Keller's conjecture

Szabó (1993), Shor (2004), and Zong (2005) give surveys of work on Keller's conjecture and related problems.

In formulating the same problem, Shor (2004) instead considers all tilings of space by congruent hypercubes and states, without proof, that the assumption that cubes are axis-parallel can be added without loss of generality.

This version of the problem is true or false for the same dimensions as its more commonly studied formulation.

[1] Hajós (1949) first reformulated Keller's conjecture in terms of factorizations of abelian groups.

He shows that if there is a counterexample to the conjecture, then it can be assumed to be a periodic tiling of cubes with an integer side length and integer vertex positions; thus, in studying the conjecture, it is sufficient to consider tilings of this special form.

[1] Szabó (1986) showed that any tiling that forms a counterexample to the conjecture can be assumed to have an even more special form: the cubes have side length a power of two and integer vertex coordinates, and the tiling is periodic with period twice the side length of the cubes in each coordinate direction.

Together with the fact that they do not overlap, this implies that the cubes placed in this way tile space without meeting face-to-face.

Subsequently, Debroni et al. (2011) showed that the Keller graph of dimension seven has a maximum clique of size 124.

Finally, a 200-gigabyte computer-assisted proof in 2019 used Keller graphs to establish that the conjecture holds true in seven dimensions.

[7] As Szabó (1993) describes, Hermann Minkowski was led to a special case of the cube-tiling conjecture from a problem in diophantine approximation.

One consequence of Minkowski's theorem is that any lattice (normalized to have determinant one) must contain a nonzero point whose Chebyshev distance to the origin is at most one.

A second related conjecture, made by Furtwängler in 1936, instead relaxes the condition that the cubes form a tiling.

Furtwängler's conjecture is true for two- and three-dimensional space, but Hajós found a four-dimensional counterexample in 1938.

[9] Once counterexamples to Keller's conjecture became known, it became of interest to ask for the maximum dimension of a shared face that can be guaranteed to exist in a cube tiling.