A large piece of paper is cut into a right isosceles triangle, where the equal sides each have length 512. The paper is folded in half by bringing the two equal sides together, yielding another right isosceles triangle. This is repeated another 12 times, for a total of 13 folds. Physical impossibilities aside, what would be the length of the longest side in the resulting triangle?
Remark: Britney Gallivan will be able to tell you the size of the paper needed to achieve an actual physical 13th fold.