Iran University of Art Journal of Architecture and Urban Planning 2717-1299 18 48 2025 09 23 Investigation and drawing of geometric patterns with heterogeneous rosettes based on Tony Lee's mathematical theories Investigation and drawing of geometric patterns with heterogeneous rosettes based on Tony Lee's mathematical theories 136 154 1436 10.30480/aup.2025.5799.2250 FA Mahdi Azizi Hamedani Ph.D. Candidate in Architecture, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran Gholamhossein Memarian Professor, Department of Architecture, School of Architecture and Environmental Design, Iran University of Science and Technology, Tehran, Iran Asghar Mohammadmoradi 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 Journal Article 2025 04 17 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. 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. https://aup.journal.art.ac.ir/article_1436_fdd6319b6d10e8cf409c0732f50b4986.pdf