The Lagrange spectrum is a classical object in number theory, defined as the set of values of liminf_{q→∞} q dist(qα,ℤ), where α runs through irrational numbers. It has a complicated structure, with the discrete part, Hall’s ray, and a transitional part in between. One can similarly define the Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. I will show that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between 0 and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum. The latter is shown to be equal to the interval between 0 and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.