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

arXiv:math.CO/0106095

- Download book version (final except for copy-editing):
- See also: Nice point sets can have nasty Delaunay triangulations

Abstract:

We construct, for any positive integern, a family ofncongruent convex polyhedra inR^{3}, such that every pair intersects in a common facet. The polyhedra are Voronoi regions of points on the helix (t, cost, sint). 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".

Publications - Jeff Erickson (jeffe@cs.uiuc.edu) 23 Jun 2003