page_title: Hagia Sophia
section_title: Architecture
hierarchical_section_title: Hagia Sophia / Architecture
caption_attribution_description: English: Hagia sophia mathematische Konstruktion
context_page_description: Hagia Sophia, officially the Hagia Sophia Grand Mosque and formerly the Church of Hagia Sophia, is a Late Antique place of worship in Istanbul. Built in 537 as the patriarchal cathedral of the imperial capital of Constantinople, it was the largest Christian church of the eastern Roman Empire and the Eastern Orthodox Church, except during the Latin Empire from 1204 to 1261, when it became the city's Roman Catholic cathedral. In 1453, after the Fall of Constantinople to the Ottoman Empire, it was converted into a mosque. In 1935 the secular Turkish Republic established it as a museum. In 2020, it re-opened as a mosque.
  Built by the eastern Roman emperor Justinian I as the Christian cathedral of Constantinople for the state church of the Roman Empire between 532 and 537, the church was then the world's largest interior space and among the first to employ a fully pendentive dome. It is considered the epitome of Byzantine architecture and is said to have "changed the history of architecture". The building was designed by the Greek geometers Isidore of Miletus and Anthemius of Tralles.
context_section_description: Hagia Sophia is one of the greatest surviving examples of Byzantine architecture. Its interior is decorated with mosaics and marble pillars and coverings of great artistic value. The temple itself was so richly and artistically decorated that according to much later legend, Justinian proclaimed, "Solomon, I have outdone thee!" (Byzantine Greek: Νενίκηκά σε Σολομών). Justinian himself had overseen the completion of the greatest cathedral ever built up to that time, and it was to remain the largest cathedral for 1,000 years up until the completion of the cathedral in Seville in Spain.
  The Hagia Sophia is of masonry construction. The structure has brick and mortar joints that are 1.5 times the width of the bricks. The mortar joints are composed of a combination of sand and minute ceramic pieces distributed evenly throughout the mortar joints. This combination of sand and potsherds was often used in Roman concrete, predecessor of modern concrete.
  Justinian's basilica was at once the culminating architectural achievement of late antiquity and the first masterpiece of Byzantine architecture. Its influence, both architecturally and liturgically, was widespread and enduring in the Eastern Christianity, Western Christianity, and Islam alike.
  The vast interior has a complex structure. The nave is covered by a central dome which at its maximum is 55.6 m (182 ft 5 in) from floor level and rests on an arcade of 40 arched windows. Repairs to its structure have left the dome somewhat elliptical, with the diameter varying between 31.24 and 30.86 m (102 ft 6 in and 101 ft 3 in).
  
  At the western entrance side and eastern liturgical side, there are arched openings extended by half domes of identical diameter to the central dome, carried on smaller semi-domed exedrae; a hierarchy of dome-headed elements built up to create a vast oblong interior crowned by the central dome, with a clear span of 76.2 m (250 ft).
  The theories of Hero of Alexandria, a Hellenistic mathematician of the 1st century AD, may have been utilized to address the challenges presented by building such an expansive dome over so large a space. The idea is that , which is not expressible as a rational number, was approximated and deliberately used as a less accurate rational value that approximates the irrational number . This is appropriate for 22/7 which was used for solving circle problems in construction. When it is ascertained that the circle circumference is U=D*22/7, several elegant solutions are produced, among them those which use a diameter which is cancelling the denominator. As thus Svenshon and Stiffel proposed that the architects used Heron's proposed values for constructing vaults. Central for the erection of the dome of the Hagia Sophia was that the central square corresponds to a diagonal, which is expressible as a rational number approximating . If this was not rightly done, all related measures to the square would not be manageable by the instruments of the surveyors of those times. As thus for the calculation of the square measurements derived from the so-called side-and-diagonal number progression were used. With its help, approximations for  can be produced, which is essential for measuring all types of square or square-related objects and surfaces (i.e. 1/1, 3/2, 7/5, 17/12, 41/29, 99/70). With this method, the squares defined by the numbers 12 and 17, whereas 12 defines the side of the square and 17 its diagonal, has been used as a standard value as early as in cuneiform Babylonian texts.
  As the great square in Hagia Sophia is 31 m long, it was previously thought, without any reassurance, that this length corresponded to 100 Byzantine feet. Yet such a round figure for the square side would lead to a diagonal with the irrational length of 141.421... because of the factor . This would mean that the square and all dimensions related to it would not be manageable. It has now been realised that in this context, the diagonal is nothing else but the diameter of the circle defined by the vault's circumference, while at the same time, as Heron's circle calcula
caption: The geometric conception is based on mathematical formulas of Heron of Alexandria. It avoids use of irrational numbers for the construction
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