تحلیل پیچیدگی نقوش معقلی در ازاره‌های مسجد حکیم اصفهان باروش فرکتال

Background: Iran’s historical architecture has always been inspired by the use of geometry and mathematical principles in its designs. One influential perspective in architecture is the theory of complex systems, which relates to fractal geometry. To date, fractal analysis has been utilized as a practical analytical tool in architecture and urban planning at various scales. The use of geometric order in design, combined with hidden geometry, repetitive patterns, and diverse self-similar motifs, has consistently been a focus in architectural decoration. Fractals are characterized by self-similarity, and the fractal dimension of fractal shapes is fractional, serving as a measure to quantify their complexity, irregularity, texture, and roughness. The Hakim Mosque in Isfahan, a masterpiece of the Safavid era, is rich in maqeli decorations. The repetition of patterns, while preserving proportions and the method of drawing maqeli based on grid networks, indicates the presence of fractal geometry within them. Thus, fractal features can be sought and demonstrated in the dado tile patterns of the Hakim Mosque. Understanding the geometric features of the dado patterns from the perspective of non-Euclidean geometry will aid in better comprehending the characteristics and intellectual roots of traditional Iranian architectural artists. Furthermore, this understanding will influence modern designs while preserving traditions. On the other hand, the visual complexity of an artwork can serve as a criterion for its aesthetics and plays an important role in organizing urban spaces. However, judging the degree of complexity requires precise calculation (Forsythe et al., 2010, pp. 49-50).
Objectives: The aim of this research is to analyze the complexity of the maqeli patterns on the dado tiles of Hakim Mosque in Isfahan using the fractal method. To this end, first, the fractal geometry features-self-similarity and fractal dimension-are examined in the patterns, and then the complexity of the maqeli designs on the dado tiles is studied.
Method: This research is applied in terms of its objective and descriptive-analytical in terms of its methodology. Data collection was conducted through fieldwork involving direct observation and photography, while documentary (library) sources were used to complete the theoretical foundations and introduce fractal structures and maqeli patterns. For analysis, the structure and principles of fractal geometry, as well as the structure of the dado tile patterns, were first described. To this end, nine representative samples of the dado tiles from Hakim Mosque were purposefully selected, all featuring similar knot designs but differing in complexity and density. The existence of fractal geometry in these patterns was then demonstrated, and the complexity of the dado tile patterns was analyzed. The patterns were initially drawn using AutoCAD software to examine their self-similarity. Then, these patterns were converted into shapefiles and vector files using ArcMap software and imported into Fractalyse software to calculate their fractal dimension. The obtained data were analyzed both qualitatively and quantitatively. After calculating the fractal dimension of the dado patterns, the complexity analysis of the maqeli patterns was conducted based on a reference motif through pattern comparison.
Result: By comparing the dado tile patterns with a common base motif and calculating their fractal dimensions, the study concluded that the maqeli patterns of the Hakim Mosque dado tiles gradually increase in complexity by expanding a base motif. For this purpose, the dado tiles were purposefully divided into three groups of three samples each. For example, in Table 3, three samples with the swastika (pili maqeli) design, which gradually become more complex and fragmented when combined with other knots-resulting in an increased fractal dimension-are compared. The next three samples, shown in Table 4, are similar to interlaced diamond designs, where variations in combination lead to differences in pattern complexity and density, causing changes in their fractal dimensions. Table 5 presents the Zahra and Chahar Selli designs combined in various ways, including with the swastika maqeli pattern, resulting in higher fragmentation, complexity, density, and consequently higher fractal dimensions.
This analysis demonstrates how the fractal properties of self-similarity and fractional dimension effectively describe the increasing visual complexity of the maqeli tile patterns in the Hakim Mosque’s dados through systematic expansion and combination of base motifs.
Conclusion: This research initially described and analyzed the maqeli tile patterns on the dadoes of Hakim Mosque in Isfahan based on fractal geometry features, self-similarity, and fractal dimension. The results showed that the maqeli patterns on the mosque’s dadoes exhibit self-similarity in both their hidden and visible geometries. Additionally, the fractal dimensions of the patterns were fractional, consistent with fractal dimensions. Therefore, it can be concluded that the dado tile patterns possess fractal characteristics.
Since fractal shapes have properties such as texture, complexity, fragmentation, smoothness, and roughness-all of which affect fractal dimension-quantifying the complexity, fragmentation, and density of the dado patterns based on their fractal dimensions and categorizing them according to base knot motifs led to the finding that the designer of the mosque’s maqeli patterns created visual harmony with the natural environment (such as mountains or plants) by using repetitive rhythms and gradually increasing complexity in the tile designs.
This reflects a simultaneous combination of order and complexity in the design, which in Islamic art is considered a measure of balance between simplicity and complexity. The fractal analysis of the dadoes showed that despite their apparent complexity, the patterns have an underlying order due to the use of a base motif and the presence of self-similarity. This analysis demonstrates that Safavid-era artists, with an intuitive understanding of complex geometry principles, succeeded in creating works that can today be described using advanced mathematical concepts like fractals.
The harmony between traditional art and modern mathematical concepts attests to the deep geometric knowledge of Iranian architects. Considering the research results, the diversity of traditional patterns, and the use of repetitive motifs, it is possible-within a non-Euclidean space and infinite repetition of self-similar patterns and quantification of visual complexity, fragmentation, and density (which serve as criteria for aesthetics and influence urban design)-to create new designs in architectural decoration.