Foldaway stars
There used to be a very interesting geometrical problem called "the problem of the foldaway container". It was based on the following observation: in two dimensions, the only rigid polygons are triangles. That is, if we suppose that polygon's sides are rigid, but they can fold at the corners, all but triangles are flexible, and in many cases (for example, a rectangle) , they can fold away to a line. In three dimensions the situation is quite different. Very few polyhedra (bodies defined by flat faces) are flexible, and none is known that can fold away to a flat surface. Maybe you're thinking some paper bags can be folded flat, but they aren't foldaway containers in three dimensions, because paper isn't rigid. If the bag had rigid faces, it coudn't be opened or closed.
 

The star folds into a cross pressing simultaneously on the four green dots. The cross can then easily be folded flat, and it looks like the picture on the right.		 I set to find a foldaway container, and I'm happy to say a found a very promising candidate. With this I mean calculations showed it isn't really foldaway, but a paper model folds quite smoothly, and besides, it can be modified in several directions, all of which fold quite well. So I thought I was getting rather close to the foldaway container. 
The container has the shape of a four-pointed star. If you want to make your own model, cut two stars and four diamond shapes as in the picture (use the grid as a guide). The black lines are the borders of the figure, blue lines are folds to the inside or "valleys" and red lines are folds to the outside or "mountains". Paste the four diamonds to the points of the star. Sides "A" and "B" in the figure should coincide. Finally, paste the second star of the model symmetrically. 
I said I thought I was getting rather close. That is... until I read an article in Scientific American explaining that foldaway containers don't exist. To be more precise, it has been proven that the volume of a polyhedron, flexible or not, can be calculated knowing the length of its sides. Flexible polyhedra always keep their volume constant. There can be several solutions - an example is my star, that can be folded flat or be a solid body - but the volume can't vary continously, as it should while it folds away.
Still, one may ask: how close can we get to the foldaway container? Nothing is perfectly rigid in real life. Given any material of known rigidity, could we make a foldaway container with it? The answer is "yes", and it's easy to find one based on my foldaway star. There are two possible paths one might follow:
1) Make the star with longer points
2) Add more points to the star
Any of them will give you a solution. So I can say I got as close as possible to the foldaway container.

If you've got any comments or suggestions about this matter, e-mail me at: lusina@redestb.es


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