Shaping thin sheets and the geometry of wavy leaves

Full Text:Gauss's famous theorem (theorema egregium) establishes the connection between intrinsic metric properties of n surface and its possible shapes in Space. This link provides a powerful mechanism, not yet studied. For the generation of complex three dimensional shapers from train elastic sheets, by - prescribing curved metrics on them-For example the edge of a torn plastic sheet is composed of an organized cascade of waves. The waves are similar in shape but differ greatly in scale. Leading to the formation of n 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, but the limitations on the embedding of such a metric in Euclidean space ''force' the sheet to wrinkle. We use environmentally responsive gels to form ''engineered sheets'' - sheets that adopts a prescribed metric upon induction by environmental conditions. With this system wave can study the shaping mechanism in a large variety of metrics. We suggest that some complex shapes of leaves and flowers might result from this buckling instability that links between simple growth and complex configuration. The complexity, in this case. Results from elasticity and not from complex growth processes, as commonly accepted