مرکز منطقه ای اطلاع رسانی علوم و فناوری مطالعات شهر ایرانی-اسلامی 2228-639X 16 61 2026 7 1 Analysis of the Visual Intertwining of Decorations in the Prayer Hall Under the Main Dome of Hakim Mosque with Emphasis on Dadoes درهم‌تنیدگی سازمان‌یافته بصری تزیینات ازاره‌ها و زیرگنبد شبستان اصلی مسجد ‌حکیم با تأکید بر هندسه نااقلیدسی 25 44 fa کیهانه رئیسی هنر اصفهان ایمان زکریایی کرمانی تربیت مدرس جلیل جوکار هنر اصفهان صمد نجارپور هنر اصفهان 2025 2 2 Organized Visual Entanglement of the Ornamentation of the Dadoes and the Main Prayer Hall’s Dome Underside in the Hakim Mosque, with an Emphasis on Non-Euclidean Geometry<a href="#_ftn1" name="_ftnref1">[1]</a> Keyhane Raeisi<a href="#_ftn2" name="_ftnref2"></a>*  Iman Zakariaee Kermani<a href="#_ftn3" name="_ftnref3">**</a>  Jalil Jokar<a href="#_ftn4" name="_ftnref4">***</a> Samad Najarpour Jabari<a href="#_ftn5" name="_ftnref5">****</a>   The criterion of entanglement in architectural ornamentation, particularly in geometric forms, enhances the richness of user experience and creates multiple semantic layers within a space by stimulating visual perception and improving the visual quality of the building. Entanglement helps ornamentation present valuable information more effectively to the audience. This criterion is examined in the Hakim Mosque of Isfahan, a Safavid masterpiece characterized by rich ornamentation and diverse dadoes. Accordingly, this research aims to analyze the visual entanglements of the ornamentation in the main dome’s underside prayer hall (shabestan) and seeks to answer the primary question: What role does the visual entanglement of the ornamentation in the main dome’s underside prayer hall of the Hakim Mosque play in the legibility of the architecture and spatial functionality? This research is applied in terms of objective and descriptive-analytical in terms of methodology. The data collection methods include library, internet, and field research. Data analysis is conducted using fractal geometry and software. After analyzing the entanglement of patterns in the dadoes of the main dome’s underside prayer hall, it was concluded that these patterns exhibit a relatively high degree of entanglement compared to the dadoes of adjacent subsidiary prayer halls. Moreover, this degree of entanglement is consistent with the entanglement of the main dome’s underside ornamentation. Consequently, this feature distinguishes the main prayer hall, adds to its grandeur, and attracts more audience attention than adjacent prayer halls. Additionally, due to their organized fractal characteristics (self-similarity) and informational quality, these patterns establish a connection between the audience and the forms and surfaces, thereby improving spatial functionality.   Keywords: Entanglement, Organized Structure, Fractal Dimension, Self-Similarity, Hakim Mosque.   Introduction Traditional architecture, by establishing a relationship between human and building, introduces ornamentation as an essential component of this relationship. The entanglement of ornamentation in historical buildings demonstrates the harmony between art, engineering, and the artisans’ skill, thereby contributing to a better understanding of space. Fractal geometry (which traditional artists have subconsciously derived from nature) provides a quantitative tool for analyzing this entanglement. The case study for this research is the Hakim Mosque of Isfahan, specifically the dadoes of the southern iwan and the main dome’s underside prayer hall. The primary objective is to analyze the role of visual entanglement of ornamentation in architectural legibility and spatial functionality. Research questions include “the exact type and degree of entanglement” and “its effects.” The significance and necessity of this research lie in analyzing entanglement, visual richness, and the characteristics of geometric patterns in traditional Iranian architectural ornamentation from the perspective of non-Euclidean geometry to better understand these patterns, their perceptual effects on the audience, and to explore hidden semantic layers and the intellectual roots of traditional artists—a fundamental issue in interdisciplinary studies of art and architecture. Studies of traditional examples within common approaches are often based on technical, semiotic, or aesthetic concepts, with less attention paid to methodologies related to hidden geometry, especially non-Euclidean geometries. Given that living in an entangled (complex and fractal-like) world requires possessing quantitative and structured tools for measuring this entanglement, applying fractal geometry as one of the quantitative analysis methods plays a significant role in understanding the visual systems governing traditional ornamentation. The methodology based on fractal geometry, while providing a theoretical and practical gateway for researchers in the field of art (especially in architectural studies), can also prove effective in contemporary modernist designs by maintaining continuity with authentic traditions, paving the way for integrated approaches between historical authenticity and innovation.   Research Methodology This research is applied in terms of objective and descriptive-analytical in terms of methodology. Data collection methods include library, internet, and field research, and software was used for data analysis. The selected samples are initially analyzed to identify the type of entangled pattern using the principles of fractal geometry (self-similarity). Subsequently, their fractal dimension is calculated using the box-counting method via Fractalyse software to quantify the degree of entanglement. This involves linearizing the patterns using AutoCAD to examine their self-similarity. The patterns are then converted into shapefiles using ArcMap software and imported as vector files into Fractalyse, where the fractal dimension is computed using the box-counting method and logarithmic formula. Before calculating the dimension in the software settings, the maximum and minimum sizes must be entered, which depend on the image dimensions. The minimum size is typically set to 1, ensuring fine details are included in the analysis. The maximum value is approximately 1/3 to 1/4 of the longest side (length or width) of the input image . In this method, a grid of boxes is overlaid on the image, the number of boxes along the grid is counted (1/s), and the number of boxes covering the image (N(s)) is also counted. The fractal dimension (Db) is calculated as follows (Bovill, 1996: 41-42): Db = (log(N(S₂)) - log(N(S₁))) / (log(1/S₂) - log(1/S₁)) In the next step, the box size is halved, doubling the number of boxes along the grid, and the boxes covering the image are counted again. The placement of the grid and the calculation of the dimension are performed automatically by the software. Sample calculations are provided in . Since fractal dimension calculation is a gradual process, for greater accuracy, the box size must be progressively reduced to approximately fit the pattern. The software continues this process down to the smallest box size that fits the pattern. In this research, four stages of this process are illustrated. A scatter plot with coordinates Log(N) and Log(1/S) is generated, and a straight line is fitted to the points using the software. The slope of this line equals the exact fractal dimension.In the manual box-counting method using the formula over four stages, three different dimension values are obtained. Based on the Log(1/S) versus Log(N) plot generated by the software, the slope of the line between points that have the least deviation from linearity is calculated. This value will be the closest to the fractal dimension (manual calculation may involve minor errors due to rounding compared to software calculation). This method is called differential box-counting (Ansari & Pandey, 2020:1473). The dimension calculation for all samples is performed using this method. The fractal dimension is a number between 1 (a straight line) and 2 (a fully filled two-dimensional plane). A pattern with a dimension midway between a line and a surface, i.e., a fractal dimension of 1.5, is termed a (mid-range) fractal. The more complex the fractal, the closer its dimension approaches 2; the less complex, the closer it approaches 1 (Salingaros, 2022: 189).   Discussion and Conclusion Entangled ornamentation plays a significant role in the quality of architectural space, the most important of which is the enhancement of qualitative attributes. Moreover, it bestows identity upon the space. With the emergence of ornamentation, many qualitative attributes of space—such as artistic quality, imagination, entanglement, and variety are elevated. Therefore, the degree and type of entanglement must be carefully examined. Entanglement introduces ambiguity into ornamentation; the greater this ambiguity, the higher the capacity for imagination and wonder evoked in the audience. The use of fractal patterns as organized entangled structures can create harmonious yet engaging spaces for the audience. Based on the analyses conducted on the dadoes of the main dome’s underside prayer hall, the plotting of the entanglement diagram for the dadoes of the southeastern and southwestern prayer halls, and their comparison, the ornamentation of the dado and the dome underside of the main prayer hall possessing high entanglement attracts the audience and imparts grandeur and visual richness to that part of the building. On the other hand, due to the organized nature of the entanglement, this ornamentation neither confuses nor fatigues the audience and does not carry superfluous information. High organized entanglement endows the ornamentation with a distinct character. According to the figure, the degree of entanglement of the dadoes in the main dome’s underside prayer hall is higher than that of adjacent prayer halls. Consequently, it possesses a greater capacity to attract the audience and induce pause within that space; the audience subconsciously spends time exploring these entanglements and acquiring information. Furthermore, this higher entanglement compared to adjacent spaces indicates the important and primary function of this space relative to other spaces within the building. In effect, the main prayer hall is distinguished from its surrounding areas. Since the fractal dimension calculation has been performed on the linear patterns of the images, image quality does not affect the dimension value (because fractal dimension is directly influenced by lines, fractures, and edges). Image scales are directly related to the maximum and minimum input thresholds of the software and thus influence the research results. Therefore, efforts have been made to account for this factor, ensuring that the sizes of images input into the software are approximately uniform. Of course, the increasing or decreasing trend of the dimension is of primary interest in this research. Among the research limitations are potential errors inherent in the box-counting method, the absence of perceptual data, and the limited generalizability from a single historical case study. <table style="float: left;"> <tbody> <tr> <td width="1">   </td> <td colspan="3" width="603"> </td> </tr> <tr> <td width="1">   </td> <td colspan="3" width="603"> The main prayer hall (shabestan) and subsidiary prayer halls </td> </tr> <tr> <td colspan="2" width="301"> </td> <td width="303"> </td> <td width="1">   </td> </tr> <tr> <td colspan="2" width="301"> Southwestern Prayer Hall Main Prayer Hall- -Southeastern Prayer Hall </td> <td width="303"> Southwestern Prayer Hall -Main Prayer Hall- Southeastern Prayer Hall </td> <td width="1">   </td> </tr> <tr> <td width="1"> </td> <td width="299"> </td> <td width="303"> </td> <td width="1"> </td> </tr> </tbody> </table>   Based on the analyses conducted on the entanglement of the dado ornamentation in the main dome’s underside prayer hall of the Hakim Mosque and the comparison of their degree of entanglement with adjacent prayer halls, it is concluded that the dadoes in this section possess a high degree of entanglement, and this degree is consistent with the entanglement of the ornamentation of the main dome’s underside. This high entanglement, in addition to demonstrating the harmony between ornamentation and building function and enhancing its visual richness, attracts the audience to this area as the main space of the iwan. Furthermore, due to the structural information resulting from the entanglement, it retains the audience within that space. On the other hand, because of the harmony between the dado and dome underside ornamentation and the organized nature of the entanglement of these decorations, it creates a sense of harmony with the architecture in the audience; living forms such as the human body constitute an organized fractal form. The human brain, responsible for information processing, itself possesses an organized entangled structure. Therefore, it effectively communicates with information similar to its own structure. Thus, entanglement, particularly in the form of fractal geometry, holds significant potential for transforming architecture both in terms of visual appeal and in promoting sustainable, user-friendly environments. This characteristic is abundantly found in Safavid architecture. Consequently, traditional architects, with sufficient skill and through the execution of robust, rich, and calculated ornamentation, have facilitated audience interaction with architectural spaces and, through ornamentation and the creation of entanglement within them, have clearly delineated spatial functions. The results and findings of this research help contemporary architects understand how to consciously employ organized entanglement not merely as a decorative element but as a tool for directing audience behavior (attraction, pause, and movement), replacing taste-based and arbitrary ornamentation with identity-based decoration.   References Alexander, C. 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This article is based on the master's thesis of “Keyhane Reisi” titled " Analysis of Fractal Geometry System in The Maqeli Patterned Tiles of Hakim Mosque in Isfahan" supervised by Dr. “Iman Zakariaee” and Dr. “Jalil Jokar” and advised by Dr. “Samad Najarpour” witch was done at the faculty of Handicrafts, Isfahan University of Art. <a href="#_ftnref2" name="_ftn2"></a>* PhD Student in Art Research, Faculty of Advanced Art Research, Isfahan University of Art. Isfahan. Iran. k.raeisi@aui.ac.ir http://orcid.org/0009-0005-7485-4716 <a href="#_ftnref3" name="_ftn3">**</a> Corresponding Author: Associate Professor, Carpet Department, Faculty of Handicrafts, Isfahan University of Art, Isfahan, Iran. <a href="i.zakariaee@aui.ac.ir">i.zakariaee@aui.ac.ir</a> http://orcid.org/0000-0001-5337-7811 <a href="#_ftnref4" name="_ftn4">***</a> Instructor, Faculty of Handicrafts, Isfahan University of Art, Isfahan, Iran.  j.jokar@aui.ac.ir http://orcid.org/0009-0004-2995-1050 <a href="#_ftnref5" name="_ftn5">****</a> Associate Professor, Department of Calligraphy and Painting, Faculty of Handicrafts, Isfahan University of Art, Isfahan, Iran. s.najarpoor@aui.ac.ir <a href="http://orcid.org/0009-0001-6213-9580">http://orcid.org/0009-0001-6213-9580</a> معیار در&zwnj;هم&zwnj;تنیدگی در تزیینات بنا بهویژه در فرم&zwnj;های هندسی با تحریک حس بصری و ارتقای کیفیت دیداری بنا باعث ارتقای غنای تجربه کاربران و ایجاد لایه&zwnj;های معنایی مختلف در فضا می&zwnj;شود. در&zwnj;هم&zwnj;تنیدگی به تزیینات کمک می&zwnj;کند تا اطلاعات واجد ارزش را به مخاطب، بهتر ارائه دهند. این معیار در مسجد حکیم اصفهان به عنوان یکی از شاهکار&zwnj;های صفوی که در بر&zwnj;دارنده غنای تزیینات و ازاره&zwnj;های متنوع است، بررسی می&zwnj;شود. از اینرو پژوهش با هدف تحلیل درهمتنیدگی&zwnj;های بصری تزیینات شبستان زیر&zwnj;گنبد اصلی مسجد، در پی پاسخ به این پرسش اصلی است که درهمتنیدگی بصری تزیینات شبستان زیرگنبد اصلی مسجد حکیم، چه نقشی در خوانایی معماری و عملکرد فضایی دارد؟ این پژوهش از نظر هدف، کاربردی و از نظر روش، توصیفی- تحلیلی است. روش داده&zwnj;اندوزی، کتابخانه&zwnj;ای، اینترنتی و میدانی است. تحلیل داده&zwnj;ها با کمک هندسه فرکتال و نرمافزار انجام می&zwnj;شود. پس از تحلیل درهم&zwnj;تنیدگی نقوش ازاره&zwnj;های شبستان اصلی زیر&zwnj;گنبد، این نتیجه حاصل شد که این نقوش، نسبت به ازاره&zwnj;های شبستان&zwnj;های فرعی مجاور، درهمتنیدگی نسبتاً بالایی دارد و این میزان درهم&zwnj;تنیدگی با درهم&zwnj;تنیدگی زیر&zwnj;گنبد اصلی نیز هماهنگ است. بنابراین این مورد باعث تمایز، شکوه و جذب مخاطب بیشتر این شبستان نسبت به شبستان&zwnj;های مجاور است. همچنین به دلیل داشتن ویژگی&zwnj;های فرکتالی (خود&zwnj;مانایی) از نوع سازمانیافته و دارای کیفیت اطلاعات، باعث برقراری ارتباط مخاطبان با فرم&zwnj;ها، سطوح و بهبود عملکرد فضایی می&zwnj;شوند.

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