The Greek cross and paper folding squares

Researches have shown that paper-folding is an invaluable approach for developing mathematical skills and logical minds. It is obvious that many geometric origami works are symmetrical in nature. This fine property serves as an effective check and reduces inevitable errors in folding, particularly for young children. Our problem under study is: Given a square piece of paper how to fold a new square one-nth of its original area in general? Formerly we have already developed a series of method based on similar triangles and circle properties. Nevertheless when we come upon the case of n=5, we find that the famous symmetrical Greek-Cross folding technique yields a neat and simple way in solving the problem. So further problems come into our minds: Can this technique be extended to other values of n? Under what constraints is this technique transferable? In this paper we shall discuss our findings and suggest some ways of putting the folding methods into classroom practices.