This is an interesting question, and Sander and Michael started putting out ideas for resources. I thought it is a good idea for a start and I would note a couple of related things.

Robert Lang
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First of all Robert Lang, the presenter in the TED video linked by Sander, is an avid Mathematica user. For example quoting Rober Lang from McGill's "Origami pteranodon" :
Just in case here is another link. Robert Lang is also an author at Wolfram Demonstrations where the source code notebooks can be freely downloaded:



You can find moch more at Demonstrations Project - just search, for example: folding or origami or paper. According to his recent interview The Art of Origami we can again quote Robert Lang:- something to keep in mind.

Yves Klett
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Another great Mathematica user Yves Klett has had presented at Wolfram Technology conference about folding - see for example: Aerospace Origami & Fractal Folding and the video An Engineer's Box of Chocolates.

Research
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The following research paper

(Non)existence of Pleated Folds: How Paper Folds Between Creases
Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, Tomohiro Tachi

has extensive discussion on creases with given Mathematica code. If you check the authors affiliations and do a little web search on them, you will find them to be very interesting creative people. Here is a snapshot from the paper:



I hope more discussion will follow with more precise approaches.

Dear Dietmar,

This is basically origami. It is all about crease-patterns, obeying several rules/constraints. And that can be solved by circle packing.

See especially this TED video:

Robert Lang: The math and magic of origami

enjoy

Hi Dietmar

A while ago (Fiac 2007-8 maybe), in a famous garden in Paris, I have seen what seems to be huge sheet of wrinkled metal, exactly like a paper sheets, the one you throw to the trash can. The artist who made that is Wang Du  ( some samples here ). I am still wondering of how he succeed to do it, is it reproduction at scale or is it made using a mechanical process ? It was metal and seems to have been made by a geant hand...

The right tool to simulate that kind of shaping is finite element, with some special hypothesis (thin material). Those kind of tools are used in the automotive industry for crash test simulation.  But even if you could access such a tool (witch usually run on a supercomputer), for artistic design, I am not sure there is a way to design any shape in an other way than experiment on scaled models. And it is maybe not even reproducable, even with a folding robot that apply repeatable constraints because of instable effects. This let randomness play a good role for art creativity, anyway.

Nicolas

Thanks for supporting. I did not yet study all, and I will spend some time with this. Up to now I have the feeling, that origami is a part, but differs. Maybe its a matter of dimension (3 instead of 2? 4 instead of 3?;-) or random or physics. This is a spontaneous try to focuse an example:



PS.: Of course I saw the "stent",- this drawing is not a 100% serious. But up to now I lack words to descibe the difference.

I heard about this guy (and tried to contact him many years ago.) Yes, it is nearby. Maybe this is it,- but I am not sure: Think of it not at a way which leads to the optimum but as an accident which goes its physical way. I am longing for the process. What happens on the way from the flat sheet to 180°-fold? How tide can you go? What about the stiffnes of the sheet and the tendency to smooth all edges? Where is the point of break, from which you need more sharp angles? (And, by the way, I would not be able to transform his program into 3d-program.)
Thank you so far.

Hi Dietmar

There are many issue with FEM, it is slow but with some simplifications i thing it could be real time on modern PC. FEM software are not designed for interactivity like 3D cad software, also the interaction with the material must be done in a mathematical way by defining deformation or effort you apply on the material. It is far to be as easy to use as 3D software. With a FEM software you could deform the material in a realistic way, and go to a final result depending of the path of your physical interactions, but you cannot design directly the final object assuming constraints of the material are respected.

What seems possible is to use your prefered CAD software and design your shape using only surface primitives,
assuming in you mind that it is a perfect sheet at the origin, which obviously requires experience. 
Then, create a polygon approximation of the shape (the CAD software is maybe able to do it for you)
Feed the result in some Mathematica code to compute the constraint that would appear by flattening your design back to the original sheet form.
For example, use a color map, to display on the polygon where the tensile effort are the biggest positive (red) and lowest negative (blue)
Then map back the colored polygon to your original shape, by reimporting in the CAD software.
Then use a design heuristic :
- where the tensile force is to big you have to deform the shape to lower it (give locally more surface to the shape by changing the surounding)
- where the tensile force is to small you have to add some wrinkled (remove locally some surface)
This is just an idea, but if you could add this feature in real time superimposed to your software,
you could have a desgin assistant to follow the perfect sheet deformation constraint...

You can also try to use a physics engine created for games, they can approximate clothes or flat material by a chain, and are realistic.

Nicolas

I thought of something like an universal fold, which can be combined to any folds and wrinkles. But I cant fix it up to now, so I close this discussion. Thanks for your ideas and suggestions. Maybe another time!
Dietmar