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A small aperiodic set of Wang tiles. (English) Zbl 0861.05017
A Wang tile is a unit square tile with colored edges. Consider tilings of the Euclidean plane by arbitrarily many copies from a given finite tile set \(T\) of Wang tiles translated to integer lattice points and such that contiguous edges have the same color. Such a tiling, determined by a function \(f:Z^2\to T\), is periodic if there exists \((a,b)\in Z^2\backslash\{(0,0)\}\) such that \(f(x,y)= f(x+a,y+b)\) for every \((x,y)\in Z^2\). A tile set \(T\) is called aperiodic if there exists a tiling, but there does not exist any periodic tiling. In 1966 R. Berger [Mem. Am. Math. Soc. 66 (1966; Zbl 0199.30802)] constructed an aperiodic tile set consisting of over 20,000 Wang tiles. The number has since been reduced, including a set consisting of 16 Wang tiles due to R. Amman. In the paper under review, the number is further reduced to 14 Wang tiles.

05B45 Combinatorial aspects of tessellation and tiling problems
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
Full Text: DOI
[1] Beatty, S.; Beatty, S., Problem 3173, Am. math. monthly, Am. math. monthly, 34, 159, (1927), solutions in · JFM 53.0198.06
[2] Berger, R., The undecidability of the domino problem, Mem. amer. math. soc., 66, (1966) · Zbl 0199.30802
[3] Grünbaum, B.; Shephard, G.C., Tilings and patterns, (1987), W.H. Freeman and Company New York · Zbl 0601.05001
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