When manufacturing Lego pieces or the pieces of a jigsaw puzzle special care is taken to assure that the pieces would snugly fit next to one another without any need to deform. However, when tissue grows, a ductile material plastically deforms or a microstructure self-assembles there is no similar mechanism to assure that the different constituents would indeed perfectly fit next to one another. The misfit of the different part of the object gives rise to geometric frustation. We are interested in understanding the equilibrium configurations such frustrated object assume. These configurations compromise between the objects' contradicting intrinsic tendencies and gives rise to highly complex structure and exotic response properties.
When a plastic bag is torn it edge gets stretched uniformly along the tear path. The amount of stretch, however, varies with the distance to the torn edge decaying rapidly into the bulk. As there are no structural variation across the thickness of the thin plastic sheet it prefers to not bend, much like thin straight plates. However, the differential stretching induces a hyperbolic non-Euclidean metric on the sheet, which cannot be realized without bending. Such structures are termed "non-Euclidean plates" and form are one of the simplest realizations of geometric frustration in thin sheets. The ruffles formed on the free edge of a torn plastic sheet are a direct outcome of the non-Euclidean metric induced on it. This complex multi-scale ruffled structure contains the minimal amount of bending among all admisible realizations of the prescribed non-Euclidean metric. We coined the term non-Euclidean plates in this PRE paper . By using responsive material one can engineer 2D metrics to generate desired three dimensional shaped in thin sheets as seen in the image above, and discussed here .