IRMACS Interdisciplinary Colloquium: "Tiling n-space by cubes"

Abstract

A lattice tiling $\mathcal{T}$ of $n$-space by cubes is a tiling where the centers of cubes in $\mathcal{T}$ form a group under the vector addition. In 1907 Minkowski conjectured that in a lattice tiling of $n$-space by unit cubes there must be a pair of cubes that share a complete $(n-1)$-dimensional face. Minkowski's problem attracted a lot of attention as it is an interface of several mathematical disciplines. In fact, Minkowski's problem, like many ideas in mathematics, can trace its roots to the Phytagorean theorem $a^{2}+b^{2}=c^{2}$.

We discuss the conjecture, its history and variations, and then we describe some problems that Minkowski's conjecture, in turn, suggested. We will focus on tilings of $n$-space by clusters of cubes, namely by spheres in Lee metric, and show how these tilings are related to the perfect error-correcting codes. The Golomb-Welch conjecture, the long-standing and the most famous conjecture in the area, will be discussed. At the end of our talk we will consider tilings by $n$-crosses, the Lee spheres of radius 1 (if we reflect an $n$-dimensional cube in all its faces, then the $2n+1$ obtained cubes form an $n$-cross). Some "unexpected" tilings by $n-$crosses will be presented.