Application of Gireh on Free Surface Pattern with Recognition of Gireh Projection on Iranian Domes Surface in the Congruence Method

Free surfaces are one of the requirements of contemporary architecture. Owing to the emergence of these surfaces, the configuration of Girehs in Islamic architecture became incompatible with them. The traditional structural principles of Girehs expand two-dimensionally, whereas free surfaces expand irregularly and three-dimensionally. Gireh-work is considered one of the original and identity-giving decorations in Islamic architecture. To identify and develop contemporary architecture and past architectural achievements, it is necessary to find a strategy for projecting Girehes on free surfaces based on historical authenticity. The Islamic architecture of Iran has had many valuable achievements that can be applied to contemporary architectural challenges. In Iranian Islamic architecture, the dome and Gireh decoration play a crucial role, so the Iranian architect has tried to match these two elements using the geometric features of the dome and Gireh. As part of a qualitative research process, two examples of Iranian domes have been selected and analyzed with a descriptive-analytical approach in order to answer the following question: how Gireh’s geometrical principles on domical surfaces can be applied to contemporary free surfaces? In order to project Gireh onto the dome, a network based on radial gore segments was used as part of the projection technique. Transferring Girehs based on this network causes congruence homogeneity. Topological models of free surfaces are based on NURBS lines, making it possible to discretize and define the points on them. This makes it possible to apply anisotropic congruence transformation to Gireh projections on free surfaces.