The lattice-simplex covering density problem aims to determine the minimal density by which lattice translates of the -simplex cover -space. Currently, the problem is completely solved in dimensions. A computer search on the problem in three dimensions gives experimental evidence that for the simplex (the convex hull of the unit basis vectors), the most effective lattice corresponds to the tile known as the -shape. The -shape tile has been shown to be a local minimum of the density function. We explain the mechanics behind an algorithm which determines the most efficient lattice in the interior of an arbitrary combinatorial type.
