From geometric optics to plants: eikonal equation for buckling

Optimal isometric embedding to three-dimensional space of exponentially growing squeezed 2D surfaces, like plant leaves, lilies or other colonies of exponentially reproducing cells, is considered in the framework of conformal approach. It is shown that a boundary profile, adopted by the growing tissue is described by the eikonal equation, which provides the geometric optic approximation for the wave front propagating in the media with inhomogeneous refraction coefficient. The detailed spatial dependence of the refraction coefficient is connected with the specific form of conformal mapping which in turn is dictated by the local growth protocol. We show that numerical patterns predicted by the eikonal equation and development of buckling instabilities are strikingly similar with those found in nature.