New polytopes from products

We construct a new 2-parameter family Emn, m, n ≥ 3, of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E44 is the 24-cell. For large m, n the f -vectors have “fatness” close to 6. The Et-construction of Paffenholz and Ziegler applied to products of polygons yields cellular spheres with the combinatorial structure of Emn. Here we prove polytopality of these spheres. More generally, we construct polytopal realisations for spheres obtained from the Et-construction applied to products of polytopes in any dimension d ≥ 3, if these polytopes satisfy some consistency conditions. We show that the projective realisation space of E33 is at least nine dimensional and that of E44 at least four dimensional. This proves that the 24-cell is not projectively unique. All Emn for relatively prime m, n ≥ 5 have automorphisms of their face lattice not induced by an affine transformation of any geometric realisation. The group Zm × Zn generated by rotations in the two polygons is a subgroup of the automorphisms of the face lattice of Emn. However, there are only five pairs (m, n) for which this subgroup is geometrically realisable.