Folding movies illustrate theorem about flat surfaces


I learned the really interesting theorem from Joe Fu that a flat polyhedral surface always has the property that the spherical polygon enclosed by all the surface normals at a vertex always encloses area zero. As a favor to Joe, I made some neat Mathematica movies showing the folding process taking a flat sheet with three folds at 120 degree angles to a square pyramid that you get by folding a pair of faces into the center of the pyramid. The resulting animations are here and in a top view here. The Mathematica notebook that I used to create them ishere; it contains some code which computes the folded shape in terms of the first n-3 bending angles (the last three are determined by the intrinsic geometry of the surface).

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