[math-fun] Origamics

I recently acquired an e-copy of this Kazuo Haga monograph, which explores (at considerable and oriental length) geometrical constructions achievable via paper-folding. In particular, he claims to have constructed all prime length ratios up to 31:1 , prompting the following question. Given a square of paper --- assumed accurately foldable to superpose two existing points such as corners, or a point upon an existing line such as an edge --- are all rational ratios constructible? Fred Lunnon