The pictures in the wikipedia article cannot be approximated by C^2 isometric embeddings, because they look strictly convex. A C^2 embedding of a flat surface has a tangent plane at each point which is tangent along an entire line (this property is actually a characterization of "developable" surfaces, as they're called). If you have any figure made by doubling any flat shape, then for any line, the distance between its endpoints on one sheet equals the the distance on the other, so they must be connected by a straight line on both sides of the cushion. Therefore the map into E^3 is the same for both sides. If you have cloth made from fabric woven in the traditional pattern, with threads running in two orthogonal directions, it can be deformed by stretching diagonally while keeping the warp and weft directions constant in length. I wouldn't be surprised if this could be done smoothly, although I'd have to think harder to figure out whether and how. Maybe that's what the computer simulation represents. Or, it could be using Nash's criterion to construct a "short" embedding, or, it could look smooth because of how computer graphics is often rendered using smooth shading that disguises the underlying features. There's a certain amount of mathematical literature on this condition for cloth, although I'm not familiar with its current status. For instance, Struik's "Lectures on Classical Differential Geometry" mentions this on p. 204 (via Google), referencing Tchebycheff "Sur la coupe des vetements", 1878, oeuvres II, p. 708. The key thing about C^2 surfaces is that the quadratic approximation (the 2nd fundamental form) gives a lot of information. For instance, wherever the principle curvatures are not equal, we get a line field. For C^1 surfaces, these line fields are not necessarily defined in any continuous way. In particular, a C^1 developable surface need not have globally defined straight lines through every point. Bill On Oct 13, 2007, at 6:01 AM, David Gale wrote:
Thanks Bill for the example of the mutilated pyramid, (lots of concavities). Question; You say C^2 is not possible (because of zero curvature condition?). See the pictures of sofa cushions in the wikipedia article. They look pretty C^2 to me. What are they?