週期決定平面網格雙色邊著色的熵

Abstract

本篇完整地分類平面網格雙色邊著色問題中的空間熵正性問題。一個著色的正方格本篇稱為磚(tile)。多個磚組成的集合若以數量最少週期地拼平面網格，我們稱最小週期生成元。磚集合以最多個最小週期生成元表示時，其空間熵完全被最小週期生成元決定。 我們更近一步的找出39個邊緣正熵(marginal positive-entropy)等價類，及18個飽和零熵(saturated zero-entropy) 等價類。

This investigation completely classifies the positivity problem of spatial entropy in plane edge coloring (Wang tiles) with two symbols. A set of Wang tiles is called a minimal cycle generator if they can periodically tile a plane and the number is minimal. Given a set of Wang tiles, write as the union of minimal cycle generators. The positivity of spatial entropy is completely determined by these minimal cycle generators. 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.