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DOI: 10.1126/Science.1135491 , 1106 (2007); 315 Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture Peter J. Lu, et al. Science 315, 1106 (2007); DOI: 10.1126/science.1135491 This copy is for your personal, non-commercial use only. If you wish to distribute this article to others, you can order high-quality copies for your colleagues, clients, or customers by clicking here. Permission to republish or repurpose articles or portions of articles can be obtained by following the guidelines here. The following resources related to this article are available online at www.sciencemag.org (this infomation is current as of January 2, 2011 ): A correction has been published for this article at: http://www.sciencemag.org/content/316/5827/982.1.full.html Updated information and services, including high-resolution figures, can be found in the online version of this article at: http://www.sciencemag.org/content/315/5815/1106.full.html Supporting Online Material can be found at: on January 2, 2011 http://www.sciencemag.org/content/suppl/2007/02/20/315.5815.1106.DC1.html A list of selected additional articles on the Science Web sites related to this article can be found at: http://www.sciencemag.org/content/315/5815/1106.full.html#related This article has been cited by 15 article(s) on the ISI Web of Science This article appears in the following subject collections: Computers, Mathematics www.sciencemag.org http://www.sciencemag.org/cgi/collection/comp_math Downloaded from Science (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by the American Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. Copyright 2007 by the American Association for the Advancement of Science; all rights reserved. The title Science is a registered trademark of AAAS. REPORTS 21. Materials and methods are available as supporting 27. N. Panagia et al., Astrophys. J. 459, L17 (1996). Supporting Online Material material on Science Online. 28. The authors would like to thank L. Nelson for providing www.sciencemag.org/cgi/content/full/315/5815/1103/DC1 22. A. Heger, N. Langer, Astron. Astrophys. 334, 210 (1998). access to the Bishop/Sherbrooke Beowulf cluster (Elix3) Materials and Methods 23. A. P. Crotts, S. R. Heathcote, Nature 350, 683 (1991). which was used to perform the interacting winds SOM Text 24. J. Xu, A. Crotts, W. Kunkel, Astrophys. J. 451, 806 (1995). calculations. The binary merger calculations were Tables S1 and S2 25. B. Sugerman, A. Crotts, W. Kunkel, S. Heathcote, performed on the UK Astrophysical Fluids Facility. References S. Lawrence, Astrophys. J. 627, 888 (2005). T.M. acknowledges support from the Research Training Movies S1 and S2 26. N. Soker, Astrophys. J., in press; preprint available online Network “Gamma-Ray Bursts: An Enigma and a Tool” 16 October 2006; accepted 15 January 2007 (http://xxx.lanl.gov/abs/astro-ph/0610655) during part of this work. 10.1126/science.1136351 be drawn using the direct strapwork method Decagonal and Quasi-Crystalline (Fig. 1, A to D). However, an alternative geometric construction can generate the same pattern (Fig. 1E, right). At the intersections Tilings in Medieval Islamic Architecture between all pairs of line segments not within a 10/3 star, bisecting the larger 108° angle yields 1 2 Peter J. Lu * and Paul J. Steinhardt line segments (dotted red in the figure) that, when extended until they intersect, form three distinct The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in polygons: the decagon decorated with a 10/3 star medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, line pattern, an elongated hexagon decorated where the lines were drafted directly with a straightedge and a compass. We show that by 1200 with a bat-shaped line pattern, and a bowtie C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations decorated by two opposite-facing quadrilaterals. of a special set of equilateral polygons (“girih tiles”) decorated with lines. These tiles enabled Applying the same procedure to a ∼15th- the creation of increasingly complex periodic girih patterns, and by the 15th century, the century pattern from the Great Mosque of tessellation approach was combined with self-similar transformations to construct nearly perfect Nayriz, Iran (fig. S2) (11) yields two additional quasi-crystalline Penrose patterns, five centuries before their discovery in the West. polygons, a pentagon with a pentagonal star pattern, and a rhombus with a bowtie line on January 2, 2011 irih patterns constitute a wide-ranging repeat the same decagonal motifs on several pattern. These five polygons (Fig. 1F), which decorative idiom throughout Islamic length scales. Individually placing and drafting we term “girih tiles,” were used to construct a Gart and architecture (1–6). Previous hundreds of such decagons with straightedge wide range of patterns with decagonal motifs studies of medieval Islamic documents de- and compass would have been both exceedingly (fig. S3) (12). The outlines of the five girih tiles scribing applications of mathematics in ar- cumbersome and likely to accumulate geometric were also drawn in ink by medieval Islamic chitecture suggest that these girih patterns distortions, which are not observed. architects in scrolls drafted to transmit architec- were constructed by drafting directly a net- On the basis of our examination of a large tural practices, such as a 15th-century Timurid- work of zigzagging lines (sometimes called number of girih patterns decorating medieval Turkmen scroll now held by the Topkapi Palace strapwork) with the use of a compass and Islamic buildings, architectural scrolls, and other Museum in Istanbul (Fig. 1G and fig. S4) (2, 13), www.sciencemag.org straightedge (3, 7). The visual impact of these forms of medieval Islamic art, we suggest that providing direct historical documentation of girih patterns is typically enhanced by rota- by 1200 C.E. there was an important break- their use. tional symmetry. However, periodic patterns through in Islamic mathematics and design: the The five girih tiles in Fig. 1F share several created by the repetition of a single “unit cell” discovery of an entirely new way to conceptual- geometric features. Every edge of each polygon motif can have only a limited set of rotational ize and construct girih line patterns as decorated has the same length, and two decorating lines symmetries, which western mathematicians first tessellations using a set of five tile types, which intersect the midpoint of every edge at 72° and proved rigorously in the 19th century C.E.: Only we call “girih tiles.” Each girih tile is decorated 108° angles. This ensures that when the edges of two-fold, three-fold, four-fold, and six-fold with lines and is sufficiently simple to be drawn two tiles are aligned in a tessellation, decorating Downloaded from rotational symmetries are allowed. In particular, using only mathematical tools documented in lines will continue across the common boundary five-fold and 10-fold symmetries are expressly medieval Islamic sources. By laying the tiles without changing direction (14). Because both forbidden (8). Thus, although pentagonal and edge-to-edge, the decorating lines connect to line intersections and tiles only contain angles decagonal motifs appear frequently in Islamic form a continuous network across the entire that are multiples of 36°, all line segments in the architectural tilings, they typically adorn a unit tiling. We further show how the girih-tile final girih strapwork pattern formed by girih-tile cell repeated in a pattern with crystallographical- approach opened the path to creating new types decorating lines will be parallel to the sides of ly allowed symmetry (3–6). of extraordinarily complex patterns, including a the regular pentagon; decagonal geometry is Although simple periodic girih patterns in- nearly perfect quasi-crystalline Penrose pattern thus enforced in a girih pattern formed by the corporating decagonal motifs can be constructed on the Darb-i Imam shrine (Isfahan, Iran, 1453 tessellation of any combination of girih tiles. using a “direct strapwork method” with a C.E.), whose underlying mathematics were not The tile decorations have different internal ro- straightedge and a compass (as illustrated in understood for another five centuries in the West. tational symmetries: the decagon, 10-fold sym- Fig. 1, A to D), far more complex decagonal As an illustration of the two approaches, metry; the pentagon, five-fold; and the hexagon, patterns also occur in medieval Islamic archi- consider the pattern in Fig. 1E from the shrine of bowtie, and rhombus, two-fold. tecture. These complex patterns can have unit Khwaja Abdullah Ansari at Gazargah in Herat, Tessellating these girih tiles provides several cells containing hundreds of decagons and may Afghanistan (1425 to 1429 C.E.) (3, 9), based on practical advantages over the direct strapwork a periodic array of unit cells containing a method, allowing simpler, faster, and more ac- 1 Department of Physics, Harvard University, Cambridge, common decagonal motif in medieval Islamic curate execution by artisans unfamiliar with MA 02138, USA. 2Department of Physics and Princeton Center for Theoretical Physics, Princeton University, architecture, the 10/3 star shown in Fig. 1A (see their mathematical properties. A few full-size Princeton, NJ 08544, USA. fig. S1 for additional examples) (1, 3–5, 10). girih tiles could serve as templates to help po- *To whom correspondence should be addressed. E-mail: Using techniques documented by medieval sition decorating lines on a building surface, [email protected] Islamic mathematicians (3, 7), each motif can allowing rapid, exact pattern generation. More- 1106 23 FEBRUARY 2007 VOL 315 SCIENCE www.sciencemag.org REPORTS over, girih tiles minimize the accumulation of (1197 C.E.) (11, 17, 18), where seven of eight standard line decoration of Fig.
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