  We give new classes of examples of orbits of the diagonal group in the space
of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit
closures, as well as examples of orbits with explicitly computable but
irregular orbit closures. We give Diophantine applications to the former, for
instance we show that if x is the cubic root of 2 then for any y,z in R liminf
|n|<nx-y><nx^2-z>=0 (as |n| goes to infinity), where <c> denotes the distance
of a real number c to the integers.
