A cross field is a locally-defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science. In this talk, I will consider the problem of generating an arbitrary n-cross field using a fourth-order Q-tensor theory that is constructed out of tensored projection matrices. Computationally, one can then use a Ginzburg-Landau relaxation towards a global projection to reliably generate n-cross fields on arbitrary Lipschitz domains. This tensor framework provides an approach to study the behavior of the singular set, i.e. the set on which the domain fails to be a cross field. In particular we can use the classical Ginzburg-Landau theory to study singularities of the associated energy. This is joint work with Dmitry Golovaty and Albert Montero.