  Cao & Yuan obtained a Blichfeldt-type result for the vertex set of the
edge-to-edge tiling of the plane by regular hexagons. Observing that every
Archimedean tiling is the union of translates of a fixed lattice, we take a
more general viewpoint and investigate basic questions for such point sets
about the homogeneous and inhomogeneous problem in the Geometry of Numbers. The
Archimedean tilings nicely exemplify our results.
