From a mainly architectural point of view, Esfahan contains one of Islam’s most significant buildings, its marvelous Masjed-e Jomeh . As Oleg Grabar puts it in his wonderful book , one might rightly consider Esfahan’s Great Mosque as the “Chartres of Iran”. In terms of importance it has been compared, for example, with the Haram al-Sharif in Jerusalem, the Umayyad Mosque in Damascus, Cairo’s Ibn Tulun Mosque, the Alhambra in Granada, Istanbul’s Süleymaniye, or the Taj Mahal of Agra.
Of course, Esfahan is rich in stunning architectural sights, and as a matter of fact, Shah Abbas’ the Great (d. 1629) celebrated and majestic Safavid ensemble of a huge square, two magnificent mosques, and the gorgeous Ali Qapu palace, which is located in the modern city center, attracts much more attention of tourists and visitors. Esfahan’s Great Mosque is located about one kilometer to the northeast and may be reached while strolling through the city’s famous main bazaar. At first glance it seems to be humble and rather hidden albeit the entire complex clearly dominates the old city, the Great Seljuq Empire’s capital.
A Few Remarkable Details
Esfahan’s Friday Mosque is special for a great number of reasons. They cannot be listed here in any detail. But some aspects, which have recently attracted my interest, may be mentioned. First, one would not get a complete overview of its huge dimensions, with 170 by 140 meters by far the largest mosque in Iran, unless from a hot-air balloon or helicopter. Its fabric completely blends into what remains of the old dwellings. There are no clear bordering walls or even easily identifiable entrances. The court of the mosque is the center of the building, the space from which it spreads out to the periphery, the surrounding urban space .
Construction work on the mosque had been done for more than one thousand years, from the 8th through the 17th and 18th centuries. Six different buildings or periods have been identified. The first mosque, built in 771 CE, was located in the Jewish quarter of the city of Jayy, or Yahudiyyah. The reason for this location remains obscure. Esfahan itself is said of having been founded by Jews who settled at the Zayandeh Rud in Central Iran, after having been freed by Cyrus the Great from their Babylonian Captivity in the late 6th century BCE. The Jewish quarter in Esfahan may have been more urbanized after the Arab conquest than other settlements in the lush oasis on the river banks. It might be that the first mosque was built on the foundations of an even older temple. It is possible that it was a Jewish, rather than Sassanid temple as Sylvia A. Matheson speculates . Thus, Yahudiyyah might have been a city built by Jews after the destruction of their temple in Jerusalem .
The still visible main complex is the Seljuq mosque of the 11th century. It is the first mosque in Iran where the four-iwan arrangement of vaulted recessions located midway the four sides of the huge court was adopted. Two are in the qibla direction towards Makkah and two are located perpendicular to this main axis. It is said that the ensemble is in the tradition of pre-Islamic brick vaults of Sassanid palaces. The mosque had been burnt by the Assassins in 1121 CE but little evidence for this incident has been found.
Very recently, a pattern on the portal’s inner sides of the western iwan of the mosque had been mentioned, which strikingly resembles the quasi-crystalline tiling found on the Darb-i Imam shrine (1453 CE) but is distinctly periodic. The Darb-i Imam is located about 300 meters west to the mosque in the old Dardasht quarter . A limited set of three girih tiles (equilateral decagons, elongated hexagons, and bowties), decagonal rotational symmetry, and application of a subdivision rule were the prerequisites enabling medieval artisans to create quasi-crystalline patterns by using self-similar transformations; 500 years earlier than mathematicians and artisans in the West. The description of this scientific sensation by Lu and Steinhardt  has found world-wide attention last year. An amazing finding was that the pattern on one spandrel of an arch at the Darb-i Imam shrine contains a few point defects where in fact Penrose matching rules for aperiodic tiling were violated. More information about this amazing discovery may be found here.
In contrast to this example of a quite small, quasi-crystalline, tiling on the now rather famous Darb-i Imam, the tiling on Esfahan’s Friday Mosque is distinctly periodic and huge, about one meter wide and extending up to the height of the arch, i.e., ten meters or more, where it can be shown to be absolutely symmetric on both sides with a rare, in fact incredible, precision. How medieval artisans were able to decorate these vast areas with highly complicated decagonal patterns at this level of precision is certainly worth studying in greater detail .
The vast building comprises an unbelievable number of 476 cupolas  but the two large brick domes, the one in the south and the smaller but exquisite in the north are very special . The southern dome in front of the mirhab was constructed by the Seljuq sultan Malikshah (d. 1092) and, in particular, his visionary vizier Nizam al-Mulk (assassinated in 1092) in 1086-87. It was the largest dome of the Islamic world at that time. Rather sturdy than elegant, the brick dome was the first which exhibited, on a grand scale, the detailed muqarna zone of transition.
One may immediately ask why there is a second major dome. The northern Gunbad-e Khāki, or “the earthly dome”, was commissioned by vizier Nizam al-Mulk’s rival and arch enemy Taj al-Mulk, and completed in 1088, one year after the construction of the southern dome had been accomplished. It comprises a squared pavillon with a dome resting on an octagonal zone of transition. While technique, style, and decorative structure of this prototype southern dome were rapidly repeated all over Iran, the northern dome has no model. The cupola itself sports a pentagon around an open oculus. Proportions of the dome have been described as basically applying the principle of the Golden ratio. This has recently been questioned, and I will discuss that interesting observation below. Inscriptions on the northern dome, which had once been conceived outside of the mosque, include a not common proclamation of cosmic power (Q7: 54) and another of extraordinary piety (Q3: 26-27), and may be contrasted to the more prosaic call for prayers on the outside of the southern dome, curiously including even a proscription of illegal sex. As Oleg Grabar  puts it, “[S]omething is going on here which does not deal simply with building a religious building we generally know as a mosque.”
Close to the northern dome, at its northeastern gate, the mosque contains another inscription from the Qur’an  which is also rarely found on a monument. It adds that the building (imarah, not masjid) was restored after a fire in 515 AH (1121-1122 CE). The burning of parts of the mosque was due to members of the heretic Batiniyyah sect, the Assassins, or Hashashiyyin. Oleg Grabar suggests that the inscription refers to some particularly revolting form of desecration that might have befallen the mosque.
Another eminent inhabitant of the Seljuq Empire’s capital in the late 11th century was Omar Khayyam (d. 1122). Due to the inaccurately re-created and romanticizing translations  of The Rubaiyat of Omar Khayyam by Edward Fitzgerald (d. 1883), Omar Khayyam is mainly known in the West as a poet. However, Khayyam (the ‘tentmaker’) is considered as one of the greatest mathematicians and astronomers of his time who has even been contributed to the eventual development of non-Euclidean geometry. In his calendar reform of 1079, he estimated the length of the year with unprecedented precision (365.24219858156 days) and he advocated heliocentrism well before Copernicus.
Omar Khayyam was born in Nishapur in Khorasan in 1048. In 1070 he moved to Samarqand where he wrote his famous Treatise on Demonstration of Problems of Algebra and several other books some of which are now lost .
On invitation of Malikshah and his vizier Nizam al-Mulk, Khayyam moved to Esfahan in 1074 in order to set up an observatory there. After his friend and benefactor Nizam al-Mulk had been assassinated in 1092, support for the observatory ceased and the calendar reform was put on hold. Khayyam came under attack from orthodox Muslims who felt that especially his open mind did not conform to Islam. He left Esfahan in 1118 for Merv in what is now Turkmenistan and died in 1122 in his home city Nishapur.
The period of 18 years during which Omar Khayyam lived in Esfahan saw certain remarkable achievements in the city’s architecture. One of these is the erection of the majestic South Dome in Esfahan’s Masjid-e Jomeh in 1086/87. And the other, only two years later, its geometrically perfect counterpart: the in a way mysterious northern dome. One might intuitively ask: has Omar Khayyam been involved in the construction of the latter?
Omar Khayyam’s observatory, the main reason for moving from Samarqand to Esfahan, had never been identified in the city. It could have been destroyed, of course, during the Mongolian conquest led by Ghengis Khan in 1218 and that of Tamerlane in 1387. Oleg Grabar  mentions that Eric Schroeder considers the North Dome as the mysterious observatory . The perfection of the North Dome led Grabar to assume the presence in Esfahan of a particularly ingenious architect. He reports on Schroeder’s argument that since its proportions were derived from the Golden ratio, and Omar Khayyam at that very time had identified properties of the pentagon, he might have been the designer .
However, the recently deceased mathematician Alpay Özdural  argues that Schroeder’s syllogism may be based on inaccurate information. Accurately surveyed dimensions of the North Dome show his proportions to be only approximations  differing by about 1% from the expected proportions.
The Mathematics of the North Dome Chamber
While Schroeder’s speculation of Omar Khayyam being possibly the designer of the North Dome may in fact be unverifiable, Özdural draws the attention to the properties of a specific right triangle described by Khayyam which may be linked to the proportions of the dome . With regard to the harmonic, arithmetic and geometric means, the most widely used ratio systems in Greek and Islamic mathematics, Omar Khayyam’s triangle has unexpected and surprising properties, which are outlined in considerable detail in Alpay Özdural’s paper . For instance, the proportion composed of arithmetic and harmonic means between two given numbers had been called the “musical proportion” and judged most perfect in ancient times. As Özudral notes, Omar Khayyam’s triangle comprises richly interrelated proportions and even presents itself provocatively as a tool suitable for architectural application. And, the musical proportion of irrational magnitudes is a most interesting property of this triangle.
Özural’s own thorough analysis of the construction of the building  is based on digitized drawings of a photogrammetric survey by the Rassad Survey Company, Tehran, which was published in 1974. The construction proved to be exceptionally precise with deviations of horizontal and vertical elements from true positions within ±0.02 m or, on average, 0.3%. The position of the vertical axis of the central arches of the dome is even more significant. “The axis crosses the apex of each arch and of the dome with an error of about 0.1%. To attain such precision aligning the dome axis with one of the arches would be a difficult task using modern technology.” Even more amazing is that the geometry of the dome appears to be generated wholly from Omar Khayyam’s triangle (and not the Golden ratio, as has frequently been reported by others) with an average deviation of 0.012 m or 0.2%. And finally, as Özdural is not growing tired to mention, “[t]he proportion that Omar Khayyam posed as the objective of his untitled treatise … appears as a constant theme underlying all musical proportions of the chamber, filling the dome with the echoes of the geometric mean between the span and the whole transition zone.” A mathematical sonata for architecture.
Supposed, work on the northern dome had started in the early 1080s and the properties of the specific right triangle had been discovered by Omar Khayyam shortly before that, is it possible that he was the designer of the North Dome? The disparities with the South Dome with its lack of precision makes it inevitable that different master-builder were involved. The architect of the South Dome was Abul’l Fath, son of Muhammad the Treasurer . Another argument may be the pentagonal star formation of interlacing transverse ribs in the dome, the earliest example of decoration on the concavity of a domical surface. “[T]he geometric schema must have been planned on a two-dimensional surface, following which the straight lines were transformed into curves. As the ribs and the dome itself were constructed together, this unprecedented idea should have been conceptualized and worked out during the design process. Who but Omar Khayyam in Isfahan could have had the vision and expertise to conceive such a revolutionary design?”
The Earthly Dome
In spite of conducting the obligatory pilgrimage to Makkah late in his life, Omar Khayyam’s intellectual skills, his revolutionary thoughts, and broad mind may have created problems with the authorities and even his sponsors. Here, we can return to the unusual inscription at the northeastern gate of Masjid-e Jome in Esfahan . As has been mentioned already, the building is rich in unusual inscriptions, but this one which, as Oleg Grabar  stresses, may in fact point to a revolting desecration, and might explain, as Özdural writes, why Omar Khayyam’s name was never mentioned as the designer of the North Dome.
The dome was known locally as Gunbad-I Khāki, or the earthly dome. Alpay Özdural closes his remarkable treatise with one of Omar Khayyam’s quatrains which might give us a final hint about the construction of the mysterious North Dome of Esfahan’s Great Mosque:
My beauty’s rare, my body fair to see,
Tall as a cypress, blooming like the tulip;
And yet I don’t know why the hand of Fate
Sent me to grace this pleasure-dome of Earth.
 Generally recognized holy places in Islam are, of course, only Al Masjid al-Haram in Makkah, Masjid an-Nabawi in Madinah, and Masjid al-Aqsa in Jerusalem, all related to the Prophet of Islam (PBUH). Shi’a holy sites include the tombs of the Imams and their relatives. The most important in Iran is the holy shrine of the Eighth Imam Ridha in Mashhad which is located close to the gorgeous Azim-e Gohar Shad from the Timurid period.
 Grabar refers to a story of an Esfahanian Jew who refused to sell his house, which was located within the space needed for the mosque unless he was paid an exorbitant price, a kind of early anti-Jewish sentiment.
 Dating of the specific decoration pattern on the Friday mosque also to the late 15th century (as that on the Darb-i Imam) might in fact be possible. Architectural artisan Sayyed Mahmud-I Naqash had signed his name, for example, to the decoration of the famous Timurid gate close to the western iwan and dated it 850 AH (1447 CE). Also the 1475/76 decorative additions on the southern iwan may stem from this exceptional artisan. The decorations, in a way marking the apogee of 15th century Islamic architecture, are most notable for their fineness in scrollwork and calligraphy featuring floral patterns and twines rather than the rigorous geometrics seen on the western iwan. According to Oleg Grabar (1990), both location and purpose of the gate are somewhat problematic. It is assumed that the gate has been moved from elsewhere into its present location within the two-storey arcades later. A similar, floral, style in decoration can be found on the main gate of the Darb-i Imam (1453) which is attributed to Sayyed Mahmud, too, close to the quasi-crystalline, self-similar, geometric patterns which have been described by Lu and Steinhardt (2007). The same style can also be found on the southern iwan of the Great Mosque in Esfahan. All seem to be designed by Sayyed Mahmud (Hutt A, HarrowL. Islamic Architecture, Iran 2. Scorpion Publication Ltd., London 1979). However, the difference in styles (largely floral vs. strictly geometric, even quasi-crystalline) may point to completely different esthetic and, most important, intellectual concepts, and may have been created by different artisans and even periods. Since the two sites with this specific, strikingly similar decagonal, on the Darb-e Imam even quasi-crystalline, pattern are, so far, the only ones reported in Esfahan (I suppose even in the entire Islamic world), they may represent an anomaly related to a different artisan than Sayyed Mahmud. Grabar  mentions, for example, that for the first time the court façade was consistently decorated with muqarnas inside the iwans and of colored bricks and tiles outside under Mongolian rule in the late 14th century.
 Lu PJ, Steinhardt PJ. Decagonal and quasi-crystalline tilings in medieval Islamic architecture. Science 2007; 315: 1106-1110.
 Unless real girih tiles are found in contemporary tile factories in Iran or Uzbekistan the assumption that medieval artisans had used them might be considered a speculation. There are constantly areas of restoration and repair in the mosque and today’s craftsmen have to fully recognize the old patterns. Errors may in fact occur, as Lu and Steinhardt were amazingly able to show in their paper . Hillenbrand describes in his article from 1987 (Hillenbrand R. Aspects of Timurid architecture in Central Asia. Republished in the EJOS VI 2003; 20: 1-37) that artisans during the Timurid period were under high pressure when decorating huge surfaces of monumental imperial and religious buildings in, for example, Tamerlane’s capital Samarqand. It is more than reasonable to assume other techniques than compass and straightedge, as Lu and Steinhardt  did, to create highly complicated geometric patterns. Hillenbrand also makes clear that the centers of medieval architectures were located in Samarqand, Bukhara, Mashhad, and Herat rather than Esfahan.
 The larger southern dome, which may be dated to 1086-88, contains two preserved inscriptions, one praising Malikshah, one of the most brilliant rulers of the Great Seljuq Empire, and his famous vizier Nizam al-Mulk. The other inscription is a remarkable piece of advice for humble and pious daily life, even mentioning sexual conduct only with those who were joined in the marriage bond.
 Q2: 114: “And who is more unjust than he who forbids that in places for the worship of God (masajid) God’s name should be celebrated, whose zeal is in fact to ruin them? It was not fitting that such (people) should themselves enter these (places of worship) except in fear. For them there is nothing but disgrace in this world and in the world to become an exceeding torment.”
 Fitzgerald clearly belongs to these Western writers of Victorian 19th century who romanticized in their descriptions of what they found in Islamic countries with a Eurocentric supremacy, not aware of the great scientific achievements of what is wrongly called the ‘medieval’ Islamic world. An especially orientalist, in fact kitschy, description of Omar Khayyam and his time (and later Iran at the turn of the century one hundred years ago) is provided ironically by Lebanese writer Amin Maalouf in his popular fairy tale Samarkand (1988). It may be perceived with very mixed feelings, too.
 Treatise on Demonstration of Problems of Algebra contains a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Khayyam claims that the contributions by earlier writers (Al-Mahani, Al-Khazin) were to translate geometric problems of the Greeks into algebraic equations, something which was essentially impossible before the work of Al-Khwarizmi. Khayyam himself seems to have been the first to conceive a general theory of cubic equations. He also realized that cubic equations may have more than one solution. In his commentary on Euclid’s Elements, in trying to prove the parallel postulate Khayyam accidentally proved properties of figures in non-euclidean geometries. In this book, he also gave important results on ratios, extending Euclid’s work to include the multiplication of ratio. See further information here.
 According to Grabar , Eric Schroeder never published this unlikely hypothesis but expressed his opinion privately. “But, as usual, the direction suggested by Schroeder toward something unusual and unexpected may well be right.” P. 84, note 25.
 “Many years ago, the late Eric Schroeder suggested to me that, in his view, it was Omar Khayyam the mathematician who was behind the conceptual thinking that created the North Dome. It was he, after all, who had at that very time identified the various properties of the pentagon. Like so many thoughts and ideas spun around Isfahan, this is unverifiable but attractive.” Ibid. p. 85, note 5.
 Alpay Özdural. A mathematical sonata for architecture. Technology and Culture 1998; 39: 699-715.
 Ibid., notes 3, 4. Özdural writes that, according to Schroeder (Schroeder E. Seljuq Architecture. In: Pope AU and Ackerman P (eds.) A Survey of Persian Art, 2nd ed., Tehran 1977, 3:1005), the interior height of the dome is twice the base, but using actual dimensions, 19.27 m ÷ 9.90 m = 1.946 m. Schroeder claims the line where the transition zone begins divides the total height into two parts whose ratio is the golden ratio (½(√5 + 1):1 = 1.618…) but 19.27 m ÷ 12.04 m = 1.601. He adds that the ratio of heights of the upper to the lower main arches is also the golden ratio, whereas 10.81 m ÷ 6.60 m = 1.638. The golden ratio or golden section was called “extreme and mean ratio” in Greek and Islamic mathematics and was used in defining the pentagon and the decagon. In the west, it became the golden ratio after Luca Pacoli’s Divina Proportione (1509).
 The problem Omar Khayyam raised in an unnamed paper published by Ali R. Amir-Moéz (Scripta Mathematica 1963; 26: 323-37): divide a quarter circle with its center at A by a point B so that if BD is drawn perpendicular to the radius AH, the ratio AH:BD equals AD:DH. He assigns arbitrarily the value of 10 to AD and x to BD and reduces the problem to x3 +200x = 20x2 + 2000. After solving the cubic equation by means of conic sections, he suggested a more practical, approximate solution using astronomical handbooks and sexagesimal arithmetic. Interestingly, another practical, verging procedure to solve the problem can be found in an important anomynous Persian work on ornamental geometry probably from the 13th century, On Interlocking Similar or Corresponding Figures. It is a mechanical equivalent of a cubic equation and might be evidence, according to Özdural, for the close contacts of medieval Islamic scientists and mathematicians with the artisans in, how he calls it, conversazione.