Arbitrarily large neighborly families of congruent symmetric convex 3- polytopes

A family of d-dimensional convex polytopes is neighborly if every pair of polytopes has a (d−1)-dimensional intersection. It has been known for centuries that a neighborly family of convex polygons (or any other connected sets) in the plane has at most four members. In 1905, Tietze [27, 28] proved that there are arbitrarily large neighborly families of 3-dimensional polytopes, answering an open question of Guthrie [17] and Stäckel [26]. Tietze’s result was independently rediscovered by Besicovitch [4], using a different construction, and generalized to higher dimensions by Rado [11] and Eggleston [22].