Sunday, 26 April 2009

Abstract Art

For some time, the Gonbad-e Qabud in Maraghah in Western Iran has attracted considerable attention. Maraghah is a small city east of Daryacheh Urumiyeh in the East Azerbaijan province of Iran. It lies about 100 km south of Tabriz close to the southeastern shores of the huge super-salty lake at the southern foot hills of 3700 meters high Kuh-e Sahand. On the other side of the mountain lies the picturesque village of Kandovan, Iran’s Cappadocia [1].

Maraghah is quite famous for its five tomb towers (four are preserved) from the Post-Seljuq and Mongolian periods (12th till early 14th centuries). Gonbad-e Qabud, the Blue Tower (1196/97), has the most elaborated and complex brick pattern which has fascinated and confused generations of explorers and tourists. It represents an octagonal tower with eight panels each crowned by a niche with a pointed, gothic, arch. The brickwork results in highly ornamental net of unglazed ribs interlaced with turquoise blue ribbons unrelated to the pentagonal geometry of the overall pattern. It can be shown that the pattern extends over two panels and therefore repeats four times.

Almost hidden in a book about Fivefold Symmetry edited by István Hargittai (World Scientific, Singapore 1992) which compiles very interesting articles on all aspects of fivefold symmetry, mineralogist Emil Makovicky at Copenhagen University has argued that the incredibly complex brick pattern which is displayed on the eight panels of the octagonal tower may in fact represent a Penrose pattern [2]:

“Aperiodic tiling with pentagonal geometry, discovered by Penrose [in 1974, 1978], have been, in its different versions, the object of intensive study by numerous mathematicians and crystallographers. The present discovery of a similar, 800-year-old tiling from (post) Saljuq Iran therefore represents a matter of considerable interest. Besides giving a surprising insight into the skills of ancient geometric artists, it also reveals some new aspects of Penrose tiling and leads toward further generalizations.”

Makovicky correctly describes the large-scale pattern of the Gonbad-e Qabud as consisting of:

“[…](a) regular pentagons; (b) complex decagons, hereafter called butterflies with convex angles of 72° and reentrant angles of 108°: (c) deltoids (“kites”) and a pair of partly overlapping pentagons that always form together a rhomb with “deltoid-marked” corners of 72° and unmarked corners of 108°; and (d) occasional nested pentagons with five spokes. “

What follows are combination rules, described as “simple”:

“[only] straight-line segments of the net intersect (at 72°), whereas all line breaks (of 108° or 144°) are outside these intersections. Polygons of the same kind do not share edges. Butterfly wings terminate in pentagons and are surrounded either by four additional pentagons or by an additional cis pair of pentagons and a cis pair of rhombs (each straddling the long diagonal).

“The entire pattern is too complex to be understood at a glance. It requires long contemplation, and almost appears to be designed by a mathematician rather than an artist. Its badly damaged lowermost portions can be safely reconstructed because of the good state of preservation of the corresponding uppermost portions.

However, “[in] a small part of the bottom portions of the pattern the artist gained the upper hand over the mathematician. The tenfold stars, which can be traced in the polygonal net on both sides of the partly overlapping nested pentagons at the bases of the corner pilasters […] were emptied of their original polygonal contents and were filled by fivefold 'rosettes.' Eye-attracting rosettes of this kind are common in Islamic wall ornaments, but those used here (only once per each side of the building) are completely foreign to the rest of the pattern.”

After his lengthy analysis of the pattern on the Gonbad-e Qabud, Makovicky concludes that it is “[b]ased on tiles that can readily be obtained by transformation of the Penrose pattern of pentagons, stars, and lozenges. It deviates from a true cartwheel Penrose tiling only in several geometric and artistic adaptations.”

No Penrose tiling

As a matter of fact, the pattern on the Gonbad-e Qabud lacks any characteristics of a Penrose tiling. First and most eminent, it is not aperiodic. And secondly, it does not implement a self-similar subdivision. The small-scale pattern seen is unrelated to the large-scale major pattern [3].

A simple method how the medieval artists (and it can be argued that in that particular case not even a mathematician was involved in the process of decoration) has been suggested by Lu and Steinhardt [4]. They discovered on what is called now the Topkapı Scroll [5], a 15th century Timurid-Turkmen scroll now in the collection of the Topkapı Palace Museum in Istanbul, that most of the highly complex geometric patterns found on buildings and paintings in the Islamic world can be created seamlessly with the aid of a set of five tiles displaying well-defined decorative ribbons, a decagon, a pentagon, an elongated hexagon, a bowtie, and a rhombus, which they called girih tiles which “[share] several geometric features: every edge of each polygon has the same length and the two decorating lines intersect the midpoint of every edge at 72° and 108° angles. This ensures that when the edges of two tiles are aligned in a tessellation, decorating lines will continue across the common boundary without changing direction. Because both line intersections and tiles only contain angles that are multiples of 36°, all line segments in the final girih strapwork pattern formed by girih-tile decorating lines will be parallel to the sides of the regular pentagon; decagonal geometry is thus enforced in the girih pattern formed by the tessellation of any combination of girih tiles. The tile decorations have different internal rotational symmetries: the decagon, 10-fold symmetry; the pentagon, five-fold; and the hexagon, bowtie, and rhombus, two-fold” [4].

Lu and Steinhardt reconstructed the pattern on the Gonbad-e Qabud with four girih tiles. I have followed the suggestion by Makovicky and have not included a decagon “rosette”.

The Maraghah pattern compared with the decagonal pattern on the west iwan of Esfahan’s Great Mosque

Another suspected site displaying allegedly a “quasi-crystalline” pattern of tesserae is the western iwan of Masjed-e Jomeh in Esfahan. The reconstruction revealed that it is not a Penrose tiling. The “dazzling” appearance turns out to be largely a rosette which can be constructed by use of a set of four girih tiles. There is no self-similar subdivision. In a way, it resembles a bit the pattern found in Maraghah, although there, some irregularities occur, as described above.

The artists who have created the decorations at either site (1197 in Maraghah, mid of the 15th century in Esfahan) did not use color but chose a high degree of abstraction. It is amazing that an intentional reduction of a piece of art to a strict geometric pattern with an unbelievable degree of precision has led to profound confusion among a large number of visitors. The perception of the artistic effort in fact confused even certain scientists who argued that medieval artists could have discovered what became famous as Penrose patterns, 500 or even 800 years before they were described and understood in the West.


[1] I have posted some pictures about trips in and around Tabriz on Salmiya.

[2] Makovicky E. 800-year-old pentagonal tiling from Marāgha, Iran, and the new varieties of aperiodic tiling it inspired. In: Istvan Hargittai (ed.) Fivefold Symmetry. World Scientific, Singapore 1992, pp. 67-86.

[3] See Lu and Steinhardt’s response to Makovicky’s comment on their paper at Science 2007; 318: 1383b.

[4] Lu PJ, Steinhardt PJ. Decagonal and quasi-crystalline tilings in medieval Islamic architecture. Science 2007; 315: 1106-1110.

[5] Necipoglu G. The Topkapı Scroll: Geometry and Ornament in Islamic Architecture. Getty Center for the History of the Art and Humanities. Santa Monica, CA, 1995.

First published at Freelance.

Thursday, 9 April 2009

Dazzling Decagonal

The most interesting tile decorations and muqarnas, or stalactite vaults, are found on the western iwan of Esfahan’s Great Mosque. While all iwans have been added to the Seljuq mosque after a fire pillaged by the Hashashiyyin sect in 1121 CE, their decorations are Timurid and early or even late Safavid (late 15th till early 17th century). Next to the western iwan the pretty famous Timurid gate had been moved and inserted into the façade. It contains signature and date of its creator Sayyid Mahmud-e Naqash, 1447. A similar, highly decorative floral style can be seen on the south iwan and on the Darb-e Imam, some 300 meters west to the mosque, which is dated 1453.

The date ۱۳۱۷ (1317) translates into 1939, by the way, when restoration had taken place. The Timurid gate near the western iwan of Masjed-e Jomeh leads to a room with a stunning dated (1310) mihrab of sultan Oljatu, the great Ilkhanid Mongolian ruler in northern Iran. The inscriptions are, according to Oleg Grabar in his book about the Great Mosque, not qur’anic, but contain traditions about mosques and about Ali. Amazing that Oljatu in fact converted to Shi’a Islam in 1310.

The western iwan and its counterpart to the east are called the sofe of the student (shāgird) and master (ustadh). Although both iwans were built at the same time as the south iwan (early 12th century), both of them are, “in their visible shape, late Safavid works of the seventeenth and, in case of the west one, even early eighteenth centuries”, as Grabar writes.

“[A] celebrated square panel in the western iwan [which] is one of the most commonly cited examples of complex geometric ornament using writing. It is easy to argue that here is a wonderful example of a simple design rotated 45 degrees which acquires two separate values, one as a carrier of geometric forms filled with (by the time of the panel) antiquarian writing, the other one as a violator of the sequence of both writing and architecture by forcing one into rare contortions to read the writing. And one could argue that here is precisely the use of geometry which gives it the high status so frequently heard and read about. In fact, however, the corner spaces contain the following rather undistinguished pious quatrain: ‘As the letter of our crime became entwined [i.e., grew so long], [they] took it and weighed it in the balance against action. Our sin was greater than that of anyone else, but we were forgiven out of the kindness of Ali.’ The central square is taken up by a signature of one of the most active craftsmen busy repairing the mosque in the seventeenth century. Even though formally related to the angular style of writing on the face of the iwan and in fact much more sophisticated in design, this panel is nothing more than a ‘plug’ for a local artisan.”

The exact construction of a similar “square from three squares” has been described in Abu’l Wafa’s (d. ca. 998) book “On the Geometric Constructions Necessary for the Artisan”. As Alpay Özdural describes it in his article “Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World” (Historia Mathematica 2000; 27: 171-201), contemporary mathematicians frequently held so-called conversazione with artisans explaining them how to create new inspiring geometric decorations.

Now let’s turn to what I've called “dazzling decagonal”. I have reported on my stunning first experience with mysterious decagonal tessellations in Esfahan’s old city several times, here on this blog as well as on Freelance. There are suggestions by Peter Lu at Harvard that there had been a breakthrough in creating (almost) Penrose tiling in the 15th century, in particular on the Darb-e Imam near Esfahan’s Great Mosque. In the supplementary material of Lu and Steinhardt’s article, you may find a picture of the western iwan where the authors suggest that the tiling can be subdivided in the same way as the Darb-e Imam pattern(s). You can easily identify the pattern at the inner sides of the iwan’s portal. It is huge, about one meter wide and up to 10 meters high. At first glance the two sites seem to be an anomaly in Esfahan. Lu and Steinhardt also suggest so-called girih tiles to facilitate the incredible precision of the tiling.

For instance, I have mirrored the right part of a picture of the arch borrowed from ArchNet (left part of the panel below) and can demonstrate (right part of the panel) that each tiny tessera on one side (as small as, say, a square centimeter) can be found in exactly the same place on the other side of the vault.