  I study the zero-temperature phase behavior of bosonic particles living on
the nodes of a regular spherical mesh ("Platonic mesh") and interacting through
an extended Bose-Hubbard Hamiltonian. Only the hard-core version of the model
is considered here, for two instances of Platonic mesh. Using the mean-field
decoupling approximation, I show that the system may exist in various ground
states, which can be regarded as analogs of gas, solid, supersolid, and
superfluid. For one mesh, by comparing the theoretical results with the outcome
of numerical diagonalization, I manage to uncover the signatures of diagonal
and off-diagonal spatial orders in a finite quantum system.
