Inscribed Squares and Relation Avoiding Paths

We develop a connection between the inscribed square problem and the question of understanding relation avoiding paths in a complex vector space. Our main theorem is that a Jordan curve with no inscribed squares would have a seemingly impossible structure which we call a square envelope. We will make some conjectures about the nature of relation avoiding paths in vector spaces, and show that these conjectures would imply the existence of inscribed squares in Jordan curves with finitely many arbitrarily complicated singularities.