Descriptive and Analytical Study of the Structure and Elements of Muqarnas according to the Fourth Article of the Ninth Chapter of Al-Kashi’s Meftah el-Hessab Treatise

Muqarnas, as one of architectural decorative art, is an expression of Iranian culture and can represent the hidden parts of Iran's cultural landscape. The complex geometry of muqarnas is made of several basic structures in successive layers. The flexible language of muqarnas in creating various types has made it difficult to understand the geometrical order of this art and therefore, it has been labeled as authorship. In the meantime, the first computational and geometrical approach to muqrans was presented by Ghiyasuddin Jamshid Kashani (758-808 AH) in the fourth article of the book Miftah al-Hasab, which today has many compositions and interpretations of this book with different points of view which provides basic information in the field of muqrans. In addition, the present research has searched for the constituent structure and the elements and units of this complex structure, by examining Kashani's approach and recognizing the productive components of muqarnas from his point of view. In this regard, we have tried to draw all the possible situations (modes) according to the realities of muqarnas, through logical reasoning.
Accordingly, this research tries to present a more accurate understanding of Al-Kashi’s treatise using 3D techniques. To do so in the first phase the original Article in Arabic has been studied consulting with Arabic language scholars in Shiraz University to better comprehend the text. Secondly, the related literatures have been studied and compared. These phases help us to know that the key point in misunderstanding and/or different interpretations of Kashi’s Muqarnas types lie on different comprehension of Muqarnas fundamental units as were described by al-Kashi. This paper introduces the components as follows: outer shell, inner shell, lower roof, upper roof, facet and facet base.  That is preventing from inaccurate use of technical vocabulary for identification of Muqarnas structures and also resulting in more accurate and concise understanding of Al-Kashi Muqarnas typologies.
Furthermore, using the introduced Muqarnas components in previous phase, a detailed identification of Al-Kashi Muqarnas typologies is presented.  In his Article Al-Kashi classified Muqarnas into four main categories namely Simple (sazej/sade), clay-plastered (motayyan), curved (ghos) and Shirazi.
In addition, the present research has searched for the constituent structure, the elements, and units of this complex structure, by examining Kashani's approach and recognizing the productive components of muqarnas from his point of view. In this regard, we have tried to draw all the possible situations and modes according to the realities of muqarnas, through logical reasoning.
In total, 16 elements can be identified as compositional examples according to the principles of construction of muqarnas. Thirteen elements have been seen exactly in real examples in Iran, and the rest of the elements have been seen in other examples of Muqarnas in the world or in later models of muqarnas, i.e. radial patterns. In addition, it was found that the muqarnas with checkerboard principles have precise principles and the existence of symmetry and the use of a series of fixed values ​​and sizes (1.1/2 √(2), 1-√(2) and 0.765367) in total.