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Tiling with arbitrary tiles. (English) Zbl 1347.05027

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 \(d\)?

MSC:

05B45 Combinatorial aspects of tessellation and tiling problems
05B50 Polyominoes

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References:

[1] Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 pp 1– (1966) · Zbl 0199.30802
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[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
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