In 1908 Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time.
Voronoi showed that any lattice with optimal packing density must correspond to a so-called perfect (quadratic)
form and his algorithm enumerates the finitely many perfect forms up to similarity in a fixed dimension.
However, the number of non-similar perfect forms grows quickly in the dimension and as a result
Voronoi’s algorithm has only been completely executed up to dimension 8.
We prove an upper bound of $\exp(O(d^2 \log(d)))$ on the number of similarity classes . The proof is mostly
geometric and concludes by a volumetric argument.
Additionally we discuss some practical challenges and progress for completing Voronoi's algorithm in dimension 9,
e.g. how to efficiently remove redundant similar quadratic forms from a large set .