### Abstract

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 Z^{2} generated by B. Given a set of Wang tiles B, write B=C_{1}∪ C_{2}∪ ... ∪ C_{k}∪N, where C_{j}, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C_{1}∪C_{2}∪∪C_{k}. Then, the positivity of spatial entropy h(B) is completely determined by C_{1}∪C_{2}∪ ... ∪C_{k}. 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 language | English |
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Article number | 022704 |

Journal | Journal of Mathematical Physics |

Volume | 57 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2016 |

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## Cite this

*Journal of Mathematical Physics*,

*57*(2), [022704]. https://doi.org/10.1063/1.4941734