About a new discovery in Islamic mathematics and its application to Islamic ornament

Apparently a new article from unlikely (though highly welcomed) authors could have made a new contribution to the study of Islamic art in general and Islamic ornament in specific. Peter Lu, from Harvard's Department of Physics, and Paul Steihardt, Princeton's Department of Physics, wrote an article in the last issue of Science (Feb, 23) arguing for a new discovery with regards to Islamic mathematics and its application on Islamic ornamental designs (a supplement with more pics could be found here). The article has made it to the headlines (unusual for studies on Islamic art) in the BBC (where we know that Lu is the major contributor and who came only by chance into Islamic art after being inspired by some buildings while wondering in Turkeminstan: "Mr Lu, who designs physics experiments for the International Space Station, was in the region in order to visit a space facility in Turkmenistan"). We find it also in the headlines of the Discovery Channel where we read notably: "But if it's right, 'this would be a hitherto undiscovered episode in the spectacular developments of geometry in central Islamic lands ... achieved by artisans probably inspired by theoretical mathematicians,' said Islamic art specialist Oleg Grabar".

This is really a two-fold argument, which is not really elaborated as it should due understandably to the authors' restricted background. I think both of them, who have done a great job as mathematicians observing a field that is not really familiar to them, suggest a promising venue with regards to the issue of the relationship between art/architecture making and contemporary mathematical literature. But this is not the first contribution on this topic. Few scholars on Islamic art have already suggested serious ideas about this (there are many who write some generic stuff but not really with any real substance). My adviser Renata Holod is among those who have written on the subject specifically on the 10th century mathematician from Baghdad Al-Buzajani (d. 998 AD) and his probable impact on Islamic architecture: Holod, Renata. 1988. “Text, Plan and Building: On the Transmission of Architectural Knowledge”. In Theories and Principles of Design in the Architecture of Islamic Societies. Margaret Bentley Sevcenko (ed). Cambridge, Massachusetts: Aga Khan Program for Islamic Architecture), which is available here (By the way both authors used her archival photographs of the Buzajani’s manuscripts). Another major contribution on the same topic that was written after a decade of Holod's article is: Ozdural, Alpay. "A Mathematical Sonata for Architecture: Omar Khayyam and the Friday Mosque of Isfahan" Technology and Culture, Vol. 39, No. 4 (Oct. 1998): pp. 699-715. Although it must be said that the principles suggested in these contributions were not accepted by all experts (as it's shown in: Bloom, J. "On the Transmission of Design in Early Islamic Architetcure" Muqarnas vol. 10 (1993): pp. 21-28), but Lu and Steinhardt's article would push further the debate towards the lines proposed by architectural historians such as Holod and Ozdural.

This is just to say that there are prior contributions with regards to this issue; and more importantly the way it was argued in Lu and Steinhardt's article could be enhanced especially when taking into account the generic question discussed among art historians notably those working on Islamic art and architecture, which is the issue of the nature of the relationship between makers of art/architecture and mathematicians or other sections of the medieval intelligentsia… Still Lu and Steinhardt’s article shows an essential thing that is the importance of interdisciplinary approaches to the fields of humanities… There is no doubt that only the eye of a mathematician that could ever see the patterns of Islamic ornament highlighted in their article.

The first part and underlying argument relates to the purely theoretical-mathematical level and shows some precedence in Islamic mathematics with regards to western-European mathematics:

1-Decagonal and pentagonal rotational symmetries are thought to be “impossible” as expressed in 19th century studies (“The visual impact of these girih patterns is typically enhanced by rotational symmetry. However, periodic patterns created by the repetition of a single “unit cell” motif can have only a limited set of rotational symmetries, which western mathematicians first proved rigorously in the 19th century C.E.: Only two-fold, three-fold, four-fold, and six-fold rotational symmetries are allowed. In particular, five-fold and 10-fold symmetries are expressly forbidden”). Then there are the “Penrose tiles” or quasi-crystalline forms, which were demonstrated in western mathematics only recently (in the 1970s). This far more complicated development that comes up from rotational symmetry (involving the decagonal and the pentagonal) but differs from it by providing the opportunity of freeing the generated forms from repetition or order is explained as follows: “A subdivision rule, combined with decagonal symmetry, is sufficient to construct perfect quasi-crystalline tilings—patterns with infinite perfect quasi-periodic translational order and crystallographically forbidden rotational symmetries, such as pentagonal or decagonal— which mathematicians and physicists have come to understand only in the past 30 years. Quasi-periodic order means that distinct tile shapes repeat with frequencies that are incommensurate; that is, the ratio of the frequencies cannot be expressed as a ratio of integers. By having quasi-periodicity rather than periodicity, the symmetry constraints of conventional crystallography can be violated, and it is possible to have pentagonal motifs that join together in a pattern with overall pentagonal and decagonal symmetry”

2-In Islamic mathematics the ways of building decagonal and pentagonal forms were “documented by Islamic mathematicians” since the 10th century (from a 10/3 star unit cell). The authors refer here (note 7) to drawings in Al-Buzjani’s manuscripts. These drawings, however, provide only the way to build unit cells and not necessarily the way to duplicate them, and thus, to make rotational symmetries. The more serious contribution appears to have happened some time later but it is discovered by the authors in ornaments found in architectural monuments beginning from the end of the 12th century: “we suggest that by 1200 C.E. there was an important breakthrough in Islamic mathematics and design: the discovery of an entirely new way to conceptualize and construct girih line patterns as decorated tessellations using a set of five tile types, which we call “girih tiles””. The main solid point to argue that this is a departure from the unit cell (10/3 star) we find in Al-Buzjani’s drawings is given in this paragraph: “Girih tiles further enable the construction of periodic decagonal-motif patterns that do not arise naturally from the direct strapwork method. One class of such patterns repeats pentagonal motifs but entirely lacks the 10/3 stars that establish the initial decagonal angles needed for direct drafting with straightedge and compass. Patterns of this type appear around 1200 C.E. on Seljuk buildings, such as the Mama Hatun Mausoleum in Tercan, Turkey (1200 C.E.; Fig. 2A)”. The set of “girih tiles” that are corresponding 5 tiles of different shapes, which enable rotational symmetries the moment are put together are: “the decagon, 10-fold symmetry; the pentagon, five-fold; and the hexagon, bowtie, and rhombus, two-fold.” (Fig. I F). But unlike Al-Buzjani where we know the actual mathematician and his manuscripts we don’t know (at least up until now) who was behind this new complicated process that could not be invented with pure practice (I’m thinking as I’m sure many others are of an exceptional mathematician like Al-Khayyam especially that he is usually mentioned in prior studies on the topic of mathematical impact on architecture and that most cases mentioned in the article are from the eastern Islamic lands where Al-Khayyam was located and more influential as wrote mainly in Persian). More importantly than this systematic way of building rotational symmetries involving the decagonal and the pentagonal led to a new development that is quasi-crystalline forms as proven in ornamental examples found on buildings from the 15th century (we don’t know it actually happened among Muslim mathematicians). Here is the explanation of this connection between the 12th century and the 15th century developments: “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… The large, thick, black line pattern consisting of a handful of decagons and bowties (Fig. 3C) is subdivided into the smaller pattern, which can also be perfectly generated by a tessellation of 231 girih tiles”…. We find in the 1970s “Penrose tiles” two approaches to construct tilings but only the second approach is proven to be used in the indicated ornaments : “The second approach is to repeatedly subdivide kites and darts into smaller kites and darts, according to the rules shown in Fig. 4, A and B. This self-similar subdivision of large tiles into small tiles can be expressed in terms of a transformation matrix whose eigenvalues are irrational, a signature of quasi-periodicity; the eigenvalues represent the ratio of tile frequencies in the limit of an infinite tiling”….

Now the second fold of the argument relates to the mathematical applications on Islamic art, which is as far as I’m concerned the more interesting subject. The authors mentioned that the drawings found in Al-Buzjani’s manuscripts could have been used as a possible written source for repeating the same process (drawing by “using the direct strapwork method” as shown in Fig. I A to D). This seems to be, however, unlikely when put into practice and here comes the first solid point of this article: “These complex patterns can have unit cells containing hundreds of decagons and may repeat the same decagonal motifs on several length scales. Individually placing and drafting hundreds of such decagons with straightedge and compass would have been both exceedingly cumbersome and likely to accumulate geometric distortions, which are not observed”. Therefore rotational symmetry using the five set forms seems to be highly useful especially with regards to the process of repetition and this seems to me the most important element that explains its use by the designers. And here comes the most important element in this whole analysis: the designers could not be improvising or repeating just by remembrance of basic formulas or even imitating images of designs, but they must have been following specific instructions probably written graphic instructions/designs. This is highly important with regard to the issue of the transmission of artistic knowledge. Obviously as higher is the level of complexity as much as the Muslim artist is in need of such written instructions. The authors say: “Our analysis indicates that Islamic designers had all the conceptual elements necessary to produce quasi-crystalline girih patterns using the self-similar transformation method: girih tiles, decagonal symmetry, and subdivision.” But needless to say that those conceptual elements could not be read, understood, and finally applied by any artists. In other words in the case of the highly complicated quasi-crystalline patterns he would not only be in need of such instructions (in the process of repetition) but he would be also skilled enough to be able to read such instructions. The most evident surviving source proving the existence of such written/drawn graphic instructions are found in the “Topkapi scroll” (series of drawings in a scroll 30 meters in length located now in Istanbul’s Topkapi Sarayi Library-Ms. 1956, but they are Persian and probably made between the 10th and 15th centuries). This is in fact an opportunity to go back to these drawings and dig more out of them notably by comparing them in a systematic way (understandably Lu and Steihardt did only a selected comparison) with surviving architectural ornaments. The excellent work by Necipoglu (Necipoglu, Gulru (1995) The Topkapi Scroll: Geometry and Ornament in Islamic Architecture. Santa Monica) has already laid down the preliminary ways to read them but with these new insights there are clearly new venues to deal with them.

Politicizing archeology: on the recent "excavations" in Jerusalem

Yuval Baruch, the managing director of the current "excavations" in the area of the Maghribi quarter in Old Jerusalem (otherwise called the "Mughrabi ramp"), on behalf of "Israel Antiquities Authorities" (IAA), wrote a statement about the "Real Story" behind the current crsis defending the "right of excavation" in the area. I actually knew of this statment through a news report in the London-based Al-Hayat newspaper (I found out later that the author of the report gave a distorted reading of Baruch's article by claiming that he uncovered the "reason of the current crisis" that is presumibly a "cover-up" (Ar. "tasattur") of a newly discovered (in 2004) small prayer hall in the area... Baruch's article mentions this discovery but he does not really say that they are trying to make a cover up.... especially that the the IAA is always finding and registring Islamic monuments).

Before commenting on Yuval Baruch's statement I must say that the IAA conducted major excavations all over Muslim Palestine and actually contributed a great deal of discoveries with regards to Islamic archeology notably with regards to the early period (Umayyad and Abbasid periods). One of the recent examples is the results of the exacavations of Bisan ("Beit Shean"), which gave an additional case proving the early developped urban sense of the new Muslim rulers of the former Byzantine territories. I must also say that I was interested in and I'm a bit faimiliar with "Jewish art" including the representations of "The Temple" throughout history (I enrolled few years ago on a course on this topic, which was after all interesting especially in the ways it shows how Jewish makers in Egypt, Yemen, and Muslim Spain were very much adopting an "Islamic visual language" when making their Bibles and Synagoges). On the other hand I'm stunned with the high level of politicisation of archeological investigation as expressed in Baruch's statement.

The major problem with Yuval Baruch's narrative is that it already assumes an ideological-poltical position from the start. Regardless of all his arguments the very attribution of toponyms such as the "Temple Mount" to the site of Al-Haram Al-Sharif is essentially against the ethics of archeological investigation. Here we have an archeologist who not only conducts an excavation in order to prove the existence of a building that would serve for an ideoloical-politcal goal (that is "Jerusalem is Jewish") but he simply talks about this building as if it really exists merely because few written sources (not contemporary with the events of foundation) have suggested in general terms the existence of this building. This is basically an anti-archeological approach since we do archeology, after all, to overcome the textual sources.

Regardless of all the political circumstances and only from an archeological point of view we have to admit that:

-First, Al-Haram Al-Sharif (that is both the Dome of the Rock and Al-Aqsa Mosque) is the currently existing archeological site, we can't deny its existence (by adopting another toponym) for ideological-political reasons.

-Secondly, all written sources and the few archeological evidences (developped since the beginning of the 20th century by Hamilton and Creswell) suggest that the Islamic buildings of Al-Haram Al-Sharif were built on rubbles or at least already destroyed structures, which means that they belong in their totality to a different and new archeological strata and thus we (as archeologists) we can't deny their original identity that gives them the right to be protected as separate and more importantly surviving archeological sites (which are rare for the Umayyad period: that is the founding period of Islamic architetcure). This simply means that any archeological investigation should never threaten their integrity even if that would mean the discovery of new archeological structures.

Besdies from a political point of view it's really difficult not to ask a very simple question: if we (all the peoples of this earth) decide to claim political authority over places (territories, buildings...) solely because each one's scripture gives a religious right of appropriation on any given site in that case, then, we can't as Muslims prevent other (stupid) Muslims to claim, for instance, Cordoba as "taken from us" since it was part of Dar Al-Islam.

The problem of the "Temple Mount" (i.e. Al-Haram Al-Sharif) is essentially the problem of Old Jerusalem; and the problem of Old Jerusalem should be (and especially because of its special religious status for many believers of various faiths) a political problem that needs to be discussed not from the perspective of premodern laws (that is the absence of laws except the absolute right of violent appropriation) but from the perspective of modern international law. Otherwise we'll only legitimize anarchy. The question here is and will always be why should the "Jewish People" given Old Jerusalem's political authority? Why is it that the value of the Jewish religious right is higher than the Christian or the Muslim religious rights? The essential political answer here is the very obvious presence of a population of Old Jerusalem that is the Palestenian population of Jeruslaem that was never given the basic right of choosing its political affiliation.

An archeologist (that is someone who knows better the meaning of human stratigraphy within historical contexts) would be better suited to understand that giving a supremacy of religious right to the Jewish People is nothing more than a premodern way of approaching the politics of Jerusalem. He also, to preserve his very archeological raison d'etre, should never be implicated in the premodern process of destroying monuments for the only reason of reviving older monuments. Archeology, after all, was possible only after the human community decided that all past is valuable AS IT STANDS. It's true that archeology is essentially a process of destruction but never is it a process of destroying surviving and active/alive buildings.

The question will never be whether these "excavations" are archeologically justified or not because simply a seemingly apolitical justification should be recognized by any archeologist not as the right explanatory context for the whole crsis. In other words it's really unethical precisely from an archeological point of view to concede that the IAA as a body belonging to an occupying force (according to modern international law) is the legitemate (thus the adequate) party that could be given the "right of excavation" (and much less the "right of arguing for the right of excavation") anywhere in Old Jerusalem.

P.S.

Just a marginal note on the way Yuval Baruch dealt with the question of the Islamic "Buraq Wall", which shows a concrete example of inadequate polticization of archeological and historical questions. To assert a certain Jewish exclusive right for holiness in the "Western Wall" he debates the "Muslim claim" of holiness of the same wall (as being the "Buraq Wall") as simply a modern invention (in the 1920 by the Mufti of Jerusalem) only to counter the Jewish attempts to buy it or control it ("After the Balfour Declaration, the Zionist institutions began to emphasize the Western Wall as a national symbol of the Jewish people, in addition to its religious significance. This action led the Mufti of Jerusalem to claim that the Jews intended to take control of the Western Wall, so he declared the Wall  with no religious or historical substantiation  a holy Moslem site. This wall of stones, to which the Muslims ascribed no importance, was thenceforth called El Buraq, after the name of the magical horse of the Prophet Mohammed"). But in a later paragraph he admits the presence of a premodern Muslim belief in the holiness of the wall (precisely as the "Buraq Wall") though he ascribes it only to the 15th century ("The first connection between El Buraq and this area can be ascribed to Mujar al-Din, a 15th century Jerusalem judge... "). In fact this connection goes back even beyond the 15th century. This is for example (and this is only after a very preleminary investigation) explicitely stated by the 13 th century Abu Yahya Al-Qazwini (d. 1283 AD) in his Athar Al-Bilad in the following paragrpah in which he describes Al-Haram Al-Sharif (sorry I don't have an English version of this):

وفي صحن المسجد مصطبة كبيرة في ارتفاع خمسة أذرع، يصعد إليه من عدة مواضع بالدرج، وفي وسط هذه المصطبة قبة عظيمة مثمنة على أعمدة رخام مسقفة برصاص، منمقة من داخل وخارج بالفسيفساء، مطبقة بالرخام الملون. وفي وسطها الصخرة التي تزار، وعلى طرفها أثر قدم النبي، عليه السلام، وتحتها مغارة ينزل إليها بعدة درج يصلى فيها. ولهذه القبة أربعة أبواب، وفي شرقيها خارج القبة قبة أخرى على أعمدة حسنة يقولون: انها قبة السلسلة. وقبة المعراج أيضاً على المصطبة، وكذلك قبة النبي، عليه السلام. كل ذلك على أعمدة مطبقة أعلاها بالرصاص، وذكر أن طول قبة الصخرة كان اثني عشر ميلاً في السماء، وكان على رأسها ياقوتة حمراء كان في ضوئها تغزل نساء أهل بلقاء.

وبها مربط البراق الذي ركبه النبي، عليه السلام، تحت ركن المسجد.

Cleopatra's Face... I don't like it!

As shown in this 32 BC roman coin Cleopatra's profile does not show really the Shakepearean idealistic image (let's remeber that she was never really described in the roman contemporary sources as beautiful)...

Just a reminder of the greatness of coins as historical sources: in the Greek-Roman-Byzantine cases the ruler's profile was a must and it was usually much more realistic than images in other media (including sculpture)....

Links: English 1, English 2 (constructing the early modern idealistic figure), Arabic, and French.