### 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”….

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.

## 5 Comments:

Information très intéréssante. peut-on avoir le texte en français ?

I don't know if I can re-write in French... it seems a lot of work for me right now...

I thought this would be of interest to you

http://www.dailymotion.com/video/x17rtx_

Thanks... it's funny how whoever posted the documentary (who is Egyptian anyways) writes (Turath): TURAS!!! I had it with the egyptians...

Thanks Tarek for your nice post!

Free PS3

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