In the Tower of Babel

 

 

 

 

 

 

 

Those who have studied Islamic art and architecture for some time inevitably have asked sooner or later the following questions: How did they do that? Apart from the application of fundamental principles in geometry, how could they create most sophisticated and highly complicated geometric designs over extended areas in this stunning precision? And then, why did Muslims in the Golden Age of Islam do that? Who had taught them, and how? Where are the books and manuscripts? When and on what occasions met and collaborated  scientists and artists in Islamic civilization?

In the early 1970s these simple questions struck a young and extraordinary talented Iraqi lady with a strong background in history and historiography when she searched for a suitable topic for a doctoral thesis at Harvard [1]. These questions weren’t obvious at that time. When Wasma’a Chorbachi had explained her preliminary proposal and her desire of finding the relevant literature which had obviously been lost during the centuries, she was rather quickly turned down. Her advisor expressed his strong opinion that there was not such a thing. There had never been. His good advise was rather to expand her list of questions in order not to fail, for instance, including questions such as: Has the interest in science or geometry been part of the average cultured person’s background in the ninth or tenth century? What practical geometry had been developed by the tenth century? What caused the growth of this phenomenon? Geographically, where did it begin and in what directions did it spread?

 

A Needle in the Haystack

Wasma’a started her search taking advantage of the extensive resources of the Harvard library system. She read through catalogues and indices of manuscript collections available in libraries throughout the world. By the end of the week she had come across Kamāl al-Dīn Yūnis bin Man’a, one of the most outstanding teachers at the main school of the early 13^th century in Mosul, Iraq (which has later been named after him, al-Madrasah al-Kamālīyah, [2]). Among his work was a commentary on an earlier work of one of the most eminent mathematicians and scientists of the Islamic world of the 10^th century, Abū’l-Wafā al-Būzjānī. He lived in Baghdad from approximately 945 CE until his death in about 987 CE. The transliterated title of the main work was also more or less the title of Wasma’a’s PhD project: “A treatise on what the artisan needs of geometric problems”, while the title of Kamāl al-Din Yunis’ commentary was “Commentary on the geometry problems.” Thus, by the third week of her search Wasma’a Chorbachi had already been successful in achieving her first aim: to locate the relevant literature as regards the teaching of medieval artisans of the Islamic world by scientists.

Wasma’a’s next step was to travel to Europe and find and read the original manuscripts, in the Victoria and Albert Museum in London and the Bibliotheque Nationale in Paris where she had located a Persian translation of Abū’l-Wafā al-Būzjānī’s manuscript of the “Treatise on what the artisan needs of geometric problems.” In Paris, she found an unnamed, undated manuscript probably from the 14^th century which clearly was of significantly greater importance than Abu’l Wafa’s work: “On interlocking similar and congruent figures.” Wasma’a writes:

“By the time I returned to Cambridge, I had located a range of written material, in the history of Islamic science and geometric design from the tenth century of the mid-nineteenth century, lying in library and museum storage rooms all over the world. In point of fact, my material turned out to be so convincing that it is now being used and propagated even by those who demonstrated such a strong sceptical attitude towards it at the beginning. Though locating the manuscripts took only two months, acquiring microfilms and/or photocopies of these documents without any backing or support took several years. Meanwhile; I was struggling to decipher the material, and to find an appropriate language in which to discuss it and describe the geometrical patterns with which it dealt.”

 

Confusing Language

Studying the right language (while noticing that different people with different background will describe what they see by using different terminology) took years for Wasma’a. It foremost included Group Theory, Crystallography and Symmetry Notation, fields with which historians and art historians are not really familiar per se. Wasma’a strictly applied scientific reasoning, though. It is interesting reading her rebuttal of ‘esoteric’ reasoning in explaining the ‘meaning’ in Islamic art which became most popular in the mid 1970s. According to proponents, the “principle of the unity of being’ was even “pushed to a point of scientific fallacy such as the claim that all geometric patterns of Islamic art are derivable through a single method of construction based on the subdivision of the circle, in order to declare this art work an example of the “Unity of Being”. ” Divine Unity, or Tawhīd, as the driving force for geometric patterns. That didn’t make sense in her opinion.

“The general public unfortunately remains unaware of this. If in these books, that are now readily available on the market, their authors had made clear that the presented views were modern understandings of old forms, turning them into symbols, there would be no reason to object. The problem lies in presenting these modern mystical views as historical truths, as if these symbols were the meanings at the time the art forms were created. The non-Islamicist who is exposed to these books [for example, I. El-Said’s Geometric Concepts in Islamic Art; L. Bakhtiar’s Sufi: Expressions of the Mystic Quest] will anachronistically assume that a modern interpretation is the historical truth. Where does one draw the line between true historical research and the creation of and attribution of symbolic meaning to forms from the past? How can we redeem the geometric shapes, forms and patterns from the shrouds of mystical interpretations in order to see the precise scientific design at their basis?” 

Describing the visual perception and linguistic or even fashionable semiotics further served only to confuse the interested layman in particular in the 1970s [3].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In a comprehensive case study Wasma’a Chorbachi deconstructs one of several amazing brick pattern on one of the two Seljuq Kharraqan tomb towers (1093 CE) in the vicinity of Qazvin in northern Iran which consists, at first sight, of V-forms, X-forms as well as dots, but which, at second sight, comprises an extremely popular geometric structure, a square within a square within another square. I have described this pattern, which can be found, for instance, several times on the western and southern iwans of Esfahan’s Great Mosque [4], and how it may be created in another posting on this blog. It’s construction in five steps had been described in a systematic, scientifically correct, way in the above mentioned, unnamed, undated Paris manuscript No. 169 “On interlocking similar and congruent figures”, Wasma’a had been working on.

What follows is another case study of the Persian manuscript folio 192b about a similar structure of a kind of pinwheel which fascinates “in its use of a strict algorithm with irrational numbers.” It shows how the principles may lead to different designs which probably have been considered from a pure esthetic point of view.

“The science of symmetry of patterns tell[s] us that there are 17 different periodic two-dimensional groups and 7 groups periodic in a singular direction (string or ribbon), also that each of these groups could have an infinite number of different designs. Ad seen, these Islamic geometric manuscripts give us samples of the infinite design variations of the basic 17 periodic groups; these documented geometric problems or examples in turn could be the basis for developing  many new sets of design.”

See Dr. Wasma’a Chorbachi homepage here.

 

Notes

[1] This posting is about a remarkable text by Wasma’a K. Chorbachi which was based on two lectures given at MIT, Cambridge, in November 1987 and had been published in Computers Math Applic 1989; 17: 751-789: In the Tower of Babel: Beyond symmetry in Islamic design. It deals with a lot of questions which I have asked myself (and many others) since I became fascinated of Islamic art and architecture in recent years.

[2] Despite his Arabic name, Wasma’a’s advisor considered Kamāl al-Dīn Yūnis a member of the Nestorian Church which had been revived in Iraq in the 12^th century. Dr. Chorbachi explains her dismay with considerable prejudices as well. I suppose it is not entirely correct that the annoying response of her supervisor reflected a general ignorant attitude towards the achievements of the Islamic world in the West after WWII, as she describes it. Ignorant supervisors are frequently found in Academia, even at Harvard. It might in fact be the case that in particular Americans are in essence Eurocentric. Not to forget that the 1970s were a decade of great technological and scientific achievements mainly coming from the US, which were very much occupied in proxy wars of the Cold War, for instance in Vietnam. Islamic art and architecture may not have been regarded a fruitful field where scientific breakthroughs had to be expected. In any way, Wasma’a continued her search and found quite a lot of information about Kamāl al-Dīn Yūnis. I have to admit that in spite of considerable search of the internet, I could not identify the scholar yet.

[3] Mystic interpretations of Islamic geometric patterns are still prevalent in many esoteric circles in the West. When trying to talk about new discoveries or searches, for instance, the search for quasi-crystalline patterns, one generally faces incomprehension among people with a general interest in Islamic art and art historians. The “meaning” of the stunning patterns is of greater importance than the question, how could it be created. And whether it has been chosen for esthetic reason only.

[4] Interestingly, Wasma’a mentions 1122 CE as construction date of the iwans, i.e., after Assassin rebels had set the mosque on fire in 1121. She also mentions that the iwans were re-decorated in 1800. In fact, restoration and repair of the structures and tessellations constantly takes place. The celebrated decoration of, for instance, the western iwan is usually considered to be Timurid (15^th century) or Safavid (16^th and 17^th century).

See ArchNet for further pictures of the two Kharraqan tomb towers.

Polygons

“He who knows not and knows not that he knows not, shun him. And he who knows not and knows that he knows not, awaken him. And he who knows and knows that he knows, follow him.”

Arabic saying

The swastika has nowadays a bad reputation but it has of course not been invented by German Nazis. Rather it is a positively connoted, sacred symbol in Hinduism and Buddhism, such as lucky charm. It is interesting to see that it has also found its way into Islamic Art, even as a sign of blessing. A famous square panel on the western iwan of Esfahan’s Great Mosque dating from the 17^th century (Shi’ite Safavid) resembles a Swastika, and its calligraphy mentions Ali [1]. It might be a beautiful 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” [2]. The southern iwan which had got additional decorations by Sayyid Mahmud-e Naqash in 1475/76 sports a similar but definitely Timurid swastika-like panel, with its ample arabesque and floral motifs [3].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A Square from Five Squares

These examples are not strict swastikas. Rather, they represent a popular Islamic geometric pattern, a square composed of three squares. In the 10^th century, artisans were thoroughly taught in a distinct academic context by mathematicians in geometry. Alpay Özdural (d. 2003) describes [4] how, for instance, Abu’l-Wafā’ al-Būzjāni (940- ca. 998), in his famous treatise Kitāb fīmā yahtāju ilayhi al-sani’ min al-a’māl al-handasiya (On the Geometric Constructions Necessary for the Artisan) teaches the right way of constructing this very combination of squares and avoid often made mistakes of the carpenter whose job involved cutting single pieces of material into parts and arranging them skillfully in attractive patterns in mosaics. Abul’l-Wāfa explains that artisans and even geometers (muhandis) often err in the assembling of the pieces, the former since they do not know the scientific proof, the latter due to lack of practice. As Özural writes, Abu’l-Wāfa’s book on Geometric Constructions was apparently motivated by meetings with practitioners and aimed in the proper advancement of Islamic Art. As a true academic, he displayed, in his book “pure geometry, familiarity with practical applications, and skill in teaching theoretical subjects to practical-minded people.”  

The figure below (from Özdural’s article) shows how, by cutting and pasting two, five and nine squares, according to Abu’l Wāfa’s theoretical solutions [5], pretty attractive patterns are created. The earliest “square from five squares” can be seen on the wooden door of the mosque of Imām Ibrāhīm in Mosul which is dated 1104 CE. And Abu’l-Wāfa also explains patiently why some popular ‘practical solutions’ were essentially wrong.

 

 

 

 

 

 

 

 

 

While between the 11^th and 15^th centuries in Iran and Central Asia, Spain and elsewhere in the Islamic World, geometric tessellations became more and more ambitious, dazzling, breakneck artistic, it is not clear how much artisans actually knew about geometry and mathematics. Özdural’s paper convincingly shows how academics such as Abu’l-Wāfa in Baghdad or later Omar Khayyām in Esfahan and Jamshīd al-Kāshī in Samarqand frequently met with artisans, architects, masons and carpenters in what he calls conversazione, i.e., seminars and practical sessions, where the then popular cut and paste technique of dividing larger material into smaller pieces was exercised and got a sound theoretical foundation. While the Golden Age of Islamic Science and Art before and around 1000 CE, in particular Persia, was brutally brought to an end by Mongol invasions after 1220, with catastrophic destruction and by and large architectural inactivity for several decades, later-on, during Ilkhanid, Timurid, and even Ottoman periods, scholars again took over in assisting those who created the most incredible geometric and arabesque tessellations. But they still noted lack of knowledge and unwillingness of master-builders to entirely rely on geometric proof but rather dealt “with geometry in their unmethodological and incorrect way three centuries after Abu’l-Wāfa.” “Yes, we have heard of it, but in essence we have not heard how science of geometry works and what it deals with.”

 

Pentagons and Decagons

Especially fascinating may be the way, artisans had tried to use pentagons and decagons in their tessellations. There have even been speculations, at least since the late 1980s, whether medieval Islamic artists had been able to create aperiodic tiling, such as those which had been described by Roger Penrose in the 1970s.

 

 

 

 

 

 

 

 

 

In studying the probably 13^th century manuscript by an anonymous author, Fī tadhākul al-ashkāl al-mutashābihah aw al-mutawāfiqa (On Interlocking Similar or Congruent Figures), which is now located in the Bibliotheque Nationale in Paris, Wasma’a K. Chorbachi and Arthur L. Loeb [6] point to the similarity of the here described problem of interlocking convex decagons and pentagonal stars (the Islamic Pentagonal Seal) with those being now popularly known as aperiodic Penrose Tiling [7].

 

 

 

 

 

 

 

 

 

In this manuscript one may find an interesting ‘practical’, albeit incorrect, solution for creating regular decagons and pentagons by cutting and pasting the kunya-5 triangle, a right-angled triangle with one angle equal to 36°. The approximation differs from 36° by only 12’22’’, i.e., 0.5% [8].

 

 

 

 

 

In particular in the 13^th century, the golden triangle (an isosceles triangle having angles of 36°, 72° and 72°; its base length equals f times its side-length, where f is the golden fraction defined by the equation phi = 1/(1+phi)), was used by Muslim scientists for the construction of regular pentagons and decagons [9]. The golden triangle can be subdivided in such a way that another golden triangle and a golden gnomon results, i.e., a isosceles triangle having angles 108°, 36° and 36°. As Chorbachi and Loeb write, artisans may actually have created the 36° angle using the (incorrect) method of constructing kunya-5.

The construction of the Pentagonal Seal in the Paris manuscript is, according to Chorbachi and Loeb, a very particular one, with its five-pointed star constituted by ten golden gnomons which exactly match the ten golden triangles which constitute the decagon. “It is historically significant that as early as the thirteenth century A.D., it was known that what we presently call the golden triangle and golden gnomon are together capable of tessellating the Euclidean plane, and that during the Middle Ages, Islamic design continued in the tradition of the Alexandrian and other eastern Mediterranean schools of mathematics. The use of this five-pointed star appears to have stimulated mathematicians to work on these practical problems in design. The importance of this problem to the Muslim scientists may be inferred by the fact that they tried over the course of several centuries to find the perfect solution.”

According to Wasma’a K. Chorbachi in “The Tower of Babel” [5], “[t]he true patron of the scientists who wrote these ancient manuscript was art. It was the artisans and the architects who called for the services of science and scientists to assist them solving the design problems that they were facing. And as in the case of Islamic art in the past, science must come to the service of the arts, whether we are talking today of Islamic art, of Western art or of art generally, today more than ever before […].” “[I]slamic tradition is so strong that, if we are in touch with the language of the present time and ground ourselves in this strong old tradition, we can arrive at an expression that is not only contemporary but could be meaningful and valid in the coming century.”

 

Notes

[1] According to Oleg Grabar in his fine book The Great Mosque of Isfahan (New York University Press 1990, p. 34) it contains in the four corners the 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.” Grabar notes that the central part of the panel is nothing else than the plug of the artisan who was diligently involved in restoring the mosque in the 17^th century, Muhammad ibn Mu’min Muhammad Amin.

[2] Ibid.

[3] Decorative brickwork on the northern iwan of the mosques also shows clockwise and counterclockwise swastikas in one of the circumferential bands.

 

 

 

 

 

 

 

 

 

[4] Özdural A. Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World. Historia Mathematica 2000; 27: 171-201.

[5] Ibid. It is the Islamic proof of the Pythagorean Theorem, which is closer to the Indian method of Bhāskara Achārya (d. 1185) than to the Greek method in Euclid’s Propositions, as is beautifully explained by Wasma’a K. Chorbachi in her eye-opening article “In the Tower of Babel: Beyond Symmetry in Islamic Design. Computers Math Applic 1989; 17: 751-789.

[6] Chorbachi WK, Loeb AL. An Islamic pentagonal seal (from scientific manuscripts of the geometry of design). In Hargittai I (ed) Fivefold symmetry. World Scientific, Singapore 1992, pp. 283-305

[7] Ibid., p. 284: “Although the approach to the generation of this pattern in the Paris manuscript is quite different from that taken by Penrose, it is notable that these ‘quasi-periodic’ patterns were already of interest at least in the thirteenth century A.D. The manuscript stresses the uniqueness of the fivefold center of rotational symmetry in the pentagonal seal, thus implying the lack of translational symmetry in the pattern, but does not explicitly deal with the matter of non-periodicity.”

[8] Ibid., p. 286f: “The construction was therefore remarkably accurate, though not correct. Kamal ad-Din Musa Ibn Yunus Ibn Man’a in his thirteenth-century commentary on Abu’l Wafa’ al Buzjani’s book on the geometry of construction, with whom this construction may well have originated, actually was quite explicit in cautioning that some of his constructions, in particular of the heptagon, were practical, but not mathematically exact. They can be used in small-scale designs without noticeable discrepancies, which however become manifest on a larger scale.” 

[9] Ibid., p. 293: “[I]n the second half of the thirteenth century (ca. 1259) in the town of Marāgha, which became a center of scientific activities and contained the famous observatory, another illustrious mathematician, Nasir ad-Din at-Tusi, wrote commentaries on Euclid, in which he made obvious use of the golden triangle. … [H]is commentaries on Euclid included a short treatise dealing with the inscription and circumscription of polygons within the circle: Sittat Maqalat min Kitab Tahrir Uqlidis: Six Books/Articles from Euclid’s Book of Elements.” As an example, see the construction below, which had been created with some guidance from Eric Broug’s booklet Islamic Geometric Patterns, Thames & Hudson, New York 2008.

 

 

 

 

 

 

 

 

See also on this blog

About difficulties of the Western perception of Islamic abstraction which might easily result in fundamental misconceptions

About decagonal tessellations on the west iwan of Esfahan’s famous Friday Mosque

About Alpay Özdural’s proof that the mysterious North Dome of Esfahan’s Great Mosque is based on Omar Khayyām’s triangle

A review of a booklet which makes complicated Islamic geometric patterns easy to reproduce

Darb-i Imam

The small Darb-i Imam shrine (1453) about 300 meters west of the Great Mosque may in fact be one of the gems of Timurid architecture in Esfahan. The site is rather hidden in the labyrinthine lanes of the northern part of the old city of Esfahan [1].  

The shrine consists of a funerary complex [2] with courtyards, shrine structures, and a small cemetery. During the centuries it had been steadily reconstructed and repaired, especially in the early and late 17^th century. Characteristic are the two closely spaced domes, one bulbous with beautiful arabesques and one more slender with floral decoration, on high drums with highly stylized calligraphy.

Its pishtaq, or porch, contains several exquisite mosaics made of glazed tiles. Some of them are said to be created by Sayyid Mahmud-I Naqash, who has also decorated the southern iwan and the celebrated Timurid gate on Esfahan’s Masjed-e Jomeh.

What has recently attracted more interest are the geometric patterns made of black glazed and unglazed terracotta pieces in several spandrels and a porch next to the mentioned main pishtaq.

 

 

 

 

 

 

 

 

It had been suggested that they represent the so far only discovered example of an almost perfect Penrose tiling which had been created 500 years before their description in the West [3]. In their meticulous reconstruction using the famous “kite-and-dart” type of Penrose tiling, Peter J. Lu and Paul J. Steinhardt very much focus on a spandrel which in fact matches almost perfectly with a Penrose tiling [4].

 

 

 

 

 

 

 

What makes this tiling so unique may be the subdivision rule painstakingly elaborated by Lu and Steinhardt:

“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.”

 

 

 

 

 

 

 

 

 

It has been questioned whether the pattern on the spandrel is really self-similar. The difference between the large and small scales is very big. In their analysis, Lu and Steinhardt create the spandrel itself by four large length scale decagons and two bowties, while the small scale consists of three girih tiles, the decagon, the bowtie and the elongated hexagon. So, where is the large-scale elongated hexagon [5]? Can it be that it has been overlooked?

 

The pattern is in fact aperiodic. There is only one small-scale area in the whole spandrel which resembles the large-scale pattern: in the upper right corner. The area with the corresponding (yellow) borders of the small-scale spandrel is shown in the picture below. Here, a part of the (green) elongated hexagon shows in the lower corner.

Thus, the large-scale spandrel may be reconstructed in a different way, shown below. Although the bold blue lines do not exactly fit, the reconstruction here seems to support the concept of self-similarity and aperiodicity of the tiling on this particular spandrel of the Darb-i Imam[6].

 

 

 

 

 

 

 

 

 

Notes

[1] The historical city with its huge bazaar had been cut by Kh. Abdorrazaqq into two halves some 40 years ago in an attempt of urbanization.

[2] The complex contains the tombs of two descendants of Imam Ali from Safavid times, Ibrahim Tabatabai and Zain al-Abedin Ali.

[3] Penrose himself had been inspired by Johannes Kepler’s Harmonices Mundi (1619), where he constructed tilings around regular pentagons which can be extended into Penrose tilings. Pentilings, i.e. arrangements of regular pentagons in the plane in which each pentagon makes edge-to-edge contact with two, three, four, or five neighbors, thereby sharing vertices in such a way that no gaps large enough to contain another pentagon are left in the array, have even been described by Albrecht Dürer in 1525.

[4] In the supporting online material for their article in Science, Lu and Steinhardt (2007) have suggested, based on a more than 40-year-old photograph that the tiling on the western iwan of Esfahan’s Friday mosque can be subdivided in the same way as that on the Darb-i Imam. Meanwhile, it has been shown that the patterns are different and that the one on the Friday mosque contains, in addition to a decagon, an elongated hexagon and a bowtie, a fourth girih tile, a rhomb (see an illustration of the girih tiles here). The pattern is, in addition, periodic, similar as the pattern on the Gonbad-e Qabud in Maraghah, which had been constructed in fact 250 years earlier.

[5] While Lu and Steinhardt had elaborated only a subdivision of a decagon and a bowtie by smaller-scale decagons, elongated hexagons and bowties (see below), P. R. Cromwell has recently presented a corresponding subdivision of the hexagon.

[6] Respective spandrels can be found everywhere in Esfahan, not only on the Darb-i Imam. They seem to have been very popular during Safavid times.

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 Urmiyeh 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 (12^th till early 14^th 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 15^th 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 15^th 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.

                                                                                

 

 

Notes

 

[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.

 

Decagonal Tesselations

The Great Seljuq Empire (1037-1194 CE) has been described as a period with stunning scientific and artistic achievements in particular in Iran. Their capital became Esfahan in central Iran under Malikshah I (d. 1092). Among the many Seljuq monuments found in Iran, Esfahan’s Great Mosque, or Masjed-e Jomeh, is probably the most remarkable. The Great Mosque’s huge courtyard of 65 by 55 meters with its four iwans , the standard model of later Iranian mosques, provides the two axes, one in the Makkah direction and the other perpendicular to it. The iwans differ considerably in their composition and decoration. The most important iwan to the south is connected to the larger of the two main domes which contains the mihrab indicating the direction of prayer. 

 

The western iwan is the most unusual and complex of all. While all iwans had been added to the Seljuq mosque after a fire pillaged by the Hashashiyyin sect in 1121 CE, their decorations are from the Timurid and early or even late Safavid periods (late 15^th till early17^th century) [1]. The western iwan and its counterpart to the east are called the sofe of the student (shāgird) and master (ustadh), respectively. Although both iwans were built at the same time as the southern iwan (early 12^th 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 in his book about the Great Mosque writes [2]. So, while dating of the specific decorations may be highly problematic if the artisan had not signed his work, there is constantly restoration work which will inevitably change the appearance of the ‘living monument’ over time. More information about Esfahan’s Great Mosque, its amazing history and stunning architecture, can be found here.

 

There were suggestions that there had been a breakthrough in creating (almost) Penrose tiling in the late 15^th century, in particular on the Darb-i Imam in the Great Mosque’s vicinity. In the supporting online material  of Peter J. Lu and Paul J. Steinhardt’s article in Science magazine, 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 respective pattern(s) on the Darb-i Imam shrine [3]. 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 especially this site seems to be an anomaly in Esfahan. Lu and Steinhardt also suggested so-called girih tiles to facilitate the incredible precision of the tiling [4].

 

As Lu and Steinhardt point out, based on a blurred picture taken from the book Design and Color in Islamic Architecture by Seherr-Thoss (Smithsonian Institution, Washington, DC 1968) the large-scale pattern consists of large decagons and bowties [5]. When reconstructing the small-scale pattern, I could identify similar but not the same subdivision rules which transform the large bowtie and decagon girih-tile pattern into the small girih-tile pattern of decagons, bowties and elongated hexagons as on the Darb-i Imam. For instance, the pentagonal areas encircled in magenta can be filled with a fourth girih-tile described by Lu and Steinhardt, the rhombus. See, for instance, the rightmost picture of the panel and, in particular, in the magnification below. So, the pattern on the western iwan of Esfahan’s Great Mosque differs from that found on the Darb-i Imam.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Much of the discussions after the paper of Lu and Steinhardt had been published were about the possibility of medieval artisans had consciously or unconsciously been able to create what has become known as Penrose tiling, five hundred years before its description in the West. It might be concluded, however, that neither the dazzling pattern on the Darb-i Imam nor that on the western iwan of Esfahan’s Great Mosque are Penrose tiling, simply, because they are not aperiodic.

 

Lu and Steinhardt had been criticized not having given due regard to extensive previous work on Islamic Art. In particular, reading an almost forgotten book about Fivefold Symmetry, edited by István Hargittai (World Scientific, Singapore 1992), might be revealing. Much of Lu and Steinhardt’s ideas and conceptions may in fact be found there, not only Emil Makovicky’s paper on the 800-years-old Gunbad-i Kabud in Marāgha in northwestern Iran. Emil Makovicky’s response to the Science article articulates that neither the Gunbad-i Kabud pattern nor that on the Darb-i Imam are aperiodic, and hence do not represent Penrose tiling. Moreover, when considering the reconstructed pattern on the Gunbad-i Kabud in both Makovicky’s (Fig. 8b ibid) and Lu and Steinhardt’s (Fig. S6 of supplementary online material) articles, it may in fact be assumed that the pattern on the western iwan of Esfahan’s Great Mosque is not entirely dissimilar to the former.

 

I suppose Islamic artisans tried their best in creating most interesting (indeed dazzling) patterns which attract the attention of visitors now for several hundred years [6] rather than producing quasi-crystals. As E. Makovicky argues, both the patterns on the Darb-i Imam and the west iwan of Esfahand’s Great Mosque are variations of the stunning decagonal pattern on the Gunbad-i Kabud in northwestern Iran, built in 1196/97 CE. “[w]e believe that the artisans were satisfied by creating a large fundamental domain without being concerned with the mathematical notion of indefinitely expandable quasiperiodic patterns. However, they understood and used yo their advantage some of the local geometric properties of the quasi-crystalline patterns they constructed.”

 

 

 

Notes

 

 

[1] For instance, 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-i Imam shrine, some 300 meters west to the mosque, which is dated 1453. By the way, on the gate the date 1317 appears which translates into 1939 when restoration work 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, not qur’anic, but contain traditions about mosques and about Ali. Amazing that Oljatu in fact converted to Shi’a Islam in 1310.

 

[2] “[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.

 

[3] A second visit, after 2007, of the Darb-i Imam shrine end of December 2008 revealed that the patterns were in fact temporarily not visible. Because of the upcoming Ashura festivities the complex was heavily decorated with religious banners and transformed into a place of observance for daily husseiniyyas.

 

[4] I have mirrored, for example, the right part of a picture of the arch borrowed from ArchNet (left part of the panel) 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.

 

 

 

 

 

 

 

 

[5] It may be of interest to note that Peter Lu visited Esfahan only after his paper in Science magazine which attracted considerable public interest worldwide. The political situation before the US American election in the end of 2008 largely complicated the procedures for issuing visas for Iran, in particular for US citizens and individual travelers. Thus, the whole article was based on the diligent work in libraries, as Peter Lu mentions in a colloquium where he reports on his amazing findings.

 

[6] Just by comparison I would not assume that Sayyid Mahmud-e Naqash, who created and notched the late Timurid, beautiful floral, decorations on the south iwan and the gate in the western façade of the mosque was the one who designed the decagonal patterns on the west iwan (and similarly different decorations on the Darb-i Imam, as well). But who knows? It would be interesting to learn how contemporary artisans repair, with incredible precision, the decorations. 

 

 

 

 

See also on this blog

 

Esfahan’s Old City. Some impressions of a cultural heritage at risk. 

 

Islamic Geometric Patterns.  A nice booklet teaching you drawing incredibly difficult patterns with compass and straightedge.

 

The Mysterious North Dome of Esfahan’s Great Mosque. About the most significant mosque (from an architectural point of view) in Iran. Pictures can be found here and here.

 

Dazzling Tesselations. Presumed almost perfect Penrose patterns in medieval Esfahan which have attracted enormous interest in 2007 after a publication by Lu and Steinhardt in Science magazine.

 

 

 

Dazzling Tessellations

 

People in the West are rightly questioning the contributions of the Islamic World for the development and implementation of human rights or humanity, which are late achievements of the Enlightenment of the West of the 18^th century; or peace in the last couple of centuries. Especially the latter contributions by the ‘West’ might of course be questioned as well. However, one should in any case constantly remember Islam’s genuine achievements when Europe underwent especially dark ages, namely the end of what people there called the Middle Ages. These medieval times were brightly illuminated in the Islamic World which extended from India, Central Asia, the Middle East, North Africa and, of course, Al Andalus, i.e., the Iberian Peninsula.

 

A Center of Islamic Art and Architecture

If interested in exploring medieval Islam, stunning emotions can certainly be expected when visiting Esfahan in Central Iran, one of the most beautiful cities of the world.  More than once I had a vision of George W. Bush and his wife Laura sitting together with, say, Mahmoud Ahmadinejad and his wife on the roof of the small tea house in the northwestern corner of Naqsh-e Jehan, the central square in Esfahan and one of the largest in the world, smoking qaliyan, or hubble-bubble, drinking sweetened tea (chay), talking about God and the World. The sophisticated Safavid architecture with the Shah mosque in the southern corner of the square, the Lotfallah mosque at the eastern side and, opposite, Aliqapoo palace, with the mountains in the background and sun setting is actually overwhelming, a vista that can hardly be topped anywhere in the world. Maybe the course of the world would change afterwards. Maybe not. People, who I told my vision, indulged me smiling but considering me completely naïve. Of course, Bush would like to bomb the mosques.

 

In October 2007, on my last visit of the city, I had a special destination. I had read with increasing interest a recent article published in Science magazine [1] about a simplifying technique employing so-called girih tiles for the outer beautification of late medieval Islamic buildings. A further breakthrough in tiling during the Timurid era in the 15^th century may be considered a cultural if not scientific sensation. Its mathematic background was thought to be properly described only 500 years later by 20^th century’s astronomer and recreational mathematician Sir Roger Penrose. According to the article, a couple of examples of quasi-crystalline tiling may in fact be found in the old city of Esfahan.

 

Islamic Tiling

In Islamic architecture animated creatures are usually not displayed for mere decoration. Instead, mosques and other buildings are over and over covered with ceramic tiles with both calligraphy from the Holy Qur’an and dazzling geometric patterns. The two authors of the Science article, Peter J. Lu from Harvard and Paul J. Steinhardt from Princeton, illustrate that most of these already incredible patterns may be drawn as zigzagging lines where the lines were directly drafted with the aid of a straightedge and a compass.

 

An important aspect in Islamic tiling may be periodic repetition of a ‘unit cell’. Rotational symmetry may be desired in some cases in order to enhance the overall impression of complexity. Lu and Steinhardt mention that it had been shown in the mid-nineteenth century by western mathematicians that only two-, three-, four-, and six-fold rotational symmetries are allowed.  Five- and ten-fold symmetries are crystallographically forbidden. While pentagonal and decagonal motifs appear frequently in a unit cell, they are usually repeated within the allowed symmetries.

 

Although possible in many cases, especially larger artwork demands simpler techniques than drawing lines with straightedge and compass. Lu and Steinhardt showed that by 1200 CE Islamic artisans rather used a set of five different types of equilateral polygonal tiles for creating an almost unlimited number of complex tessellations: a decagon, a pentagon, a hexagon, a bowtie, and a rhombus, which they called girih tiles. Each tile had decorating lines which intersect with the midpoints of every edge at angles of 72 and 108 degrees.

 

Lu and Steinhardt provide strong evidence that girih tessellating was the only way to create the almost incredible patterns on the (periodic but on a much larger scale) wall panels of the famous octagonal tomb tower Gonbad-e Kabud (Blue Tower) in Maragha (1197 CE), in the Azerbaijan province of Iran (see figure 1 below). While the decagonal pattern in each panel does not repeat, the ‘unit cell’ of this explicitly periodic tiling spans the length of two complete panels, in fact several meters. Furthermore, while the main decorative brick pattern follows the above mentioned decorating lines, another set of smaller lines conforms to the internal rotational symmetry of each individual girih tile without adhering to pentagonal angles. Within each region occupied by a hexagon, bowtie, or rhombus, the smaller line decoration has a two-fold rotational symmetry. I must admit that ‘seeing’ the girih tile pattern in Lu and Steinhardt’s pictures took me some time.

 

Quasi-crystalline Structures

There was a further and in fact hardly to believe breakthrough in tessellations when artisans of the Timurid period in the late 15^th century applied self-similarity transformations in girih tiles. Larger girih tiles were subdivided into smaller ones. Thus, overlapping patterns at two different length scales were created in which each pattern is generated by the same girih tile shapes.

 

A stunning and revealing example of self-similarity (where, in addition, outlines of girih tiles are clearly given in lighter ink) can be seen on a Timurid-Turkmen scroll belonging to the collection of the Topkapi Museum in Istanbul which has first been described by Professor Gülru Necipoglu, Aga Khan Program of Islamic Architecture at Harvard.

 

Another example was published in Lu and Steinhardt’s paper. It may be found in one spandrel of the Darb-e Imam shrine (figure 2) complex (1453 CE) in the old city of Esfahan [2]. The authors clearly indicate that, for instance, the basis of the right spandrel (as indicated by a large, thick, black line pattern) of a kind of portal is formed, on a much larger scale, by four decagons and two bowties. It can, however, be subdivided into smaller patterns, which can be perfectly generated by a tessellation of 231 girih tiles. The authors have even identified the subdivision rule to generate this pattern on one this particular spandrel at the Darb-e Imam shrine. And there are other examples of self-similarity there which may be studied there.

 

The Scientific Sensation

A subdivision rule in combination with decagonal symmetry has apparently allowed these medieval artisans to construct (consciously or not) perfect patterns with an infinite quasi-periodic translational order and crystallographically forbidden pentagonal or decagonal rotational symmetries. The sensation here is that western scientists have understood this principle only about 500 years later. So-called quasi-crystals are structural forms that are ordered but in essence nonperiodic, meaning a shifted copy will not ever match with the original. Although not being the first describing nonperiodic sets of tiles, since the mid 1970s, astronomer Roger Penrose described a set of six tiles, famous ‘kite-and-dart’ tiles, and another set consisting of two rhombuses with equal sides but different shapes which allow aperiodic tiling and five-fold symmetry. Lu and Steinhardt now indicate, in their 2007 paper, that the tile pattern of the particular spandrel in the Darb-e Imam can be mapped directly into the kite-and-dart Penrose tiles using a self-similar subdivision of large girih tiles into smaller ones. Thus, all prerequisites for the creation of infinite quasiperiodic patterns were available to the medieval artisans in Esfahan: girih tiles, decagonal rotational symmetry and self-similar subdivision. In their supporting online material, Lu and Steinhardt identify local point defects in the right spandrel, where the Penrose matching rules were violated. According to their interpretation, these mismatches and other facts point to the fact that the artisans did not really understand what they had actually achieved.

 

 

 

 

Another Example of a Decagonal Pattern

Not more than 300 meters to the east of Darb-e Imam lies the much more famous, magnificent Masjed Jomeh, Esfahan’s Friday mosque (see an aerial view in figure 3 and the main, southern iwan in figure 4a). Different parts of this wonderful mosque, which is considered by Arthur Pope as one of the world’s most awesome Islamic buildings, cover several centuries of medieval architecture. Amazingly, decagonal tessellations have been described by Lu and Steinhardt only at the portal of the western iwan. Lu and Steinhardt present a picture of the pattern in the supporting online material only which very much resembles that on the Darb-e Imam (see figure 4b and compare with figure 2c). Why has that decagonal tile pattern not been applied in other areas of the just wonderful mosque with its several styles from the Seljuq, Timurid and even Safavid periods?

 

The origins of the first mosque date back to the early 8^th century. Legend tells that the first mosque had been erected at the site of a Sassanid fire temple. Under the Abbasid Caliphs the first building was considerably enlarged in the 9^th century. The main structure which is still visible today was erected by the Seljuqs who had chosen Esfahan as their capital in 1051. As Henri Stierlin writes in his monumental work on Islamic art and architecture [3], during the vizierate of Nizam al Molk (d. 1092) architecture especially in Esfahan developed a distinctive form and the full range of its modes of expression thanks to the technological innovations of the Seljuq period.

 

It is important to note that ceramic tile decorations at Esfahan’s Masjed-e Jomeh were added considerably later. The main axis of the mosque with the two enormous domes is oriented in the Makkah direction. The large southwestern main dome has a diameter of 14 meters. The smaller one in the northeastern corner was built by Nizam al Molk’s political rival Taj al Molk in 1088 possibly based on blueprints of mathematician Omar Khayyam (d. 1122) who then lived in Esfahan [4]. The dome with a diameter of about ten meters, called Gunbad-e Khaki (Dome of the Earth) is widely considered to be the finest brick dome ever built. It is mathematically perfect and has survived dozens of earthquakes the country is plagued so much, for more than 900 years.

 

The four iwans of the Masjed-e Jomeh illustrate impressively the varieties of decorative arts of Esfahanian architects of the Seljuq period in late 11^th century. While the northern iwan has a simple pointed tunnel vault only, three iwans present with apses with elaborate muqarnas, or honeycomb-like structures, and are beautifully decorated with kufic calligraphy and geometric patterns (figures 4a, b). There are several large hollow spherical triangles derived from the “squinch, a constructional device that was rigorously and rationally formulated under the Seljuqs” as Henri Stierlin writes (p. 30).

 

 

 

With regard to the present article the western iwan deserves special attention. It is the most unusual and complex of all. First, as is also shown in Henri Stierlin’s marvelous book (p. 31), it presents, at its back view, an extraordinary ribbed structure. “The technique used in the 11^th century to support large honeycomb vaults grew out of an intensive study of the resistance of brick arches carried out in Seljuq Iran.” It presents with “… a large-scale muqarnas vault in which the juxtaposed spherical triangles are not only decorative but functional in a very original way, making up a system as ingenious as that of Buckminster Fuller’s 20^th-century geodesic domes” (ibid, p. 215). And, as mentioned before, similar tile patterns as the quasi-crystalline ones on the Darb-e Imam can be found on the inner sides of the portal [5]. The same subdivision rule applies here as on the tiling on the Darb-e Imam shrine nearby.

 

Esfahan’s Masjed-e Jomeh shows many more interesting details on its four iwans. The tessellations on the western portal may be one interesting scientific issue. While wandering through this 1300 year-old unique testimony of Islamic history one gets, however, more and more excited about its perfection in both its overall architecture and even the smallest details (see, in particular, Henri Stierlin, pp. 214).   

 

The Seljuq Dynasty

The Great Seljuq Empire stretched from Central Asia to Syria. Seljuqs were Sunni Muslims. The founder of the dynasty, Seljuq himself, had converted his Central Asian Turkish tribes to Islam in the mid 10^th century. Under Toghrϊl Beg (d. 1063) the Abbasid Capital Baghdad was taken from the Shi’a Buyid dynasty in 1055. Under the reign of, in particular, Alp Arslan (d. 1072) and Malik Shah I (d. 1092) remarkable monuments were created. Their perspicacious and inspirational vizier Nizam al Mulk had a great influence on the development of Esfahan as the center of Persian power. Art and science were flourishing and major architectural projects were encouraged by the Seljuq rulers.

 

On tomb towers (gunbad) and mosques, complicated geometric brick and stucco patterns were used for ornamental beautifications. One famous example is the already mentioned octagonal Blue Tower (Gunbad-e Kabud) in Maragah with its periodic albeit immensely dazzling network of glazed bricks on its eight panels. Maragah was famous for at least five gunbads, and four can still be seen there. Key feature of the mosques were four conch-shaped iwans facing each other and marking the two axes of the building which met in the ablutions fountain. This typical Persian outline of a mosque, its four iwans with recessed space covered with either a pointed or hemispherical vault, may in fact be a further development of the Sassanid royal hall (see Henri Stierlin, p. 298).

 

Further Achievements during the Timurid Period

While under the Seljuq rulers Persia experienced an early renaissance which cannot be overstated, it must also be emphasized that, as always in its so long history, what is called Persianization of the Turkish intruders from Central Asia took place. The same was true when Tamerlane (Timur Lenk, d. 1405) had conquered Iran and wreaked havoc and incredible massacres among civilians including building pyramids of sculls of killed inhabitants of the conquered cities, in particular in Esfahan in 1387. Under the Timurid rulers, science and art flourished again in Iran. Samarqand and Bukhara in Transoxiana as well as Esfahan and, for example, Mashhad experienced an incredible proliferation of geometric decorations on mosques, madrassahs, and mausoleums. In addition to rigorous geometry (including the above quasi-crystalline patterns) and kufic and, in particular, thuluth calligraphy, now plenty of interwoven vine twines and flowers occur.

 

The true peak of splendid virtuosity of Timurid decoration in Iran may again be found at Esfahan’s Friday Mosque, especially on the southern iwan with its elegant vases of flowers and on the doorway north to the western iwan, facing the courtyard. The latter was built by Sayyed Mahmud in 1447 [6].

 

An especially fine example of great Timurid architecture in present day Iran can be found in Mashhad in Khorasan. The Holy Shrine of the Eighth Shi’a Imam, Reza, and the fine Azim-e Gohar Shad Mosque (1416) were commissioned by the wife of Tamerlan’s eldest son Shah Rokh. Under the influences of Gohar Shad and, in particular, her son Ulugh Beg (d. 1449), Persian language, art and science became central elements of the Timurid dynasty. The exceptionally fine ceramic cladding as well as sturdy drum and bulbous dome are clearly related to more Timurid masterpieces in Transoxiana, in particular, in Samarqand and Bukhara. On photographs taken in the late 19^th century, the splendor of façade decoration but also interior designs in these almost mystical cities on the legendary Silk Road are obvious [7].

 

Some Concluding Remarks

Coming back to decagonal and quasi-crystalline tilings as a further scientific breakthrough of medieval artisans one has to ask why they have been so rarely implemented among the huge variety of just periodic tiling, vine and flowers, and calligraphy. It is quite obvious that these late medieval artisans and architects commanded the whole repertoire of techniques, periodic or nonperiodic, with great proficiency. Based on the evidence presented by Lu and Steinhardt, it may be a mere speculation whether or not they really understood what they did. If occurrence of quasi-crystalline tiling can in fact be traced to the Timurid period at the end of the 15^th century, first, more examples have to be detected and studied. The top contemporary mathematicians may in fact be found at Samarqand’s University which was founded in 1420, for example Ulugh Beg and Ghiyath al Kashi (d. 1429) and their group.

 

The scientific dispute [8] after Lu and Steinhardt’s paper about when aperiodic patterns have been introduced in Islamic tiling may be considered even immature. It may provide some insights in profound misconceptions and even dogmatism which might point again to the incredible superior sovereignty of artisans living several hundred years ago, who probably did not care to much about ‘scientific breakthroughs’ when decorating mosques in the praise of the Almighty.  

 

Time for further research is running. I have found only one short note by Iranian news media after the publication of Lu and Steinhardt’s paper which otherwise aroused considerable interest worldwide. Whether there are current investigations by Iranian scientists at the Darb-e Imam shrine is not known. Renovations in November 2007 may actually have prevented me from detecting one of the spandrels displayed in Lu and Steinhardt’s paper. The pattern on one looked similar (see figure 2d), but kufic descriptions at the margins essentially differ from the published picture.

 

And finally, the political situation in present day Iran doesn’t make things easier.

 

 

Notes

 

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

 

[2] In fact, Peter J. Lu identified decagonally symmetric motifs on two different length scales, a telltale sign of what is called a quasi-crystal, on a photograph of the Darb-e Imam shrine only, as John Bohannon writes in a short editorial of Science (2007; 315:1066). He visited the site in Esfahan earlier this year (personal communication).

 

[3] Stierlin H. Islamic Art and Architecture. Thames & Hudson Inc., New York 2002, pp. 29ff.

 

[4] Architecture in Medieval Iran is generally considered ‘Islamic’. One has to keep in mind, however, that Iran was inhabited by rather great subpopulations of Christians, Jews, and Zoroastrians as well. Nowadays, that does not hold anymore, of course. The most gorgeous monuments are in fact religious, Islamic, buildings. Howver, Omar Khayyam, for instance, who was involved in the planning of the northeastern dome of Esfahan’s Friday Mosque, was not an avowed Muslim but rather a free spirit. A universal genius, who had not only contributions in poetry (for what he is well-known in the West after Edward Fitzgerald’s (d. 1883) translation of the Rubaiyat), but also and even more in astronomy and mathematics, who has laid the foundation of non-Euclidean geometry several hundred years before its final conception in the 19th century. Iran’s solar calendar, introduced in 1925 in an attempt by Reza Shah (d. 1944) to secularize the country, is based on his calculations, too. It is more precise than the Gregorian calendar which is used in the West. Omar Khayyam’s very special, almost modernistic tomb (in fact not ‘Islamic’ at all) is located in Nishapur in Khorasan, Eastern Iran.

 

[5] The decagonal but distinctly periodic structures are only shown in Lu and Steinhardt’s supporting online material. They date the mosque (or the tessellations on the western iwan’s portal only?) to the late 15th century. As has been mentioned already, the mosque is much older. Its origin can in fact be assumed in the 2nd century AH.

 

[6] The respective sanctuary contains an extraordinary example of a stucco mirhab built in 1310 by the Mongolian Il-Khan Öljeitü.

 

[7] Naumkin V. Caught in Time: Great Photographic Archives. Samarkand. Garnet Publishing Ltd. Reading 1992. In the same series and by the same author, a book with early photographs of Bukhara appeared in 1993. Most of the photos had not been published before 1992. Among the Samarqand photos, few seem in fact to indicate quasi-crystalline tiling patterns(?). See, for example, Fig. 3, the spandrel of the portal at the Shah-i Zindah complex (p. 6); at the famous Registan (Figs. 82, 83 on pp. 120f); or at Ishrat Khana (Figs. 87, 88 on pp. 128f).

 

[8] See, in particular, Makovicky E. Science 2007; 318: 1383.

 

[9] This posting is dedicated to Ms. Dora Fischer-Barnicol whose profound knowledge about and sympathy for foreign cultures may be one of the main reasons for my constant interest in the Middle East and its people.