AbstractE. Cartan's famous isoparametric hypersurface inS4with three distinct constant principal curvatures is geometrically a parallel hypersurface of the Veronese surface, and topologically it is an 8‐fold quotient of the 3‐sphere. In the present paper we describe a polyhedral analogue with only 15 vertices. Combinatorially this is an 8‐fold quotient of the boundary complex of the 600‐cell, and geometrically it is a quite regular subcomplex of a certain almost convex simplicial 4‐sphere inE5. The euclidean symmetry group of this embedding is isomorphic to the icosahedral groupA5acting transitively on the 15 vertices.