Theory of edges of leaves

– We performed experiments in which tearing pieces of plastic produced a fractal boundary. Similar patterns are commonly observed at the edges of leaves. These patterns can be reproduced by imposing metrics upon thin sheets. We present an energy functional that provides a numerical test-bed for this idea, and derive a continuum theory from it. We find ordinary differential equations that provide minimum energy solutions for long thin strips with linear gradients in metric, and verify both numerically and experimentally the correctness of the solutions. Introduction. – If one takes a thin piece of plastic, such as a garbage bag, and rips it in half, the edge takes on a complicated rippled form. We recently performed a careful experimental study of torn sheets and found the edge can have a fractal character, with waves superimposed upon waves over many generations [1]. Our aim in this letter is to provide some explanations for the waves appearing in these systems. We do not believe that details of how plastic tears are important, and we will not discuss them. All that appears to be essential is that the act of tearing imposes a new metric upon the thin sheet of plastic, one in which the plastic is uniformly elongated as one moves toward the newly formed edge. Similar elongation can be created by completely different physical processes, such as growth laws in plants. Therefore, it is not surprising that the edges of many leaves and flowers exhibit very similar shapes. We present in fig. 1 a comparison between torn plastic sheets, leaves, and some numerical experiments that motivated our studies.