Spatial chaos of Wang tiles with two symbols

Source 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)≠0̸ and P(B′)=0̸ 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.