We present an experimental study of the buckling cascades that are formed along the edge of a torn plastic sheet. The edge of the free sheet 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. Equilibrium shapes are expected to obey this metric, in order to reduce the elastic energy of the sheet. However, it is not clear whether such smooth surfaces exist in Euclidean space. Indeed, our data show that the precise scaling of the cascades is not given by geometry alone. It depends on the sheet thickness as well, indicating the relevance of bending-stretching competition at all scales. This might be an indication for the absence of a smooth imbedding of the generated metrics in Euclidean space. Similar geometrical features (Similar metrics) could result from very simple growth mechanisms. We, thus, suggest that some of the complex wavy shapes of leaves and flowers might result from this spontaneous buckling instability. We support this suggestion by presenting measurements of metrics of naturally-wavy leaves and by artificially inducing buckling in naturally-flat leaves.