On periodically-cyclic Gale 4-polytopes

Based upon a labelling (total ordering) of vertices, a 4-polytope is Gale if the vertices satisfy a part of Gale's Evenness Condition and it is periodically-cyclic if there is an integer k such that every subset of k successive vertices generates a cyclic 4-polytope. Among the bi-cyclic 4-polytopes introduced by Smilansky, we determine which are Gale or periodically-cyclic or both. (C) 2001 Elsevier Science B.V. All rights reserved.