A longstanding question of Gromov asks whether every one-ended
word-hyperbolic group contains a subgroup isomorphic to the fundamental group
of a closed hyperbolic surface. An infinite family of word-hyperbolic groups
can be obtained by taking doubles of free groups amalgamated along words that
are not proper powers. We define the set of polygonal words in a free group of
finite rank, and prove that polygonality of the amalgamating word guarantees
that the associated square complex virtually contains a $\pi_1$-injective
closed surface.