Spatial chaos of Wang tiles with two symbols

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∪ ... ∪ C_k∪N, where C_j, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C₁∪C₂∪∪C_k. Then, the positivity of spatial entropy h(B) is completely determined by C₁∪C₂∪ ... ∪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.