Inspired by the origami question, the only new suggestion that I can make is to mention the origamist Robert J. Lang. According to the biography on his web site, he is also a physicist. There was an open problem called the napkin-folding problem: Does there exist a contractive map from the square to the plane that increases its perimeter? Lang found a counterexample using his experience in origami.
Actually, I just discovered that you (Anton) already know this story, following a link on the Wikipedia page. However, my remaining comment is that maybe Lang is a good reference or expert for this new question.