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.

 

What Next?

In response to a lawsuit and exactly five years after the Abu Ghuraib prison scandal the Pentagon is now going to release dozens if not hundred of photos which have been taken to document abuse or alleged abuse of terror suspects by U.S. military personnel in Iraq and Afghanistan, as the Washington Post reports today. What do we have to expect and, honestly, why have they been taken if not for reasons of pure sadism? Did the abuse of detainees go on despite former President Bush’s claim of being “un-American”?

 

Amrit Singh, an American Civil Liberties Union (ACLU) staff attorney involved in the 2004 Freedom of Information Act lawsuit that led to the promise to release the photos, said:

 

“[The photos] show[s] that the abuse of prisoners at Abu Ghraib was not aberrational but was systemic and widespread.

 

“This will underscore calls for accountability for that abuse.”

 

It is in fact not clear what will finally be shown. An anonymous Pentagon official disputes that the photographs would prove systematic abuse in prisons run by U.S. military personnel in Iraq and Afghanistan. The images in questions have been investigated in 60(!) of the military’s own investigations of abuse allegations.  

                                                                             

“What it demonstrates is that when we find credible allegations of abuse, we investigate them.”

 

This claim is once more not very trustworthy. According to the Washington Post, Secretary of Defense Robert M. Gates has said yesterday:

 

“There is a certain inevitability, I believe, that much of this (!) will eventually come out. Much has already come out.”

 

Mr. Gates also expressed concern that the release of photos and interrogation memos may cause unrest and create further problems for U.S. troops serving in Iraq, Afghanistan and elsewhere.

 

The former Bush administration has argued a section of the Geneva Convention might be violated when photos of prisoners are shown to the public. But a three-judge panel of the U. S. Court of Appeals for the 2^nd Circuit had rejected such arguments in September 2008. There is in fact a significant public interest in potential government misconduct.

Damage Control

Not entirely unprecedented, Iranian President Mahmoud Ahmadinejad interfered today with yesterday’s verdict of the Revolutionary Court sending Iranian-American journalist Roxana Saberi to jail for 8 years for charges of espionage. In his letter to Tehran’s prosecutor he interestingly also mentioned Iranian-Canadian blogger Hossein Derakhshan who has been detained since October last year for spying for Israel. Mr. Ahmadinejad called on the prosecutor to precisely handle the case, observe administration of justice and ensure that the accused persons (Ms Saberi and Mr Derakhshan) can freely and legally defend themselves.

 

“Please, personally observe the process to ensure that the defendants are allowed all legal rights and freedom in defending themselves and that their rights are not violated even by one iota,” reported Iran’s official government news agency IRNA.

 

This is a quick response to an apparent attempt of torpedoing the just initiated process of rapprochement between Iran and the new U.S. administration after decades of enmity.

 

 

 

Enhanced Interrogation Techniques

It is generally hold that the former Bush administration and its allies had lost the war in Iraq on the day when the pictures from the Abu Ghuraib prison in Baghdad had been aired. It was exactly five years ago: detainees in extremely humiliating positions, piled naked bodies forced to almost homosexual practices, interrogation of shackled detainees in front of barking shepherd dogs. Private Lynndie England holding a naked prisoner in a leash and stupidly leering into the camera of her boyfriend Charles A. Graner Jr. Un-American, as G. W. Bush declared with disbelieving disgust. Only one week later, on May 7, 2004, American citizen Nicholas Berg, who had been kidnapped in Baghdad, was beheaded in front of a video camera by the murderer Abu Musab al-Zarqawi. His killer claimed that his death was carried out to avenge the abuse of U.S. soldiers at Abu Ghuraib prison.

 

The two culprits at Abu Ghuraib prison, England and Graner, and a handful of others, all at low military ranks, have later been convicted guilty of conspiracy, maltreating and committing “indecent acts”. The term “torture” has not been mentioned, though.

 

Not un-American must then apparently be regarded what has now been released by the Obama administration: four memos detailing “tough interrogation techniques”, including “walling”, facial slap, cramped confinement, wall standing, stress positions, insects placed in a box confinement, and the notorious water boarding, which have obviously been applied to culprits in Guantánamo Bay and secret CIA camps since at least 2002.

 

The New York Review of Books will publish the International Committee of the Red Cross report on the treatment of fourteen “High Value Detainees” in CIA custody later this month. A pdf file can be downloaded here.

 

President Barack Obama, who has promised to close down the Guantánamo camp in Cuba within 18 months after taking office on January 20, has banned the use of, for instance, sleep deprivation and simulated drowning but has made clear that he would not prosecute CIA agents who have been involved in the mentioned “enhanced” interrogation procedures. “[T]hose who carried out their duties relying in good faith upon legal advice from the Department of Justice… will not be subject to prosecution.” A grave mistake. Obama’s decree will definitely not prevent others from conducting torture. Torture is to be called torture. It is illegal under both American and international laws. Prosecution is in fact mandatory.

 

 

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.

 

 

 

The Anointed

The messianic demeanor of the presidential hopeful Barack Obama in his “The World that Stands as One” at Berlin’s Victory Column in July 2008 in front of a crowd of 200’000 rejoicing Germans had been criticized both in the US as well as in Europe, even Germany. How can a man who, at that time, had not even a clear program in case of his election, attract and enthuse so many ingenuous citizens of an, admittedly at least allied, foreign country? Is it typical German subservience? It might be not. After his American ta’arouf to the Iranian people on the occasion of Nowruz, his brilliant speeches in Strassburg and Prague, now Ankara.

 

In fact, Obama finds the right words. This approach lets him succeed even in the extremely difficult task to prevail in the controversial case of the Danish Prime Minister as NATO’s next Secretary General when European leaders in general fail.

 

I read Obama is going to give a speech in an Islamic capital next (Ankara doesn’t count). Tehran would be impossible. The mullahs would be swept away in minute.

 

 

Floss or Die

New Orleans was still the Big Easy. Hurricane Katrina devastated the southern metropolis about a decade later, and my attendance of the annual meeting of my main professional society was a nice relief of the daily monotony at my dental school at home. I met many of my colleagues and I had a fantastic dinner on a Mississippi paddle steamer which was sponsored by a company with which I was working at that time.

 

It was one of the very first occasions that scientists in my discipline talked about a possible relationship between poor oral health and dangerous chronic conditions such as cardiovascular and cerebrovascular diseases. One of our most distinguished Professors in the field commenced his presentation with the message “Floss or Die”, a really frivolous statement. He later put the alleged relative risk for getting an infarction or stroke to 1.3 or so in case of more severe oral disease. It was the beginning of an endless discussion which mostly interested the numerous practitioners who, whithout any knowledge about odds or risk ratios, bothered their patients during the next decade with the menace of myocardial infarction, ischemic stroke, low birth weight, etc., if their oral diseases had not properly been treated. Most of this has vanished in the meantime and more realistic views have emerged.

 

What I didn’t know at that time was that the distinguished Professor in Oral Biology had just received an award in Economics. Not the real Nobel Prize but the “IgNobel Prize”. It stands for ignoble, and it is in fact a parody for achievements that “first make people laugh, and then make them think.” The chief sponsor of the prize is the Annals of Improbable Research which is a successor to a periodical that satirized scientific publications. That particular year, Jacques Chirac was among the laureates who, as the French President, had just launched a series of atomic bomb tests in the Pacific Ocean while the world was observing the 50^th anniversary of the Hiroshima massacre. It was the Peace prize, of course. Understandably, he didn’t show at the ceremony in Harvard University’s Memorial Hall where the prizes were conferred about 2 months before the real ones each year since 1991.

 

Our distinguished Professor got the prize in Economics for his remarkable discovery that “financial strain is a risk indicator for destructive periodontal disease.” Of course, he didn’t accept the prize either but rather ignored the event. It was long before the financial markets went out of control, somewhat before the dot-com bubble emerged and finally burst in 2000. People at risk for gum disease (according to our Professor those in financial trouble) have not participated in either bubble, I suppose.

 

All of this is certainly not related to the decline of periodontal disease which has been observed over the last decade or so. A slide with “Floss or Die” is still in use to attract the interest of my undergraduate and graduate students. But as ever, more as a joke than a serious suggestion.