On unknown polytopes with given combinatorial lateral surface lattices (Q2763057)

In his PhD Thesis [Univ. Birmingham (1968)], the reviewer proved that, if the symmetry group of a (convex) \(d\)-polytope \(P\) is transitive on the \(j\)-faces of \(P\) for each \(j=0,\dots, d-1\), then \(P\) is regular. He also asked whether the combinatorial analogue of this result holds. In pursuit of this problem, he devised what the author of the work under review calls the ``McMullen sphere'' \(S_{McM}\), which is defined as follows. Project the boundary complex of the 4-dimensional regular 600-cell (with 120 vertices) from its centre (taken to be the origin) onto the unit sphere. The vertices fall on 12 spherical decagons (they are Clifford parallels). Delete the edges of these decagons and the triangles to which they belong, thus grouping the tetrahedral cells in fives to form pentagonal bipyramids. Then \(S_{McM}\) is self-dual, and has 120 vertices, 600 edges, 600 triangular faces and 120 cells, and its symmetry group (of order 1200) is transitive on incident vertex-edge and face-cell pairs, and thus, in particular, on its faces of each dimension. With full symmetry, \(S_{McM}\) cannot be isomorphic to the boundary complex of a 4-polytope; the question (directly germane to that asked above) is whether it is polytopal at all (that is, can be convexly realized).NEWLINENEWLINENEWLINEIn her thesis, the author tackles this problem, and a related one involving a 240-cell \(S_{240}\), obtained from \(S_{McM}\) by splitting each bipyramid into two pentagonal pyramids; the 240-cell is (fairly clearly) realizable as a non-convex PL-sphere. She proves that both spheres satisfy all the standard necessary properties to be polytopal-shellable, 4-connected graph, and so on. Her main result is that \(S_{McM}\) is not convexly realizable with a 5-fold symmetry.NEWLINENEWLINENEWLINEThe original question remains open.