Thin sheets, such as plant leaves, cell membranes and atomically thin materials, can show complex and surprising behaviors when allowed to grow and deform in three dimensions. One such behavior is the purely elastic shape memory found in disordered spring networks and, to a good approximation, crumpled paper. In this talk, I will introduce a simplified model for this effect, in which we locally swell a periodic array of points on an elastic sheet. When the local expansion is sufficiently large, or the sheet is sufficiently flexible, the regions of dilation will prefer to buckle out of the plane. We find that we can understand the ground state as well as the finite temperature behavior of this system if we assume that the buckled dilations behave like spins in an Ising model, a simple model for magnetism and phase transitions.