Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes

We construct, for any positive integer n, a family of n congruent convex polyhedra in R³, such that every pair intersects in a common facet. The polyhedra are Voronoi regions of points on the helix (t, cos t, sin t). The largest previously published example of such a family contains only eight polytopes. With a simple modification, we can ensure that each polyhedron in the family has a point, a line, and a plane of symmetry. We also generalize our construction to higher dimensions and introduce a new family of cyclic polytopes.

Note: The construction described in this paper (or a linear transformation theoreof) was first discovered by Scott Kim in the late 1980s using different methods. Except for a brief announcement by Martin Gardner, Scott's construction was never published. I independently rediscovered the same construction in 2001.
Note: Lemma 1 in this paper was independently proven by Daciana Bochis and Francisco Santos. See Lemma 4.2 of their paper "On the number of facets of three-dimensional Dirichlet stereohedra II: Non-cubic groups".