From geometric optics to plants: eikonal equation for buckling

TL;DR

This paper models the shape of growing biological tissues like leaves and cell colonies using the 2D eikonal equation, linking geometric optics principles to biological growth patterns.

Contribution

It introduces a conformal approach to describe optimal embedding of exponentially growing surfaces via the eikonal equation, connecting growth protocols to surface geometry.

Findings

Boundary profiles follow the 2D eikonal equation.

Optimal surface shapes depend on spatial refraction variations.

Growth protocols influence the embedded surface geometry.

Abstract

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.