تبیین تطبیقی نظام‌های هندسه فراکتال در سازماندهی معماری و آرایه‌های مسجد نصیرالملک

Iranian geometry is the result of the architect's view of philosophical aspects, natural sciences, and applied mathematics. The architecture and application of ornamentation, derived from nature’s geometry patterns in organizing all kinds of Karbandies and celestial motifs, express the ontology and aesthetic concepts of the patterns. In this regard, from modern cosmological sciences, fractal knowledge is the study of the structure and geometry of nature, in which a direct interaction between architecture, function, application of arts, and fractal geometry can be raised. The purposes of this research are 1. Explaination of geometric orders and adaption of the functional properties in the fractal structure; 2. Description of the composition and hierarchy of connections in Nasir Al-Mulk Mosque from the system architecture; 3. Evaluation of fractal quality in celestial motifs in the vaults of Nasir Al-Mulk Mosque based on comprehensive fractal patterns. Therefore, the questions are 1. Can the geometrical layouts of nature, explain the architectural organization of Nasir Al-Mulk Mosque? 2. How are architectural elements, composition, and ornamentation in Nasir Al-Mulk Mosque compared to the characteristics of fractal geometry? This applied research employs the method of correlation in applications of geometry conducting a qualitative, phenomenological view. Collecting information is based on library studies, scientific reference and electronic sources. The analysis is done in formal and functional geometry modeling. In this study, seven geometric stages in organizing the structure of a fractal sample are elaborated and categorized. The free-form and centralized geometry layouts in the fractal, as a self-organizing structure, show the authentic trend in the architecture and ornamentation of the Nasir Al-Mulk Mosque. Fractal geometry features, based on self-similarity, self-organization, and unity, are described in evaluating the fractal quality of the Nasir Al-Mulk Mosque using a comparative-descriptive model.  Based on the research results, nature’s geometry layouts have been applied as an efficient idea for creating the structure, the elemental composition, the hierarchy of connections, and aesthetic concepts in the architecture and the arrays of Nasir Al-Mulk Mosque.
The flourishing of fractals and patterns in the Mandelbrot set was illustrated and proposed by the French mathematician Benoit Mandelbrot. In this theory, Mandelbrot explains how natural phenomena have fractal dimensions and patterns, and nature uses fractal characteristics in creating different forms. In this accord, fractal geometry is self-similar and self-organized, meaning that the components, on a certain scale, represent a part of the whole and in a certain order, show unity. Comparing the characteristics of self-similarity and self-organization in fractals with the style of Persian architecture can express that self-similarity has been created in different scales and proportions in traditional architecture. Iranian patterns use the principle of geometric exchanges to perform levels of connections and branches in the structure of the building. These patterns, which are seen in various architectural layouts of Iran, show harmonious proportions and have cosmological and aesthetic concepts. In a study of the application of fractal geometry in architecture, it is observed that Iranian geometry is inspired by cosmological sciences and is derived from nature. With the knowledge of philosophy and mathematics, the architect drew the abstract geometry of nature and used the dimensions of natural patterns in architecture. While architects and thinkers have proposed theories in the expression of fractals and geometry of nature, a comprehensive explanation of nature’s geometry layouts and features in Iranian architecture as a sustainable system has been considered in this research. The importance of research is based on fractal characteristics and their application in architectural elements, in which the required parameters of architecture are described under the language of geometry and its expansions in three systems. The influence of these parameters is considered in the process of organization in architecture, the expression of the aesthetic concepts of ornamentation, and the building of engineering system. It can be concluded that applying nature’s geometry as an efficient idea in the spatial structure of this building is mainly associated with the use of self-similarity, self-organization, and unity, which show fractal qualities in organizing the architecture and ornamentation of this building. To answer the first question, on how to use nature’s geometry layouts in organizing the architecture of Nasir Al-Mulk Mosque, the geometric order in the fractal sample, explains the seven-stage process. They are: 1. Rotation, transformation, conjunction, and branching in the structural system; 2. Hierarchy of connections and geometric progression in the functional system; 3. Unity and plurality in the unity system. To answer the second question, the research expresses that the functions of fractal geometry in compositional elements, the hierarchy of connections, and the use of ornamentation follow fractal features according to the following: 1. Self-similarity in the structural system creates the main elements and structural elements; 2. Self-organization in the functional system coordinates the hierarchy of connections; 3. Unity in plurality in the physical system unifies the structural elements of architecture with the related elements.