The edge-transitive polytopes that are not vertex-transitive

In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and triacontahedron, only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show these polyhedra do not have higher-dimensional analogues, is, in dimension d ? 4, edge-transitivity of convex implies vertex-transitivity.More generally, we give a classification all which at same time edges length, an edge in-sphere bipartite edge-graph. any such polytope 4 is