In geometry, Keller's conjecture is the conjecture introduced by Ott-Heinrich Keller () 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 was shown to be true in dimensions at most 6 by Perron (, ). However, for higher dimensions it is false, as was shown in dimensions at least 10 by Lagarias and Shor () and in dimensions at least 8 by , using 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 di