From 2D Conformal Growth to 3D Morphogenesis: The Case of Fresh Leaf Development

The morphogenesis of living systems is governed by stresses generated through growth. Predicting these complex morphologies remains a significant challenge, as existing models are largely constrained to idealized geometries. This work presents a unified theoretical framework that overcomes this limitation by describing the morphological evolution of thin biological tissues through two sequential regimes. A key innovation of our framework lies in its establishment of a direct link between stress-free configurations and the time evolution of a conformal mapping. This approach, for the first time, directly links this profound mathematical concept to shape transformation, enabling the accurate reconstruction of diverse planar features—from entire leaf contours to complex topological events like fenestration in Monstera deliciosa.

We further demonstrate how sustained growth disrupts stress-free state, driving a transition into three-dimensional configurations. By uniquely integrating complex analysis with the Föppl-von Kármán theory, our framework captures a wide array of natural 3D patterns of conformal geometry. It quantitatively resolves how spatial growth anisotropy determines the final form, recapitulating phenomena such as doming and curling in complex leaf shapes. While validated on botanical systems, this framework provides a general principle for understanding morphogenesis in soft biological matters.