The effect of algebraic and geometrical approach on the optimization of 3D-Girih development in the architecture of Iran during the Sajlughi era

Mathematics in Islamic architecture, particularly algebra, and geometric algebra, has significantly influenced modern mathematics, including the work of Descartes. This study, therefore, seeks to close this gap through a comparison study design and mixed historical-critical research that employs qualitative content analysis, inductive, and an intertwining comparative paradigm that categorizes mathematics fully in architecture. the study analyzes the geometric algebraic approach in architecture. The research question is as follows: What is the attitude in architectural mathematics? This research question is crucial to understanding the algebraic and geometric attitudes in architectural mathematics and the influence it has on Seljuk decorations. Moreover, this study might illustrate how this attitude can be utilized in contemporary architecture. This might create an opportunity to revive some of the pioneering developments through an in-depth examination of the attitude and its long-term effects on Islamic architecture and its modern developments. For this reason, the research aims to explore algebraic and geometric layers and their application in Seljuk decorations, focusing on identifying mathematical complexities in the decorative elements of this period. The achievement of this research has been to illustrate how algebra and geometry are integrated into the designs of three-dimensional and decorations of the Seljuk period. To facilitate individuals with different mathematical knowledge, computational methods, and algebra are employed in the hidden layer, while geometry is utilized in the visible layer. Therefore, it is expected that mathematics in any field encompasses four main areas: optimization methods, design methods and expression, concepts, and the possibility of interaction with architecture.