Wittenʹs lectures on crumpling

Crumpling is a distortion brought about by a strong compression of a surface in which energy condenses in extremely small regions rather than being stored uniformly. Scaling laws describe how much energy goes into how little space. As crumpling of a sheet proceeds, a network of ridges and vertices develops. The precise nature of these geometrical singularities depends on the stretching and bending characteristics. It is shown that for two-dimensional surfaces in three-dimensional space energy mainly condenses in stretching ridges. The scaling arguments are put on a firm basis by using the mathematical description of an elastic membrane through the von Kármán equations, and the universal properties of ridges are discussed. From this theory one can understand that crumpling confers strength. Finally, it is sketched how energy condensation and its scaling laws depend on the dimensions of the confined sheet and embedding space.