مرکز منطقه ای اطلاع رسانی علوم و فناوری مطالعات شهر ایرانی-اسلامی 2228-639X 16 59 2025 12 27 A Meta-Analysis of Hidden Proportional Patterns in Dome Interiors (An Inquiry into the Pentagonal System, Spatial Cubes, and Horizontal Inscriptions) فراتحلیلی بر الگوهای پنهان تناسب‌بخش به فضای گنبدخانه‌ها (با جستاری بر نظام پنج‌ضلعی، مکعب فضایی و کتیبه‌های افقی) 29 45 fa سید مجید هاشمی طغرالجردی دانشگاه تربیت دبیر شهید رجایی مجید کفایی قائنی دانشگاه تربیت دبیر شهید رجایی سحر اسدپور دانشگاه تربیت دبیر شهید رجایی 2024 12 12 A Meta-Analysis of Hidden Proportional Patterns in Dome Interiors (An Inquiry into the Pentagonal System, Spatial Cubes, and Horizontal Inscriptions)   Seyed Majid Hashemi Toghroljerdi<a href="#_ftn1" name="_ftnref1">*</a>  Majid Kafaei Ghayeni<a href="#_ftn2" name="_ftnref2">**</a>  Sahar Asadpoor<a href="#_ftn3" name="_ftnref3">***</a>   Identification of methods for creating spatial proportions in dome chambers is one of the fundamental issues in Iranian architectural studies. The purpose of this research was to discover the hidden proportional patterns in the interior space of square-planned religious dome chambers and to analyze the role of visual elements in establishing spatial balance. The study was qualitative–exploratory and based on geometric analysis, field observations, and semi-structured interviews with traditional architects. The findings indicated that the height of the inner dome in many cases corresponds to the height of a pentagon proportional to the plan dimensions, and that horizontal elements such as inscriptions corresponding with the boundaries of the spatial cube play a key role in creating spatial balance. These results, while revealing the hidden geometric system, also demonstrate applicability in contemporary architectural design.   Keywords: hidden pattern; dome chamber proportion; horizontal inscription; spatial cube; pentagonal system.   Introduction In Islamic–Iranian architecture, dome chambers, as the central core of religious spaces, play a fundamental role in creating visual and spatial balance. Previous studies indicate that achieving harmonious proportions in these spaces requires the utilization of geometric principles and hidden patterns (Ardalan, 2001; Pirnia, 2006; Navai & Haji-Ghasemi, 2011). Furthermore, examining historical case Studies demonstrates that specific elements such as the interior height of the dome (āhiyāneh), the exterior dome, the spatial cube, and the placement of inscriptions have a determining role in creating a sense of spatial balance and harmony between the plan form and the dome form (Soltanzadeh & Montazer, 2018; Rahimnia et al., 2022). Nevertheless, existing research has rarely analyzed the pentagon, spatial cube, and placement of horizontal elements simultaneously, and there is insufficient evidence to explain the geometric system in shaping the interior space of dome chambers.   Problem statement Considering the importance of geometric and visual proportions in designing dome chambers, the present study addresses the following question: “which geometric systems are the dome chambers which were designed to create a sense of spatial balance based on? and which spatial and visual elements have architects made use of to achieve these proportions?” In this study, the research variables include the interior height of the dome (āhiyāneh), the spatial cube, placement of inscriptions, and pentagonal proportions. The target population of the research consists of traditional architects and architecture students experienced in spatial analysis, and the study context includes historical dome chambers in Iran following the Razi, Azari, and Isfahani styles.   Type and necessity of the Study This research is of a qualitative–exploratory type and analyzes hidden patterns in the spatial proportions of dome chambers through data collected from written sources, field observations, and semi-structured interviews with traditional architects. The necessity of this research lies in the fact that a precise understanding of geometric systems and spatial element placement can contribute to the revival of authentic principles of Islamic–Iranian architecture and provide a framework for contemporary design.   Research Hypotheses The interior height of the dome (āhiyāneh) is adjusted according to the height of a pentagon corresponding to the plan dimensions to achieve spatial balance. Rationale: Case studies indicate that many dome chambers set their interior height in relation to pentagonal ratios of the plan, and this geometric system creates a sense of balance and spatial harmony (Ardalan, 2001; Navai & Haji-Ghasemi, 2011). Horizontal elements within the dome chamber, especially inscriptions, play a primary role in creating spatial balance when placed in specific geometric positions. Rationale: Inscriptions and other horizontal elements, in addition to their decorative function, act as tools to organize the spatial cube and create visual equilibrium throughout the dome’s height (Rahimnia et al., 2022; Hayati & Nafchi, 2024). Considering the importance of this field, the present study aims to identify and analyze hidden proportional patterns in dome chambers, so that in addition to advancing historical–architectural knowledge, the application of these principles in contemporary design is also facilitated.   Research Methodology This research is of a qualitative–exploratory type, and the data were collected through a combination of field observations, measurement of plan dimensions and dome height, study of written sources, and semi-structured interviews with traditional architects and university instructors. The research Cases included 39 historical dome chambers in Razi, Azari, and Isfahani styles, selected based on historical significance, researcher access, and the extent in which the building represented the style (the better-known status of the building). Geometric ratios, including the pentagonal proportion and spatial cube, were calculated and analyzed. The data were examined through comparison of field observations, drawings from analytical sources, and interviews, using meta-analysis and case study methods, in order to identify hidden patterns and elements influencing spatial balance in the dome chambers.   Findings Figure 1: Different types of spatial proportions in Architecture styles (Source: The authors) <table style="float: left;"> <tbody> <tr> <td width="155"> Type of proportion </td> <td width="94"> Razi style </td> <td width="95"> Azari style </td> <td width="113"> Isfahani style </td> <td width="52"> Total </td> <td width="90"> Percentage </td> </tr> <tr> <td width="155"> Pentagonal </td> <td width="94"> 3 </td> <td width="95"> 2 </td> <td width="113"> 2 </td> <td width="52"> 7 </td> <td width="90"> 18% </td> </tr> <tr> <td width="155"> Spatial cube </td> <td width="94"> 8 </td> <td width="95"> 4 </td> <td width="113"> 4 </td> <td width="52"> 16 </td> <td width="90"> 41% </td> </tr> <tr> <td width="155"> Combined </td> <td width="94"> 3 </td> <td width="95"> 3 </td> <td width="113"> 6 </td> <td width="52"> 12 </td> <td width="90"> 31% </td> </tr> </tbody> </table> Discussion and Conclusion: The present study was conducted to discover and analyze the geometric systems and proportions of dome chambers in Iranian Islamic architecture. The findings indicate that traditional architects apply a coherent geometric system to achieve spatial balance. The spatial cube was applied most frequently, accounting for 41% of the cases, and by aligning the plan dimensions and positioning horizontal inscriptions, a balanced space was created. The combination of the pentagon and the spatial cube was observed in 31% of the samples, indicating the simultaneous use of multiple geometric systems to enhance visual harmony. The simple pentagon was used less frequently, with an 18% share, yet it demonstrates the role of Platonic geometry in determining the internal height of the dome chamber. The analysis of four scenarios regarding the ratio of internal plan dimensions to dome height showed that matching the dome height with the height of the pentagon or using the spatial cube plays a significant role in creating a sense of balance and spatial harmony. Moreover, horizontal elements such as inscriptions and the cube edges play a key role in organizing the space and establishing visual proportion. The results indicate that this geometric system originated in the 4th century AH, and patterns such as the spatial cube combined with horizontal inscriptions became a replicable model that can also be applied in contemporary design. Considering the limitations of the sample and access to execution documents, it is recommended that future studies expand the sample range and integrate geometric analysis with structural and symbolic aspects to better understand the spatial system of dome chambers.   References Afshin Mehr Vahid, G., Ghafari, A., & Zamani, R. (2021) Assessment of acoustic quality in the dome chamber of Nizam al-Mulk, Jameh Mosque of Isfahan. Islamic Architecture Studies, 9(2), 123–140. <a href="https://doi.org/10.52547/jria.9.2.123">https://doi.org/10.52547/jria.9.2.123</a> Ardalan, N. (2001) The art of architecture in Iran. 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Journal of Architectural Engineering Technology, 4, 137. <a href="https://doi.org/10.4172/2168-9717.1000137">https://doi.org/10.4172/2168-9717.1000137</a>           <a href="#_ftnref1" name="_ftn1">*</a> Corresponding Author: Assistant Professor, Department of Architecture, Shahid Rajaee Teacher Training University, Tehran, Iran. <a href="mailto:Sm.hashemi50@gmail.com">Sm.hashemi50@gmail.com</a> ORCID: 0000-0003-2268-6982 <a href="#_ftnref2" name="_ftn2">**</a> Visiting Professor, Department of Architectural Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran. Majid_kafaee@yahoo.com ORCID: 0009-0003-4975-0777       <a href="#_ftnref3" name="_ftn3">***</a> M.A Student, Department of Architecture, Shahid Rajaee Teacher Training University, Tehran, Iran. <a href="mailto:Sahar.asadpoor@gmail.com">Sahar.asadpoor@gmail.com</a> ORCID: 0009-0008-3560-4070 بازشناسی شیوه&zwnj;های ایجاد تناسبات خوش&zwnj;آیند در گنبدخانه&zwnj;ها، از موضوعات مهم پژوهش&zwnj;های معماری ایرانی به شمار می&zwnj;رود. مطالعات نشان می&zwnj;دهد که تناسبات پنج&zwnj;ضلعی، نقشی تعیین&zwnj;کننده در شکل&zwnj;گیری ارتفاع گنبدخانه داخلی (آسمانه) و گنبد بیرونی (خُود) داشته&zwnj;اند. تحلیل&zwnj;ها بیانگر آن است که بسیاری از شیوه&zwnj;های پنهان سامان&zwnj;دهی فضاهای ایرانی با بهره&zwnj;گیری از دایره به &zwnj;عنوان مبنا، تناسبات شکلی یا فضایی را تعریف کرده&zwnj;اند. هدف این مقاله، کشف الگوهای پنهان تناسب&zwnj;بخش در فضای داخلی گنبدخانه&zwnj; فضاهای مذهبی با پلان مربعی و تبیین عناصر بصری و فضایی مؤثر در ایجاد حس تعادل فضایی است. روش گردآوری داده&zwnj;ها از منابع مکتوب، مشاهدههای میدانی و مصاحبه&zwnj;های نیمه&zwnj;ساختاریافته با معماران سنتی است که با رویکرد تطبیقی و بهره&zwnj;گیری از تجربه پژوهشگر، تحلیل شده است. بررسی پلان و مقطع گنبدخانه&zwnj;هایی در شیوه&zwnj;های رازی، آذری و اصفهانی بیانگر آن است که در برخی موارد ارتفاع (آسمانه) با ارتفاع پنج&zwnj;ضلعی متناظر با ابعاد پلان، تطابق دارد. همچنین شکل&zwnj;گیری مکعب فضایی متناظر با پلان و جای&zwnj;گذاری کتیبه&zwnj;های افقی در تراز بالای یال&zwnj;های مکعب و نقطه آغاز خیز گنبد، به ایجاد تعادل فضایی انجامیده است. نتایج پژوهش نشان می&zwnj;دهد که این رویکرد با بهره&zwnj;گیری از هندسۀ عملی اندیشمندانی چون ابوالوفا بوزجانی از قرن چهارم هجری آغاز شده و نخستین نمونه&zwnj;های آن در مقبره دوازده امام یزد مشاهده می&zwnj;شود. تطبیق نمونه&zwnj;ها نشان می&zwnj;دهد که کاربرد عناصری مانند کتیبه افقی برای تعریف مکعب فضایی، به الگویی تکرارپذیر بدل شده است که در معماری معاصر نیز قابلیت بهره&zwnj;برداری دارد.

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