“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.”
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 17th century (Shi’ite Safavid) resembles a Swastika, and its calligraphy mentions Ali . 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” . 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 .
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 10th century, artisans were thoroughly taught in a distinct academic context by mathematicians in geometry. Alpay Özdural (d. 2003) describes  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 , 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 11th and 15th 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 13th 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  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 .
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% .
In particular in the 13th 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 . 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” , “[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.”
 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 17th century, Muhammad ibn Mu’min Muhammad Amin.
 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.
 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.”
 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.”
 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.