We investigate ground state configurations for atomic potentials including both two- and three-body
nearest-neighbor interaction terms. The aim is to prove that such potentials may describe
crystallization in different lattice geometries, such as the typical hexagonal periodicity
of carbon nanostructures. We give conditions in order to prove that planar energy minimizers
are periodic and to quantify the lower-order surface energy contribution. We discuss the
optimality of ground states in terms of discrete isoperimetric inequalities and we study
their aspect ratio as the number of particles grows. Finally, by recasting the minimization
problem in three-space dimensions, we remark some properties about the geometry of carbon
rolled up structures and fullerenes. (Joint works with Paolo Piovano and Ulisse Stefanelli).