Comparison of Timurid vault and dome using computational methods of Al’Kashi models: the case of Goharshad Mosque, Mashhad

The application of computational formulas has a major role in the geometry of Iranian architecture in the Timurid era.  What is interesting about the architecture of this period is the use of the ideas of Al’Kashi --a ninth century mathematician-- in measuring, calculating, and formulating the methods of constructing vaults and domes. One of the most important achievement of Ghiaseddin is related to triangulation. It was Mirza Abu Turab Natanzi, a mathematician of the Qajar era, who later dealt with this issue through a geometric method equivalent to Al’Kashi’s algebraic method.Research Question: This research explores whether Al’Kashi has accurately considered computational operations in architecture or his thesis is just a practical and theoretical exercise. We also investigate whether Al’Kashi’s models are precise descriptions of existing vaults and whether the builder has carefully paid attention to Al’Kashi’s recommendations. To address these issues, we aim at discovering the relationships and explain the new computational rules proposed in Al’Kashi’s works.

 The research method is based on logical reasoning through precise mathematical propositions. The analytical tool includes Python programming language to be run in Rhino software, which could link computational rules and the Shape Grammars in Al’Kashi models.  The results of this evaluation are identified in the architectural elements of Goharshad Mosque, which corresponds to Al’Kashi 's presence in the region of Khorasan at the time the mosque was built. Computational exploration of architectural elements requires modeling and physical encoding. The predominant feature of Al’Kashi’s works is approaching architectural elements from a descriptive-analytical perspective. This method involves component-based classification, defining the parameters of the elements, and describing their resulting hierarchy. Therefore, our method is based on modeling in Rhino software. In addition, the Grasshopper extension and Python programming language were used to examine and determine computational relationships, the language of shapes between architectural elements (vault and dome), and Al’Kashi models based on the geometric system of Timurid architecture.  We first investigate how accurately Al’Kashi calculated the coefficients of the vaults; next, by referring to the dimensions and sizes of the vault and dome in the related building, we search for the roots of their geometric system in order to find the numerical correspondence of these ratios in Al’Kashi models.

 The results suggested that Al’Kashi’s calculations and drawings are not merely a practical and theoretical exercise; rather, they have correctly used computational operations to develop architectural elements. Besides, there is a relationship between the computational rules of Al’Kashi models and the geometric system of the vault and dome of Goharshad Mosque. Indeed, Ghavam al-Din Shirazi, the architect of the Timurid court, has appropriately deployed the geometric system and computational rules of his time in the construction of this building. These findings reinforce the hypothesis that the ‘captains of industries’ used to obtain their mathematical and geometric knowledge through dialogue with contemporary intellectuals. Goharshad Mosque showcases the harmony and balance between the geometric and computational system of the Timurid era.