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December, 2020 Nonemptiness Problems of Wang Cubes with Two Colors
Hung-Hsun Chen, Wen-Guei Hu, Song-Sun Lin
Taiwanese J. Math. 24(6): 1529-1549 (December, 2020). DOI: 10.11650/tjm/200304


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.


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Hung-Hsun Chen. Wen-Guei Hu. Song-Sun Lin. "Nonemptiness Problems of Wang Cubes with Two Colors." Taiwanese J. Math. 24 (6) 1529 - 1549, December, 2020.


Received: 18 September 2018; Revised: 18 July 2019; Accepted: 19 March 2020; Published: December, 2020
First available in Project Euclid: 19 November 2020

MathSciNet: MR4176885
Digital Object Identifier: 10.11650/tjm/200304

Primary: 05B45 , 37B50 , 52C22 , 52C23

Keywords: aperiodic set , nonemptiness problem , Wang cube , Wang tile , Wang's conjecture

Rights: Copyright © 2020 The Mathematical Society of the Republic of China


Vol.24 • No. 6 • December, 2020
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