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In geometry, Keller's conjecture is the conjecture introduced by 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 . However, for higher dimensions it is false, as was shown in dimensions at least 10 by 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 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. , , and give surveys of work on Keller's conjecture and related problems. ==Definitions== 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 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. 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. An ''n''-dimensional hypercube has 2''n'' faces of dimension ''n'' − 1, that are themselves hypercubes; for instance, a square has four edges, and a three-dimensional cube has six square faces. Two tiles in a cube tiling (defined in either of the above ways) meet ''face-to-face'' if there is an (''n'' − 1)-dimensional hypercube that is a face of both of them. Keller's conjecture is the statement that every cube tiling has at least one pair of tiles that meet face-to-face in this way. The original version of the conjecture stated by Keller was for a stronger statement, that every cube tiling has a column of cubes all meeting face to face. As with the weaker statement more commonly studied in subsequent research, this is true for dimensions up to six, false for dimensions eight or greater, and remains open for seven dimensions It is a necessary part of the conjecture that the cubes in the tiling all be congruent to each other, for if similar but not congruent cubes are allowed then the Pythagorean tiling would form a trivial counterexample in two dimensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Keller's conjecture」の詳細全文を読む スポンサード リンク
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