貢中元的部落格

The five-fold symmetrical tile that can be self-coupled to build up a five-fold tiling,periodic and four-way symmetry tiles (crystals) to infinity

Some smaller unit cells that form different decagonal two-dimensional crystals

Four unit cells that intercepted from a translational decagonal style crystal and form four different crystals.

Tessellation of unlimited equilateral rhombuses

 If the acute angle of a rhombus is 90/N degrees, and N is an integer, then there can be N equilateral rhombuses whose acute angles are integer multiples of 90/N respectively. A standard drawing method with numerous variables, in which N equilateral rhombus can be tessellated into an infinite number of groups (kinds) of combinations of infinite pattern. There are four general rules of drawing, which are described hereafter.

new

tellation of three triangles

asymmetry  Penrose binary  tiling  不對稱彭羅斯平鋪 

1 one may start from any conventionally defined five-fold symmtry (around a specific center, center of blue star,)or on-five fold symmerric  penrose tile , either pentagonal or circular shape.as shown in fig 1

All  aperoidic and periodicPenrose (Kung)  tiling  (# one set全集之2)

,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

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

   慶祝發現彭羅斯瓷磚五十週年 Celebrating the fiftieth anniversary of the discovery of the Penrose tile 貢中元   Chung yuan Kung

主要的工作簡述如下

((  Penrose tiles, they often have a pleasing and even mesmerizing quality, and their regularity often exhibits a certain natural harmony((((9彭羅斯瓷磚, 它們通常具有令人愉悅甚至令人著迷Enchanting的品質，它們的規律性通常表現出某種自然和諧，我們大多數人都知道，只有三角形、正方形和六邊形才能用單一類型的正多邊形瓷磚平鋪平面.

。激發您的想像力，所有這些各種圓形和五邊形的基本彭羅斯拼塊都可以以不同方式拉伸，尤其是通過耦合到無窮大。Boost your imagination, all these various circular and pentagonal elementary Penrose tiles can stretch differently, especially by coupling going to infinity.

Penrose tiling size of infinite using Kung's clusters (album)

The Penrose tiling is one of the most famous and complex tessellation patterns, and has been drawn and expressed in many different ways.(  ) This fivefold symmetrical tiling is associated with the discovery of quasicrystals and has ignited numerous studies in the field..    In this study, the author just concentrated on the variation of Penrose tiling and how to extended tiles to infinite size.