Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling-up often occurs when boundaries are not fixed. In this talk, I explain this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two- and three-body terms. We prove that all energy minimizers are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes. For suspended samples we refine the analysis further and prove the emergence of wave patterning. Specifically, we show that almost minimizers of the configurational energy develop waves with certain wavelength, independently of the size of the sample. The talk is based on joint work with Ulisse Stefanelli.