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- Q4802474 subject Q10185811.
- Q4802474 subject Q6248542.
- Q4802474 subject Q7347245.
- Q4802474 subject Q7463500.
- Q4802474 abstract "Template:ForIn geometry, Keller's conjecture is the conjecture introduced by Ott-Heinrich Keller (1930) 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 (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), 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.Szabó (1993), Shor (2004), and Zong (2005) give surveys of work on Keller's conjecture and related problems.".
- Q4802474 thumbnail Shifted_square_tiling.svg?width=300.
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- Q4802474 comment "Template:ForIn geometry, Keller's conjecture is the conjecture introduced by Ott-Heinrich Keller (1930) 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 (1940a, 1940b).".
- Q4802474 label "Keller's conjecture".
- Q4802474 depiction Shifted_square_tiling.svg.