三角形邊著色的決定性問題

Abstract

這個研究是關於用邊著色的正三角形與倒三角形拼湊整個平面。如果對每個正三角形與倒三角形相對應的邊都有相同的顏色，則這兩個三角形可以放在相鄰的位置。在這篇論文，我們考慮邊上著兩色與三色的三角形。我們研究的問題為：是否任意可佈滿整個平面的正三角形集合必存在週期性的拼法覆蓋整個平面。我們使用演算法來研究這個問題，然後藉由電腦計算得到結果。最後，這篇論文的主要結果為：在著兩色及三色的前提下，如果整個平面可以被邊著色的三角形拼滿，則整個平面就存在週期性的拼法覆蓋整個平面，反之亦然。

This investigation is about tiling the whole plane with upper triangles and lower triangles which have colors on edges. Upper and lower triangles can be placed side by side if each of the intersections has the same color. In this paper, we consider upper and lower triangle with two and three colors on edges. The problem we studied is that: any set of triangle that can fill with the whole plane whether it can cover the whole plane periodically. We use an algorithm to do the problem and get the result by computers. Finally, the main result of this paper is that the whole plane can be tiling by triangle with two and three colors if and only if the whole plane is covered by the local pattern periodically.