  We focus on two important classes of lattices, the well-rounded and the
cyclic. We show that every well-rounded lattice in the plane is similar to a
cyclic lattice, and use this cyclic parameterization to count planar
well-rounded similarity classes defined over a fixed number field with respect
to height. We then investigate cyclic properties of the irreducible root
lattices in arbitrary dimensions, in particular classifying those that are
simple cyclic, i.e. generated by rotation shifts of a single vector. Finally,
we classify cyclic, simple cyclic and well-rounded cyclic lattices coming from
rings of integers of Galois algebraic number fields.
