Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes

Written with Scott Kim.

Discrete Geometry: In Honor of W. Kuperberg's 60th Birthday (András Bezdek, editor), Marcel-Dekker, 2003, pp. 267-278.

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.

Publications - Jeff Erickson ( 23 Jun 2003