Investigation and drawing of geometric patterns with heterogeneous rosettes based on Tony Lee's mathematical theories

Document Type : Original Article

Authors

1 Ph.D. Candidate in Architecture, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran

2 Professor, Department of Architecture, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran

3 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

Abstract

Geometric patterns in Islamic art and their mathematical aspects have been a subject of keen interest for researchers in this field. Individuals such as Tony Lee took significant strides in this direction with their manuscripts in 1967, and this work has continued to the present day. Composite geometric patterns with heterogeneous rosettes are a type of geometric design observed in historical sources such as the treatise Risāla fi Tadākhul al- Ashkāl al- Mutashābiha wa al-Muwāfaqa (On the Interlocking of Similar and Complementary Shapes) from the 9th century Hijri, as well as in historical structures across the Islamic world. These patterns have consistently captivated traditional artisans and contemporary researchers due to the diversity of rosettes and their geometric complexities. In this article, we aim to explore the mathematical relationship between two heterogeneous rosettes, as articulated by Tony Lee, and apply this understanding to the drawing of geometric patterns. To render these patterns, the polygonal-in-contact (PIC) edge method has been employed. This study reveals that implementing heterogeneous patterns—despite their mathematical compatibility for adjacency—requires distinct contextual frameworks for each, as they are inherently dissimilar. Furthermore, by analyzing the symmetry groups of geometric patterns, a field explored by mathematicians since 1977, the requisite symmetry group for knots with heterogeneous rosettes has been identified. The research demonstrates that mathematical analysis alone is insufficient to establish the similarity of these patterns; instead, differences in the required symmetry groups for rosettes become apparent through their practical illustration. In examining the drawing of interlace patterns (gereh) with heterogeneous rosettes, we demonstrated that a single symmetry logic applicable to ten-fold patterns cannot be generalized to other configurations. Instead, three types of symmetry frameworks—square, rectangular, and triangular—are required for rendering rosettes. For composite eight- and twelve-fold patterns, a quarter-square framework can be constructed by reflecting a right-angled triangle, mirroring the methodology used in ten-fold patterns. However, this logic does not extend to other composite patterns. For instance, (9-12) and (18-6) knots necessitate triangular symmetry frameworks, while (14-7) and (18-8) knots require adjustments to the background repetition and rosette positioning relative to one another. Additionally, similar to eight- and twelve-fold patterns, these rosettes can be re-rendered in symmetry frameworks with diverse configurations. This study investigates the mathematical relationship between heterogeneous rosettes and proportional symmetry frameworks in Islamic geometric patterns, grounded in Tony Lee’s theories. Key findings reveal a specific mathematical correlation between the number of rosette vertices (m,n) and radial division parameters (p,q), expressed by the formula : (1/2 =q/n + p/m )This formula enables the proportional design of heterogeneous rosettes while preserving geometric harmony. However, practical implementation demands alignment with the 17 crystallographic symmetry groups (e.g., p6m, p4m). Our work highlights that rendering patterns via mathematical logic within radial grid infrastructures (the standard method) requires modifications to rosette placement, background elements, and auxiliary motifs to achieve space-filling configurations. This principle also applies to patterns incorporating more than two rosettes.

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References

امین‌پور، احمد، اولیا، محمدرضا، ابوئی، رضا و حاجبی، بیتا (۱۳۹۴). پیشنهاد مدلی برای ترسیم گره در تزیینات وابسته به معماری اسلامی. نامه معماری و شهرسازی، 8(15)، 57–78.
حلی، سید علی اکبر(1365). گره و قوس‌ها در معماری اسلامی. قم: چاپخانه مهر.
دهشتی، مجید، خوش‌نژاد، مهدی و منان‌رئیسی، محمد (1398). روشی نو در ترسیم نقش‌مایه‌های گره ده تند و کند. نگره، 51 (14)، 53–64.
رئیس‌زاده، مهناز (1398). احیای هنر‌های از یاد رفته معماری اسلامی (جلد اول). تهران: مولی.
زمرشیدی، حسین (1365). گره چینی در معماری اسلامی و هنرهای سنتی. تهران: مرکز نشر دانشگاهی.
شاکرسلمان، عامر(1383). هم‌آراستگی در نگاره‌های اسلامی (مترجم: مریم آقاربیع). تهران: به نشر.
شعرباف، اصغر (1385). گره و کاربندی. تهران: انتشارات سازمان میراث فرهنگی کشور.
شفایی، جواد (1399). هنر گره‌سازی در معماری و درودگری (دوره 3 جلدی). تهران: انجمن مفاخر فرهنگی.
شیروانی، محمد‌رضا (1401). «حَسبا» شیوه ترسیم نقوش هندسی اسلامی مراکش در مقایسه با شیوه شعاعی ایرانیان. مبانی نظری هنرهای تجسمی،  ۱۴(7)، 28–39.
گلیار، محمد (1399). دفتر گره: روش طراحی گره‌ها در معماری اسلامی ایران. تهران: میراث اهل قلم.
ماهرالنقش، محمود (1361). کاشی‌کاری در ایران دوره اسلامی. تهران: شرکت افست.
نژاد ابراهیمی، احد و عزیزپور شوبی، عارف (۱۴۰۲). کاربرد لایه پنهان درتوسعه گره‌چینی بر مبنای مستندات تاریخی در ایران. مبانی نظری هنرهای تجسمی،  ۸(1)، 101–116.
ولی‌بیگ، نیما، نظریه، نوشین و رهروی، ساناز (۱۳۹۶). مطالعۀ مقایسه‌ای گره مادر در گسترۀ شیوه‌های ترسیم با ارائه و معرفی شیوه‌ای نامکتوب. تاریخ علم، ۱۵(2)، 251–274.
Abas, S.J., & Salman, A.S. (1994). Symmetries Of Islamic Geometrical Patterns. World Scientific.
Aljamali, A.M., & Banissi, E. (2004). Grid Method Classification of Islamic Geometric Patterns. In M. Sarfraz (Ed.), Geometric Modeling: Techniques, Applications, Systems and Tools (pp. 233–254). Springer Netherlands. https://doi.org/10.1007/978-94-017-1689-5_13
Berglund, J., & Kaplan, C. (2021). Beyond the Great 96. https://isohedral.ca/other/Beyond96/
Bodner, B.L. (2012). From Sultaniyeh to Tashkent Scrolls: Euclidean Constructions of Two Nine- and Twelve-Pointed Interlocking Star Polygon Designs. Nexus Network Journal, 14(2), 307–332. https://doi.org/10.1007/s00004-012-0111-y
Bonner, J. (2017). Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction (1st ed.). Springer. http://gen.lib.rus.ec/book/index.php?md5=5a5e5059510cc44d7305c0fe537e07ae
Broug, E. (2013). Islamic Geometric Design (1st ed.). Thames & Hudson.
Castera, J.M. (2016). Persian Variations. Nexus Network Journal, 18(1), 223–274. https://doi.org/10.1007/s00004-015-0281-5
Critchlow, K. (1976). Islamic patterns: An analytical and cosmological approach. Schocken Books.
Cromwell, P. (2009). The search for quasi-periodicity in Islamic 5-fold ornament. The Mathematical Intelligencer, 31, 36–56. https://doi.org/10.1007/s00283-008-9018-6
Cromwell, P.R. (2016). Modularity and Hierarchy in Persian Geometric Ornament. Nexus Network Journal, 18(1), 7–54. https://doi.org/10.1007/s00004-015-0280-6
El-Said, I., & Parman, A. (1976). Geometric Concepts in Islamic Art. Dale Seymour Pubn.
González, J.M., Albert, F., Carretero, C., & Gomis, J.M. (2004). Structural description of textile and tile pattern designs using image processing. Proceedings of the 17th International Conference on Pattern Recognition. https://doi.org/10.1109/ICPR.2004.1334175
Hajebi, B., & Hajebi, P. (2024). Improving recognition of deteriorated historical Persian geometric patterns by fusion decision methods. Neural Computing and Applications, 36(20), 11809–11831. https://doi.org/10.1007/s00521-024-09932-3
Hajebi, B., Aminpoor, A., Olia, M.R., & Abouei, R. (2016). Drawing Parametric Polygonal Pointed Star Patterns. Soffeh, 26(1), 109–124. https://soffeh.sbu.ac.ir/article_100303_en.html
Hankin, E.H. (1925). Drawing of Geometric patterns in Saracenic Art. http://archive.org/details/HankinSaracenicArt
Lee, T., & August, A.S. (2014). The Geometric Rosette: Analysis of an Islamic decorative motif. https://www.semanticscholar.org/paper/The-Geometric-Rosette-%3A-analysis-of-an-Islamic-Lee-August/29f33d97a25d88ff187585277edb8d4f208d152a
Makovicky, E. (1986). Symmetrology of art: Coloured and generalized symmetries. Computers & Mathematics with Applications, 12(3–4), 949–980. https://doi.org/10.1016/0898-1221(86)90435-9
Necipoğlu, G. (2017). The Arts of Ornamental Geometry. In Persian Compendium on Similar and Complementary Interlocking Figures. Brill. https://brill.com/display/title/32181
Wichmann, B., & Wade, D. (2017). Islamic Design: A Mathematical Approach. Birkhäuser Basel. https://doi.org/10.1007/978-3-319-69977-6
Journal of Architecture and Urban Planning
Volume 18, Issue 48
September 2025
Pages 136-154

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