Faugal Find an arbitrary point C on the second ray and drop perpendiculars from C meieval the sides of angle A counter-clockwise. International Journal of Islamic Architecture. Steinhardt suggested that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilingspredating them by five centuries. Lu and Paul J. Arabic architecture Berber architecture Iranian architecture Islamic architecture Mughal architecture Ottoman architecture. We show that by C.

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F G Fig. 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. 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.

The girih tiles were then tessellated, with the regular line pattern expressed in large raised brick on the tower and the second set of lines expressed in smaller bricks. The dual-layer nature of line patterns on the Maragha tower thus adds strong evidence that the pattern was generated by tessellating with the girih tiles in Fig. Perhaps the most striking innovation arising from the application of girih tiles was the use of self-similarity transformation the subdivision of large girih tiles into smaller ones to create overlapping patterns at two different length scales, in which each pattern is generated by the same girih tile shapes.

Examples of subdivision can be found in the Topkapi scroll e. A spandrel from the Darb-i Imam shrine is shown in Fig. The large, thick, black line pattern consisting of a handful of decagons and bowties Fig.

We have identified the subdivision rule used to generate the Darb-i Imam spandrel pattern Fig. A subdivision rule, combined with decagonal symmetry, is sufficient to construct perfect quasi-crystalline tilings—patterns with infinite perfect quasi-periodic translational order and crystallographically forbidden rotational symmetries, such as pentagonal or decagonal— which mathematicians and physicists have come to understand only in the past 30 years 20, Quasi-periodic order means that distinct tile shapes repeat with frequencies that are incommensurate; that is, the ratio of the frequencies cannot be expressed as a ratio of integers.

By having quasi-periodicity rather than periodicity, the symmetry constraints of conventional crystallography can be violated, and it is possible to have pentagonal motifs that join together in a pattern with overall pentagonal and decagonal symmetry The most famous example of a quasicrystalline tiling is the Penrose tiling 20, 22 , a two-tile tessellation with long-range quasiperiodic translational order and five-fold symmetry.

The Penrose tiles can have various shapes. A convenient choice for comparison with medieval Islamic architectural decoration is the kite and dart shown on the left side of Fig. For the matching rules, the kite and dart can each be decorated with red and blue stripes Fig. We see no evidence that Islamic designers used the matching-rule approach. The second approach is to repeatedly subdivide kites and darts into smaller kites and darts, according to the rules shown in Fig.

This self-similar subdivision of large tiles into small tiles can be expressed in terms of a transformation matrix whose eigenvalues are irrational, a signature of quasi-periodicity; the eigenvalues represent the ratio of tile frequencies in the limit of an infinite tiling Our analysis indicates that Islamic designers had all the conceptual elements necessary to produce quasi-crystalline girih patterns using the self-similar transformation method: girih tiles, decagonal symmetry, and subdivision.

The pattern on the Darb-i Imam shrine is a remarkable example of how these principles were applied. Using the self-similar subdivision of large girih tiles into small ones shown in Fig. The asymptotic ratio of hexagons to bowties approaches the golden ratio t the same ratio as kites to darts in a Penrose tiling , an irrational ratio that shows explicitly that the pattern is quasi-periodic.

Moreover, the Darb-i Imam tile pattern can be mapped directly into Penrose tiles following the prescription for the hexagon, bowtie 22 , and decagon given in Fig. Using these substitutions, both the large Fig. Note that the mapping shown in Fig.

Therefore, the mapping is completed by using this freedom to eliminate Penrose tile edge mismatches to the maximum degree possible. Note that, unlike previous comparisons in the literature between Islamic designs with decagonal motifs and Penrose tiles 18, 24 , the Darb-i Imam tessellation is not embedded in a periodic framework and can, in principle, be extended into an infinite quasiperiodic pattern. Although the Darb-i Imam pattern illustrates that Islamic designers had all the elements needed to construct perfect quasi-crystalline patterns, we nonetheless find indications that the designers had an incomplete understanding of these elements.

First, we have no evidence that they ever developed the alternative matching-rule approach. Second, there are a small number of tile mismatches, local imperfections in the Darb-i Imam tiling. These can be visualized by mapping the tiling into the Penrose tiles and identifying the mismatches. However, there are only a few of them—11 mismatches out of Penrose tiles—and every mismatch is point-like, removable with a local rearrangement of a few tiles without affecting the rest of the pattern Fig.

This is the kind of defect that an artisan could have made inadvertently in constructing or repairing a complex pattern. Third, the designers did not begin with a single girih tile, but rather with a small arrangement of large tiles that does not appear in the subdivided pattern. This arbitrary and unnecessary choice means that, strictly speaking, the tiling is not self-similar, although repeated application of the subdivision rule would nonetheless lead to the same irrational t ratio of hexagons to bowties.

Our work suggests several avenues for further investigation. Although the examples we have studied thus far fall just short of being perfect quasi-crystals, there may be more interesting examples yet to be discovered, including perfectly quasi-periodic decagonal patterns.

The subdivision analysis outlined above establishes a procedure for identifying quasi-periodic patterns A C D Fig. B Photograph by A. 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.

Also, analogous girih tiles may exist for other noncrystallographic symmetries, and similar dotted tile outlines for nondecagonal patterns appear in the Topkapi scroll. Finally, although our analysis shows that complex decagonal tilings were being A made by C. Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran C. 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.

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. El-Said, A. Abas, A. 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.

Bravais, J. Golombek, D. Press, Princeton, NJ, , pp. Ettinghausen, O. Grabar, M. Press, New Haven, CT, , p. Additional architectural examples of patterns that can be reconstructed with girih tiles, shown in fig. Similar patterns also appear in the Mamluk Qurans of Sandal to C. 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. S2 and S3A 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.

A similar convention was used to mark the same girih tiles in other panels e. A and B The A kite A and dart B Penrose tile shapes are shown at the left of the arrows with red and blue ribbons that match conC tinuously across the edges in a perfect Penrose tiling.

F Mapping of a region of small girih tiles to Penrose tiles, corresponding to the area marked by the white rectangle in Fig. F 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.

Makovicky, in Fivefold Symmetry, I. Hargittai, Ed. World Scientific, Singapore, , pp. Sarhangi, N. Friedman, Eds. Penrose, Bull. Levine, P. Steinhardt, Phys. B 34, 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 Makovicky has connected the Maragha Gunbad-i Kabud pattern in Fig.

S6 that the pattern is periodic, so by definition it cannot be a properly quasiperiodic Penrose tiling. Ozdural, Hist. Hillenbrand, Islamic Architecture Columbia Univ. Press, New York, , pp. Chorbachi, Comp. We thank G. Necipoglu and J. Spurr, without whose multifaceted assistance this paper would not have been possible. We also thank R. Holod and K. Eniff for permission to reproduce their photographs in Figs. Tam and E. Simon-Brown for logistical assistance in Uzbekistan; S.

Siavoshi and A. Blair, J. Bloom, C.

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