A Mathematical Sonata for Architecture: Omar Khayyam and the Friday Mosque of Isfahan

Figure 1.

The North Dome Chamber. (Photo courtesy of Renata Holod.)

On the north end of the main axis of the Masjid-i Jami’ (Friday Mosque) of Isfahan stands the elegant North Dome Chamber of Terkan Khatun, built in 1088–89. It corresponds to the majestic, slightly earlier dome of Nizam al-Mulk on the south. Although the dome chamber is relatively modest in size and austere in materials, the art historian Eric Schroeder has judged it to marry genius and tradition more elegantly than many other famous domed structures (fig. 1). ¹ The identity of the designer of this remarkable building is a mystery that has intrigued many architectural historians.

The perfection of the North Dome led Oleg Grabar to assume the presence in Isfahan of a particularly inventive designer. ² Grabar reports Schroeder’s argument that since its proportions were derived from the golden section and the poet and mathematician Omar Khayyam at that very time had identified properties of the pentagon, he might have been the designer. ³ But Schroeder’s syllogism is based on inaccurate information: accurately surveyed dimensions of the North Dome show his proportions to be only rough approximations. ⁴ Also, although Omar Khayyam understood the properties of the pentagon, he did not report them in his works. ⁵

These defects do not mean Schroeder’s hypothesis is misconceived. Many Islamic documentary sources refer to special meetings—which I call conversazioni for want of a better word—between artisans and mathematicians, suggesting their frequent collaboration. ⁶ Ornamental patterns based on cubic equations, or conic sections, bear further witness to their association, since these powerful conceptual tools were available only to mathematicians at the time. Renata Holod has argued that a thorough awareness of engineering history may be necessary to explain aesthetic, structural, or spatial innovations in major Islamic centers. ⁷ Her case hinges on a critical word, muhandis, “engineer” in modern Arabic, Persian, and Turkish. Its original Arabic meaning was “geometer,” or “mathematician” in a broader sense. Modern historians usually translate references to muhandis involved in building activities as referring to engineers, whereas the original sense of the word would be more accurate. I believe muhandis evolved to mean “engineer” as more mathematicians became involved with architecture through conversazioni.

Schroeder’s attribution generally has been considered attractive but unverifiable. An untitled treatise by Omar Khayyam, first published in 1960, on a right triangle he discovered can shed new light on this issue. I shall readdress the question whether Omar Khayyam was the designer of the North Dome Chamber after analyzing the properties of his triangle and the proportions of the building.

Omar Khayyam’s Triangle

Figure 2.

Omar Khayyam’s solution of the problem by using a right triangle.

In his untitled treatise, Omar Khayyam proposes this objective: 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 (fig. 2). ⁸ He attempts first a solution based on conic sections but leaves that as an exercise for the student to complete. He then proposes an alternative method for readers not knowledgeable in conics. He defines the right triangle ABC as one whose hypotenuse equals the sum of the shorter side and the perpendicular to the hypotenuse ( AC = AB + BD ), and then proves that BC = AB + AD. He then assigns the arbitrary value 10 to AD and x to BD and reduces the problem to the cubic equation x³ + 200 x = 20 x² + 2000. ⁹

After solving the cubic equation by means of conic sections, Omar Khayyam offers another practical, approximate solution. Using astronomical handbooks and sexagesimal arithmetic he estimates the angle BAC as 57°, its sine as 0.833, its cosine as 0.544, and its versed sine (1 - cosine) as 0.455. ¹⁰ Computing the values with modern tools after assigning to the hypotenuse AC a length 1, we obtain angle...