Spatial chaos of Wang tiles with two symbols

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) not equal empty set and P(B\') = empty set whenever B\' not subset of 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 boolean OR C-2 boolean OR ... boolean OR C-k boolean OR N, where C-j, 1 <= j <= k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C-1 boolean OR C-2 boolean OR ... boolean OR C-k. Then, the positivity of spatial entropy h(B) is completely determined by C-1 boolean OR C-2 boolean OR ... boolean OR 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. (C) 2016 AIP Publishing LLC.