Multiple Lattice Tilings in Euclidean Spaces

Abstract

This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane. If a centrally symmetric octagon can form a lattice multiple tiling, then the multiplicity is at least seven. However, there are decagons which can form five-fold $($or six-fold$)$ lattice tilings. Consequently, whenever $n\ge 3$, there are non-parallelohedral polytopes which can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

6 pages, 2 figures

References in corpus (2)

• Exhaustive search of convex pentagons which tile the plane
• The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices