Recent experimental and theoretical evidence suggests that the structures of many grain boundaries are quasicrystals, consisting of an aperiodic sequence of structural units which have a deflation symmetry. We determine the conditions for a grain boundary to be deflatable and therefore quasicrystalline, in two and three dimensions. We also show why it is possible for two- or three-dimensional quasicrystal particles to grow on one- or two-dimensional grain boundaries, respectively. This work suggests that a quasicrystal is the structure of minimal Gibbs free energy under specific, but not uncommon, boundary conditions.