Geometrically frustrated rose petals

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.