Roses, long admired for their beauty and symbolic richness, owe their iconic petal shape to a mechanical process that has remained largely mysterious—until now. According to a new study, the pointed cusps that gradually form at the edge of rose petals as they grow are shaped not by the well-known mechanics behind wavy leaves, but by a distinct geometric frustration called Mainardi-Codazzi-Peterson (MCP) incompatibility. According to the findings, this stress-focusing phenomenon not only sculpts the rose's form but also feeds back to influence how the petal grows, offering new insights into the mechanics of nature, and potential inspiration for the design of bio-inspired materials. The intricate curves and curls of leaves and flower petals often arise from the interplay between natural growth and geometry. In elastic materials, like plant tissues, growth can create a mismatch between the material's natural geometric preference and what is physically possible, resulting in inherent stresses known as geometric incompatibilities. As these stresses accumulate, they can result in shape changes – an effect known as Gauss incompatibility. This explains features like the rippling edges of leaves and petals. However, the distinctive, sharply pointed cusps along the edges of rose petals stand apart from the soft, wavy patterns seen in many other flowers – features that cannot be explained by traditional Gauss incompatibility.
Here, Yafei Zhang and colleagues combined theoretical analysis, computational modeling, and experimental fabrication of synthetic disc petals to investigate growth-induced mechanical instabilities in rose petals. Zhang et al. discovered that the unique shapes of rose petals are not governed by Gauss incompatibility, but instead by a unique type of geometric frustration known as Mainardi-Codazzi-Peterson (MCP) incompatibility. Unlike traditional shape changes driven by Gauss-type mismatches, this mechanism concentrates stress in highly localized areas, giving rise to the sharply defined cusps seen in roses. Moreover, the authors show that the intense concentration of stress at petal cusps influences how the surrounding tissue grows and takes shape, revealing a powerful feedback loop between biological growth, geometric constraints, and mechanical forces. "Identifying Mainardi-Codazzi-Peterson incompatibility as a shaping mechanism is not only an important milestone in morphogenesis research but also an inspiration for new designs of shape-morphing materials and structures," write Qinghao Cui and Lishuai Jin in a related Perspective. "Combining Gauss and Minardi-Codazzi-Peterson incompatibilities could give rise to deformation behaviors that have yet to be seen."
