平面雙色著色的週期花樣生成問題

Abstract

本文部分地分類平面網格雙色點著色問題中有無新週期拼法生成，並探索其生成規則。一個著色的正方網格本篇稱為磚(tile)。多個磚組成的集合以數量最少週期地拼平面網格，我們稱之為最小週期生成元。

若以兩個不同的最小週期生成元混合，形成一組基礎集合，是否能生成新的混合週期拼法是關注的焦點。

This investigation partially studies the periodic patterns in plane vertex coloring with two symbols. A set of tiles $\mathcal{B}$ is called a minimal cycle generator if $\mathcal{P(B)} \neq \emptyset$ and $P(B^{ rime})=\emptyset$ whenever $\mathcal{B}^{ rime} \subsetneqq \mathcal{B},$ where $\mathcal{P(B)}$ is the set of all periodic patterns on $\mathbb{Z}^2$ generated by $\mathcal{B}.$

Given a set of tiles $\mathcal{B},$ write $\mathcal{B} = \mathcal{C}_i \cup \mathcal{C}_j,$ where $\mathcal{C}_i$ and $\mathcal{C}_j$ are mutually different minimal cycle generators for $1\leq i \leq j \leq M.$ $M$ is the number of minimal cycle generators. When $\mathcal{B}$ contains no minimal cycle generator except $\mathcal{C}_i$ and $\mathcal{C}_j,$ is called the two minimal cycles union. Then, this thesis studies whether or not $\mathcal{B}$ can generate new periodic patterns.