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