Tiling $n$-space by unit cubes.

Peter Horak

University of Washington, Tacoma

Abstract

Tiling problems belong to the oldest problems in mathematics. They attracted attention of many famous mathematicians. Even one of the Hilbert's problems is devoted to the topic. The interest in tilings by unit cubes originated with a conjecture raised by Minkowski in 1907.

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

We discuss this conjecture, its history and variations, and then we describe some problems that Minkowski's conjecture, in turn, suggested. We