Further notes on quasi-crystal tilings.

We appreciate that Science has clarified its news article (“Quasi-crystal conundrum opens a tiling can of worms,” News of the Week, J. Bohannon, 23 Feb., p. [1066][1]; see Corrections and Clarifications on page [982][2] in this issue) regarding our paper “Decagonal and quasi-crystalline tilings in medieval Islamic architecture” (Reports, 23 Feb., p. [1106][3]). We certainly recognized that our study builds on earlier work, as acknowledged in our references ( 3 -6 , 14 , 18-19 ), and citations therein, although more can be said. Many authors from Hankin in 1925 [( 14 ) in our Report] to Wade ([1][4]), Critchlow ([2][5]), and Kaplan ([3][6]) have related Islamic geometric patterns to configurations of polygons, including some with the same outlines as the decorated girih tiles introduced in our paper. Bonner [( 19 ) in our Report] has applied these ideas to self-similar geometric patterns with five-fold and other symmetries. Makovicky [( 18 ) in our Report], and previously Zaslavsky et al. ([4][7]) and Chorbachi [( 31 ) in our Report], suggested relations between certain historic Islamic tilings and Penrose tilings based on studies of small isolated motifs or fragments embedded within manifestly periodic patterns. We gladly acknowledge all these contributions, which complement our own. However, we wish to emphasize a few distinctions here. First, our approach was founded on the historical record, particularly the Topkapi scroll first understood and published by Gulru Necipoglu (Harvard University), who guided us. Insisting on exact reconstructions of historical monuments resulted in some differences from previous work; for example, our analysis of the Gunbad-i Kabud tomb tower (Figs. 2 and S6), based directly on archival photographs, differs systematically from the transcription used in reference ( 18 ) and reveals plainly the intentional periodicity and regular deviations from a true Penrose tiling. Second, our explanation of these patterns clearly differs from earlier ideas: We propose that historical designers constructed a wide range of patterns by tessellating with the same five units (“girih tiles”) described in our paper, not merely polygons but shapes with specific interior line decorations that form the pattern when the tiles are joined together. Constructing patterns by laying these girih tiles edge to edge this way is simpler than other proposed methods; we have observed young children successfully applying it in the classroom. Moreover, other methods generate many patterns that do not appear historically; by contrast, we presented a series of patterns from historically significant buildings, scrolls, and Qurans throughout the medieval Islamic world that can all be constructed from the same five girih tiles (including their decorations). Third, our analysis of the Darb-i Imam shrine revealed two other novel elements—the explicit subdivision of these girih tiles into smaller girih tiles of the same shape, and a large fragment based on decagonal symmetry that is not embedded in a periodic matrix, properties sufficient to transform the Darb-i Imam shrine pattern into an infinite quasi-crystalline tiling. Our conclusions were guarded, concurring with the remarks by Socolar and Levine in the accompanying news article, suggesting that evidence beyond a single large fragment is needed to prove that the designers understood this possibility. We hope our small contribution, combined with the earlier works, will lead to further explorations of these impressive works of art and mathematics. 1. 1.[↵][8] 1. D. Wade , Pattern in Islamic Art (Overlook, Woodstock, NY, 1976). 2. 2.[↵][9] 1. K. Critchlow , Islamic Patterns (Thames & Hudson, London, 1976). 3. 3.[↵][10] 1. C. S. Kaplan , Graphics Interface Conference Proceedings 177, 186 (2005). 4. 4.[↵][11] 1. G. M. Zaslavsky, 2. R. Z. Sagdeev, 3. D. A. Usikov, 4. A. A. Chernikov , Weak Chaos and Quasi-Regular Patterns (Cambridge Univ. Press, Cambridge, 1991). [1]: /lookup/doi/10.1126/science.315.5815.1066 [2]: /lookup/doi/10.1126/science.316.5827.982a [3]: /lookup/doi/10.1126/science.1135491 [4]: #ref-1 [5]: #ref-2 [6]: #ref-3 [7]: #ref-4 [8]: #xref-ref-1-1 "View reference 1. in text" [9]: #xref-ref-2-1 "View reference 2. in text" [10]: #xref-ref-3-1 "View reference 3. in text" [11]: #xref-ref-4-1 "View reference 4. in text"