  We introduce Artinian Gorenstein algebras defined by the face posets of
regular polyhedra. We consider the strong Lefschetz property and Hodge--Riemann
relation for the algebras. We show the strong Lefschetz property of the
algebras for all Platonic solids. On the other hand, for some Platonic solids,
we show that the algebras do not satisfy the Hodge--Riemann relation with
respect to some strong Lefschetz elements.
