  Given a noncompact disconnected complete periodic curve $\Gamma$ with no self
intersection in $\mathbb R^3$, it is proved that there exists a noncompact
simply connected periodic minimal surface spanning $\Gamma$. As an application
it is shown that for any tetrahedron $T$ with dihedral angles $\leq90^\circ$
there exist four embedded minimal annuli in $T$ which are perpendicular to
$\partial T$ along their boundary. It is also proved that every Platonic solid
of $\mathbb R^3$ contains three types of free boundary embedded minimal
surfaces of genus zero intersecting every face of the solid along convex closed
congruent curves.
