# On the Plants Leaves Boundary, ‘Jupe à Godets’ and Conformal Embeddings

### Sergei K. Nechaev 1, 2, Raphael Voituriez 1

#### Journal of Physics A 34 (2001) 11069-11082

he stable profile of the boundary of a plant's leaf fluctuating in the direction transversal to the leaf's surface is described in the framework of a model called a 'surface à godets'. It is shown that the information on the profile is encoded in the Jacobian of a conformal mapping (the coefficient of deformation) corresponding to an isometric embedding of a uniform Cayley tree into the 3D Euclidean space. The geometric characteristics of the leaf's boundary (like the perimeter and the height) are calculated. In addition a symbolic language allowing to investigate statistical properties of a 'surface à godets' with annealed random defects of curvature of density $q$ is developed. It is found that at $q=1$ the surface exhibits a phase transition with critical exponent $\alpha=1/2$ from the exponentially growing to the flat structure.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. L D Landau Institute for Theoretical Physics,
Landau Institute for Theoretical Physics