Let $G$ be a three-dimensional $d_1\times d_2\times d_3$ cubic lattice. I project $G$ onto a plane to generate a two-dimensional set of lattice points $G_p$. How should I orient $G$ to minimize the number of lattice points in $G_p$ (sets of overlapping points are counted as a single point), and how well can I do as a function of the dimensions of the original cubic lattice?
If you project it along the axis of $\max(d_1,d_2,d_3)$, you will do the best. This is the maximum number of points you can get on each line, and each line will have that many. If $d_1$ is the greatest, you will have $d_2d_3$ points.