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. Our polyhedra are Voronoi regions of evenly distributed 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.