Spatial chaos of Wang tiles with two symbols

TL;DR

This paper classifies when Wang tile sets with two symbols exhibit spatial chaos, based on their entropy, identifying minimal cycle generators and equivalent classes of sets with positive or zero entropy.

Contribution

It provides a complete classification of spatial chaos in two-symbol Wang tiles, introducing minimal cycle generators and characterizing entropy positivity through specific tile set classes.

Findings

39 classes of marginal positive-entropy sets identified

18 classes of saturated zero-entropy sets identified

Spatial entropy positivity determined by minimal cycle generators

Abstract

This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles $\mathcal{B}$, spatial chaos occurs when the spatial entropy $h(\mathcal{B})$ is positive. $\mathcal{B}$ is called a minimal cycle generator if $\mathcal{P}(\mathcal{B})\neq\emptyset$ and $\mathcal{P}(\mathcal{B}')=\emptyset$ whenever $\mathcal{B}'\subsetneqq \mathcal{B}$, where $\mathcal{P}(\mathcal{B})$ is the set of all periodic patterns on $\mathbb{Z}^{2}$ generated by $\mathcal{B}$. Given a set of Wang tiles $\mathcal{B}$, write $\mathcal{B}=C_{1}\cup C_{2} \cup\cdots \cup C_{k} \cup N$, where $C_{j}$, $1\leq j\leq k$, are minimal cycle generators and $\mathcal{B}$ contains no minimal cycle generator except those contained in $C_{1}\cup C_{2} \cup\cdots \cup C_{k}$. Then, the positivity of spatial entropy $h(\mathcal{B})$ is completely determined by $C_{1}\cup C_{2} \cup\cdots \cup C_{k}$. Furthermore, there are 39 equivalent classes of marginal positive-entropy (MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy (SZE) sets of Wang tiles. For a set of Wang tiles $\mathcal{B}$, $h(\mathcal{B})$ is positive if and only if $\mathcal{B}$ contains an MPE set, and $h(\mathcal{B})$ is zero if and only if $\mathcal{B}$ is a subset of an SZE set.