A thorough study of regular and quasi-regular polyhedra shows that the
symmetries of these polyhedra identically describe the quantization of orbital
angular momentum, of spin, and of total angular momentum, a fact which
permits one to assign quantum states at the vertices of these polyhedra assumed
as the average particle positions.
Furthermore, if the particles are fermions, their wave function is
anti-symmetric and its maxima are identically the same as those of repulsive
particles, e.g., on a sphere like the spherical shape of closed shells, which implies equilibrium of these
particles having average positions at the aforementioned maxima. Such
equilibria on a sphere are solely satisfied at the vertices of regular and
quasi-regular polyhedra which can be associated with the most probable forms of
shells both in Nuclear Physics and in Atomic Cluster Physics when the
constituent atoms possess half integer spins. If the average sizes of the
constituent particles are known, then the average sizes of the resulting shells
become known as well. This association of Symmetry with Quantum Mechanics leads
to many applications and excellent results.

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