Advanced geometry of Islamic art
(http://news.bbc.co.uk/2/hi/middle_east/6389157.stm)
A study of medieval Islamic art has shown some of its geometric patterns use principles established centuries later by modern mathematicians.
Researchers in the US have found 15th Century examples that use the concept of quasicrystalline geometry.
This indicates intuitive understanding of complex mathematical formulae, even if the artisans had not worked out the underlying theory, the study says.
The discovery is published in the journal Science.
The research shows an important breakthrough had occurred in Islamic mathematics and design by 1200.
"It's absolutely stunning," Harvard's Peter Lu said in an interview.
Pattern from 15th Century archway of Darb-i Imam shrine, Isfahan, Iran [Image courtesy of Peter Lu/Science]
"They made tilings that reflect mathematics that were so sophisticated that we didn't figure it out until the last 20 or 30 years."
The Islamic designs echo quasicrystalline geometry in that both use symmetrical polygonal shapes to create patterns that can be extended indefinitely.
Until now, the conventional view was that the complicated star-and-polygon patterns of Islamic design were conceived as zigzagging lines drafted using straightedge rulers and compasses.
"You can go through and see the evolution of increasing geometric sophistication. So they start out with simple patterns, and they get more complex," Mr Lu added.
He became interested in the subject while travelling in Uzbekistan, where he noticed a 16th Century Islamic building with decagonal motif tiling.
Mr Lu, who designs physics experiments for the International Space Station, was in the region in order to visit a space facility in Turkmenistan.
Islamic art traditionally uses a mixture of calligraphy, geometric and floral designs because of a prohibition on the portrayal of the human form.
- Re: Advanced geometry of Islamic art (BBC)posted on 02/25/2007
FuSheng:
I am writing a story of art and math, check it tomorrow. I am just thinking I write it for u :)
I got your email, Haa - posted on 02/25/2007
把原文里的图转过来 -
Peter J. Lu and Paul J. Steinhardt “Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture,” Science 315, 1106 (2007)
Fig. 1. Direct strapwork and girih-tile construction of 10/3 decagonal patterns. (A to D) Generation of a common 10/3 star pattern by the direct strapwork method. (A) A circle is divided equally into 10, and every third vertex is connected by a straight line to create the 10/3 star that (B) is centered in a rectangle whose width is the circle's diameter. In each step, new lines drafted are indicated in blue, lines to be deleted are in red, and purple construction lines not in the final pattern are in dashed purple. (E) Periodic pattern at the Timurid shrine of Khwaja Abdullah Ansari at Gazargah in Herat, Afghanistan (1425 to 1429 C.E.), where the unit cell pattern (D) is indicated by the yellow rectangle. The same pattern can be obtained by tessellating girih tiles (overlaid at right). (F) The complete set of girih tiles: decagon, pentagon, hexagon, bowtie, and rhombus. (G) Ink outlines for these five girih tiles appear in panel 28 of the Topkapi scroll, where we have colored one of each girih tile according to the color scheme in (F).
Fig. 2. (A) Periodic girih pattern from the Seljuk Mama Hatun Mausoleum in Tercan, Turkey (~1200 C.E.), where all lines are parallel to the sides of a regular pentagon, even though no decagon star is present; reconstruction overlaid at right with the hexagon and bowtie girih tiles of Fig. 1F. (B) Photograph by A. Sevruguin (~1870s) of the octagonal Gunbad-i Kabud tomb tower in Maragha, Iran (1197 C.E.), with the girih-tile reconstruction of one panel overlaid. (C) Close-up of the area marked by the dotted yellow rectangle in (B). (D) Hexagon, bowtie, and rhombus girih tiles with additional small-brick pattern reconstruction (indicated in white) that conforms not to the pentagonal geometry of the overall pattern, but to the internal two-fold rotational symmetry of the individual girih tiles.
Fig. 3. Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran (1453 C.E.). (A) Photograph of the right half of the spandrel. (B) Reconstruction of the smaller-scale pattern using girih tiles where the blue-line decoration in Fig. 1F has been filled in with solid color. (C) Reconstruction of the larger-scale thick line pattern with larger girih tiles, overlaid on the building photograph. (D and E) Graphical depiction of the subdivision rules transforming the large bowtie (D) and decagon (E) girih-tile pattern into the small girih-tile pattern on tilings from the Darb-i Imam shrine and Friday Mosque of Isfahan.
Fig. 4. (A and B) The kite (A) and dart (B) Penrose tile shapes are shown at the left of the arrows with red and blue ribbons that match continuously across the edges in a perfect Penrose tiling. Given a finite tiling fragment, each tile can be subdivided according to the "inflation rules" into smaller kites and darts (at the right of the arrows) that join together to form a perfect fragment with more tiles. (C to E) Mappings between girih tiles and Penrose tiles for elongated hexagon (C), bowtie (D), and decagon (E). (F) Mapping of a region of small girih tiles to Penrose tiles, corresponding to the area marked by the white rectangle in Fig. 3B, from the Darb-i Imam shrine. At the left is a region mapped to Penrose tiles following the rules in (C) to (E). The pair of colored tiles outlined in purple have a point defect (the Penrose edge mismatches are indicated with yellow dotted lines) that can be removed by flipping positions of the bowtie and hexagon, as shown on the right, yielding a perfect, defect-free Penrose tiling.
- posted on 02/25/2007
这条线是很有意思的。
伊斯兰严守教律,不得涂画动物与人形,只得在花叶和几何形体上飞
进,就象他们守酒戒,却在嗅觉和他种味觉上飞进。
我去清真寺是留意到许多对称型的,还有能三维。这些对称型,如果
整合成理论就是对称群了,平面对称,点群,空间群。
上回提爱因斯坦的对称性,其实犹太人也守教律,不偶像的。
说到对称群,伽罗华了不起。开拓了现代数学的思维,爱因斯坦的相
对论,是伽罗华在物理中小小的一个应用。
看到这些图型,我就象吃了阿细细,胡乱思绪一气。 - Re: Advanced geometry of Islamic art (BBC)posted on 02/25/2007
Bach would have loved Allah better, I suppose.
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