Spatial chaos of Wang tiles with two symbols

Jin Yu Chen, Yu Jie Chen, Wen Guei Hu, Song-Sun Lin

Research output: Contribution to journalArticle


This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles B, spatial chaos occurs when the spatial entropy h(B) is positive. B is called a minimal cycle generator if P(B)≠∅ and P(B')=∅ whenever B'⊂B, where P(B) is the set of all periodic patterns on Z2 generated by B. Given a set of Wang tiles B, write B=C1∪ C2∪ ... ∪ Ck∪N, where Cj, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C1∪C2∪∪Ck. Then, the positivity of spatial entropy h(B) is completely determined by C1∪C2∪ ... ∪Ck. Furthermore, there are 39 equivalence classes of marginal positive-entropy sets of Wang tiles and 18 equivalence classes of saturated zero-entropy sets of Wang tiles. For a set of Wang tiles B, h(B) is positive if and only if B contains a MPE set, and h(B) is zero if and only if B is a subset of a SZE set.

Original languageEnglish
Article number022704
JournalJournal of Mathematical Physics
Issue number2
StatePublished - 1 Feb 2016

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    Chen, J. Y., Chen, Y. J., Hu, W. G., & Lin, S-S. (2016). Spatial chaos of Wang tiles with two symbols. Journal of Mathematical Physics, 57(2), [022704].