The Local Minima in the Lattice-Simplex Covering Problem

The lattice-simplex covering density problem aims to determine the minimal density by which lattice translates of the \( n \)-simplex cover \( n \)-space. Currently, the problem is completely solved in \( 2 \) dimensions. A computer search on the problem in three dimensions gives experimental evidence that for the simplex \( D \) (the convex hull of the unit basis vectors), the most effective lattice corresponds to the tile known as the \( 84 \)-shape. The \( 84 \)-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.