Publication Date:
2016-02-20
Description:
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 ) ≠ 0̸ and P ( B ′ ) = 0̸ 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.
Print ISSN:
0022-2488
Electronic ISSN:
1089-7658
Topics:
Mathematics
,
Physics
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