Geodesic
The edge-transitive polytopes that are not vertex-transitive
Ars Mathematica Contemporanea, Tome 23 (2023) no. 2, article no. 01, 29 p.

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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, edge-transitivity 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.
DOI : 10.26493/1855-3974.2712.6be
Keywords: Convex polytopes, symmetry of polytopes, vertex-transitive, edge-transitive 
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Frank Göring; Martin Winter. The edge-transitive polytopes that are not vertex-transitive. Ars Mathematica Contemporanea, Tome 23 (2023) no. 2, article  no. 01, 29 p. doi : 10.26493/1855-3974.2712.6be. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2712.6be/

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