The Edge-transitive Polytopes that are not Vertex-transitive

TL;DR

This paper proves that in dimensions four and higher, convex polytopes with uniform edges, an inscribed sphere, and bipartite edge graphs are necessarily vertex-transitive, extending known 3D exceptions.

Contribution

It establishes that higher-dimensional analogues of certain 3D edge-transitive but not vertex-transitive polyhedra do not exist, and classifies convex polytopes with specific symmetry properties.

Findings

In dimensions d ≥ 4, such polytopes are always vertex-transitive.

The known 3D exceptions do not extend to higher dimensions.

Classified all convex polytopes with equal edge lengths, inscribed sphere, and bipartite edge graph.

Abstract

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 $d\ge 4$, 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 $d\ge 4$ is vertex-transitive.