Nonemptiness Problems of Plane Square Tiling with Two Colors

Abstract. This investigation studies nonemptiness problems of plane square tiling. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges of p colors are arranged side by side such that adjacent tiles have the same colors. Given a set of Wang tiles B, the nonemptiness problem is to determine whether or not Σ(B) = ∅, where Σ(B) is the set of all global patterns on Z2 that can be constructed from the Wang tiles in B. When p ≥ 5, the problem is well known to be undecidable. This work proves that when p = 2, the problem is decidable. P(B) is the set of all periodic patterns on Z2 that can be generated by B. If P(B) = ∅, then B has a subset B′ of minimal cycle generator such that P(B′) = ∅ and P(B′′) = ∅ for B′′ B′. This study demonstrates that the set of all minimal cycle generators C(2) contains 38 elements. N (2) is the set of all maximal noncycle generators: if B ∈ N (2), then P(B) = ∅ and ̃ B B implies P( ̃ B) = ∅. N (2) has eight elements. That Σ(B) = ∅ for any B ∈ N (2) is proven, implying that if Σ(B) = ∅, then P(B) = ∅. The problem is decidable for p = 2: Σ(B) = ∅ if and only if B has a subset of minimal cycle generators. The approach can be applied to corner coloring with a slight modification, and similar results hold.