The use of Euclidean geometry in the Parthian Empire urbanism, ancient round city of Darabgerd , Iran (Kermani, 2012) 

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... mathematics in architecture through emphasizing the role of mathematicians in writing mathematical texts which are related to architecture (Denny, Necipoglu, & al-Asad, 1998). Recently Iranian scholars, Such as Taheri and Hejazi also, have proceeded on studying and investigation in this regard, providing contents in the form of books and valid articles (Taheri, 2008). Hejazi shows that A comprehensive utilisation of proportions in Persian architecture, such as in the design of plans, elevations, geometric and architectural patterns, and mechanical and structural features, can be proved through geometrical analysis of Persian historical buildings (Hejazi, 2005). Taheri showed that how the mathematical knowledge of a Persian scientist was utilized in architecture and craftsmanship, he shed some light on the hidden layers of Kâshânî ’s architectural life, who until now has been considered only a mathematician and astronomer (Taheri, 2011). In order to get familiar with Iranian people's knowledge of Geometry specially applied geometry in art and investigating geometrical developments used in Iranian Architecture before Islam, we will present several historical eras, here, to show samples indicating achieving developments in applied geometry in architecture, scientists' knowledge of geometry and its feedback in architecture art of that era. Evidence of architecture and urban development in this era indicate that using geometry and architecture was not ineffective in urban design and structures of this era. According to Herodotus Note 7 , Ecbatana Note 8 city (contemporary Hamad ā n or Hamed ā n ), capital city of Medians , was completely in circular form and the city was surrounded by seven concentric walls. Moreover, these walls had been various balanced pathways and structures were constructed in its extension (Zarei, 2000, p. 77). In This period, Iranian architecture makes use of abundant symbolic geometry, using pure forms such as circles and squares, and plans are based on often symmetrical layouts featuring rectangular courtyards and hall. Achaemenian were among the greatest architects and geometers of Iranian History. Here is Shahkarami 's note about Persepolis Note 10 : experts' investigations on structures of especially its columns and its conical forms indicate that: Achaemenian were using p-value 2500 years before mathematics scientists. They were completely familiar with pure mathematics and engineering. They identified the secret of p-value to build conical masses. Geometry is latent in Achaemenian's architectural decorations with simple or frequent repetition of natural or abstract motifs, which were based on axial symmetry, motif's places and their distance of each other with specified geometrical structures (Kharazmi & Sarhangi, 2013). In decoration part related to architecture, Lotus design or vortex curl was administrated using simple geometrical divisions relying on spiral development based on such divisions (Hejazi, 2005, p. 83) √2 ratios were used in Achaemenian's places specially in Apadana Note 11 palace, in a way that hall involves 36 central columns as It's basic square and its diameters illuminate three sides of palace in width of columned balconies (See Figure 1) (Bemanian, Okhovat, & Baghaei, 2011, p. 175). In Pasargadae complex Note 12 and royal garden, also, the geometrical network has been used in garden designing which is according to some scholars the origin of advent of Iranian Chaharbagh Note 13 (Naima, 2006, p. 27). So it can be concluded that geometry was influential from micro scales e.g. decorations to macro scales, urban designing and royal complexes. Part tribes were fully skilled in using various designs, advanced techniques of arch and dome, through applying local materials (Zarei, 2000, p. 106). This requires applying strong geometry. The architecture has major Greek influences, while some Iranian elements from the beginning are observable. Architects of Parthian Empire started new innovations in the field of art especially in architecture, trying to revive Iranian design symbolic features and Iranian art through combining Iranian and Greek elements (Pope, 1971, p. 47). In this Empire art, motifs Note 15 are more geometric and structures are more complicated due to influences of Greek art (Kharazmi, Afhami, & Tavoosi, 2012). Creating border and placing motifs inside specific cadres are the main influences Greek art on Iranian art of Parthian Empire (Godard, 1965). For example the architectural decorations in Nisa are usually purely Greek. There were Ionic and Corinthian capitals proportion with acanthus leaves. Nisa 's battlements and fortifications, however, have their origin in a more Iranian fashion. Moreover through developments in urban fields of Sassanid and Parthian Empire, the tradition of making circular cities in monuments of Darabgerd , City of Goor and Khoreh Ardeshir Note 16 cities continued. This indicated applied usage of geometry in large scales (See Figure 2 & 3). On the rise of Sassanid dynasty, a national government was established, which provided the bed for great artistic changes through taking Achaemenian's art as their model and using experiences obtained from influences of Rome and Greek civilizations on Iran (Ghirshman, 1954). In Sassanid art change in method is from employing geometry from latent state to elaborate geometrical structures, increase in rhythms from frequent repetition to elaborate rhythms with various motifs and utilizing central symmetry, which provided the bed for great changes in architectural changes of next Empire specially changes in geometrical motifs of Islamic Architecture (Kharazmi, Afhami, & Tavoosi, 2012). In this Empire, geometric feature far from former art is eliminated, gradually, and we come across with free motifs. With such organizing, all motifs rely on using geometric infrastructure (Ibid). Dome has been established on square base in dome structures of Sassanid Empire with specific techniques of Iranian architecture, which starts with tightening four conical three-corner on four corners of square to convert it to octagon ( √2 ratios) and sixteen-gone till at final phase this main tetrahedron transfer converts to be multilateral with corners highly close to circle (Bemanian, Okhovat, & Baghaei, 2011, p. 172). Moreover, central porch of Kasra palace follows √3 ratios. In a way that mentioned saloon with length and width of 60.34 and 30.25 has the √3 ratios (Ibid, p. 176). The unique characteristic of Sassanid architecture, was its distinctive use of space. The Sassanid architect conceived his building in terms of geometric masses and surfaces; hence the use of massive walls of brick decorated with molded or carved stucco. All of these samples from the essence of the hypothesis which are emphasizing the issue of geometry in Iranian Architecture. Islamic era architect one of whose condition to step into this profession is being familiar with principles of mathematics and geometry, is completely familiar with principles of employing scales in structure and using a scale, he could use ratios of scientific geometry artistically, depending on environmental conditions (Sadeghi & Ahmadi, 2011, p. 107). Doing so he could start unique creativities, through using his interest, genius on one hand, and employing geometrical component one the other hands who are generally considered as support of his work appropriateness. In early Islam's era, with gradual development in mathematics, the art of architecture developed too. The reason for reliability of this issue can be investigated with environmental evidence of Topkap ı Scroll Note 19 and relating his geometrical language to historical developments of mathematics. Unlike Latin West, Islamic World achieved the book of principles of Greek mathematicians at early 2 nd century. At that time, this was had been translated for Caliph Harun al-Rashid . (Denny, Necipoglu, & al-Asad, 1998). New writings have been published in different fields of mathematics, astronomy, geometry, optics, algebra, and trigonometry. These publications lead geometry to the world which was unknown earlier, resulting in employing it in new affairs, including architecture and decoration arts (such as dome and arch form, geometrical and Muqarnas motifs (Ibid). Doing so with an increase in Muslim's knowledge and development and progress in geometry, especially in applied and theoretical sections, an appropriate ground was appeared and formed for using theoretical knowledge, mathematics and geometry in applied fields of architecture. In 4 th century, an unprecedented emphasis of Baghdad school Note 20 on balancing theoretical sciences with practical work has had an undeniable effect on visual culture of Islam. One of the most important and the oldest book which was written about close relationship between theoretical and applied sciences, is the book Ihsa' al-'ulum written by Al-Farabi Note 21 in 4 th century. This book is a pseudo- encyclopedia of theology and sciences, explaining all sciences in a fluent and simple word, mentioning branches of each science (al-Farabi, 2005). Moreover the books Nej ā rat and geometrical acts were written by Ab ū al-Waf ā ' B ū zj ā n ī Note 22 . B ū zj ā n ī had a prominent place in applied geometry and architecture. Especially he is of higher position due to his issue of dividing a planet with the aid of grand circles, which requires higher understanding of spatial imagination and its application in architecture (Tavassoli, 2004, p. 25). On the other hand, geometry has stepped into the decorative fields of architectural monuments in a way that the origin of views limited to geometry in Iranian architecture i.e. 4 century brickwork school is related to North and North East of Iran (Bemanian, Okhovat, & Baghaei, 2011, p. 194). Renata Hold , master of art history, Islamic Architecture and town history, points to the book " Iranian geometry: Application of Geometry in ...