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- Kellers_conjecture abstract "In 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.".
- Kellers_conjecture thumbnail Shifted_square_tiling.svg?width=300.
- Kellers_conjecture wikiPageExternalLink b34h1sza.ps.gz.
- Kellers_conjecture wikiPageID "33297462".
- Kellers_conjecture wikiPageLength "20491".
- Kellers_conjecture wikiPageOutDegree "55".
- Kellers_conjecture wikiPageRevisionID "630599460".
- Kellers_conjecture wikiPageWikiLink Abelian_group.
- Kellers_conjecture wikiPageWikiLink Benchmark_(computing).
- Kellers_conjecture wikiPageWikiLink Branko_Grünbaum.
- Kellers_conjecture wikiPageWikiLink Bulletin_of_the_American_Mathematical_Society.
- Kellers_conjecture wikiPageWikiLink Category:Conjectures.
- Kellers_conjecture wikiPageWikiLink Category:Cubes.
- Kellers_conjecture wikiPageWikiLink Category:Parametric_families_of_graphs.
- Kellers_conjecture wikiPageWikiLink Category:Tessellation.
- Kellers_conjecture wikiPageWikiLink Chebyshev_distance.
- Kellers_conjecture wikiPageWikiLink Clique_(graph_theory).
- Kellers_conjecture wikiPageWikiLink Clique_problem.
- Kellers_conjecture wikiPageWikiLink Closed_set.
- Kellers_conjecture wikiPageWikiLink Crelles_Journal.
- Kellers_conjecture wikiPageWikiLink Cyclic_group.
- Kellers_conjecture wikiPageWikiLink Determinant.
- Kellers_conjecture wikiPageWikiLink Diophantine_approximation.
- Kellers_conjecture wikiPageWikiLink Discrete_and_Computational_Geometry.
- Kellers_conjecture wikiPageWikiLink Duke_Mathematical_Journal.
- Kellers_conjecture wikiPageWikiLink Euclidean_space.
- Kellers_conjecture wikiPageWikiLink Euclidean_tilings_by_convex_regular_polygons.
- Kellers_conjecture wikiPageWikiLink European_Journal_of_Combinatorics.
- Kellers_conjecture wikiPageWikiLink Geoffrey_Colin_Shephard.
- Kellers_conjecture wikiPageWikiLink Graph_theory.
- Kellers_conjecture wikiPageWikiLink Group_theory.
- Kellers_conjecture wikiPageWikiLink György_Hajós.
- Kellers_conjecture wikiPageWikiLink Hajxc3xb3ss_theorem.
- Kellers_conjecture wikiPageWikiLink Hermann_Minkowski.
- Kellers_conjecture wikiPageWikiLink Hypercube.
- Kellers_conjecture wikiPageWikiLink Lattice_(group).
- Kellers_conjecture wikiPageWikiLink Mathematische_Zeitschrift.
- Kellers_conjecture wikiPageWikiLink Minkowskis_theorem.
- Kellers_conjecture wikiPageWikiLink Parallelepiped.
- Kellers_conjecture wikiPageWikiLink Pythagorean_tiling.
- Kellers_conjecture wikiPageWikiLink Spectral_theory.
- Kellers_conjecture wikiPageWikiLink Square-integrable_function.
- Kellers_conjecture wikiPageWikiLink Sumset.
- Kellers_conjecture wikiPageWikiLink Tessellation.
- Kellers_conjecture wikiPageWikiLink Translation_(geometry).
- Kellers_conjecture wikiPageWikiLink Triangular_matrix.
- Kellers_conjecture wikiPageWikiLink File:Keller-labeled_Clebsch.svg.
- Kellers_conjecture wikiPageWikiLink File:Shifted_square_tiling.svg.
- Kellers_conjecture wikiPageWikiLinkText "Keller graph".
- Kellers_conjecture wikiPageWikiLinkText "Keller's conjecture".
- Kellers_conjecture author1Link "Jeffrey Lagarias".
- Kellers_conjecture author2Link "Peter Shor".
- Kellers_conjecture authorlink "Oskar Perron".
- Kellers_conjecture authorlink "Ott-Heinrich Keller".
- Kellers_conjecture first "Ott-Heinrich".
- Kellers_conjecture last "Keller".
- Kellers_conjecture last "Lagarias".
- Kellers_conjecture last "Perron".
- Kellers_conjecture last "Przesławski".
- Kellers_conjecture last "Shor".
- Kellers_conjecture last "Łysakowska".
- Kellers_conjecture wikiPageUsesTemplate Template:Citation.
- Kellers_conjecture wikiPageUsesTemplate Template:For.
- Kellers_conjecture wikiPageUsesTemplate Template:Harv.
- Kellers_conjecture wikiPageUsesTemplate Template:Harvs.
- Kellers_conjecture wikiPageUsesTemplate Template:Harvtxt.
- Kellers_conjecture wikiPageUsesTemplate Template:Tessellation.
- Kellers_conjecture year "1930".
- Kellers_conjecture year "1940".
- Kellers_conjecture year "1992".
- Kellers_conjecture year "2008".
- Kellers_conjecture year "2011".
- Kellers_conjecture subject Category:Conjectures.
- Kellers_conjecture subject Category:Cubes.
- Kellers_conjecture subject Category:Parametric_families_of_graphs.
- Kellers_conjecture subject Category:Tessellation.
- Kellers_conjecture hypernym Conjecture.
- Kellers_conjecture type Conjecture.
- Kellers_conjecture type Pattern.
- Kellers_conjecture type Polytope.
- Kellers_conjecture type Statement.
- Kellers_conjecture type Statement.
- Kellers_conjecture comment "In 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).".
- Kellers_conjecture label "Keller's conjecture".
- Kellers_conjecture sameAs Q4802474.
- Kellers_conjecture sameAs Conjecture_de_Keller.
- Kellers_conjecture sameAs m.0h7px1w.
- Kellers_conjecture sameAs Q4802474.
- Kellers_conjecture wasDerivedFrom Kellers_conjecture?oldid=630599460.
- Kellers_conjecture depiction Shifted_square_tiling.svg.
- Kellers_conjecture isPrimaryTopicOf Kellers_conjecture.
