Let 0 < n ∈ Z . In the unit distance graph of Z n ⊂ R n , a perfect dominating set is understood as having induced components not necessarily trivial. A modification of that is proposed: a rainbow perfect dominating set, or RPDS, imitates a perfect-distance dominating set via a truncated metric; this has a distance involving at most once each coordinate direction taken as an edge color. Then, lattice-like RPDS s are built with their induced components C having: i vertex sets V ( C ) whose convex hulls are n-parallelotopes (resp., both ( n − 1 )- and 0-cubes) and ii each V ( C ) contained in a corresponding rainbow sphere centered at C with radius n (resp., radii 1 and n − 2 ).