From geometric optics to plants: eikonal equation for buckling

Optimal embedding in the three-dimensional space of exponentially growing squeezed surfaces, like plants leaves, or 2D colonies of exponentially reproducing cells, is considered in the framework of conformal approach. It is shown that the boundary profile of a growing tissue is described by the 2D eikonal equation, which provides the geometric optic approximation for the wave front propagating in the media with inhomogeneous refraction coefficient. The 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.