Formation of the periodic translational symmetry Penrose-tile crystal *possible" ???形成周期性平移對稱Penrose-tile晶體，*可能"

The formation of the periodic translational Penrose tiles (crystal) and their corresponding unit cells 週期性平移彭羅斯瓷磚（晶體）及其相應晶胞的形成

Chung Yan Kung , retired Professor , Chung Hsing University, Taichung, Taiwan, ROC

Key words: decagon, Penrose tiling, binary tiling system, circular shape, translational periodic Penrose tiles, unit cell

Copied from WiKi : A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.

彭羅斯平舖是最著名和最複雜的鑲嵌圖案之一，並且曾以多種不同的方式繪製和表達. 這幅五重對稱鋪磚圖與準晶體的發現有關，並引發了該領域的許多研究.. ..在這篇論文中，作者只是集中討論了 Pentose tiles 以及如何將 tiles 擴展到無限大。. 以下是繪製無缺陷彭羅斯平舖的一些最系統的程序，最重要的是它可以擴展到無限且無缺陷。

主要的工作簡述如下

1.二個等邊菱形，可以形成六個基本的正方形組態，如圖1。Two equilateral rhombuses can form six basic regular ten-square configurations, as shown in Figure 1.

2. 根據彭羅斯發現的原始構型，製作了一個正五邊形圖案，每邊有五個十邊形，包括圖1中 type-a、 type-b、type-c的三種正十邊形。除了中央十邊形標記為深藍色並用於查找位置的框架外，所有類型 a 十邊形都標記為紅色

Based on the original configuration discovered by Penrose, a regular pentagonal pattern is made, each side contains five decagons , including three kinds of regular ten-squares of type -a, -b, -c showing in Figure 1. All type-a decagons are marked in red except the central decagon which is marked in dark blue and is used to find the frame of the location.

3. 同時製作了幾十種類似彭羅斯拼塊的五邊形圖案，但內部結構不同，每一個正五邊形的所有頂角都被指定為a型十邊形，並特別用紅色標記為紅星，以便為日後後面的耦合對齊對位打底。雖然這五個頂點可以換成其他類型的十邊形。

in the meantime, dozens of pentagonal patterns similar to Penrose tiles have been made, but with different internal structures, and all the vertex corner of each regular pentagon are designed to be the type-a decagon and specially red-marked as red star for the coupling alignment later (In order to lay the groundwork for future coupling alignment) Although these five vertices can be replaced with other types of decagons.

4 並分為四類and all clusters divided into four categories

1.無缺陷微胖型 1. Non-defective bloated type

2.無缺陷微瘦型 2. Degree free slim type

3.無缺陷正常型 3 Defect free Normal type

4.內部有缺陷型 4. With inner defect

3. 4. Internal defective type

. 簇中的內部缺陷定義為任何五邊形簇中的所有菱形必須包含在上述六個正十邊形中的任何一個中，否則，計算作為缺陷簇。

The internal defects in the cluster are defined as that all rhombuses in any pentagon shape cluster must be involved in any one of the above-mentioned six regular decagons, other-wise, counted as defect cluster.

5 .嘗試不同的耦合方案展開彭羅斯瓷磚，自平移耦合，斜移位移耦合，共軛五重旋轉耦合

Try different coupling schemes to expand Penrose tiles, self-translation coupling, skew-shift (slanted shift) displacement coupling, conjugate quintuple(five-fold )rotation coupling

6，五倍旋轉耦合後可能會產生重疊區域的缺陷，在大多數情況下，可以去除這些耦合缺陷以形成零缺陷的更大彭羅斯瓷磚並遵循相同的方案可以將瓷磚擴展到具有可以想像的無限大圖案，

Five-fold rotational coupling may create defects in overlapping regions, in most cases these coupling defects can be removed to form larger Penrose tiles with zero defects and following the same scheme the tiles can be extended to imaginable infinite pattern.

7兩種不同類型簇相互耦合的複雜性,,

兩種不同類型團簇的深度互耦合，甚至同類型團簇大面積重疊的自耦合，複雜度難以想像。the depth mutual coupling of two different types of clusters and even the self coupling of the same types with large area overlapped, ,the complexity is hard to image ..

8 較小的圓形圖案和五邊形彭羅斯圖案都可以從1974年發現的原始彭羅斯瓷磚中提取出來。也可以製成（）。每邊只有四三個正方形。甚至更小的圓形圖案和五邊形彭羅斯圖案也可以通過鑲嵌或耦合技巧使用的 fig 1六個十邊形來製作。同樣，使用耦合方案，可以證明平移鏡像對稱週期性彭羅斯平舖的存在性，，同樣，使用耦合方案，可以證明平移鏡像對稱週期彭羅斯平舖的存在

Both the smaller circular pattern and the pentagonal Penrose pattern can be extracted from the original Penrose tile discovered in 1974. There are only four or three squares on each side. Even a smaller circular patterns and pentagonal Penrose patterns can also be made by tessellation or coupling skill using the six decagons.. Similarly, using the coupling scheme, the existence of translational mirror symmetric periodic Penrose tiling can be proved

從Penrose 平鋪看來,是由Fig 1 六個其中三個正十角形構成的標準五重密鋪, 如圖 2a。應該就是最正的五重性 Penrose 潘若斯圖,轉72度角可以回到原點…

我從原子,分子的觀念來闡述胖瘦二種菱形形成六種正十角形(類分子)、然後再用六種正十角形,以高分子簇(Clusters)的概念規範了幾類五邊形(像Penrose tiles的簇(cluster)類,同時也定義了在簇裡面缺陷的含義。並且利用不同顏色去定位五星狀的位置而突顯出簇類結構上的差異。在早期作圖時,藉由顏色的不同,容易判斷圖案的正確性。再利用群簇耦合的概念,簇邊界的分子耦合的轉換,製繪無限大的 defect free Penrose鋪磚圖。這種簇邊界的分子化學反應的概念, 希望有助於了解五重性合金quasicrystal準晶体的形成。耦合與鑲嵌的意思大体不太相同,在這裡耦合是圖案有部份的疊加,有時需要改變重疊部份的正十角形(類分子,但不影響其他鄰近分子的原型,以滿足零缺陷的要求。如果不在乎缺陷的話,Penrose 鋪磚,花俏的到處可見。在這裡盡可能的有系統的畫出不同的組熊,僖可能的說明如可可達到無‧限大..