
Abstract
Growth and form are deeply interconnected, in a manner often mediated by mechanical instabilities arising from geometric incompatibilities. Although Gauss incompatibility has long been recognized as the source of morphing in naturally growing slender organs, here we show that the growth profile of rose petals remains Gauss compatible. Their distinctive shape emerges from a different type of geometric incompatibility, the Mainardi-Codazzi-Peterson (MCP) incompatibility, which leads to the formation of localized cusps along the petal margins. We validated this mechanism in model disc petals theoretically, computationally, and experimentally. Our study reveals distinct morphological regimes, ranging from smooth edges to cusp-forming configurations, and demonstrates how stress focusing at cusps influences subsequent petal growth. These findings position MCP incompatibility as a generic mechanism for cusp formation in both natural and manmade self-morphing sheets.
Related Perspective
The mechanics behind the beauty of roses
By Qinghao Cui, Lishuai JinScience1 May 2025
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Published In
Science
Volume 388 | Issue 6746
1 May 2025
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Copyright © 2025 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works.
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Received: 10 September 2024
Accepted: 25 February 2025
Published in print: 1 May 2025
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Acknowledgments
Funding:
Y.F.Z. is supported by the Israel Academy of Sciences and Humanities and Council for Higher Education Excellence Fellowship Program for International Postdoctoral Researchers. This work was funded by the United States–Israel Binational Science Foundation (grant 2020739) and by the Israel Science Foundation (grants 2437/20 and 1441/19).
Author Contributions:
E.S. designed and initiated the research. Y.F.Z. conducted the theoretical, numerical, and experimental studies. M.M. conducted the theoretical and numerical studies. O.Y.C. designed the 3D-printed discs. Y.F.Z., M.M., and E.S. wrote the paper. E.S. and M.M. jointly supervised the work. All authors contributed to the analyses and discussions of the results.
Competing Interests:
The authors declare that they have no competing interests.
Data and Materials Availability:
All data are available in the main text and/or the supplementary materials.
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Copyright © 2025 the authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original US government works. https://www.science.org/about/science-licenses-journal-article-reuse
Authors
Affiliations
Yafei Zhang https://orcid.org/0000-0003-1057-2400
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel.
Roles: Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Validation, Visualization, Writing - original draft, and Writing - review & editing.
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Omri Y. Cohen https://orcid.org/0009-0008-1527-6372
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel.
Roles: Methodology, Resources, Validation, and Writing - original draft.
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Michael Moshe* https://orcid.org/0000-0003-1746-6619 [emailprotected]
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel.
Roles: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Writing - original draft, and Writing - review & editing.
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Eran Sharon* https://orcid.org/0000-0001-6311-1099 [emailprotected]
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel.
Roles: Conceptualization, Funding acquisition, Methodology, Project administration, Resources, Supervision, Validation, Writing - original draft, and Writing - review & editing.
View all articles by this author
Funding Information
Israel Science Foundation: 2437/20
Israel Science Foundation: 1441/19
Notes
*
Corresponding author. Email: [emailprotected] (M.M.); [emailprotected] (E.S.)
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Geometrically frustrated rose petals.Science388,520-524(2025).DOI:10.1126/science.adt0672
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