Wrapping a sphere: stress relaxation by wrinkling

The low energy deformations of thin elastic sheets are isometries because these incur no stretching energy while the cost of bending is small. Since there is no isometric map of a flat sheet, i.e. a developable surface, onto the surface of a sphere, it is natural to suspect that any such map must cost finite stretching energy. However, I will show that there are an enormous number of almost isometric mappings which approximate a sphere with arbitrary accuracy and with arbitrarily small stretching energy. I will construct an example using multiscale analysis of a radial wrinkle pattern in a thin elastic sheet bent over a sphere. These techniques could be applied to other wrinkling problems and to problems connected to developable surfaces, e.g. textures in smectic liquid crystals.