On the Existence of Similar Sublattices
  Partial answers are given to two questions. When does a lattice Lambda contain a sublattice Lambda' of index N that is geometrically similar to Lambda? When is the sublattice "clean", in the sense that the boundary of the Voronoi cells for Lambda ' do not intersect Lambda? 
J.H. Conway, E.M. Rains, N.J.A. Sloane