Arbitrarily Large Neighborly Families of Congruent Symmetric Convex 3-Polytopes

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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. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi regions of evenly distributed points on the helix (t, cos t, sin t). 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.