New polytopes from products

We construct a new 2-parameter family E m n , m , n ⩾ 3 , of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E 44 is the 24-cell. For large m , n the f-vectors have “fatness” close to 6. The E t -construction of Paffenholz and Ziegler applied to products of polygons yields cellular spheres with the combinatorial structure of E m n . Here we prove polytopality of these spheres. More generally, we construct polytopal realisations for spheres obtained from the E t -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 E 33 is at least nine-dimensional and that of E 44 at least four-dimensional. This proves that the 24-cell is not projectively unique. All E m n for relatively prime m , n ⩾ 5 have automorphisms of their face lattice not induced by an affine transformation of any geometric realisation. The group Z m × Z n generated by rotations in the two polygons is a subgroup of the automorphisms of the face lattice of E m n . However, there are only five pairs ( m , n ) for which this subgroup is geometrically realisable.