  Over the years, studies of collective properties of medium and heavy mass
nuclei in the framework of the Interacting Boson Approximation (IBA) model have
focused on finite boson numbers, corresponding to valence nucleon pairs in
specific nuclei. Attention to large boson numbers has been motivated by the
study of shape/phase transitions from one limiting symmetry of IBA to another,
which become sharper in the large boson number limit, revealing in parallel
regularities previously unnoticed, although they survive to a large extent for
finite boson numbers as well. Several of these regularities will be discussed.
It will be shown that in all of the three limiting symmetries of the IBA [U(5),
SU(3), and O(6)], energies of 0+ states grow linearly with their ordinal
number. Furthermore, it will be proved that the narrow transition region
separating the symmetry triangle of the IBA into a spherical and a deformed
region is described quite well by the degeneracies E(0_2^+)=E(6_1^+), E(0_3^+)
=E(10_1^+), E(0_4^+)=E(14_1^+), the energy ratio E(6_1^+) /E(0_2^+) turning out
to be a simple, empirical, easy-to-measure effective order parameter,
distinguishing between first- and second-order transitions. The energies of 0+
states near the point of the first order shape/phase transition between U(5)
and SU(3) will be shown to grow as n(n+3), where n is their ordinal number, in
agreement with the rule dictated by the relevant critical point symmetries
studied in the framework of special solutions of the Bohr Hamiltonian. The
underlying dynamical and quasi-dynamical symmetries are also discussed.
