  We describe an algorithm to enumerate polytopes. This algorithm is then
implemented to give a complete classification of combinatorial spheres of
dimension 3 with 9 vertices and decide polytopality of those spheres. In
particular, we completely enumerate all combinatorial types of 4-dimensional
polytopes with 9 vertices. It is shown that all of those combinatorial types
are rational: They can be realized with rational coordinates. We find 316014
combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the
boundary complex of a four-dimensional polytope and the remaining 41866 are
non-polytopal.
