Celebrating Penrose Tiling 50th Anniversary 慶祝彭羅斯平鋪 50 週年 (I) How to define a Penrose Kung cluster 如何定義彭羅斯-平鋪, cykung 貢中元－貢中元的部落格

celebration for 50th anniversary Penrose' tiling 慶祝彭羅斯平鋪 50 週年,

你一定會想知道為什麼Paul Steinhardt (Author) 作者用這個書名 The Second Kind of Impossible: The Extraordinary Quest for a New Form of Matter Hardcover – 8 1 月 2019 ,而i且乎與Penrose tiling 有關係,... 我花了一點時間在畫Penrose 圖,, 一開始總覺得有什麼不對勁,, 後來我學到一個名詞同調 ( Cohomology ) ,俗氣的說就是有那麼一點相同的味道.. 但是用在Penrose tiling ,我還是不能給你答案,。這句話就算是前言

,貢中元 cykung

How to define a Penrose Tiling 如何定義彭羅斯平鋪

兩個角度分別為36度與72度角的兩個菱形,稱之瘦原子和胖原子，可以組成6種不同基本組態的正十方形, 稱之為六個分子. 如圖 1 所示。1974年潘洛斯首先定了一条平铺规则」從Penrose 平鋪看來,是由六個其中三個正十角形構成的標準五重密鋪, 如圖 2。

真正嚴格定義下的彭洛斯鋪磚, 須要所有的菱形磚塊必須都要包含在六個其中任何一個正十角形內。深入分析密鋪作圖後有助於了解其結構秩序..

事實上這個五重密鋪是延伸到無限大,不容易看懂(完全沒有缺陷的)細節的,,也不太容易作圖，如果不是嚴格定義的話，所做出來的圖外表神似，但是內部藏有大量非基本組態定義下的缺陷,但是仍然‧合於五重性對稱。1986年發表的彭洛斯鋪磚中心是有缺陷的,或是他們說的dislocation去填補..

..,

Fig 2 . 在此我展出其他‧十幾,二十多種很像 Penrose tiling 具有五重對稱性的圖 . 最原始的 Penrose tiling 為 1A type , 其他的分別為 swollen, slim, and normal type., 可以達到上百種type, 這些簇內的結-構,可以有缺陷,仍然可以堆疊到五重性對稱,並可延伸到無限大 , ,,,特性稍後再描述 .. 為了以後容易分辨起見,我特別的把所有看起來像五星形狀圖標示出來.

上述五角形,的五個邊的結楫型態是一樣的 ,swollen type and slim type 相互補, E 型的邊是白己倒向互補。這種特性是密鋪的基礎, 在下一blog,如何有序的擴張再評述`o需要i定醒解斛釋的.D圖藍色星是在中心,它與上面那個黃色結瑙是互補的, 圖中,淺藍色的結構是第一次出現,,,,,,直到所有六個基本十角形出現才算完整的一輪`,

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

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

所謂二維密鋪,是要可以延沿到無限大的, 如何証明司一個能都衍生到無限大嗎? 如果是的話,有好多類Penrose tiling 可以衍生到無限大?,.

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..

Fig with the red tiles not in the one of the six basic configurations

Fig 2 Penrose tiling 1974

Fig 3

There is an even larger extend tilings contains more than 25 Penrose tiling.

Two rhombus with angles of 36 degrees and 72 degrees can form 6 different basic decagon configurations, as shown in Figure 1. 1974, Penrose first delineated a standard five-fold tiling consisting of three different basic decagons. Penrose tiling under the really strict definition requires that all tiles must be included in any one of the six decagons.

Anylysis of Penrose , five Corner star, 25 circular edge decagons , only three decagons configurations to perform mosaics 只需要使用這三種類型來執行鑲嵌,

if once employed another configuration,

Fig 4 This is a third layer of penrose tiling with no defect, but using four decagoncon figurations ,, please check symmetry marked ny different color. For the easy of viewing the figures, We mark the position of all stars shapes. Blue star for the center of penrose tiling, ,,,red star for the number one decagon tiles , Green star with tiles in three three third configurations, A type contains one Blue star.20 red stars and five Green stars.. Type B contains one Blue star, 15 stars and five Green stars .Type C contains one Blue star,15 stars and five Green stars.. One may watch the location difference of Green stars and the edge of Penrose between A type and B type, whire the B type and C Type are the same on the edge structure but different in the inner structure.. B type and C type can be mutual replaced when making an extended Penrose tiling. Hint , there could be only three types one so defined Penrose tiling as described above ( with my best trying).

These may be a different type (second way of drawing a Penrose tiling ) may be (counted as second or third type penrose tiling) or just an extended Penrose tiling.Extended Panrose tiling (second zone or third ) . 彭羅斯密鋪的擴展（第二個區域或第三個）可能是（算作第二或第三類型彭羅斯拼接）或只是一個擴展彭羅斯拼圖