ON A SELF DUAL 3-SPHERE OF PETER MCMULLEN

We study an equifacetted self dual 3-sphere SMcM of Peter McMullen, [10], in particular its automorphism group A(SMcM) and its relation to the Coxeter group H4 of the 600-cell. A closely related equifacetted polyhedral 3-sphere (240-cell) with 240 facets and 120 vertices has the same automorphism group. Both these 3-spheres and the polar dual of the last one cannot occur as the boundary complex of a (convex) 4-polytope with A(SMcM) as their full Euclidean symmetry. It is an open problem, whether there exist one of these three 4-polytopes at all. Their combinatorial symmetry would differ from their Euclidean one within their whole realization space, similar to the example given in [3], see also [2]. Tackling these problems with methods from computational synthetic geometry [5] fail because of the large problem size. Therefore, a partial Euclidean symmetry assumption for the questionable polytope is natural. On the other hand, we show that even a certain subgroup of order 5 of the full combinatorial symmetry group A(SMcM) of order 1200 cannot occur as a Euclidean symmetry for McMullen's questionable polytope.