Friday, February 23, 2007


The secret of Islamic patterns
This is the pre-edited version of my latest piece for news@nature. The online version acquired some small errors that may or may not be put right. But what a great paper!

Muslim artists may have used a sophisticated tiling scheme to design their geometric decorations

The complex geometrical designs used for decoration by Islamic artists in the Middle Ages, as seen in buildings such as the Alhambra palace in southern Spain, were planned using a sophisticated tiling system that enabled them to make patterns not known in the West until 20 years ago, two physicists have claimed.

By studying many Islamic designs, Peter Lu of Harvard University in Cambridge, Massachusetts, and Paul Steinhardt of Princeton University in New Jersey have decided they were put together not using a compass and ruler, as previously assumed, but by tessellating a small number of different tiles with complex shapes.

The researchers think that this technique was developed around the start of the thirteenth century, and that by the fifteenth century it had become advanced enough to generate complex patterns now known as quasiperiodic. These were 'discovered' in the 1970s by the British mathematical physicist Roger Penrose, and were later found to account for puzzling materials called quasicrystals. Discovered in 1984 in metal alloys, quasicrystals initially foxed scientists because they seemed to break the geometric rules that govern regular (crystalline) packing of atoms.

The findings provide a further illustration of how advanced Islamic mathematics was in comparison with the medieval West. From around the eleventh century, much of the understanding of science and maths in the Christian West came from Islamic sources. Arabic and Persian scholars preserved the learning of the ancient Greeks, such as Aristotle, Ptolemy and Euclid, in translations and commentaries.

The Muslim writers also made original contributions to these fields. Western scholars learnt Arabic and travelled to the East to make Latin translations of the Islamic books. Among the mathematical innovations of the Islamic world were the use of algebra, algorithms (both of which are words derived from Arabic) and the use of numerals now known as 'Arabic' (although derived in turn from Indian notation).

The mathematical complexity of Islamic decoration has long been admired. The artists used such motifs because representational art was discouraged by the Koran. “The buildings decorated this way were among the most monumental structures in the society, combining both political and religious functions”, says Lu. “There was a great interest, then, in using these structures to broadcast the power and sophistication of the controlling elite, and therefore to make the ornament and decoration equally monumental.”

Lu and Steinhardt now propose that these designs were created in a previously unsuspected way. They say that the patterns known as girih, consisting of geometric polygon and star shapes interlaced with zigzagging lines, were produced from a set of just a handful of tiling shapes ranging from pentagons and decagons (regular ten-sided polygons) to bow-ties, which can be pieced together in many different ways. The two physicists show how these tiles could themselves be drawn using geometric constructions with compasses that were known by medieval Islamic mathematicians.

Some scrolls written by Islamic artists to explain their design methods show tiles with these shapes explicitly, confirming that they were used as 'conceptual building blocks' in making the design. Lu says that they’ve found no evidence that the tiles were actually made as physical objects. “But we speculate they were”, he adds, “so as to be used as templates in laying out the actual tiling on the side of a building.”

Lu and Steinhardt say that designing this way was simpler and faster than starting with the zigzag lines themselves: packing them together in different regular arrays automatically generates the complex patterns. “Once you have the tiles, you can make complicated patterns, even quasicrystalline ones, by following a few simple rules”, says Lu.

The researchers have shown that many patterns on Islamic buildings can be built up from the girih tiles. The resulting patterns are usually periodic – they repeat again and again, and so can be perfectly superimposed on themselves when shifted by a particular distance – but this regularity can be hard to spot, compared say with that of a hexagonal honeycomb pattern.

The patterns also contain many shapes, such as polygons with 5, 10 and 12 sides, that cannot themselves be packed together periodically without leaving gaps. This property of the polygons means that scientists long believed that it was impossible for crystals to show five- ten- and twelvefold symmetries, such that rotating them by a fifth, tenth or twelfth of a full circle would allow them to be superimposed on themselves.

So when 'crystals' that appeared to have these symmetries were discovered in 1984, they seemed to violate the basic rules of geometry. But it became clear that these quasicrystals aren't perfectly periodic. In the same year, Steinhardt pointed out how patterns with the same geometric properties as quasicrystyals could be constructed from the tiling scheme devised by Penrose.

Steinhardt and Lu say that, while there is no sign that the Islamic artists knew of the Penrose tiling, their girih tiling method provides an alternative way to make the same quasicrystalline patterns. The researchers say that a design on the Darb-i-Imam shrine in Isfahan, Iran, made in 1453, is virtually equivalent to a Penrose tiling. One of the mesmerizing features of this pattern is that, like a true quasicrystal, it looks regular but never repeats exactly.

“I’d conjecture that this was quite deliberate”, says Lu. “They wanted to extend the pattern without it repeating. While they were not likely aware of the mathematical properties and consequences of the construction rule they devised, they did end up with something that would lead to what we understand today to be a quasicrystal.”

Reference
Lu, P. J. & Steinhardt, P. J. Science 315, 1106 - 1110 (2007).

Postscript
I have received some comments from Roger Penrose on this work, sadly too late for inclusion in the Nature piece but which provide some valuable perspective on the discovery. This is what he says:
"The patterns are fascinating, and very beautiful, and it is remarkable how much these ancient architects were able to anticipate concerning 5-fold quasi-symmetric organization. But, as Steinhardt (and, in effect, Lu) have confirmed directly with me, the Islamic patterns are not the same as my patterns (on several counts: different basic shapes, no matching rules, no evidence that they used anything like a "Penrose pattern" to guide them, the hierarchical structure indicated by their subdivision of large shapes into smaller ones is not strictly followed, and would not, in any case, enable the patterns to map precisely to a "Penrose tiling"). I do, however, regard this work of Steinhardt and Lu as a most intriguing and significant discovery, and one wonders what more the ancient Islamic designers may have known about such things. I should perhaps add that the great Astronomer Johannes Kepler, in his Harmonice Mundi (vol.2), published in 1619, had independently produced a regular pentagon tiling that is much closer to my own tilings than anything that I have seen so far in this admittedly wonderful Islamic work."

Peter Lu, incidentally, has indicated that he agrees with everything that Penrose says here. The relationship between the Darb-i-Imam pattern and a Penrose tiling is subtle - much more so, it seems, than media reports of this work have tended to imply.

2 comments:

uhfdf said...

歐美a免費線上看,熊貓貼圖區,ec成人,聊天室080,aaa片免費看短片,dodo豆豆聊天室,一對一電話視訊聊天,自拍圖片集,走光露點,123456免費電影,本土自拍,美女裸體寫真,影片轉檔程式,成人視訊聊天,貼圖俱樂部,辣妹自拍影片,自拍電影免費下載,電話辣妹視訊,情色自拍貼圖,卡通做愛影片下載,日本辣妹自拍全裸,美女裸體模特兒,showlive影音聊天網,日本美女寫真,色情網,台灣自拍貼圖,情色貼圖貼片,百分百成人圖片 ,情色網站,a片網站,ukiss聊天室,卡通成人網,3級女星寫真,080 苗栗人聊天室,成人情色小說,免費成人片觀賞,

傑克論壇,維納斯成人用品,免費漫畫,內衣廣告美女,免費成人影城,a漫,國中女孩寫真自拍照片,ut男同志聊天室,女優,網友自拍,aa片免費看影片,玩美女人短片試看片,草莓論壇,kiss911貼圖片區,免費電影,免費成人,歐美 性感 美女 桌布,視訊交友高雄網,工藤靜香寫真集,金瓶梅免費影片,成人圖片 ,女明星裸體寫真,台灣處女貼圖貼片區,成人小遊戲,布蘭妮貼圖片區,美女視訊聊天,免費情色卡通短片,免費av18禁影片,小高聊天室,小老鼠論壇,免費a長片線上看,真愛love777聊天室,聊天ukiss,情色自拍貼圖,寵物女孩自拍網,免費a片下載,日本情色寫真,美女內衣秀,色情網,

liwo said...

av自拍,臺灣18歲成人免費,avon,正妹強力牆,免費線上成人影片,免費遊戲,a片貼圖,正妹圖片,3d美女圖,杜蕾斯免費a片,蓬萊仙山寫真集,a片網站,哈拉網路成人區,sex女優王國,性感美女,自拍密錄館,18禁卡通,爽翻天成人網,go2av,網拍模特兒應徵,台灣18成人,制服美女,小老鼠成人,成人光碟,金瓶影片交流區,85cc免費影城,成人交友,蓬萊仙山寫真集,無碼,正妹強力牆,嘟嘟情色網,影片轉檔程式,免費成人片觀賞,拓網交友,松島楓免費影片,色美眉部落格,18成人avooo,美腿論壇,辣媽辣妹,露點寫真,哈雷聊天室,18禁影片,看a片,美女工廠,影音城論壇,美女影片,免費遊戲,免費算,小魔女貼影片,a片貼圖,美腿褲襪高跟鞋,av女優王國,觀月雛乃影片,性感美女,

女優王國,免費無碼a片,0800a片區,免費線上遊戲,無名正妹牆,成人圖片,寫真美女,av1688影音娛樂網,dodo豆豆聊天室,網拍模特兒,成人文學,免費試看a片,a片免費看,成人情色小說,美腿絲襪,影片下載,美女a片,人體寫真模特兒,熊貓成人貼,kiss情色,美女遊戲區,104 貼圖區,線上看,aaa片免費看影片,天堂情色,躺伯虎聊天室,洪爺情色網,kiss情色網,貼影區,雄貓貼圖,080苗栗人聊天室,都都成人站,尋夢園聊天室,a片線上觀看,無碼影片,情慾自拍,免費成人片,影音城論壇,情色成人,最新免費線上遊戲,a383影音城,美腿,色情寫真,xxx383成人視訊,視訊交友90739,av女優影片,