مرکز منطقه ای اطلاع رسانی علوم و فناوری مطالعات شهر ایرانی-اسلامی 2228-639X 15 57 2025 7 12 Comparative Analysis of Geometric Pattern Drawing Methods in Islamic Art and Designing an Optimal Method for Redrawing Artifacts تحلیل تطبیقی شیوه‌های ترسیم نقوش هندسی در هنر اسلامی و طراحی روشی بهینه در بازترسیم آثار 1 23 fa مهدی عزیزی همدانی علم و صنعت ایران غلامحسین معماریان علم و صنعت ایران اصغر محمدمرادی علم و صنعت ایران 2024 11 15 Comparative Analysis of Geometric Pattern Drawing Methods in Islamic Art and Designing an Optimal Method for Redrawing Artifacts   Mahdi Azizi Hamedani<a href="#_ftn1" name="_ftnref1">*</a> Gholamhossein Memarian<a href="#_ftn2" name="_ftnref2">**</a> Asghar Mohammadmoradi<a href="#_ftn3" name="_ftnref3">***</a>   This article proposes an optimal method for recognizing and redrawing geometric patterns in Islamic and Iranian architecture to facilitate their application in contemporary design and knowledge transfer to future generations. By analyzing four primary pattern-drawing methods—polygonal, radial substructure, The Point-Joining Technique, and grid-based approaches—the strengths and weaknesses of each method were evaluated through historical resource analysis, interviews with master artisans, and criteria such as geometric accuracy, generalizability, and ease of instruction. Findings reveal that the radial substructure method, relying on radiating grids and concentric circles, is optimal due to its comprehensive coverage of diverse pattern types, reduced drawing errors, and adaptability to various contexts. This method not only enables precise reproduction of historical patterns but also allows the creation of new designs while preserving geometric authenticity. Combining technical precision and practical simplicity, the proposed method offers an effective solution for preserving artistic heritage and redrawing geometric patterns.   Keywords: Geometric patterns, Interlace (Gereh), Optimal method, Polygonal method, Grid-based method, Radial substructure.     Introduction Geometric patterns, as a hallmark of Islamic and Iranian art, hold a unique place in historical architectural ornamentation. These patterns reflect both the aesthetic sensibilities of past artists and intricate geometric systems requiring precise knowledge of construction rules. However, many of these rules remain obscure or inefficient due to the lack of systematic knowledge transfer from masters to apprentices. Previous studies have focused on historical descriptions or aesthetic analyses but rarely provided practical methods for redrawing and teaching these patterns. Existing methods fail to cover all variations of geometric patterns and often impose excessive complexity. This article addresses this gap by evaluating four primary pattern-drawing methods and proposing an optimal approach for understanding historical patterns, effective teaching, and contemporary design applications.   Research Methodology This descriptive-analytical study combines qualitative and quantitative methods: Literature Review: Examination of historical resources (e.g., Topkapı Scrolls), traditional artisan manuals (e.g., works of Sharbaf and Lurzadeh), and contemporary research (e.g., Bonner and Majewski). Empirical Learning: Apprenticeship with master artisans and researchers to understand practical drawing techniques. Field Analysis: Site visits to historical monuments (e.g., Isfahan Jameh Mosque, Fatima Masumeh Shrine) for pattern documentation. Criteria Evaluation: Assessment of four methods based on geometric accuracy, flexibility, teachability, and innovation potential. Comparative Analysis: Tabular comparison of methods across seven key criteria.   Results The four methods—polygonal, radial substructure (artisan tradition), The Point-Joining Technique, and grid-based—were analyzed in detail. The radial substructure method, grounded in foundational geometry (radii and concentric circles), demonstrated superior versatility for reproducing historical patterns and creating new designs. Key findings include: Grid-based methods excel in conventional patterns (e.g.,6- or 8-rosette) but falter with complex configurations (e.g., 10- rosettes). The polygonal method, though diverse, demands advanced geometric knowledge and struggles with irregular rosettes. The Point-Joining method lacks flexibility for peripheral elements and distorts composite patterns. The radial substructure method minimizes manual errors, supports diverse contexts (square, hexagonal), and provides clear, reproducible steps for learners.   Discussion The study of the advantages and disadvantages of four methods of drawing geometric patterns resulted in an optimal method for studying geometric patterns. This method is based on the radial substructure method. A number of masters have used this method in drawing some patterns, but it has not been the basis of their practice. Radial substructure drawings are generally drawn in a frame requiring a four-way reflection, and the shamas used in geometric patterns are drawn as a quarter. Auxiliary lines for drawing patterns are generally determined by code lines, and in some corners of the rectangular background, the method of drawing knot devices is not explained, or tasteful drawings are made. In this method, an attempt is made to identify the repeatable background in a geometric pattern in the first stage. In drawing the background of the work, the background is used in a circumscribed circle and the shamas are placed in the center of the circle. If the knot is a combination and includes two or more types of shamas, one of the shamas is designed in the center. In the second stage, the rays are drawn and extended to the point of contact with the rays of the surrounding rays, and in the next stage, an attempt is made to draw the network of the geometric pattern infrastructure. In the next stage, based on drawing the lines of the bergamot and their extension, an attempt is made to complete the geometric pattern. In the fourth stage, the pattern will be reproduced on the surface solely with translational symmetry and without the use of reflection or rotation.   Conclusion This study analyzed four geometric pattern-drawing methods—grid-based, polygonal, radial substructure, and The Point-Joining —evaluating their strengths and weaknesses against criteria such as interpretative capability for diverse geometric patterns, clarity and logicality of drawing steps, error reduction, potential for creating new interlaced patterns (Gereh), teachability, and ease of knowledge transfer. Based on these evaluations, an optimal method for understanding and redrawing geometric patterns in historical architecture was proposed, structured into four stages to systematically facilitate comprehension and reconstruction. The proposed method proves more efficient for manual drawing and practical engagement with geometric patterns in historical buildings, making these patterns accessible to enthusiasts. While other methods may excel in specific contexts—for example, the grid-based method is faster for 8-pointed rosettes—they lack the versatility required for complex or varied patterns. Similarly, the polygonal method offers flexibility for diverse designs but often fails to align with traditional artisans’ standards for accuracy. Although methods like the polygonal and grid-based approaches may simplify substructure creation for new patterns, mastery of the radial substructure or the proposed optimal method allows artists to innovate within authentic geometric frameworks. The advantages of this method include reducing drawing errors by using radial grids and concentric circles, the ability to adapt to a variety of backgrounds (square, hexagonal) and combined designs, providing transparent and reproducible steps for students, and creating a platform for designing new designs while maintaining the geometric framework, which can be used in redrawing new works and designs.   Table: Comparison of Geometric Pattern-Drawing Methods   <table width="87%"> <tbody> <tr> <td rowspan="2" width="34%"> Index for examining the method of drawing geometric patterns </td> <td colspan="5" width="65%"> Geometric Pattern-Drawing Methods </td> </tr> <tr> <td width="14%"> Polygonal </td> <td width="17%"> Radial Substructure </td> <td width="11%"> Point-Joining </td> <td width="9%"> Grid-Based </td> <td width="12%"> Optimal Method </td> </tr> <tr> <td width="34%"> Reduction of manual drawing errors </td> <td width="14%">   </td> <td width="17%"> ¢ </td> <td width="11%">   </td> <td width="9%">   </td> <td width="12%"> ¢ </td> </tr> <tr> <td width="34%"> Facilitating creative design </td> <td width="14%"> ¢ </td> <td width="17%">   </td> <td width="11%">   </td> <td width="9%"> ¢ </td> <td width="12%"> ¢ </td> </tr> <tr> <td width="34%"> Ease of learning and teaching speed </td> <td width="14%">   </td> <td width="17%"> ¢ </td> <td width="11%"> ¢ </td> <td width="9%"> ¢ </td> <td width="12%"> ¢ </td> </tr> <tr> <td width="34%"> Versatility in pattern diversity </td> <td width="14%"> ¢ </td> <td width="17%">   </td> <td width="11%">   </td> <td width="9%"> ¢ </td> <td width="12%">   </td> </tr> <tr> <td width="34%"> Logical and clear drawing steps </td> <td width="14%">   </td> <td width="17%"> ¢ </td> <td width="11%">   </td> <td width="9%">   </td> <td width="12%"> ¢ </td> </tr> <tr> <td width="34%"> Ability to create new patterns </td> <td width="14%"> ¢ </td> <td width="17%"> ¢ </td> <td width="11%">   </td> <td width="9%"> ¢ </td> <td width="12%"> ¢ </td> </tr> <tr> <td width="34%"> Ease of manual/software implementation </td> <td width="14%">   </td> <td width="17%"> ¢ </td> <td width="11%"> ¢ </td> <td width="9%">   </td> <td width="12%"> ¢ </td> </tr> </tbody> </table>   References Abdullahi, Yahya& Mohamed Rashid Bin Embi (2013) “Evolution of Islamic geometric patterns. 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Birkhäuser Basel. <a href="https://doi.org/10.1007/978-3-319-69977-6">https://doi.org/10.1007/978-3-319-69977-6.</a>           <a href="#_ftnref1" name="_ftn1">*</a> Ph.D Student in Architecture, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran. mahdyhamedany@gmail.com <a href="#_ftnref2" name="_ftn2">**</a> Corresponding Author: Professor, Department of Architecture, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran. corresponding Author. memarian@iust.ac.ir <a href="#_ftnref3" name="_ftn3">***</a> Professor, Department of Restoration and Rehabilitation of Historic Buildings and Sites, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran. <a href="mailto:m_moradi@iust.ac.ir">m_moradi@iust.ac.ir</a> نقوش هندسی با قدمت بیش از هزار سال شاخه&zwnj;ای از هنرهای اسلامی است. در قرن اخیر پژوهشگران و شرق&zwnj;شناسان علاقه&zwnj;مند به هنرهای اسلامی سعی در بیان شیوه&zwnj;های متفاوتی برای فهم این نقوش نموده&zwnj;اند. هدف این مقاله ارائه روشی بهینه برای شناخت و بازآفرینی نقوش هندسی در معماری اسلامی و ایرانی است تا زمینه به&zwnj;کارگیری این الگوها در طراحی امروزی و انتقال دانش به نسل آینده را فراهم سازد. بدین&zwnj;منظور با بررسی چهار شیوه اصلی ترسیم نقوش شامل شیوه&zwnj;های چندضلعی، زیرساخت شعاعی، شمسه مرکزی و شبکه&zwnj;ای، مزایا و محدودیت&zwnj;های هر شیوه از طریق مطالعه متون تاریخی، کتب استادکاران و ارزیابی معیارهایی مانند دقت هندسی، انعطاف&zwnj;پذیری و سهولت آموزش مورد بررسی قرار گرفت. یافته&zwnj;ها نشان می&zwnj;دهد روش&zwnj;های ترسیمی عموماً امکان ترسیم تعدادی از نقوش هندسی را فراهم می&zwnj;کنند و تنها بازترسیم دقیق طرح&zwnj;های تاریخی را ممکن می&zwnj;کنند، اما بستری برای ابداع نقوش جدید با حفظ چارچوب هندسی اصیل فراهم نمی&zwnj;سازند. اگرچه شیوه&zwnj;هایی مانند شبکه&zwnj;ای برای الگوهای ساده یا چندضلعی برای تنوع طرح&zwnj;ها مفیدند، اما در مواجهه با نقوش پیچیده و ترکیبی انعطاف کمتری دارند. در انتها با بیان هدف اصلی شیوه ترسیم نقوش هندسی که فهم و بازترسیم نمونه&zwnj;های موجود در بناهای تاریخی است، روشی بهینه در فهم و باز ترسیم این نقوش ارائه گردید. بدین&zwnj;منظور در چهار مرحله شامل شناخت و رابطه شمسه&zwnj;ها و زمینه قابل تکرار و زوایای به&zwnj;کار رفته در گره و اتصال فاصله بین شمسه&zwnj;ها شیوه بهینه تبیین شده و در انتها با سایر شیوه&zwnj;ها مقایسه می&zwnj;گردد.

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