A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. So, and correct me if I'm wrong, you can say a simplex integer lattice is a generalization of a lattice built from the end points of simplexes in arbitrary dimensions.
But what about the geometric shape square in 2D and cube in 3D? Is there a word for generalizing them in arbitrary dimensions? What if you use them in a lattice? Simpletic honeycomb comes to mind, but obviously doesn't fit.