貢中元的部落格

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。.

75歲自序  字祚自壽

從十多年前在電腦前, 從一張照片改製第一張圖畫開始，我就自以為是無師自通、自作自授的畫家。得到最友善最佳的評語，「抽象的莫名其妙」，不知作品好歹的自稱畫家, 還默默地參加了幾次畫展。後來還真的自作自售，賣出了一批友情贊助的畫作。在電腦前面是有各種不同的學習研究工作的, 也要常常注意到學生的論文進展。多年來, 晨昏顛倒,樂此(電腦)不疲。一時興起又恣作眥受恣意畫了許多號稱超級抽象，自己也看不懂的作品。

,把type-a,type-d,type-e,type-f中的胖菱形分別做成紅色、粉紅色、黃色和綠藍色，以利於後期大圖的對齊和校正 ,make the fat rhombic in type-a, type-d, type-e and type-f as red, pink, yellow and green  blue color respectively, in order for the  benifit of the alignment and correction for the drawing of later on large penrose tiles drawing  .with the middle line of the leftmost type-a as the reference for mirror symmetrical coupling, the result in Figure 4a is obtained. Similarly, for the second leftmost, the third leftmost, the fourth leftmost and the fifth leftmost midline as the reference, the graphics obtained by mirror image coupling are shown in Figure 4b, 4c, 4d, 4e, 4f.  For the simplicity, here, just use the graph in fig 4 e as an example to continue the coupling in y-direction. the result of the specular deepest and specular shallowest are demonstrated in fig 6a and 7a, In the figure, many complex and chaotic completely indistinguishable patterns can be seen.. The correction results of the graphics in Fig and fig are shown in Figure b and fig b. After processing, a clear Penrose tiling diagram is obtained. It can be easily seen that the position of the defect belonging to the decagonal shape is easy to repair. Because, the non-matching position has a decagonal shape, it is easy to replace it with a proper decagon showing in fig 1b. , The other , as shown in the picture in Figure b, the defects are messy and take some time to process

In the two examples mentioned above, the revised tiles are no longer Penrose tiles under the traditional definition , because they don't have five-symmetry and mirror symmetry. They can be called Penrose-like tiles. After all, they are based on the original pattern of Penrose through a symmetrical coupling treatment, changed from. .After all, they are all based on Penrose's original model and transformed through symmetrical coupling processing. 

Four way  symmetric  tiling and its elementary unit cell (paper 3)

製作四向對稱平鋪及其基本晶胞 The formation of the four-way symmetric translational tiles (crystal) and corresponding unit cells

virus couplimg

virus coupling ，圓形（球形）形狀總是與病毒有關，下面演示兩種不同的皮膚耦合，一種相對容易，另一種需要將鍵合鏈溶解並重建另一種十邊形.

祝彭羅斯瓷磚發現五十週年  Celebrating the fiftieth anniversary of the discovery of the Penrose tile cykung

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.

Penrose tiling contains all six decagons and have a great chence to be extended to infinity and defect free

世界上第一個包含所有六個不同十邊形的大型彭羅斯瓷磚，並且有很大的機會擴展到無窮大。world first large Penrose tiles that contains all six different decagons, and has a great chance to be expanded to infinity. Be noted first, fig 2,3,4, hide a correctable defect. 如果不考慮上面定義的關於缺陷的限制定義， 如果不考慮先前定義的關於缺陷的極端限制性定義，六個十邊形可以共存於彭羅斯瓷磚到無窮大，見最後圖 9  Six decagons can co-exist in a Penrose tiling to infinity if the previously defined extremely restrictive definition of defects is disregarded, see the last figure

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

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