Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture

We thank Mr. Cohn for his E-Letter and for pointing us to the charming book, The Language of Pattern. We note, though, that this book is described by its authors as a study of pattern-making inspired by Islamic design, and it does not provide a natural explanation of how the Vedic square, an isolated motif, can lead to a large quasi-crystalline fragment. To be sure, the book's purpose is aesthetic, whereas our purpose was to reveal the actual construction procedure used by Islamic designers at the time. Both viewpoints contribute to the appreciation of these remarkable designs.

Peter J. Lu

Department of Physics, Harvard University, Cambridge, MA 02138, USA.

Paul J. Steinhardt

Department of Physics and Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544, USA.

We certainly share Dr. Cintas' admiration for the magnificent periodic and spiral patterns found at the Alhambra and other Islamic sites, which demonstrate a very high degree of aesthetic and geometric sophistication. In fact, as discussed in our paper, girih tiles were commonly used as a method to construct very complex periodic patterns, such as the Gunbad-i Kabud in Maragha, Iran. Periodic patterns, however, including the use of all 17 plane groups, can be traced back much further back in time: Hermann Weyl, in his wonderful 1952 book Symmetry, pointed out that all 17 groups were exploited in ancient Egyptian ornamentation (1).

The large tile fragment at Darb-i Imam is remarkable because it breaks from this tradition. Its tile edges all lie along tenfold symmetry directions, a symmetry that is incompatible with periodicity, the fragment is not embedded within a larger periodic pattern, and the pattern exhibits explicitly a subdivision rule—key elements sufficient to construct an infinite quasicrystal pattern. Our conclusions were measured, though. Contrary to the suggestion in the accompanying News article (23 Feb., p. 1066), we did not claim that the designers recognized the meaning of quasi-crystals or demonstrated a clear understanding of how to extend a quasi-crystal pattern indefinitely. Rather, as Dr. Cintas suggests, we concluded that more proof, such as written design instructions or more extensive patterns, is needed. Nevertheless, the Darb-i Imam pattern and the elements it incorporates are impressive because they bring Islamic design to the threshold of generating patterns that were not produced or understood in the West until 500 years later.

Peter J. Lu

Department of Physics, Harvard University, Cambridge, MA 02138, USA.

Paul J. Steinhardt

Department of Physics and Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544, USA.

References

1. H. Weyl, Symmetry (Princeton UP, Princeton, 1952), p. 103.

After reading the appealing article by P. J. Lu and P. J. Steinhardt on quasi-crystalline tilings in Islamic architecture (23 Feb., p. 1106), I could not resist introducing a few considerations. The authors state that there was an important conceptual breakthrough in Islamic mathematics and hence in architecture by the 13th century—girih tiles—that evolved into more sophisticated quasi-crystal arrangements by the 15th century. In contrast to this dazzling discovery of five- and ten-fold forbidden patterns, the rest of the symmetries are considered as the mere repetition of a single unit cell. These views, although plausible, are rather historically and scientifically inaccurate (Islamic design was mature enough at least two centuries earlier). It is dubious that Islamic artisans truly understood the meaning of quasi-crystals as noted by other scholars and quoted by Bohannon in the accompanying News of the Week article (23 Feb., p. 1066).

Since human figures as well as the exact representation of nature are forbidden by Islam, the Muslims mastered geometry and symbolic design, and by means of mathematical concepts, including trigonometry and algebra, they were able to identify all the regular patterns that can be put together without gaps on a flat surface. The paradigmatic example is maybe provided by the Alhambra palace of Granada, built mainly between 1238 and 1358 under the Nasrid dynasty. A mathematical study of its ornamentations reveals that they exactly match the 17 plane groups possible for all two-dimensional patterns (1). Such arrangements are generated by combining all the translations inherent in the five plane lattices with the symmetry elements found in the ten crystallographic plane point groups along with a glide element. In a logical extension, perhaps not so aesthetic and less exacting than those producing a regular tiling, it is also possible to create periodical tesselations using equilateral, not regular, pentagons, which will tile the surface by translation alone (2). Nonperiodic tesselations are not unusual in Islamic art (e.g., spiral tilings that cannot be created by the repeated translation of a unit cell of the pattern); in fact, they can also be observed in the Alhambra arabescos, although they do not fit a Penrose tiling, the only one with relevance to crystallography. Thus, the variety of the geometric tesselations covering the wall of Moorish buildings appears to be more consistent with the search for extra vistas of beauty, rather than inspired by a sudden mathematical discovery. As an additional feature, tilings in the Alhambra are also reminiscent of the superlattices into which some nanoparticles self-assemble (3). But it would be exaggerated to claim that Islamic scholars intentionally created such a pattern. The interesting lesson is that, after all, science and art may be extraordinarily close.

Pedro Cintas

Department of Organic and Inorganic Chemistry, University of Extremadura, E-06071 Badajoz, Spain

References

1. A. F. Costa Gonzalez, B. Gomez Garcia, Arabescos y Geometria (UNED, Madrid, 1995).

2. R. Tilley, Crystal and Crystal Structures (Wiley, Chichester, 2006).

3. I. Amato, Chem. Eng. News, 21 Aug. 2006, p. 45.

On reading P. J. Lu and P. J. Steinhardt's article "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture" (Reports, p. 1106, 23 Feb. 2007), I was immediately reminded of a Christmas gift presented to me in 1975 by my dear departed friend, the London architect Colin Dollimore, of a small book titled, The Language of Pattern by Keith Albarn, Jenny Miall Smith, Stanford Steele, and Dinah Walker (published by Thames and Hudson, 1974).

This book introduced me to the world of Islamic decoration, with particular focus on the mathematical structure of its pattern formations, the Vedic square. Not only did this mathematical model and the accompanying line drawings explain the seemingly intractable geometries, but—according to the authors—the Vedic square served a higher purpose, as a numerical model of the universe. The pattern could be viewed in cosmological terms; that is, not merely as decoration, but as serving a purpose. According to their thesis, one becomes aware of the whole (in essence, the cosmos) by understanding that the pattern grows into something larger, all based on the same mathematical construct, and conversely, one can abstract conceptual subsets from this organizational whole. Think one step beyond fractals, to understand a philosophical purpose of mathematics, pattern, and decoration. William Blake summarized it best, "To see a world in a grain of sand, And a heaven in a wild flower..."

The diagrams in "Pattern" illustrate how girih tiles and quasi-crystalline Penrose patterns may have been constructed using a Vedic square template. If that is the case, it would explain how 15th century craftsmen could create such intricacies, and the Vedic square being incorporated by Muslims into their own culture before 800 AD. It may be of interest to you, and to your readers, to learn more about this earlier study, and the fascinating mathematics of the Vedic square that seem to explain the beautiful complexities of Islamic pattern.

Ian J. Cohn

Architect

In our paper, we demonstrated that a sequence of Islamic patterns ranging from simple to complex can all be obtained by taking the same set of five line-decorated polygons (girih tiles) and close-packing them in different configurations. Other constructions methods are possible, in principle; several suggestions include those in the literature referenced in our paper (refs. 3-6, 14, 18-19), and the one presented by Dr. Aboulfotouh, in which one first constructs a hidden grid, decorates it with lines, and then removes the original grid. So, we appreciate Dr. Aboulfotouh's suggestion.

For several reasons, though, we believe the girih tile approach is more likely to be the construction procedure used historically. First, close-packing of repeated tiles is simpler: We have observed young children putting together our girih tiles to make impressively complex patterns, whereas the grid construction described above clearly describes more training and skill. As a practical matter, the girih tile approach is also easier to adjust as one goes along so that errors do not build up across the pattern. Second, the hidden grid approach is too general: It produces an enormous range of patterns, most of which cannot be resolved into tessellations of girih tiles. The tessellation approach is strongly suggested by the historical record: Namely, many distinct patterns on buildings, in Qurans, and in architectural scrolls can be resolved (as we demonstrated in our paper and supporting online material) into girih tile tessellations, and the Topkapi scroll contains the outlines and decorations of our five specific tiles. Moreover, some patterns explicitly highlight the girih tile decomposition in relief with outlines, or with colors corresponding to the internal girih-tile decoration (e.g., the I'timad al-Daula Mausoleum in Agra, India, in Figs. S1D and S3D). The complex fine-scale decoration on the Gunbad-i Kabud in Maragha, Iran, is especially difficult to obtain by a grid approach because it consists of curvilinear segments in a nonpentagonal pattern; yet, in the girih tile picture, it corresponds simply to adding a specific fine decoration to each tile type. Finally, on the Darb-i Imam shrine, the explicit subdivision of large girih tiles into smaller ones strongly supports the idea that the pattern was conceived as a tessellation.

Peter J. Lu, Department of Physics, Harvard University, Cambridge, MA 02138, USA.

Paul J. Steinhardt, Department of Physics and Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544, USA.

In their Report "Decagonal and quasi-crystalline tilings in medieval Islamic architecture" (1), Peter J. Lu et al. suggest that the decagonal girih patterns on the Darb-i Imam shrine are quasi-periodic and were constructed by tessellation, using a set of five tile types. Contrary to the approaches of the mathematicians towards understanding the process of designing and implementing these patterns, architects design them manually; using only a scale, T-square (2), and various types of triangles, and produce its working drawings on 1:1 scale for the artisans. Since these patterns don't include circular curves, they draw them swiftly without using the slow-compass. For the decagonal stars (1, 3, 4), two triangles were used instead: 18°/72° and 36°/54°, which are not produced today, but the triangle of various angles is an alternative. To draw a decagonal-star (Fig. 1A to D), if x is the radius of its inner circle that the 10 sides of the inner decagon are its tangents, we draw a square that its side equals 4x, and draw the perpendicular grid 4x*4x inside it. Using the triangle 18°/72°, and starting from the "center of the square," we draw the second but inclined grid on x intervals, and its tilt equals 72° on both sides. Similarly, using the triangle 36°/54°, we draw the third but inclined grid on x intervals, and its tilt equals 36° on both sides. Then, we can observe the perimeters of the decagonal-star and its surroundings, i.e., one of the multiple design outputs of the three hidden-grids. If we repeat the 4x*4x unit vertically, we produce a vertical strip of repeated stars, every 4x. On the horizontal direction, there are various options. While the horizontal grid repeats on 4x intervals, the vertical grid may follow various continuous rhythms, e.g., 4x-1.5x-4x-1.5x or 1.4x-2x-1.4x- 2x-1.4x. The rhythm 4x-1.5x generates a fifth decagonal-star in the middle of each four stars, creating the stagger-shape (Fig. 3C), within the repeating thematic-unit of 8x*9.5x. The rhythm 1.4x-2x generates the grid-shape (Fig. 1E) and the repeating thematic-unit of 4x*3.4x. The shown girih patterns are small portions of repeated thematic-units, that its vertical but hidden grid-sides were rotated, inclined, and/or located outside the domains of the design-motifs (Figs. 1G and 2C), and the inner-grids were subdivided in order to design the second and third level of inner-details (Fig. 3A). Therefore, designing and implementing these patterns without tessellation were and still are not difficult tasks, and architecturally the term quasi-periodic is valid only within the hidden domain of the thematic-unit of the decagonal girih patterns.

Hossam Aboulfotouh,

Assistant Professor,

Department of Architecture, Faculty of Fine Arts, Minia University, Egypt

References

1. P. J. Lu, P. J. Steinhardt, Science, 315, 1106 (2007).

2. Encyclopedia Britannica, Drafting practice and equipments, pp.12-15 (2007), http://www.britannica.com/eb/article-59495/drafting#213051.hook.

3. J.-M. Castéra, Arabesques, Decorative Art in Morcco, ACR Edition, (International Courbevoie, Paris, 1999), pp. 250-256.

4. Y. Korbendaq, L'Architecture Sacrée de L'Islam, ACR Edition, (International Courbevoie, Paris, 1997), p.177.