Summary: We determine the joint distribution of the lengths of, and angles between, the $N$ shortest lattice vectors in a random $n$-dimensional lattice as $n \rightarrow \infty $. Moreover, we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally, we discuss the limit distribution of any finite number of successive minima of a random $n$-dimensional lattice as $n\rightarrow \infty $.