Combinatorially two-orbit convex polytopes

TL;DR

This paper classifies all convex polytopes with a two-orbit automorphism group, showing they are all finite, 3-dimensional, and correspond to well-known Archimedean solids and their tilings.

Contribution

It proves that all combinatorially two-orbit convex polytopes are isomorphic to known Archimedean solids and their tilings, providing a complete classification.

Findings

All such polytopes are 3-dimensional Archimedean solids.

They are isomorphic to the cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic triacontahedron.

The classification extends to two-orbit face-to-face tilings by convex polytopes.

Abstract

Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group coincide.) Hence, a combinatorially two-orbit convex polytope is isomorphic to one of a known finite list, all of which are 3-dimensional: the cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic triacontahedron. The same is true of combinatorially two-orbit normal face-to-face tilings by convex polytopes.