The complete set of noble polyhedra

Abstract

We provide a complete enumeration of the finite polyhedra that are noble, that is, polyhedra that are vertex transitive and facet transitive, through a computer-assisted proof which relates the possible existence of noble polyhedra to the roots of certain univariate or bivariate cubic polynomials. This is done through a parametrization of the orbits of point groups in $\mathbb{R}^3$, which naturally leads to a notion of criticality for certain orbits of a given point group, and a related equivalence relation $\equiv_c$ on the set of orbits. We show that within a given equivalence class of orbits under this relation, noble facetings are equivalent, and furthermore that there are a finite number of such equivalence classes. This gives a finite number of test cases, which may be tested through computer search. In total, we find that there are exactly 146 noble polyhedra in addition to the previously known infinite sets of stephanoids (crown polyhedra) and disphenoids.