We prove that there is only one way to 'desingularize' the intersection of two planes in space and to obtain a periodic minimal surface as a result. The proof is mostly an exercise in, and an introduction to, basic Teichmüller theory: we translate the geometry of minimal surface in space into a statement about a moduli space of flat structures on Riemann surfaces, and then study deformation theory and degenerations in this moduli space to prove the result. We remark on the general (non-periodic) case.