  Spherical nanoclusters and nanoparticles are rising materials whose
functional design provides many useful applications ranging from catalysis,
molecular sensing, gas storage to drug targeting and delivery. Here, we develop
phenomenological crystallization theory of such spherical structures with
octahedral symmetries O and O_h. Within the developed theory, we propose a
method, which is based on constructing irreducible octahedral density functions
and allows to predict the positions of structural units in the spherical
nanoobjects. The proposed theory explains the structures of the simplest known
metal nanoclusters, some metal-organic polyhedra and membrane protein
polyhedral nanoparticles, and also predicts more complex chiral spherical
structures and achiral assemblies characterized by the geometry of semiregular
polyhedra. A relationship between the constructed irreducible octahedral
functions and spherical lattices, obtained by mapping a plane hexagonal order
onto a spherical surface through an octahedron net, is discussed as well.
