Lattice-Simplex Coverings and the 84-Shape
Rodney W. Forcade, Jack W. Lamoreaux · SIAM Journal on Discrete Mathematics · 2000
It is known that diameters of abelian Cayley graphs with n generators are related to Manhattan diameters of a particular type of lattice tile called a Cayley tile and that determining the lower bound of those diameters is related to the density of lattice-simplex coverings of R n . We construct a natural tile associated with each lattice-simplex covering and show that the 84-shape (discovered in [R. Dougherty and V. Faber, The Degree-Diameter Problem for Several Varieties of Cayley Graphs, http://www.c3.lanl.gov/dm/pub/laces.html (1994) and C.M. Fiduccia, J.S. Zito, and E. Mann, Network Interconnection Architectures and Translational Tilings, Tech. report, Center for Computing Science, Bowie, MD, 1994]) represents a local mininum of the density of lattice coverings of R 3 by a particular simplex D (the convex hull of the unit basis vectors).