Buckling Cascades in Free Sheets and the Geometry of Wavy Leaves and Flowers
by
Eran Sharon
The Hebrew University of Jerusalem

We present an experimental study of the buckling cascades that are formed along the edge of a torn plastic sheet. The edge is composed of an organized cascade with up to six generations of waves. The waves are similar in shape but differ greatly in scale, leading to the formation of a fractal edge as an equilibrium configuration. We show that the tearing process prescribes a hyperbolic metric near the edge of the sheet. This metric should be satisfied in order to reduce the stretching energy. We suggest, however that no smooth surfaces exist in a 3D Euclidean space with the prescribed metric. The free sheet, thus, undergoes a "geometrically driven" wrinkling instability. Our data support this picture, showing that the scaling of wavelengths in the cascades depends explicitly on the sheet thickness.

Similar geometrical features (similar metrics) could result from very simple growth mechanisms. We, thus, suggest that some of the complex shapes of leaves and flowers might result from this buckling instability. The complexity, in this case, results from elasticity and not from complex growth processes, as commonly accepted. Finally, I will present results from experiments in plants and environmentally responsive gels.

Date received: October 6, 2003