×

The structure of cube tilings. (English) Zbl 1486.52054

Consider tilings of the \(d\)-dimensional Euclidean space by unit cubes. The paper is related to Keller’s conjecture in the stronger form that each cube tiling of \(\mathbb{R}^d\) contains a column. A counterexample to Keller’s conjecture in dimension 8 was found by J. Mackey in 2002. In 2012, M. Lysakowska and K. Przeslawski showed that Keller’s conjecture is true for \(d \le 6\). Keller’s conjecture is still open for dimension 7.
In the paper under review, the structure of cube tilings is investigated. Based on a lemma that in every 4-dimensional cube tiling each cylinder contains a 2-column, some new methods are proposed. A new proof of Keller’s conjecture about columns for dimension \(d \le 4\) is given. The new methods seem to make possible investigation of the structure of cube tilings in higher dimensions.

MSC:

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
05B45 Combinatorial aspects of tessellation and tiling problems
PDFBibTeX XMLCite
Full Text: Link

References:

[1] J. Brakensiek, M. Heule, J. Mackey and D. Narváez, The Resolution of Keller’s Conjecture,arXiv:1910.03740v3 [math.CO] 7 Apr 2020.
[2] J. Debroni, Aj. D. Eblen, M. A. Langston, W. Myrvold, P. W. Shor and D. Weerapurage, A complete resolution of the Keller maximum clique problem,SODA (2011), 129-135. · Zbl 1376.05109
[3] M. Dutour Sikirić and M. Łysakowska, On the structure of two-periodic cube tilings of the4-dimensional Euclidean space,Utilitas Math.(to appear). · Zbl 1455.52019
[4] G. Hajós, Über einfache und mehrfache Bedeckung desn-dimensionalen Raumes mit einem Würfelgitter,Math. Z.47(1941), 427-467. · JFM 67.0137.04
[5] O. H. Keller, Über die lückenlose Erfülung des Raumes Würfeln,J. Reine Angew. Math.163(1930), 231-248. · JFM 56.1120.01
[6] A. P. Kisielewicz, On the structure of cube tilings ofR3andR4,J. Combin. Theory Ser. A120(2013), 1-10. · Zbl 1253.05061
[7] A. P. Kisielewicz, Rigid polyboxes and Keller’s conjecture,Adv. Geom.17 (2017), 203-230. · Zbl 1388.52016
[8] A. P. Kisielewicz and M. Łysakowska, On Keller’s Conjecture in Dimension Seven,Electron. J. Comb.22(1) (2015), #P1.16. · Zbl 1305.05042
[9] J. C. Lagarias and P. W. Shor, Keller’s cube-tiling conjecture is false in high dimensions,Bull. Amer. Math. Soc.27(1992), 279-287. · Zbl 0759.52013
[10] M. Łysakowska and K. Przesławski, On the structure of cube tilings and unextendible systems of cubes in low dimension,European J. Combin.32(2011), 1417-1427. · Zbl 1241.52009
[11] M. Łysakowska and K. Przesławski, Keller’s conjecture on the existence of columns in cube tilings ofRn,Adv. Geom.12(2012), 329-352. · Zbl 1248.52002
[12] J. Mackey, A cube tiling of dimension eight with no facesharing,Discr. Comput. Geom.28(2002), 275-279. · Zbl 1018.52019
[13] H. Minkowski,Geometrie der Zahlen, Taubner, Leipzig, 1896. · Zbl 0050.04807
[14] O. Perron, Über lückenlose Ausfüllung desn-dimensionalen Raumes durch kongruente Würfel I, II,Math. Z.46(1940), 1-26, 161-180 · JFM 66.0179.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.