貢中元的部落格

a variety of depth couplingtheir translational Penrose tiles and corresponding unit cell ,糾正深度耦合對有多難，  chung yuan Kung 貢中元

The translational periodic Penrose tiles (crystal) made by two different types of Pentagonal shape Penrose tiles.

 Chung Yuan Kung , retired Professor , Chung Hsing University, 250, Kuo Kuang rd. 402,Taichung, Taiwan, R.O.C.

San Ling Young, Professor,

Key words: decagon, Penrose tiling, Pentagon shape , translational periodic Penrose tiles, unit cell

潘路斯密舖是非週期密鋪的例子，密鋪指以不重疊的多邊形或其它形狀覆蓋平面，非週期意味將有這些形狀的任何密鋪移動任何有限距離而不旋轉，不會產生相同的密鋪。然而，儘管不是平移對稱，潘路斯密鋪可能同時是反射對稱和五重旋轉對稱。潘路斯密鋪以1970年代研究潘路斯密鋪的數學家和物理學家潘路斯命名。by penrose wiki

Abstract: We constructed two different types of pentagonal Penrose tiles, And using them to make a translational periodic Penrose tiles (crystals)

The Penrose tiling is one of the most famous and complex tessellations and has been drawn and represented in many different ways, most of them are in artistic form. This five-fold symmetric Penrose tiling diagram helps to explain the five-fold nature of quasicrystals and has sparked many studies in this field... In this paper, the authors only focus on strictly defined defect-free Pentose tiles and how to systematically and orderly extend tiles to infinity. The following are one of some systematic procedures for drawing defect-free Penrose tiles, the most important of which are perceptible repeating zero-defect internal structures, and can be extended to infinity,

Two rhombuses with acute angle 36 and 72 degree can make six different decagons Fig 1. In 1974 almost fifty years ago Penrose (1) using these two rhombic created a famous Penrose-tile which can be extended unlimitedly, as shown in Fig 2a. According to the original pattern configuration found by Penrose, several decagonal shapes were added outwards, and a regular pentagonal pattern with five decagonal sides was produced, as shown in Figure 2b. The pattern contains three decagons of a, type-b, and type-c. The pentagon shape is little bit bloated. At the same time, a counterpart to a bulky Penrose tile was found, with a slightly slimmer pentagonal shape. As shown in fig 2c. All the vertices of each regular pentagon Penrose tiles are designated as a-type decagons, which are specially marked with red as red stars for type-a  decagons in the dilated pentagon and mark green color for type-a decagon on the slim one here, so as to lay the groundwork for future coupling alignment. Here, it must be noted that these five vertices can be replaced by other types of decagons, which may produce extensions in different ways. The central regions for both bloated (dilated) and slim Pentagons are marked in dark blue for coupling alignment.​​​

The details of these two types of coupling are demonstrated below. the results are shown in fig 3  , Several examples of pair coupling are shown below as in the fig 2  and fog 3 : some of them still retain the remaining mismatched regions to show the difficulty of making defect-free Penrose tiles. The different coupling mentioned above may result defects in the overlapping seam area, in most cases, these coupling defects can be easily removed to accomplish a larger zero-defect Penrose tiles

 All these coupled pairs can be translational (or slanted shift) coupled by their self as shown in fig 5  a ,b.,  In this paper, only fig 5a is employed to be self-coupled to infinity and from there on extracted the basic unit cell as shown in fig 6. And this basic unit cell can be tessellated to construct a translational periodic Penrose tiling.

Summary:  In this short article, two different types Pentagon shape of Penrose tiles are constructed based in original Penrose tiling pattern and it is found that these  two types of Pentagonal shapes can be well coupled in different depth and defect free.  An elementary unit cell can be extracted from repeated coupling results, and this unit cell can be recombined (mosaiced) to build translational periodic Penrose tiles.

**Coupling defined here is different from tessellation. It refers to two graphics with the same or different internal structure configurations, which overlap to form a larger graphic by partial matching; The misaligned area of the seam zone after coupling can be replaced or modified by any of the six decagons in Figure 1 without affecting nearby existing structures.

下面顯示了幾個配對耦合的例子：其中一些仍然保留了剩餘的不匹配區域，以顯示製作無缺陷彭羅斯瓷磚的難度. 這兩種類型的耦合的詳細信息如下所示。結果如圖 3 所示，所有這些耦合對都可以平移（或斜移）自身耦合。 在本文中，僅使用圖 5a 自耦合到無窮遠，並從那裡提取基本晶胞，如圖 6 所示。並且可以對這個基本晶胞進行鑲嵌，以構建平移週期性 Penrose 平鋪。摘要：在這篇短文中，基於原始 Penrose 瓷磚圖案構建了兩種不同類型的五角形 Penrose 瓷磚，發現這兩種類型的五角形可以在不同深度和無缺陷的情況下完美耦合。可以從重複的耦合結果中提取基本晶胞，並且可以重新組合（鑲嵌）該晶胞以構建平移週期彭羅斯平鋪。

@@，在上面提到的不同耦合，結果可能會在重疊的接縫區域產生缺陷，在大多數情況下，這些耦合缺陷可以很容易的被去除,以完成更大的零缺陷彭羅斯瓷磚，並且可預知的彭羅斯瓷磚圖案按照相同或不同的技術方案交互使用,逐步擴展到無窮大

,** defects may be created in overlapping seam 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 step to step with precognitive pattern to infinity. ( Penrose tiles can be progressively expanded to infinity by predictable pattern (perceptible patterns)

 @@ 耦合定義 – 兩種不同類型的簇在深度上相互耦合，即使是同類型的大面積重疊的自耦合，其複雜的缺陷修正程度也是難以想像的，有時可能會一眼就忽略其可修復性而不去糾正。

內部缺陷定義 Definition of Internal defect  ----.   在此，簇中的內部缺陷定義比較嚴格: 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。在藕合中經常會出現一兩倆個缺陷，會特別標注之

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. 這裡從一對膨脹的五邊形和細長的五邊形開始，如圖1所示。

How difficult to correct a depth coupling pair, Here start  with a pair of Bloated pentagon and slim pentagon,  as shown in fig 1.

Several examples of pair coupling are shown below as in the fig 2  and fig 3 : some of them still retain the remaining mismatched regions to show the difficulty of making defect-free Penrose tiles.

下面顯示了幾個配對耦合的例子：其中一些仍然保留了剩餘的不匹配區域，以顯示製作無缺陷彭羅斯瓷磚的難度。

All these coupled pairs can self-couple to form translational periodic Penrose tiles as shown in fig 4, and create a unit cell for themselves 所有這些耦合對都可以自耦合形成平移週期彭羅斯瓷磚，並為自己創建一個晶胞.For this case in fig 3d, the coupling area is  cover almost all the area of the slim cluster,this is the maximum case of a reasonable  defined coupling observed 對於圖 3d 中的這種情況，耦合區域幾乎覆蓋了 slim cluster 的所有區域,這是觀察到的合理定義耦合的最大情況，

an easy example is shown in fig 5 一個簡單的例子如圖 5 所示