Not Everybody’s Darling

 

The original sources for the detailed descriptions of legends and fairy tales which circulate among both ordinary people in the Islamic world and, for example, Sufis since Muhammad’s and his followers’ conquest of much of the world have never been described in a scientific way. When reading, for instance, Eliot Weinberger’s Muhammad (Verso, London 2006), which is, according to the author, mainly based on the Holy Qur’an and ahadīth, or the traditions of the Prophet, one may ask the question how many generations of people have, over the centuries, embellished so nicely the historical facts (?) so that an attractive legend was created which fascinates even sober, contemporary Westerners, the main target audience of Weinberger’s nice booklet.    

Allah’s Darling (or Allahs Liebling, the original title of the book which has, so far, been published only in German) is the attempt of the renowned German Orientalist Tilman Nagel, a professor emeritus of the University of Göttingen, to explain the origins and manifestations of the belief in the founder of Islam, Muhammad. The book is sort of a spin-off of Nagel’s opus maximum, his voluminous biography of the Prophet, mainly praised but also heavily criticized by others.

When having read the subtitle of “Allah’s Darling” (“Ursprung und Erscheinungsformen des Mohammedglaubens”), I was wondering whether the author wants to make the point that Islam is not an extreme form of monotheism, as claimed in particular by Sunni Muslims, but rather that Muslims are “Mohammedans”, a pretty frivolous, Orientalist, conception. He frankly admits that everyone who would undertake the task of highlighting the circumstances under which a faith could emerge which was essentially based on prefabricated “eternal” knowledge, ever-valid for any area of life; a faith in an ever-competent messenger of Allah, would inevitably face the “foolish” charge of Orientalism or Essentialism. He may be right, but whether the charge is in fact foolish was not clear to me after having read the book.

The seemingly sound construction of what one may describe as the House of Islam is, however, not different from that of other, older, world religions. That, after the Age of Enlightenment, fundamentalist Christianity, for instance, has largely (unfortunately not entirely, though) been repelled in modern, determined secular, societies may have something to do with the foundation of Christianity as the author correctly claims, but not with its Church(es), as it (they) developed in century-long processes, with its (their), for example, heated arguments regarding the “nature” of Jesus, the World’s Redeemer; or strange beliefs in the Virgin Mary. There is no difference in overall absurdity. It is self-evident that, in order to write a credible, in particular scientific, treatise or even book on one of the world religions authors should make clear in the very beginning that they are not religious! That is unfortunately not the case here.   

Several times Nagel points to the huge problems of Integrationspolitik, i.e. how Muslims may be integrated in Western societies. He stresses that the time and again overpowering (erdrückende) majority of Muslims still live their fatalism due to strong beliefs in the believer’s general inability of getting hold of his own lives. For Nagel it seems to be clear that Mohammedanism should be regarded the main reason for the widely observed (in comparison) developmental retardation in Islamic societies. His plenty arguments, however, are taken from medieval authors commenting on ahadīth [1]; notoriously unreliable, as it becomes clear time and time again in Nagel’s narrative. The realm of medieval Islam (note, that the Middle Ages describe the dark ages of European cultures and societies when, at the same time, the Islamic world was bright and pretty enlightened) was huge, though, and spanned from Spain to Central Asia, from North Africa to parts of India. Islam, as Nagel describes it using accounts of numerous medieval authors, Andalusian, Cairene, Damascene, or Iranian [2], is not, and never has been, a monolithic entity. There are four prominent Sunni schools of fiqh, or Islamic jurisprudence, and two schools for the Shi’a, which are not covered in Nagel’s book.

In his epilogue, Nagel concludes with the description of his pretty unjustified dismay about the publication of now, since in 1981, eight volumes of Muhammad. Encyclopedia of Seerah (The Muslim Schools Trust, London, 2^nd ed. 1985), clearly a sort of personality cult. He might not even be aware of comparably voluminous works of contemporary authors about Shi’a Imams with a similar, of course questionable, approach [3]. That currently by the majority of the faithful practiced Islam won’t fit into a rapidly changing, now again flat, world with its traffic, world wide web, demands of intercultural competence etc, is commonplace. Professor Nagel acknowledges, in the preface of Allahs Liebling, one of his co-workers for introducing him to and solving emerging problems with electronic data processing. So, even he might not have arrived yet in modern times.

 

Notes

[1] When introducing the reader to his text, Nagel describes the pretty bizarre “fly” hadīth: The Prophet once narrated: “If a fly falls into one of your containers (of food or drink), immerse it completely (falyaghmis-hu kullahu) before removing it, for under one of its wings there is venom and under another there is its antidote.” The purpose here is clearly defamatory, not realizing that Christian salvation history is full of similar absurdities, not mentioning the Jewish Tanakh.

[2] As regards the latter, I am not even sure. Iran, a center of medieval Islam, seems not to be covered at all. Moreover, Nagel rarely informs the reader about the specific background of the authors he extensively quotes: the historical circumstances during the periods they lived when they created their scriptures. That, of course, raises questions about the targeted audience. Is it politicians, a lay audience? The book is not a reference text. In contrast to his claims, I would not even regard it a sound scientific study. Too copious, even biased, in its descriptions of absurdities (see [1]) which may have led eventually to his (or our) perceived totalitarian Mohammedanism of the Islamic world.

[3] I own, for instance, an English translation by Jasim al-Rasheed of the 1926 book by Baqir Sharif al-Qarashi’s The life of Imam Ali bin Musa al Rida; Ansariyan Publications, Qum 2001, which was a personal gift by Kuwaiti Shi’ites on the occasion of their pilgrimage to the Holy Shrine of Imam Ridha in Mashhad in 2006, when I was invited to join the group. Much of Nagel’s descriptions of the Prophet’s reported excellence, for example of his physics, his manners, his generosity etc., which elevated him from ordinary people, may be found in the description of Imam Ridha as well. It would have been even more interesting to study the deeply rooted piety of ordinary, say, Iranian people in rural areas, including their legends and personality cults as regards Ali, Husayn, the numerous Imami Shi’a Saints, etc. In particular ahadīth related to Ali, the Nahj al-Balagha, may prove that Allah may have just another darling besides Muhammad.

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.

Benford’s Law

Astronomer Boudewijn F. Roukema at the Centre for Astronomy of Nicolaus Copernicus University Torun in Poland has launched an analysis of vote counts of 366 voting areas, which had been published by the Iranian Ministry of Interior, and has applied Benford’s Law in order to detect election fraud. According to these calculations, “the null hypothesis that the vote count distributions satisfy these distributions is rejected at a significance of p ≤ 0.007, based on the presence of 41 vote counts for candidate K (Mehdi Karroubi) that starts with the digit 7, compared to an expected 21.2-22 occurrences expected for the null hypothesis. A less significant anomaly suggested by Benford’s Law could be interpreted as an overestimate of candidate A’s (Mahmoud Ahmadinejad’s) total vote count by several million votes.” 

The study is not completely convincing. Still, the observed anomalies may be explained by chance alone. Moreover, figures 5 and 6 (pp. 5, 6) may have erroneously been exchanged. The motivation of conducting such an analysis is definitely driven by the assumption that the incumbent Iranian president was in urgent need for massive manipulation to become re-elected. That might not even be the case, as I have argued before. The brutal abolition of demonstrations in the previous two weeks have shown the true face of this regime which cannot easily been overthrown, in particular if one has to assume massive western support of  a “Green Revolution” with now rather deleterious outcome, as we experience these days. Sad to say but Iran may be on its way to a police state.

The online manuscript by Dr. Roukema (as of June 16, 2009, i.e., 4 days after the election!) which has been submitted to the Annals of Applied Statistics can be found here.

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

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.

 

Pope Benedict

Pope Bededict’s remark on his first Apostolic Journey to Africa (Cameroon and Angola) that the continent’s fight against HIV/AIDS is a problem that “cannot be solved by the distribution of condoms: on the contrary, it will increase it”, has led to a fierce editorial in the prestigious medical journal The Lancet. Irrespective of whether the Pope’s error was due to ignorance or because of a deliberate attempt to exact Catholic ideology, it had led to sharp criticism among several European governments and international health organizations. Now the scientific community condemns disastrous remarks as well, in particular as the Vatican is not withdrawing the devastating message but maneuvers with different versions and interpretations.

 

What can be found on the web site of the Holy See looks, in fact, a bit different than what had been reported at first:

 

Q. – Your Holiness, among the many ills that beset Africa, one of the most pressing is the spread of Aids. The position of the Catholic Church on the way to fight it is often considered unrealistic and ineffective. Will you address this theme during the journey? Holy Father, would you be able to respond in French to this question?

 

A. – I would say the opposite. I think that the most efficient, most truly present player in the fight against Aids is the Catholic Church herself, with her movements and her various organizations. I think of the Sant’Egidio community that does so much, visibly and also behind the scenes, in the struggle against Aids, I think of the Camillians, and so much more besides, I think of all the Sisters who take care of the sick. I would say that this problem of Aids cannot be overcome merely with money, necessary though it is. If there is no human dimension, if Africans do not help [by responsible behavior], the problem cannot be overcome by the distribution of prophylactics: on the contrary, they increase it. The solution must have two elements: firstly, bringing out the human dimension of sexuality, that is to say a spiritual and human renewal that would bring with it a new way of behaving towards others, and secondly, true friendship offered above all to those who are suffering, a willingness to make sacrifices and to practice self-denial, to be alongside the suffering. And so these are the factors that help and that lead to real progress: our twofold effort to renew humanity inwardly, to give spiritual and human strength for proper conduct towards our bodies and those of others, and this capacity to suffer with those who are suffering, to remain present in situations of trial. It seems to me that this is the proper response, and the Church does this, thereby offering an enormous and important contribution. We thank all who do so. (Emphasis added.)

 

Did the Pope talk about condoms or what is meant by prophylactics? Does he weaken his first condemnation of condoms or is he rather worsening the message by referring now to ‘prophylactics’? Difficult to tell, indeed. The script on the web page of the Holy See of his infamous lecture in Regensburg in September 2006 now contains also numerous rectifying footnotes diluting the rude and insulting first remarks on Islam and its Prophet which has led to outrageous reactions in the Muslim world and the death of at least one nun in Somalia.

 

It is a pity that the 81-year-old Pope, a professor of Catholic Theology with an immense reputation, has proved again and again that he had not effectively changed since the times of Joseph Ratzinger: a merciless exponent of the former Roman Inquisition. As a matter of fact his pontificate has been a series of scandalous speeches, remarks and deeds; a rather recent and especially incomprehensible example being his pardon (later withdrawn upon international pressure) of Holocaust denier Richard Williamson.

 

The Lancet’s condemnation today will not lead to a change in the Vatican’s policies. Life is shed with and without an organization which might vanish in due time anyway.

 

 

See also on this blog

 

Out of Control.  Pope Benedict’s scandalous pardon of Holocaust denier Richard Williamson.

 

 

 

No Peer Review

Earlier this month, Scientific American reported on a serious case of scientific fraud in the field of pain therapy. Scott Reuben, a fifty-year old Professor in Anesthesiology and Pain Medicine at Baystate Medical Center in Springfield, Massachusetts, has fabricated the data in 21 studies. If that has not yet been accomplished, they have to be retracted, so that practitioners who heavily rely on medical information transfer through databases are not further misled. An unidentified number of patients might have been died from adverse effects of certain potentially dangerous COX2 inhibitors such as Merck’s Vioxx (rofecoxib), and Pfizer’s Bextra (valdexocib) and Celebrex (celecoxib), some having been pulled from the market in 2004 since independent research has reported on greater risks than expected.

 

Another case of scientific misconduct which had been widely discussed in the media only a couple of months ago is that of the Egyptian ‘mathematician’ Mohamed El Naschie, former Editor-in-Chief of Elsevier’s Chaos, Solition and Fractals [1]. The Editor had published in ‘his’ journal more than 300 own papers since 1993 [2] and it was assumed that at least those did not undergo the usual peer review process. In The n-Category Café  physician John Baez made the point that much of this has to be considered complete nonsense [3]. Why did nobody point to that earlier? The ‘Journal’ enjoys a rather high scientific impact factor of 3.025, which is in fact higher than those of other periodicals in Mathematics [4].

 

The dependence of journals, editors, authors, and academic hopefuls on Thomson Reuter’s Impact Factor [5] has certainly excited greed among most scientists. Is it possible that the currently observed so-called information explosion in Science and Medicine is based, at least in part, on the publication of plenty of nonsense or even undiscovered, wide-spread fraud?

 

Having been observing this ‘business’ for almost 30 years, in a small clinical discipline though, the evolution from publishing definitive results of exciting, frequently ‘ground breaking’, research toward an obvious trend of publishing whatever had been done in the laboratory is obvious. Whether the, in theory, highly efficient peer review process still works in practice, in particular in the more hidden scientific habitats, might in fact be questioned [6].

 

The world has seen, during the past half a year or so, the credit crunch in the U.S. developing into a global financial and economic crisis which will keep us concerned and busy for the next couple of years. It might not be too far-fetched when comparing the recent annoying scandals in Science and Medicine with the burst of the bubble in the financial markets. Both, at first sight annoying, incidents may have positive effects on the systems in the long run.

 

 

 

 

Notes

 

[1] Publisher’s note in the latest issue: “The Founding Editor for Chaos, Solitons and Fractals Dr El Naschie has retired as Editor-in-Chief. The publisher will work with the editorial board and other advisors to identify a new editor. This is likely to also lead to revision of the aims and scope of the journal, as well as the editorial policies and submission arrangements. Prospective authors can keep informed of the progress on this through the journal’s homepage.”

 

[2] Whenever editors publish their works in their ‘own’ journal there should be suspicion regarding considerable bias: an orderly peer review process may not have been achieved. As an example, in a prestigious journal in my rather small field within Medicine, 1368 articles had been published (according to PubMed, accessed March 26) since 1990. The founding editor-in-chief, who kept this position for now more than 20 years, is authoring/co-authoring 12.7% of these, an estimate of 25% of all of his publications since 1970.

 

[3] It might be revealing that the link to the original posting is now broken. Baez’s arguments can still be found here.

 

[4] Since the impact factor of a journal is the average number of citations in a year given to those papers in a journal that were published during the two preceding years, it is likely that El Naschie has manipulated, i.e., unduly inflated, it as well, simply by citing his own ‘work’.

 

[5] The Institute of Scientific Information (ISI) had been founded already in 1960 by Eugene Garfield. The offered bibliographic databases enabled calculating measures such as a journal’s Impact Factor. Their publication in the Journal Citation Report in June is heavily awaited for by most members of the academic community.

 

[6] Editors and reviewers should be aware of attempts of fraud due to the ever increasing pressure of ‘publish or perish’. The former should encourage that all authors declare their contributions to the paper. Long traditions of adding department chairs automatically to the list of authors should of course be scrutinized. Conflicts of interest are most probably present whenever company sponsoring had been sought for; etc., pp.

 

 

 

Peer Review

In a recent article on AntiWar.com by Muhammad Sahimi an apparent information leakage of the International Atomic Energy Agency (IAEA) and David Albright’s Institute of Science and International Security (ISIS) was linked to Olli Heinonen, IAEA’s deputy director for safeguards. Albright’s analysis of the February 19 IAEA report on Iran’s nuclear program has led to much speculation. In the original IAEA report it was stated that Iran does not possess so far any uranium with an enrichment level suitable for use in nuclear weapons. ISIS, on the other hand, speculated that ‘breakout capability’ has been achieved already. Such speculations in a more and more confused public may in fact lead to irresponsible responses. There was, for a couple of days, even evidence that the current Chairman of the Joint Chiefs of Staff, Admiral Mike Mullen, and both his Secretary of Defense Robert Gates and National Intelligence Director Dennis Blair would disagree over the issue whether Iran has exceeded the limits or not. 

 

A further analysis of the latest IAEA report with a clear statement on Iran’s ability to make a nuclear weapon was done earlier this month by R. Scott Kemp and Alexander Glaser at Princeton University who concluded that “[u]nless Iran makes significant modifications to its centrifuge cascades, the claims being made overestimate the amount of weapon-usable uranium that could be produced from Iran’s low-enriched uranium stocks by a factor of three.” They estimate that it would take Iran roughly a year to make a “significant quantity” of weapon-grade uranium and that a more realistic estimate is three years”. In contrast, ISIS assumes when criticizing the statement that Iran is most probably running further, covert, uranium enrichment facilities in addition to the well-known Natanz plant. But that would make any reliable estimates impossible. They are mere speculations and should be regarded as such. So far, there is no evidence for this assumption. As a matter of fact, a respective discovery would most likely lead to an immediate attack by Israel and possibly the US. Scott and Glaser point to that but when reading both their response and Albright’s comment on their original paper, one might ask, whose opinions will finally prevail in the public?

 

Peer review is essential on such a sensitive issue but some reviewers (in fact opinion leaders) are frequently heavily biased, which has especially become true in case of the respected David Albright.

 

 

See also on this blog

 

No New Concern? The February 19, 2009 IAEA report on Iran’s nuclear program. 

 

Shut Down for Maintenance. ISIS on Iran’s uranium hexafluoride conversion.

 

Joe Biden in Munich.

 

Not Inevitable. ISIS’s reasoning on Iran’s nuclear program.

 

In a Timely Manner. ISIS report on Iran’s nuclear program during the presidential transition period.

 

 

 

Opportunities

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Two weeks ago, Iran had launched its first satellite into orbit. It is hoped (and Iranian officials are not getting tired to emphasize) that Hope (the name of the satellite, omid in Farsi) will increase our knowledge and eventually lead to more harmony and peace on Earth.

Its inhabitants do well (especially when considering the most recent underestimations of what is called global warming) when taking a search for alternative places to live into serious consideration. One extraordinary and most beautiful area (although barren and yet icy cold) might be the Victoria Crater on Mars, an impact crater at the Meridiani Planum near the equator of the planet. The crater is about 800 meters in diameter and had been visiting by Mars roboter Exploration Rover Opportunity. That little sojourner of our second nearest neighbor in space can actually be seen on the image TRA_000873_180 which was taken by the High Resolution Imaging Science Experiment (HiRISE) camera onboard the Mars Reconnaissance Orbiter spacecraft on October 3, 2006. After 2 ½ years, Opportunity (the robot) had just arrived at the rim of the Victoria crater after a drive of more than 9 kilometers. Provided high resolution, it can be seen roughly at the “10 o’clock” position along the rim of the crater. Whether it has crashed into the crater in the meantime is not known to me.

 

Crater Victoria has a distinctive scalloped shape of its rim. Erosion and material having fallen down the crater walls can be seen on the picture. The very special sand dunes in the center of the crater remind of similar structures in the Rub’ Al Khali of the Arabian Peninsula or the Dashte Lut in Iran.

 

Thanks to John Baez for attracting my attention to the beautiful crater.

 

 

69˚40’N 18˚56’E

 

One of the numerous peculiarities of the Arctic is, of course, the midnight sun and the phenomenon that, north to the Arctic Circle, sun does not rise in the winter, at least on the 21^st of December. Living 400 km north of the Arctic Circle, the dark period here in North-Norway lasts almost two months. Fortunately, spectacular dawn/dusk colors can be seen around noon in the far south, and sometimes, northern lights are illuminating the sky.

 

Another phenomenon is not so well-known. During the two months when the sun is never rising, there are usually two full moons. And if the changeable weather allows it, it can be noticed that it circulates in huge waves around the sky, in fact, never setting. Thus, similarly to the common, or solar, Arctic Circle which marks the latitude when the sun is not setting for one day in June (and not rising for one day in December) there is also a lunar Arctic Circle, which is not fixed like the solar analog. Due to the moon’s precession (i.e., the change in the direction of its axis), there is an 18.6-year cycle when the moon is sometimes higher in the sky and sometimes lower.

 

Tromsø lies way north to that band and a non-setting moon might be observed each winter twice, provided the skies are clear.