Abstract
The architect of Timurid have four systems or sets of proportions that could have combined them, but in general, a system had a major role. These systems can easily be equated to musical steps that although they are run by geometric progression, they are repeated. Each system is set based on an integer, but it is balanced by the root of that integer, according to geometric principles. Integers 2, 3, and 5 are each associated with geometric shapes determining the other elements of the scale. It was indeed Farabi who could discover the kinship of architecture with music, and then it was discussed by Bolatov. The nature of this system is well illustrated according to Farabi stating that the side of square and sector of circle in architecture is used as measuring instruments, and is similar to the theorem in logic and refrain in poetry, as well as poetry rhythms. It is precisely because of such an issue that Bolatov’s view can be accepted. Accordingly, the geometry of the design is not comparable with Western imaginations of proportion, which deals with the repetition of similar or related forms. In addition to its practical value, as an implementation procedure, the Islamic system provided the coordination and harmony for all parts, thereby the entire parts were associated with a single nature, like the relation of the parts of squares, triangles and pentagons with each other.
Keywords: Applied Geometry, Buildings Abubakr Taybadi, Goharshad Mosque, School of Ghiasieh Khargerd.
Introduction
A feature of the Timurid architecture is the use of geometry, which is manifested in the magnificence and verticality of the Timurid buildings. Also, during this period, exact sciences and geometry significantly developed. These sciences were practically applied in architecture and used as a pattern in the construction of buildings. The application of geometric formulas had a major role in coordinating proportions, balancing and overall balance and its elements, which are the features of the Timurid architecture. After preparing the design, the architect has been entered the practical research stage. Therefore, the following questions are raised in this study: What is the pattern and geometrical ratio used in the design of these buildings? The architect of the Timurid Period which geometrical principles have used in the design of these buildings?
Research Findings
Timurids architects applied all of the geometric systems used since the tenth century, including:
Rule 1: Square and its derivatives, the most important of which are the diameter of √2 of half and its double, and the side of an octagon (√ 21).
Rules 2 and 3: Equilateral triangle and its derivatives, i.e. sides and height (√3/2). Such triangles play a role in dodecagonal figures (the sides are equal,) (2√3). Sometimes, the geometry of square and equilateral triangle were combined, as it is seen in (√ 2: √3) rectangles whose height is half the generatrix square. Bearer (√3) was often used. The size of this dimension can be drawn by encircling a pentagon and extending its radii.
Rules 6, 5, 4, and 7: A halfsquare is usually formed by dividing the square of a room into halves, so that by drawing diameters of two halfsquares, a square in the center is obtained whose side is 1/ √5. The diameter itself (√5/2) plays a major role especially in determining the heights. Another method to get a rectangle with the same proportions yields a triangle with a ratio of 2:3√5. A common form associated with the halfsquare was a triangle made of diameters and two sides that could be used to separate the proportional parts of the line 1√5 and (√51) 1√5. This was done in a way that an arc was separated along the chord, the radius which was a triangle height, was then entered through this point to the base. These proportions were sometimes used for designs of façades.
Rules 8 and 9: The √5 of a rectangle or the √5 of orthogonal: using a halfsquare, the base could be divided in another way, previously known to the Greeks as “moderate limit” which is involved in the construction of the “golden section”. This is done in such a way that an arc with the length of the height along the chord is separated like the previous state, then a secondary arc whose center is smaller in the angle is drawn at the point on the chord. In the point where this arc intercepts the triangle the line is divided into two parts that one is larger {(√51)/2} represented by M, and another one is smaller {3√5/2}} represented by m. The multiples of this section are usually used in the design of interior and exterior façades and many other spaces. A number of decreasing triangles were applied in the same way as used for the diameter of the minarets {2 / (√5 37); 4 / √537) ...}.
Rule 10: A decagon encircled in a circle with a radius of 2 has a side equal to √51. The golden rectangular was made by adding unit 1 to the larger part M {1: 2 / (√5+1)}. Using the halfsquare as a base, a rectangle can be easily drawn.
Conclosion
Abubakr Taybadi’s Tomb: The use of the rules 2 and 3, i.e. the equilateral triangle and its derivatives, like in the height of the triangle. The use of rule 5 of halfsquare, i.e. √5 and its derivatives and the rule 9, namely, √5 of the rectangle. The use of rule 4, i.e. halfsquare. Dividing the square of a room which is divided into halves. The ratio was used in this rule.
Goharshad Mosque: The combined is use of the rule 5 of halfsquare, namely √5 and its derivatives, and the rule 9, i.e. √5 of rectangle.
School of Ghiasieh Khargerd: The use of the rule 1, namely √2 and its derivatives such as By analyzing such patterns and the system of proportions used in the construction of Ghiasieh Khargerd School, Goharshad Mosque and Zayn alDin monument, the present study describes the role of mathematicians and the application of geometry knowledge by architects such as Qavameddin Shirazi in the development process of architecture during the Timurid period. 
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