是，所有六個十邊形都在一個彭羅斯瓷磚中，沒有缺陷，並且有很大的潛力擴展到無限而不產生缺陷。.
因為早期的彭羅斯瓷磚是由兩種菱形形狀組成的，或者後期由另外兩種形狀組成的，風箏和矛：所有這些彭羅斯鑲嵌都不需要擔心馬賽克過程中產生的缺陷。然而，在我們的 Penrose 研究 不同的十邊形組成，並聚集在一起形成五邊形。將所有不同五邊形組合在一起的耦合方法總是會發現一些不匹配的缺陷區域，例如單個菱形不屬於三個十邊形中的任何一個，或者十邊形缺少一個菱形。特別是，當需要證明 Penrose tiles 可以擴展到無缺陷的 infinite tiles 時。需要精確定義缺陷，缺陷的潛在位置，並在開始時了解缺陷周圍十邊形的狀態，還需要通過耦合或鑲嵌在每個擴展步驟中檢查和修正缺陷的潛在產生。
Because the early Penrose tiles consisted of two rhombus shapes, or later two other shapes, the kite and the spear: all these Penrose tilings did not need to worry about imperfections created during the mosaic process. However, in our Penrose study, the tiles consisted of six different decagons that came together to form pentagons. A coupled approach that combines all the different pentagons will always find some mismatched defect regions, such as a single rhombus that doesn't belong to any of the three decagons, or a decagon missing a rhombus.
In particular, when it is required to demonstrate that Penrose tiles can scale to defect-free infinite tiles. Defects, their potential locations need to be precisely point out , and the state of the decagon around the defect needs to be known at the start, and the potential generation of defects needs to be checked and corrected at each expansion step through coupling or tessellation.
It is my fortunate, this is a Penrose tile with all six decagons together in one while defect free (defined tranditonally)and has a great potential to be extended to infinite without generating further new def ect. , 在 Kung 的嚴格定義下,under the Kung' strict definitionone should not able to find any rhombic (brick) that not belong any of the six decagons, but not in this case )
第一次讓所有
In my retired years, most notably, having the privilege of reinterpreting the Penrose tiling, resulting in over fifty more or less similar regular pentagonal and circular Penrose tiling. And all these patterns can be properly coupled to form infinite Penrose tiles.. Here, a new chemical termilogy term “ cluster” must be applied to distinguish the simility or difference of these hundrds Penrose tiles.
With all this self-taught painting experience on the computer, I've had the privilege of repainting and reorganizing, regrouping hundreds of Penrose tiles by my own. The smallest circle or pentagon has only 10 or more decagons, and the largest has nearly 10,000 decagons, which may not be clearly visible in a 5000×5000 pixel picture. With hundreds of Penrose tiles with different shapes, it is necessary to find a new term to category these Penrose tiles at outset for easy to explain following coupling procedures .
Here, a new chemical terminology term “ cluster” must be applied to distinguish the simility or difference of these hundreds Penrose tiles. Since , Penrose tile were reshaped to a Pentagonal shape with five decagons on each side, these "cluster" with each side five decagons were employed as standard to make the example of coupling.
In fact, It has been noticed that using a smaller cluster , each side contains four or three decagons is much easy to handle the cluster coupling extension. A circular shape Penrose tile could be better to reveal a symmetric characteristic of coupling behavior.
One of a relatively easy of the tricks for infinite expansion of Penrose tiles is demonstrated below. It guarantees an defect free extension just by checking the decagons in seam region .
在我退休的歲月裡，最值得一提的是，有幸重新詮釋了彭羅斯瓷磚（Penrose tiling），導致產生了五十多個或多或少相似的正五邊形和圓形的彭羅斯瓷磚圖案。並且所有這些圖案可以適當地耦合以形成無窮大的彭羅斯瓷磚。
憑藉所有這些在計算機上自學的繪畫經驗，我有幸自己一個人能夠重新繪製重組了數百塊 Penrose 瓷磚。最小的圓或五邊形只有10個或更多的十邊形，最大的有近萬個十邊形，在5000×5000像素的圖片中可能看不清楚。有數百個不同形狀的 Penrose tiles，有必要在一開始就找到一個新的術語來對這些 Penrose tiles 進行分類，以便於解釋以下耦合過程在這裡我們使用一個化學術語“簇”來描述這種彭羅斯瓷磚在明確形狀中的相似性。事實上，由於在一開始研究時，彭羅斯瓦被重塑修改為每邊有五個十邊形的五邊形，因此一開始將這些每邊有五個十邊形的“簇”作為標準來製作耦合示例。基本上，它們在外觀上具有多變的成分（十邊形），同時揭示了簇內部的不同。事實上，已經發現，使用較小的簇，每邊包含四個或三個十邊形很容易處理簇耦合擴展。事實上，圓形 Penrose 瓦片可以更好地揭示耦合行為的對稱特徵。
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The origin penrose tile is made by..
下面演示了彭羅斯瓷磚無限擴展的 技巧之一。它僅通過檢查接縫區域中的十邊形來保證無缺陷擴展
所有六個十邊形都在一個彭羅斯瓷磚中，沒有缺陷，並且有很大的潛力擴展到無限而不產生缺陷。.
To put all six Decagons in i\one Penrose tile is not difficulty job, as swown in fig 1 and fig 2 , Herere we use a term cluster to describes the simility of this P
It is my fortunate, this is a Penrose tile with all six decagons together in one while defect free and has a great potential to be extended to infinite without generating defect. .If one can find any rhombic (brick) that not belong any of the six decagons,
Here, a new chemical terminology term “ cluster” is necessary to be applied to distinguish the simility or difference of these hundrds Penrose tiles. Since Penrose tile were reshaped to a Pentegon shape with five decagons on each side, these "cluster" with each side five decagons were employed as standard to make the example of coupling. In fact, It has been found , using a smaller cluster , each side contains four or three decagons is much easy to handle the cluster coupling expension
To put all six Decagons in one Penrose tile is not difficulty job, as shown in fig 1. and fig 2
有必要使用一個新的術語來描述定義描述形狀明確的彭羅斯平鋪，以便在一開始就簡單明了地描述耦合過程.. 這裡化學術語“簇”表示可以在局部或一定程度上大面積進行化學反應，使“簇”擴展到更大，並且內部成分可變。
It is necessary to use a new term to describe well-defined Penrose tiles in order to describe the coupling process at the outset easily and clearly.，Here, a chemical term "Cluster" indicates that a chemical reaction can be performed locally or to some degree large area to extend the "Cluster" to a even big one , and with an inner composition changeable..
幸運的是，第一次將所有六個十邊形都集中在一個彭羅斯瓷磚中，並且有很大的潛力可以延伸到無窮大 ,
在常規繪製彭羅斯瓷磚時，無需擔心缺陷問題 .讓我們盡情享受彭羅斯磚的美麗結構.
Fortunately, for the first time, all six decagons are packed into a single Penrose tile, and can be extend to infinity ... In traditional Penrose tiles, there is no need to worry about defects, let's enjoy the beautiful structure of Penrose mosaic tiles **But under Kung's strict definition, there are five small defects in the different zone regions).(According to Kung's strict definition, any rhombus (brick) that does not belong to any of the six decagons is considered as defects),
One of the tricks for infinite expansion of Penrose tiles is demonstrated below It guarantees an defect free extension just by checking the decagons in seam regions .下面演示了彭羅斯瓷磚無限擴展的技巧之一。它僅通過檢查接縫區域中的十邊形來保證無缺陷擴展
將所有六個十邊形放在一個彭羅斯瓷磚中並不困難，如圖 1 和圖 2 所示。
It is necessary to apply a new terminology to describe the well shape-defined Penrose tiles for the convenience of describing the coupling procedures. Here a chemical term "cluster" to describes the similarity of this Penrose tiles with a well- defined shape, and basically, they have a changeable compositions (decagons) on the out-side look, while reveals different inside inner part of clusters. Since in the beginning of Penrose tiling study, Penrose tile was modified to be a so defined pentagon shaped -cluster, each side of cluster consisted of five decagons.
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