The edge-transitive polytopes that are not vertex-transitive - Göring, Frank (Ars mathematica contemporanea, 2023)
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension ?$d ≥ 4$?, edgetransitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such polytope in dimension ?$d ≥ 4$? is vertex-transitive.
Keywords: convex polytopes; edge-transitive; konveksni politopi; povezavno tranzitiven; simetrija politopov; symmetry of polytopes; točkovno tranzitiven; vertex-transitive