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

On Artinian Gorenstein algebras associated to the face posets of regular polyhedra