Gruslys, Vytautas; Leader, Imre; Tan, Ta Sheng Tiling with arbitrary tiles. (English) Zbl 1347.05027 Proc. Lond. Math. Soc. (3) 112, No. 6, 1019-1039 (2016). A tile \(T\) is a finite non-empty set of \(\mathbb{Z}^n\), for some natural \(n\). The authors prove the conjecture of D. A. Chalcraft [“Two tiling problems”, http://mathforum.org/kb/message.jspa?messageID=6223965; “Does every polyomino tile \(\mathbb R^n\) for some \(n\)?”, http://mathoverflow.net/questions/49915/does-every-polyomino-tile-rn-for-some-n], i.e., they prove that any tile \(T\) in \(\mathbb{Z}^n\) tiles \(\mathbb{Z}^d\), for some \(d\). Related to this question, the authors also present some open problems; for instance: How to optimize the dimension \(d\)? Reviewer: Altino Manuel Folgado Dos Santos (Vila Real) Cited in 3 ReviewsCited in 9 Documents MathOverflow Questions: Does every polyomino tile R^n for some n? Does every polyomino tile R^n for some n? MSC: 05B45 Combinatorial aspects of tessellation and tiling problems 05B50 Polyominoes Keywords:Chalcraft’s conjecture Software:MathOverflow PDFBibTeX XMLCite \textit{V. Gruslys} et al., Proc. Lond. Math. Soc. (3) 112, No. 6, 1019--1039 (2016; Zbl 1347.05027) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Coordination sequence for the XXOXX tiling with respect to a tile of type R. Coordination sequence for the XXOXX tiling with respect to a tile of type G. Coordination sequence for the XXOXX tiling with respect to a tile of type B. References: [1] Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 pp 1– (1966) · Zbl 0199.30802 [2] DOI: 10.1016/0097-3165(90)90057-4 · Zbl 0741.05019 · doi:10.1016/0097-3165(90)90057-4 [3] DOI: 10.1016/S0021-9800(66)80033-9 · Zbl 0143.44202 · doi:10.1016/S0021-9800(66)80033-9 [4] DOI: 10.1016/S0021-9800(70)80055-2 · Zbl 0319.05019 · doi:10.1016/S0021-9800(70)80055-2 [5] DOI: 10.1016/0097-3165(89)90082-4 · Zbl 0723.05041 · doi:10.1016/0097-3165(89)90082-4 [6] B. Grünbaum and G. C. Shephard , Tilings and patterns, Dover Books on Mathematics Series (Dover Publications, Incorporated, 2013). · Zbl 0601.05001 [7] DOI: 10.1016/S0021-9800(69)80044-X · Zbl 0174.04102 · doi:10.1016/S0021-9800(69)80044-X [8] MathOverflow, ’Does every polyomino tile \(\mathbb {R}^n\) for some \(n\) ?’ http://mathoverflow.net/questions/49915/does-every-polyomino-tile-rn-for-some-n (retrieved on 15 April 2015). [9] The Math Forum, ’Two tiling problems’, http://mathforum.org/kb/message.jspa?messageID=6223965 (retrieved on 15 April 2015). [10] DOI: 10.1016/S0019-9958(84)80007-8 · Zbl 0592.05017 · doi:10.1016/S0019-9958(84)80007-8 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.