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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Abstract

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

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

Dates
First available in Project Euclid: 27 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1585274419

Digital Object Identifier
doi:10.11650/tjm/200304

Subjects
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

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

Citation

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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References

  • J.-C. Ban, W.-G. Hu, S.-S. Lin and Y.-H. Lin, Zeta functions for two-dimensional shifts of finite type, Mem. Amer. Math. Soc. 221 (2013), no. 1037, 60 pp.
  • J.-C. Ban, S.-S. Lin and Y.-H. Lin, Patterns generation and spatial entropy in two-dimensional lattice models, Asian J. Math. 11 (2007), no. 3, 497–534.
  • R. J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Lett. 26 (1971), no. 14, 832–833.
  • ––––, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
  • R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 (1966), 72 pp.
  • H.-H. Chen, W.-G. Hu, D.-J. Lai and S.-S. Lin, Nonemptiness problems of Wang tiles with three colors, Theoret. Comput. Sci. 547 (2014), 34–45.
  • J.-Y. Chen, Y.-J. Chen, W.-G. Hu and S.-S. Lin, Spatial chaos of Wang tiles with two symbols, J. Math. Phys. 57 (2015), no. 2, 022704, 13 pp.
  • K. Culik II, An aperiodic set of $13$ Wang tiles, Discrete Math. 160 (1996), no. 1-3, 245–251.
  • K. Culik II and J. Kari, An aperiodic set of Wang cubes, J.UCS 1 (1995), no. 10, 675–686.
  • B. Grünbaum and G. C. Shephard, Tilings and Patterns, New York, 1987.
  • W.-G. Hu and S.-S. Lin, Zeta functions for higher-dimensional shifts of finite type, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), no. 11, 3671–3689.
  • ––––, Nonemptiness problems of plane square tiling with two colors, Proc. Amer. Math. Soc. 139 (2011), no. 3, 1045–1059.
  • E. Jeandel and M. Rao, An aperiodic set of 11 Wang tiles, arXiv:1506.06492.
  • J. Kari, A small aperiodic set of Wang tiles, Discrete Math. 160 (1996), no. 1-3, 259–264.
  • A. Lagae, J. Kari and P. Dutré, Aperiodic sets of square tiles with colored corners, Report CW 460, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, August 2006.
  • E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Lett. 18 (1967), no. 17, 692–694.
  • ––––, Residual entropy of square ice, Phys. Rev. 162 (1967), no. 1, 162–172.
  • R. Penrose, The role of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10 (1974), 266–271.
  • R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177–209.
  • H. Wang, Proving theorems by pattern recognition–-II, Bell System Tech. J. 40 (1961), no. 1, 1–41.