Voronoi defined a tiling of the space of positive definite quadratic forms on $R^n$.  Each tile is determined by a perfect lattice.  Examples of perfect lattices include the root lattices $A_n$ and $D_n$.  We will focus on the tiles that are neighbors of a $D_n$ tile.  These are determined by some simple combinatorics, but there are exponentially many different $D_n$ neighbors, and only a few families of them have been understood for all n.  I will report on two recent results, one due to myself, and one joint work with Noam Elkies.