The edge-transitive polytopes that are not vertex-transitive

Authors

DOI:

https://doi.org/10.26493/1855-3974.2712.6be

Keywords:

Convex polytopes, symmetry of polytopes, vertex-transitive, edge-transitive

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 ≥ 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.

Published

2022-10-28

Issue

Section

Articles