Chorna's PhD's thesis is an impressive attempt at systematization of origami art. I do design 3D and also other systems of origami and would like to make the following comments on 3D origami for discussion.
3D origami can be divided into 2 systems - one with intrinsically straight creases and the other with intrinsically curved creases combined with or without straight creases. Each may have closed or open surfaces. With intrinsically straight creases and closed surface, we basically represent a curved (undevelopable) surface with a polyhedral one. The more facets we have, the closer the surface simulates the actual curved one. Besides Tachi's method, there are other ways of folding closed polyhedral models. Here are two of mine: http://www.flickr.com/photos/chengchit/8178246675/in/photostream http://www.flickr.com/photos/chengchit/6236286619/in/photostream Curved surfaces except conical ones are undevelopable but of course we can use curved creases to simulate curved surfaces. The surface of the model can also be closed or open. I have a paper "Simulation of Nonzero Gaussian Curvature in Origami by Curved-Crease Couplets" on this, published in Origami^5. The closed surface examples are: http://www.flickr.com/photos/chengchit/920489454/in/photostream http://www.flickr.com/photos/chengchit/920461244/in/photostream http://www.flickr.com/photos/chengchit/4202093599/in/photostream Robert Lang has his "flanged pots" which are also closed "conical section" surface. In my paper I call the flanges "couplets". The couplets as you can see are on the surface or on the other side of the surface. There are also models with couplets, where the features are defined by the (extrinsically) curved creases. These surfaces are open. Examples of these are: http://www.flickr.com/photos/chengchit/5441384165/in/photostream http://www.flickr.com/photos/chengchit/5210404627/in/photostream Cheng Chit
