A lattice of rank k in n-dimensional Euclidean space has a shortest basis, which possesses many important properties and figures prominently in discrete optimization and theoretical computer science. In particular, it must satisfy a certain "near-orthogonality" condition: the angle between every pair of vectors in this basis must be between 60 and 120 degrees. This fact goes back to the work of Lagrange and Gauss. More generally, consider a collection of m \leq k vectors from a shortest basis, and let A be the solid angle that they span. How small can A be? Same as in the case m=2, there are reasons to believe that perhaps it cannot be too small, which means that lattices should have relatively "fat" layers, in a certain sense. While easily accessible for m=2, this question turns out to be much more difficult when m > 2. I will discuss some recent results in this direction when m=3, and exhibit a connection of this question to the classical kissing number problem of Gregory and Newton. This is joint work with Sinai Robins.