Keller cube-tiling conjecture

In geometry, Keller's conjecture is the conjecture that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, as shown in the illustration, in any tiling of the plane by identical squares, some two squares must meet edge to edge.

This conjecture was introduced by Ott-Heinrich Keller (1930), after whom it is named. It was shown to be true in dimensions at most 6 by Perron (1940a, 1940b). However, for higher dimensions it is false, as was shown in dimensions at least 10 by Lagarias and Shor (1992) and in dimensions at least 8 by Mackey (2002). These disproofs used a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. Although this graph-theoretic version of the conjecture is now resolved for all dimensions, Keller's original cube-tiling conjecture remains open in dimension 7.

The related Minkowski lattice cube-tiling conjecture states that, whenever a tiling of space by identical cubes has the additional property that the cube centers form a lattice, some cubes must meet face to face. It was proved by György Hajós in 1942.

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

A family of closed sets called tiles forms a tessellation or tiling of a Euclidean space if their union is the whole space and every two distinct sets in the family have disjoint interiors. A tiling is said to be monohedral if all of the tiles are congruent to each other. Keller's conjecture concerns monohedral tilings in which all of the tiles are hypercubes of the same dimension as the space. As Szabó (1986) formulates the problem, a cube tiling is a tiling by congruent hypercubes in which the tiles are additionally required to all be translations of each other, without any rotation, or equivalently to have all of their sides parallel to the coordinate axes of the space. Not every tiling by congruent cubes has this property: for instance, three-dimensional space may be tiled by two-dimensional sheets of cubes that are twisted at arbitrary angles with respect to each other. Shor (2004) instead defines a cube tiling to be any tiling of space by congruent hypercubes, and states without proof that the assumption that cubes are axis-parallel can be added without loss of generality.

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