Document Type : Research Paper
Authors
1
PhD. Student, Department of Islamic Architecture, Faculty of Architecture and Urbanism, Tabriz Islamic Art University, Tabriz, Iran.
2
Professor, faculty of Architecture and Urbanism, Tabriz Islamic Art University, Iran,Corresponding Author.
Abstract
The dynamic relationship between culture and art influences both domains significantly. Within various cultures, the adoption of decorative patterns holds particular significance. The selection and evolution of decorative patterns are influenced by each culture's context. Historical artifacts reveal that geometric decorations have long played a central role in human culture, undergoing progressive development over time. A subfield of geometry known as "Girehes" has extensively permeated Iranian-Islamic architecture, directly influencing its aesthetic foundation. The realm of Girehes intertwines mathematics and aesthetics, originating from a context of robust mathematical inquiry. Although historical records are scarce, they suggest that adept practitioners of practical geometry orchestrated the design of Girehes found on historical monuments. However, these designers rarely sought to establish a purely mathematical or theoretical basis for their designs.
Gireh's principles, rooted in mathematical foundations, have captivated contemporary researchers worldwide, owing to their mathematical underpinnings. Surprisingly, adherence to geometric and mathematical principles has not stifled the diversity within design and structure. On the contrary, artists during the Islamic era expanded the scope of geometric patterns with remarkable diversity. Notably, one prominent classification system within Girehes revolves around their respective families. This taxonomy encompasses Acute, Median, Obtuse, and Two-point categories, based on the angle formed by the lines of the "Shamseh," a central motif that reverberates across the entire Gireh design.
However, it's noteworthy that not all Girehes adhere to a specific angle formed by the Shamseh lines. Nevertheless, a substantial majority of researchers tend to rely on a categorization hinging on angles of 36 degrees (Acute), 72 degrees (Median), 108 degrees (Obtuse), and Two-point, due to their prevalence. It's essential to recognize that the angles attributed to Gireh families are not universally applicable. Variations include Girehes featuring angles of 30°, 60°, 90°, and Two-point, or angles of 45°, 90°, 135°, and Two-point.
To address questions surrounding the diverse angles between star lines leading to Gireh classification into various pattern families and the structural rationale behind the modular arrangement of angles within these families, this study employs a descriptive-analytical approach. It leverages library resources to delve into the structure of six-, eight-, ten-, and twelve-point Girehes, aiming to unveil the geometric underpinnings of Gireh families and the correlation between angles in different families and the Girehes' structures.
Girehes, part of the Islamic artistic tradition, comprise tessellations formed by regular geometric shapes harmoniously arranged to maintain uniformity. Known as "Gireh-Work," this form of geometric decoration often juxtaposes the "Shamseh" motif with polygonal elements to create a balanced composition. Traditional masters and modern research corroborate the notion that Gireh hinges upon a harmonious interplay of components. This interplay, however, is not arbitrary; beneath the Gireh motifs lies a concealed layer, termed the "underlying generative tessellation," elucidating the overall order of Gireh compositions. This generative tessellation forms the basis for drawing Girehes, allowing the placement of various Gireh motifs. Given Shamseh's critical role, artists and masters assign diverse names to it based on the number of Shamseh points. This motif profoundly impacts the overall design, shaping its structure. The foundational stars of Gireh often stem from this process, frequently defined by the connection of midpoints in the underlying polygons, which in turn determines pattern lines.
Shamseh's significance extends beyond the individual motif to dictate the broader Gireh structure symbolically. Consequently, the angle of the Shamseh influences all Gireh motifs and determines their respective families. Three main categories of classification emerge based on the points' arrangement, encompassing eight-pointed, six- or twelve-pointed, and ten-pointed Girehes. Classification by Shamseh angles yields four categories: Acute, Median, Obtuse, and Two-point. In earlier times, artists relied on drawing rules grounded in circle divisions, lacking modern tools to measure line angles accurately. Consequently, Gireh angles were often determined through methods aligned with the number of Shamseh feathers or multiples thereof.
The connection between Shamseh points and Gireh's structure is profound; assuming a Shamseh point count corresponds to a radial division of the circle, the number of Shamseh points shapes the angle structure of the entire pattern family. With α and β representing the angles of Shamseh lines, and P denoting the number of divisions in the pattern family, while F signifies the division number modulus in the Gireh family, the relationship can be expressed as follows:
α = β = F/P × 360° = A°
This circle division modulation closely corresponds with the fundamental polygons of Girehs. Classification of Gireh families rests on the base polygon, relying on connections of line segments between points on the polygon's sides. Regular polygons can be circumscribed within a circle, aligning the modularization of node families with radial divisions. This arrangement is evident in six-, eight-, ten-, and twelve-point Girehs. If the modularization equals the distance of one unit from the middle points in the base polygon, it's classified as Acute. Similarly, if the distance equals two units, it's Median. For a three-unit gap, it's considered Obtuse. Finally, the placement of Gireh lines at 1/4 of the polygon's sides or vertices, instead of the middle point, signifies the Two-point family.
Evidence of Gireh's historical reliance on polygon techniques is further demonstrated by the alignment of angles in Gireh families with the angles formed by lines at the polygons' midpoints. Gireh designers effectively harnessed polygons' geometric structure to craft intricate Gireh patterns within an encompassing framework known as the underlying generative tessellation. The division of the circle directly corresponds to the base polygon's grid system. The number of fold divisions dictates the base polygon's sides – a 12-fold division corresponds to a dodecagon, a 10-fold division to a decagon, and an 8-fold division to an octagon. Acute, Median, and Obtuse families are harmoniously woven using lines connecting midpoints in the primary polygon through a generative tessellation. Correspondingly, the Two-point family neatly aligns with the polygonal structure, deviating from the middle point to formulate patterns based on distances from middle points to vertices.
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