Celebrating the fiftieth anniversary of the discovery of the Penrose tile Chung yuan Kung 祝彭羅斯瓷磚發現五十週年 using Kung's clusters (album) 彭羅斯(非週期)貼磚專輯－貢中元的部落格

-Chung yuan Kung由於視力不佳，中文打字有很多錯誤，敬請諒解

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 many sparked research studies in the field. In this paper, the author just concentrated on the Penrose tiling and how to extended tiles to infinite size.

Here are some of most systematic procedures to draw defect free Penrose tiling , most of all it can be extended to infinite and defect free.

.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。

?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 configuration ,僖可能的說明如可可達到無‧限大..

• 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。以便滿足零缺陷的要求, .

Fig 1 36度與72度角的兩個胖瘦等邊菱形(視為二個不同的原子.每個原子由四個等距鏈鍵)，可以組成6種不同基本組態的正十方形(視為分子), 如圖 1 所示。真正嚴格定義下的彭洛斯鋪磚, 須要所有的胖瘦菱形(原子,磚塊), 都必須要包含在6種之中任何一個正十角形(分子)內。 -每一個簇 cluster 都是由每邊五個正十方形組成的類似 Penrose tiling 正五邊形或者每一個邊上有十八個鏈鍵。

Fig 2 A set of Kungs' Cluster, 臃腫型 # 1 ( swollen type), #1a 是最原始的 Famous (original) Penrose tiling , it contains 66 molecules (regular decagon) and defect free; #1 b to #1h #1a 的變異型 , #2 是纖瘦型 slim type, it contains 55 molecules (regular decagon); 是臃腫型的匹配對應(共軛)組態 ( is the counter part , or conjugaye part of #1 ), 是否可以考慮為reciprocal 倒晶格態, 則還需要慎思, 因為內部的正十方形數量不一樣, 相差了11個。臃腫型 #1 and 纖瘦型 # 2 can be coupled together to form a new (may not be five folded symmetric) tiling, as shown in next (簡稱共軛態) next next. #4 may be counted as new type of Penrose tiling, contains 60 molecules (regular decagon) Most of the clusters in # 1, #2, # 3 can be self-coupled to form extended defect free Penrose tiling, while # 4 could not make a defect free self-extension。 Light blue marks d -type 正十方形 yellow makes e-type正十方形, using d-, e-, f- type decagons is just a gimmick (噱頭) in the drawing。 In fact, they (d-, f- type) can only survive on the border, for e-type decagon that not on the border can always be replaced by b- or c- type decagon。Defects defined here as a bricks ( atom) that not be contained into any of six decagons (正十方形) are marked as pink color。在最初作圖時,用不同顏色的定位,有助於分析整体結構的走向,並判新可能的缺陷。 So on we may find a systematic process to draw an unlimited defect free Penrose tiles . 現在己經可以有系統的畫製大圖..

Celebrating the fiftieth anniv

fig origin penrose tiling (left) and modified penrose tiling to a pentagon pattern, as 1a 1b

this above shows only oentagon shape Penrose tiling, circular shape Penrose tiling will be shown latter ,

Celebrating the fiftieth anniv

第一次看到所有六個不同的分子（十邊形）無缺陷的在一個簇中 see all six molecults (decagon) are in one cluster defect free first

Penrose tiling, not five-fold symmetic, containing all six different decagons..and can be further extended

沒有五重對稱彭羅斯拼貼，包含所有六個不同的十邊形

Celebrating the fiftieth anniv

time n this above one also shows a pentagon shape without typee-a decagon

Celebrating the fiftieth anniv

Clusters with inner defect marks as pink color

臃腫型 # 1a ( swollen type five fold symmetric), contains 66 molecules (regular decagon); 21 a- type moleculesdecagon) . 35 b- type moleculesdecagon). 10 c-type moleculesdecagon):

纖瘦型 #2a ( slim type five fold symmetric) , contains 55 molecules (regular decagon); 15 a- type moleculesdecagon) . 35 b- type moleculesdecagon). 5 c-type moleculesdecagon):

#2 b (slim type not five fold symmetry ), contains 55 molecules (regular decagon); 16 a- type moleculesdecagon) . 29 b- type moleculesdecagon). 10 c-type moleculesdecagon):

Normal Type #4 (five fold symmetric), contains 60 molecules (regular decagon); 20 a- type moleculesdecagon) . 15 b- type moleculesdecagon). 25 f-type moleculesdecagon

Small tyoe contains 49 molecules (regular decagon); a- type moleculesdecagon) . b- type moleculesdecagon). -type moleculesdecagon

Small tyoe

(第 # 3 cluster 每邊有六個正十方形16個鍵. is part of #1 cluster) Celebrating the fiftieth anniv

Celebrating the fiftieth anniv

Fig 3 每一種簇類分子數是一樣的,但是分子種類數不一樣

cluster couple 之後會變成更大的五邊形,臃腫型仍然是,臃腫型,纖瘦型仍然是纖瘦型, cluster couple 重合部份可能會有一些缺陷,但是可以很容易的修正.. 可以再按照上一步描述的程序直到無限. The following clusters #1 and #2 have a great chance to be extended to infinite without generating defects , we will demonstrate some exemples.

Fig 4 demonstrates three basic schems to extend Penrose tiling , (A) self couple ( included modified multiple couple), (B) multual swollen-slim surrounding couple, (D ) multiple couple,,

a modifications on the boundary of cluster are necessary to fulfill defect free,.

self couple (swollen-swollen couple - slim - slim couple Celebrating the fiftieth anniv

Celebrating the fiftieth anniv

swollen - slim couple (藕合, 暫時定義,,,疊合面積超過了邊界分子部份

(A) scheme swollen- slim couple (匹配式耦合) 形成一個有78 個正十角形(分子) 類六角形, 中閒有 43 分子(大部份面積重合的)耦合重複 , 而這類六角形的 "新簇 " (是

值得注意的),可以tessellation (鑲嵌), (BUT NEED to eliminate border defects,),也可以用類似mutual sourround couple.

Celebrating the fiftieth anniv

Fig demonstrate coupling of different clusters, here, author intends to leave some decagons with defect for reader to fix them (easy). This kind of mixed coupling may perform an unlimited Penrose tiling combinaion. 有 slanted 平移週期性 for so defined unit cell.

Celebrating the fiftieth anniv

(B) 共軛態 Scheme - mutual surrounding couple -邊界上有五個分子overlaped 疊,,、二個簇之間空白處可以貼補上六個正十方形(分子),形成一個含有346?個分子具有五重對稱恬的瘦五角形含有346?個分子 vacancy can be filled by patching,(在此暫眸定義, 共軛態為疊合地區只限於邊界的

Celebrating the fiftieth anniv

Circular to circular coupling：只有兩種類型的十邊形可以將五重對稱圓形Penrose-tile擴展到無限大

Circular to circular coupling: only two types if decagon can extend a five fold symmetric circular shape Penrose -tile to infinite (but not by coupling)

Celebrating the fiftieth anniv

Fig show how to fill the inter vacancy area" of a共軛態. Since the extended Penrose tiles more or less reveals a s;anted periodics. one mayy easily some suitable area tp patch the vacancy area.

Celebrating the fiftieth anniv

Fig a strange six and five basic type wonderful arrangement! (居然有六方形的簇態‧)

Fig Basic extension of Penrose tiling, there are many different combinations to extend Pamrose tiling. self coupling and surrounding couple at different stages with different Kungs' Clusters this following one has 230 Decagons

許多不同的組合來擴展彭羅斯平舖的基本擴展 - 有許多不同的組合來擴展帕姆羅斯平鋪。不同Kungs' Cluster不同階段的自耦合和周圍耦合,如此可以有無限多種無限大的Penrose tiles.

Fig The defect area can ce easily replaced by one of six 正十方形, the vacancy area may be patched by an area in the counter part.

Fig #1 self cou[ling and #2 self coupling共軛態(只有邊界部份耦合, they may both self coupling again and again to extending to infinite or they can couple to each other and extend to infinite.

fig above is extended by origin Penrose tiling that can be extended to unlimited using only a b c正十角形 three decagons

Fig this one is has two defects.

Fig #3 abd # 4 self coupling may also make an ibfinite Penrose tiling defect free. Celebrating the fiftieth anniv

Fig. This one has no defect, but has a great chance to be extended further without generating defects. In fact, #2 and #3 are identical in the inner part m should considered as same one, as , because , edge can be well repaired to meet requrement. and contain #6 and # 5 正十方形

Fig, # 3 and #2 can be well coupled , may not generate defects, see pink area ? reader can try to elimited them ,