Reciprocal polyhedra and the Euler relationship: cage hydrocarbons, CnHn and closo-boranes [BxHx]2−

The closo-boranes B(x)H(x) (+2), or their corresponding anions [B(x)H(x)](2-) (where x = 5 through 12) and polycycloalkanes C(n)H(n) (where n represents even numbers from 6 through 20) exhibit a complementary relationship whereby the structures of the corresponding molecules, e.g., [B(6)H(6)](2-) and C(8)H(8) (cubane), are based on reciprocal polyhedra. The vertices in the closo-boranes correspond to faces in its polycyclic hydrocarbon …