  We present an algorithm that enumerates and classifies all edge-to-edge
gluings of unit squares that correspond to convex polyhedra. We show that the
number of such gluings of $n$ squares is polynomial in $n$, and the algorithm
runs in time polynomial in $n$ (pseudopolynomial if $n$ is considered the only
input). Our technique can be applied in several similar settings, including
gluings of regular hexagons and triangles.
