On periodically-cyclic Gale 4-polytopes Original Research Article

Based upon a labelling (total ordering) of vertices, a 4-polytope is Gale if the vertices satisfy a part of Galeʹs 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.