This web content explores the possibilities for shapes that a two-dimensional material can adopt in three-dimensional space, depending on its geometry. The author investigates the cases of Euclidean, spherical, and hyperbolic geometries. The content explains the assumptions made about the material, such as being infinitesimally thin and perfectly flexible. It then discusses Gaussian curvature and its role in quantifying the departure from Euclidean geometry. The content further explores intrinsically flat surfaces, including planes, cylinders, cones, and tangent surfaces. The author includes mathematical explanations and provides examples to illustrate different concepts.
https://www.gregegan.net/SCIENCE/PSP/PSP.html