Decagonal and quasi-crystalline tilings in medieval Islamic architecture

Science

Volumn 315, Issue 5815, 2007, Pages 1106-1110

  Lu, Peter J ^a   Steinhardt, Paul J ^b  

^a HARVARD UNIVERSITY   (United States)

^b PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARCHITECTURE; MEDIEVAL; POLYGON;

ARCHITECTURE; ARTICLE; BUILDING; BUILDING MATERIAL; GEOMETRY; IRAN; MATHEMATICS; MOSLEM; PRIORITY JOURNAL; RELIGION; TURKEY (REPUBLIC);

EID: 33847263544     PISSN: 00368075     EISSN: 10959203     Source Type: Journal    
DOI: 10.1126/science.1135491     Document Type: Article

References (32)

1. J. Bourgoin, in Les elements de l'art arabe; le trait des entrelocs (Firmin-Didot, Paris, 1879), p. 176.
2. G. Necipoglu, The Topkapi Scroll: Geometry and Ornament in Islamic Architecture (Getty Center for the History of Art and the Humanities, Santa Monica, CA, 1995).
3. I. El-Said, A. Parman, in Geometric Concepts in Islamic Art (World of Islam Festival, London, 1976), pp. 85-87.
4. S. J. Abas, A. S. Salman, in Symmetries of Islamic Geometrical Patterns (World Scientific, Singapore, 1995), p. 95.
5. Y. Demiriz, in Islam Sonatinda Geometrik Susleme (Lebib Yalkin, Istanbul, 2000), pp. 27, 128-129.
6. G. Schneider, in Geometrische Bauornamente der Seldschuken in Kleinosien (Reichert, Wiesbaden, Germany, 1980), pp. 136-139, plate 3.
7. Abu'l-Wafa al-Buzjani's (940-998 C.E.) treatise On the Geometric Constructions Necessary for the Artisan, and an anonymous manuscript appended to a Persian translation of al-Buzjani and likely dating from the 13th century, On Interlocks of Similar or Corresponding Figures (2), document specific techniques for architecturally related mathematical constructions (2, 25). The mathematical tools needed to construct the girih tiles are entirely contained in these two manuscripts-specifically, bisection, division of a circle into five equal parts, and cutting and rearrangement of paper tiles to create geometric patterns.
8. A. Bravais, J. Ec. Polytech. 33, 1 (1850).
9. L. Golombek, D. Wilber, in The Timurid Architecture of Iran and Turan (Princeton Univ. Press, Princeton, NJ, 1988), pp. 246-250, 308-309, 384-386, 389, color plates IV, IXb, plates 46, 374.
10. Additional examples of this particular 10/3 decagonal pattern, shown in fig. S1: the Seljuk Congregational Mosque in Ardistan, Iran (∼1160 C.E.) (16); the Timurid Tuman Aqa Mausoleum in the Shah-i Zinda complex in Samarkand, Uzbekistan (1405 C.E.) (9, 16); the Darb-i Kushk shrine in Isfahan, Iran (1496 C.E.) (2, 9, 17); and the Mughal l'timad al-Daula Mausoleum in Agra, India (∼1622 CE.) (25).
11. R. Ettinghausen, O. Grabar, M. Jenkins-Madina, in Islamic Art and Architecture 650-1250 (Yale Univ. Press, New Haven, CT, 2001), p. 109.
12. Additional architectural examples of patterns that can be reconstructed with girih tiles, shown in fig. S3: the Abbasid Al-Mustansiriyya Madrasa in Baghdad, Iraq (1227 to 1234 C.E.) (26); the Ilkhanid Uljaytu Mausoleum in Sultaniya, Iran (1304 C.E.) (17); the Ottoman Green Mosque in Bursa, Turkey (1424 C.E.) (27); and the Mughal I'timad al-Daula Mausoleum in Agra, India (∼1622 C.E.) (28). Similar patterns also appear in the Mamluk Qurans of Sandal (1306 to 1315 C.E.) and of Aydughdi ibn Abdallah al-Badri (1313 C.E.) (29). Note that the girih-tile paradigm can make pattern design structure more clear. For example, all of the spandrels with decagonal girih patterns we have thus far examined (including Fig. 3C and figs. 52 and 53A) follow the same prescription to place decagons: Partial decagons are centered at the four external corners and on the top edge directly above the apex of the arch.
13. A similar convention was used to mark the same girih tiles in other panels (e.g., 28, 50, 52, and 62) in the Topkapi scroll (fig. 54) (2).
14. E. H. Hankin, The Drawing of Geometric Patterns in Saracenic Art (Government of India Central Publications Branch, Calcutta, 1925), p. 4.
15. This pattern type also occurs on the Great Mosque in Malatya, Turkey (∼1200 C.E.) (6), and the madrasa in Zuzan, Iran (1219 C.E.) (30) (fig. 55), as well as on a carved wooden double door from a Seljuk building in Konya (∼13th century C.E.), in the Museum of Islamic Art in Berlin (Inv. Nr. 1.2672).
16. D. Hill, O. Grabar, in Islamic Architecture and Its Decoration, A.D. 800-1500 (Univ. of Chicago Press, Chicago, 1964), pp. 53, 62, 65, plates 38, 276, 346, 348.
17. S. P. Seherr-Thoss, Design and Color in Islamic Architecture (Smithsonian Institution, Washington, DC, 1968), plates 34-36, 40, 84, 90.
18. E. Makovicky, in Fivefold Symmetry, I. Hargittai, Ed. (World Scientific, Singapore, 1992), pp. 67-86.
19. J. F. Bonner, in ISAMA/Bridges Conference Proceedings, R. Sarhangi, N. Friedman, Eds. (Univ. of Granada, Granada, Spain, 2003), pp. 1-12.
20. R. Penrose, Bull. Inst. Math. Appl. 10, 266 (1974).
21. D. Levine, P. J. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
22. M. Gardner, in Penrose Tiles to Trapdoor Ciphers (Freeman, New York, 1989), pp. 1-29.
23. D. Levine, P. J. Steinhardt, Phys. Rev. B 34, 596 (1986).
24. A single figure, part of a geometric proof from On Interlocks of Similar or Corresponding Figures, has been related to the outlines of individual Penrose tiles, but there is no evidence whatsoever for tessellation (31). Makovicky has connected the Maragha Gunbad-i Kabud pattern in Fig. 2 with the Penrose tiling (18), but explicitly states (as we show in fig. 56) that the pattern is periodic, so by definition it cannot be a properly quasi-periodic Penrose tiling.
25. A. Ozdural, Hist. Math. 27, 171 (2000).
26. H. Schmid, Die Madrasa des Kalifen Al-Mustansir in Baghdad (Zabern, Mainz, Germany, 1980), plates 15, 87.
27. G. Goodwin, A History of Ottoman Architecture (Thames and Hudson, London, 1971), pp. 58-65.
28. Y. Ishimoto, Islam: Space and Design (Shinshindo, Kyoto, 1980), plates 378, 380, 382.
29. D. James, Qur'ans of the Mamluks (Thames and Hudson, New York, 1988), pp. 54, 57-59.
30. R. Hillenbrand, Islamic Architecture (Columbia Univ. Press, New York, 1994), pp. 182-183.
31. W. K. Chorbachi, Comp. Math. Appl. 17, 751 (1989).
32. We thank G. Nedpoglu and J. Spurr, without whose multifaceted assistance this paper would not have been possible. We also thank R. Holod and K. Dudley/M. Eniff for permission to reproduce their photographs in Figs. 2C and 3A, respectively; C. Tam and E. Simon-Brown for logistical assistance in Uzbekistan; S. Siavoshi and A. Tafvizi for motivating the exploration of Isfahan's sights; and S. Blair, J. Bloom, C. Eisenmann, T. Lentz, and I. Winter for manuscript comments. Photographs in Fig. 2, A and B, and in the online figures courtesy of the Fine Arts Library, Harvard College Library. Supported by C. and F. Lu and by the Aga Khan Program for Islamic Architecture at Harvard University.