  We construct, for any positive integer n, a family of n congruent convex
polyhedra in R^3, 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.
