Taiwanese Journal of Mathematics

Nonemptiness Problems of Wang Cubes with Two Colors

Hung-Hsun Chen, Wen-Guei Hu, and Song-Sun Lin

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This investigation studies the nonemptiness problems of Wang cubes with two colors. Wang cubes are unit cubes with colored faces, which are generalized from Wang tiles. For a set $B$ of Wang cubs, $\Sigma(B)$ is the set of all global patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. The nonemptiness problem is to determine whether $\Sigma(B) \neq \emptyset$ or not. Denote by $\mathcal{P}(B)$ the set of all periodic patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. For Wang cubes, the corresponding Wang's conjecture is that if $\Sigma(B) \neq \emptyset$, then $\mathcal{P}(B) \neq \emptyset$.

We introduce the transition matrices and trace operators to determine whether $\Sigma(B) \neq \emptyset$ and $\mathcal{P}(B) \neq \emptyset$ or not, respectively. A basic set $B$ is called a minimal cycle generator if $\mathcal{P}(B) \neq \emptyset$ but $\mathcal{P}(B') = \emptyset$ for all $B' \subsetneqq B$. By computer computation, there exist $86$ equivalence classes of minimal cycle generators with two colors. By verifying that the basic sets $B$ that contains no minimal cycle generators satisfy $\Sigma(B) = \emptyset$, we prove that the Wang's conjecture for Wang cubes with two colors is true.

Article information

Taiwanese J. Math., Advance publication (2020), 21 pages.

First available in Project Euclid: 27 March 2020

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Primary: 37B50: Multi-dimensional shifts of finite type, tiling dynamics 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 52C22: Tilings in $n$ dimensions [See also 05B45, 51M20] 52C23: Quasicrystals, aperiodic tilings

nonemptiness problem Wang cube Wang's conjecture aperiodic set Wang tile


Chen, Hung-Hsun; Hu, Wen-Guei; Lin, Song-Sun. Nonemptiness Problems of Wang Cubes with Two Colors. Taiwanese J. Math., advance publication, 27 March 2020. doi:10.11650/tjm/200304. https://projecteuclid.org/euclid.twjm/1585274419

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