  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.
