I do not love origami, but Mitsuko gave me an idea for a extremely hard and (not that?) beautiful puzzle. I'm really curious whether anyone here can solve it.
So here's the puzzle. You are given a large perfectly square piece of paper with no marks on it. With this square, you have to make a square of exactly one-third the area of the original square. You are given no tools such as a ruler or scissors, and all you can do is fold the paper. How do you solve this?
EDIT: Though I did not explicitly ask it, I would like a 'beautiful' solution, which for me means just using geometry not algebra. I just realized that such a solution is possible. A simple explanation, a centered solution and little folds are also a plus i.m.o. but if I soon get a correct geometry based solution (like in the one-fifth problem I based this one on), I will accept that one instead of one of the correct algebra based ones I already have or may get.
Note: There are several answers now with different approaches. I can only accept one, but I'd suggest reading a few least, if you are interested in ways to solve this.