Spiraling and Folding: The Topological View

For every $n$, we construct two curves in the plane that intersect at least $n$ times and do not form spirals. The construction is in three stages: we first exhibit closed curves on the torus that do not form double spirals, then arcs on the torus that do not form spirals, and finally pairs of planar arcs that do not form spirals. These curves provide a counterexample to a proof of Pach and T\'{o}th concerning string graphs.