Multiple Lattice Tilings in Euclidean Spaces

TL;DR

This paper characterizes convex domains that can form multiple lattice tilings in Euclidean spaces, revealing limitations on shapes and multiplicities, and extending results to higher dimensions.

Contribution

It provides a complete classification of convex domains capable of forming multiple lattice tilings in the plane and extends the concept to higher dimensions.

Findings

Only parallelograms and centrally symmetric hexagons form 2-4 fold tilings.

Octagons can form tilings with multiplicity at least seven.

Decagons can form 5- or 6-fold lattice tilings.

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.