A class of symmetric polytopes (Q917927)

For a polytope P in \(E^ n\) one says that it possesses a pair of facets which can be interchanged, if there exists an automorphism of the face lattice of P which interchanges two facets. The authors show that every polytope P in \(E^ n\) either has no pairs which can be interchanged or possesses at least two of such pairs. They also show that this property is typical for \(E^ 3\) i.e. that for every \(n\leq 4\) there exists a polytope in \(E^ n\) which possesses exactly one pair of facets which can be interchanged.