(Click for a bigger version. Guitar pick included for scale.)
As I said I would, I created the second Brillouin Zone of the fcc structure using modular origami. It's a shape that's related to some crystal structures we study in condensed matter physics, though truth be told it is not a very useful shape. It's pretty though.
It took 96 squares of paper, and four different kinds of modules. It holds together, but I wouldn't want to toss it around like some of the more stable shapes.
It turns out that designing your own modular origami is nontrivial. I started out by making the shapes exactly as they were in my origami book, and then I started making small modifications. But the first few things I tried (with simpler shapes) were not very successful. It's difficult to make them sufficiently stable.
Among the things I tried was the second Brillouin Zone of the bcc structure. I have a small one pictured below, but it doesn't count because I cheated. It completely fell apart and I had to cover it in tape. I could probably just create 48 triangle modules, but I'm trying to think of a more elegant way.