From geometric optics to plants: the eikonal equation for buckling

Abstract

Optimal buckling of a tissue, e.g. a plant leaf, growing by means of exponential division of its peripheral cells, is considered in the framework of a conformal approach. It is shown that the boundary profile of a tissue is described by the 2D eikonal equation, which provides the geometric optic approximation for the wavefront propagating in a medium with an inhomogeneous refraction coefficient. A variety of optimal surfaces embedded in 3D is controlled by spatial dependence of the refraction coefficient which, in turn, is dictated by the local growth protocol.