A Recognition Technique for the Generative Tessellations of Geometric Patterns in Islamic Architectural Ornaments; Case Study: Southern Iwan of the Grand Mosque of Varamin

Abstract

The ornamentation of historical buildings in Iran often features geometric patterns, which hold cultural and architectural significance. These patterns, rooted in Islamic tradition, are widely used in contemporary Middle Eastern architecture. By employing regular polygons, intricate designs emerge, forming interconnected tessellations and repeating modules. This paper focuses on uncovering hidden tessellations and geometric patterns within the southern Iwan of the Grand Mosque of Varamin. Through photography and field measurements, 8².4 and 3⁶ tessellations were identified. Using the Revit 2024 program, a novel method was introduced to model these patterns. By manipulating repeating units, designers can create diverse geometric latticework, preserving Islamic architectural heritage. Furthermore, these patterns offer practical applications beyond ornamentation. They can serve as architectural elements in urban environments, such as fences or enclosures, enhancing privacy in residential spaces and contributing to urban aesthetics. This approach facilitates the integration of historical patterns into contemporary architectural designs, enriching both cultural identity and urban landscapes and is a step toward smart cities.

1. Introduction

In the contemporary era, the digital preservation of 3D objects has become a compelling aspect of overall preservation strategies. This modern approach involves archiving digital models and the reverse engineering process applied to existing or lost objects [1]. Islamic geometric patterns are one of the critical characteristics of Islamic architecture in many cultural traditions of Islamic countries [2]. Geometric proportions are used by Muslim designers as the most valuable tool of the design process to produce ordered patterns that govern aesthetic beauty in designed spaces, surfaces, and objects [3]. For centuries, Islamic geometric patterns have been used as decorative elements on the walls, ceilings, doors, domes, and minarets of historical buildings in Iran. However, the absence of guidelines regarding the application of these decorative elements has often led to their inappropriate use in terms of precision, scale, and architectural style compatibility [4]. Today, in the Middle East, there is a tendency to use Islamic geometric patterns (IGP_S) as an important element of identity and culture in building design [2]. As time passes, the traditional methods of transmission of this ancestral art may be threatened or disrupted. In this context, its preservation and development will be impacted [5].

One of the methods for drawing geometric patterns is the polygonal technique for constructing Islamic geometric patterns [6,7,8,9]. This method, also known as the Hankin method, was first introduced by Hankin based on his discoveries of semi-finished drawings of geometric patterns in a series of articles [10,11]. This method creates a tessellation polygon, and then the geometric patterns are generated within them. Various efforts have been made in the field of generating geometric patterns. Azizpour et al. did a morphological investigation into Islamic geometric patterns (IGP_S) and applied formal grammar to the computational modeling of IGPs [12]. The findings have been implemented within a Grasshopper add-on, offering a versatile framework for pattern generation via strings and parameter control. Through an examination of various Gereh and their authenticity in terms of configuration, this study introduces a technique to make the studied Gereh deformable. The method uses the change in the angle of the Gereh elements in underlying generative tessellation based on the family of the Gereh (acute, median, and obtuse) [10]. They introduce a notation system, which employs generative parameters to create Gereh within the 7-fold system and various systems categorized into four groups by Bonner: acute, middle, obtuse, and two-point. They have written code within the Rhinoceros 6 software and the Grasshopper plugin [13]. This research leverages formal grammar and computer science to propose a novel approach for digitally visualizing existing Islamic Geometric Patterns (IGPs), specifically focusing on star patterns. The presented method of IGP visualization is developed as a C#-based add-on for Grasshopper in Rhino3D. Mutaz et al. center on the utilization of parametric modeling through the Rhinoceros 6 Grasshopper program for an Islamic historical geometric pattern found on the gate of the Wasit School. The investigation necessitates a spatial approach, and the program is implemented with the support of algorithmic parameters inherent to the parametric design technique [14]. It aims to produce novel Islamic geometric patterns derived from an obsolete reference pattern used in contemporary urban design. The objective is to enrich and infuse a sense of cultural inclusivity and architectural uniqueness into the urban architecture of local cities.

Hajebi and Hajebi introduced a novel approach for the intelligent restoration of parametric geometric patterns [15]. This method allows for the automatic reconstruction of missing components when provided with an image of the existing patterns. Their approach is based on image processing by detecting boundaries of deterioration, finding every individual element, and extracting features of detected patterns via Zernike moments. In [16], they introduce a novel technique known as the “Âark method” for generating fresh geometric patterns through the utilization of the “Hasba” (measurement) method, a widely practiced approach among Moroccan artisans. This method, which relies on the principles of symmetry, enables the creation of many patterns through systematic and dynamic procedures. Symmetrical patterns are derived from an asymmetric component known as the “fundamental region” by employing reflections and rotations. Khamjane et al., introduced an approach grounded in symmetry group theory for creating Islamic star patterns [17]. This method constructs star patterns using one or more variations of stars/rosettes taken from a fundamental region associated with a specific symmetry group, including a few straightforward parameters. The paper elucidates the steps in organizing and forming the stars/rosettes within the fundamental region.

Given the rapid development in construction and the pursuit of smart and digital cities, new approaches to construction have been considered. One such approach is Building Information Modeling (BIM) compared to CAD systems [18]. BIM is utilized not only in the construction of new buildings but also in the fields of cultural heritage preservation and documentation [19]. Various software programs exist for 3D modeling, such as 3ds Max, Rhino, AutoCAD 3D, and SketchUp, which primarily perform graphical 3D modeling. However, with the Revit 2024 software, building information is integrated with the model. Nowadays, this software is used both for constructing new buildings [18,20] and for the maintenance and modeling of historic structures [19]. One of the reasons this research presents a method for modeling geometric patterns using this software is its capability to meet the contemporary needs of designers and builders. In the conceptual design environment of Revit software, there is a workspace for creating tessellations [21]. However, the number of available options is limited, leading to constraints when seeking diversity in their usage. This paper’s primary objective is to discover hidden tessellations in the geometric patterns of the southern Iwan of the Grand Mosque of Varamin, one of the magnificent historical monuments in the city of Varamin, Iran, where Islamic geometric patterns have been extensively employed. Then, a novel method is introduced, incorporating the capabilities of the Revit software for novel geometric pattern generation and its application in architectural projects as part of the windows family.

Theoretical Background

Geometry in ancient Iran means specifying the dimensions and refers to the knowledge related to measuring methods of forms and spaces. The antiquity of the science of measurement (geometry) in Iran dates back thousands of years before the Common Era, and the discovered evidence indicates that geometry existed both as pure knowledge and as an applied science. The ancient East significantly contributed to the evolution of mathematical and geometric knowledge. The Sumerians and later the Babylonians, along with the scribes of the Susa temples, had established rules related to the determination of the surface area of cultivated lands, the measurement of stone and brick masses, and the volume of excavation for canals. Various inscriptions from these lands have demonstrated the resolution of various geometric problems using numerical and geometric methods [22].

The emergence of geometric patterns in Islamic countries coincided with the arrival of Islam in these regions. Muslim artists, instead of using human and animal motifs, endeavored to use plant and geometric patterns in architectural spaces. The reason for this is based on the belief of Muslims that the human figure should not be used in religious places. Therefore, they utilized abstract patterns such as geometric designs to symbolize the divine [22]. The history of Islamic geometric design can be considered an evolutionary progression from simplicity to complexity. This new form of ornamentation was characterized by a general geometric matrix with simple star shapes or regular polygons placed on the vertices of repetitive tessellations [23]. These patterns are seen in architectural spaces in conjunction with brickwork, mirrorwork, stucco, and similar techniques, as well as in traditional arts, including ceramics, lattice works, carving, marquetry, carpets, bookbinding, and more.

Islamic geometric patterns are graphic designs observed in today’s spaces, furniture, and objects. Artists from Muslim countries follow these geometric patterns, along with the culture and traditions of each country, in architectural spaces. Nowadays, with the advancement of technology, graphic and parametric software can be used to advance architectural goals and designs, allowing for the creation of a wide range of patterns by altering the parameters. Islamic geometric patterns are also parametric and graphic and are influenced by various parameters [10].

The Grand Mosque of Varamin is an exemplary traditional mosque in Iran, notable for its significance in showcasing the evolution of Islamic architecture. This mosque belongs to the category of four-Iwan mosques, and its construction began in AD 1322 during the reign of Sultan Mohammad Khodabandeh. It was completed four years later during the time of Abu Saeed.

The main entrance portal of the Grand Mosque of Varamin, located on its northern side, consists of impressive vaulted chambers and alcoves both inside and outside. Within the mosque, one can witness the complexity of the geometric patterns, especially notable due to the use of bricks or a combination of bricks and tiles (muqarnas). Geometric arrays in coarse and fine scales often manifest as narrow bands in the prayer hall and the main portal, with their prevalence diminishing towards the Mihrab area [24].

The octagonal star motif and the shames (Persian for octagonal star motif ) patterns are widely employed in the southern prayer hall and the Grand Mosque of Varamin portal. These patterns convey diverse meanings and concepts within Islamic art. Figure 1 highlights the location of the southern Iwan of the Grand Mosque of Varamin.

2. Method of Regular Polygons in Drawing Geometric Pattern

This method was first identified by Hankin [11,13]. When studying this tradition, a fundamental rule is that all patterns must be described by their geometric symmetry. Most Islamic geometric patterns exhibit three-fold, four-fold, or five-fold symmetry. What directly relates to the symmetry of a pattern is the module. Islamic geometric patterns can continuously fill a surface using only a single repeating unit (module). These modules always consist of small pieces of a pattern that can uniformly fill the surface through translational symmetry edge-to-edge.

A regular polygon with equal angles and lengths has been chosen so that only three types of regular polygons can individually cover a surface: the triangle, square, and hexagon. Figure 2 displays an isometric grid constructed from equilateral triangles along with their dual hexagons and a grid of squares and hexagons. Most Islamic geometric patterns are repeated on an orthogonal or isometric grid. Each vertex of the isometric grid has sixfold symmetry composed of six acute angles of 60 degrees, where six equilateral triangles are continuously connected edge-to-edge. The duals of the isometric grid are a grid of regular hexagons, which has three-fold symmetry resulting from three regular hexagons aligned at 120 degree angles at each vertex continuously. The vertices of the orthogonal grid have four-fold symmetry resulting from four squares meeting at a 90 degree angle at each vertex, and the dual of the orthogonal grid is a similar orthogonal grid that uses squares at the center of each module, known as quadrilaterals [23].

When drawing geometric patterns using the regular polygons in tessellations, regular polygons are placed to complete the IGP. Pattern lines are arranged at different angles inside the applied regular polygons on the tessellations. Ultimately, the final geometric shape is obtained by considering the arrangement of these pattern lines side-by-side.

Tessellation works have been used as both decorative and structural elements in various civilizations throughout history, including Byzantine, Egyptian, Greek, Roman, Iranian, Japanese, Chinese, and Arab cultures. In 1619, Johannes Kepler systematically studied polygonal patterns and defined sets of tessellations in regular and semi-regular forms [7].

Some of the geometric patterns used in the eastern and western sides of the southern Iwan of the Grand Mosque of Varamin (Figure 3) include the “Chalipa” pattern [24] and 8-pointed star motifs created with brick materials. On the eastern side of the south Iwan, the brickwork in the geometric patterns has experienced more erosion. “Chalipa” has four directions and four corners, and the number four has always held sanctity and respect both before and after Islam, symbolizing balance and harmony [27]. The “Shamse” (8-pointed star) is considered a symbol of the divine [28]. The eight-pointed star motif used in the mosque is an evolved form of the circle and “Chalipa” patterns, symbolizing the sun in various spatial and temporal contexts. The eight-pointed star shape is formed by rotating two squares relative to each other. Since ancient times, the number eight has been regarded as a symbolic representation of the sun across Europe, Asia, and Africa [29].

As seen in Figure 4, the tessellation work used in the design and construction of this pattern is created by 8-sided polygons (octagons) that are placed alongside each other, forming squares in between. The 8².4 tessellation work is used in this pattern.

In Figure 5, the process of drawing the pattern in the southern Iwan of the Grand Mosque of Varamin is illustrated. It is created using 8².4 tessellations and drawing lines at a 90 degree angle from the center of each side of the regular octagons, and these lines are connected to form an 8-pointed star (Figure 6). The desired pattern will be achieved by repeating these lines at 90 degree angles relative to each other across the midpoints of the octagons’ sides and removing the red-colored tessellation lines.

The 12-pointed star is another pattern in the southern Iwan of the Grand Mosque of Varamin, constructed using turquoise-colored tiles and bricks [30]. The use of color is a characteristic of the Iranian culture, and each color holds a spiritual meaning. Turquoise blue is a relaxing and focusing color that symbolizes calmness and clarity. It is visible throughout the day in nature, and its brightness changes [31]. The mosaic background of the 12-pointed star pattern consists of 36 tiles, which, through repetition, create this beautiful pattern. Figure 7 illustrates how the 12-pointed star pattern created using red tessellations works.

After determining the isometric tessellation with equilateral triangles in the 12-pointed star pattern, lines with an angle of 60 degrees are drawn from the middle of the sides of the triangles in the tessellation. They extend to intersect each other. Once all the lines are completed, a 6-pointed star is formed (orange lines). This star is then copied relative to the center of the 6-pointed star with a 30-degree rotation (forming green lines) (Figure 8).

In Figure 8, two six-pointed stars are shown in green and orange colors. The orange six-pointed star is drawn at a 60 degree angle from the center of the triangle’s sides and, by extending all the 60-degree angle lines outward, a six-pointed star is formed. Then, with a 30-degree rotation relative to the center of the orange star, the green star is formed. By combining both six-pointed stars, a 12-pointed star is created. By replicating this 12-pointed star at the vertices of the tessellation triangles, the geometric pattern present on the eastern side of the southern Iwan of the Jame Mosque of Varamin will be realized (Figure 3B).

3. Analysis and Proposed Method

Selecting an appropriate approach for creating patterns and considering symmetry details, especially in isometric and orthogonal tessellations, is one of the most critical factors for precise and aesthetically pleasing pattern drawings. The systematic approach for regular polygons, known as the Hankin method, is a prominent approach for drawing geometric patterns. This article examined and modeled two geometric patterns and their tessellations (8².4 and 3⁶), based on the geometric patterns found in the Varamin Grand Mosque. By utilizing these patterns and creating them in the Revit 2024 software, derivative patterns will be designed and substituted for the original ones in tessellation designs. This process will lead to the creation of diverse and innovative geometric patterns and the preservation and documentation of ancient buildings.

In the preceding sections, we have emphasized that our approach in this paper is grounded in the polygonal techniques for generating Islamic geometric patterns. Considering the need for tessellation patterns of 8².4 and 3⁶ for creating these geometric patterns, and due to the limitations of the available patterns in Revit’s curtain pattern panel base environment, the decision was made to construct these tessellations in the generic model environment. Initially, hexagons with six triangles (A), squares (B) and regular octagons (C) inside them were parametrically designed using detail lines. Parameters were assigned to all sides and angles to enable dimension adjustments when needed (Figure 9).

In a new generic model environment, regular polygons were parameterized and placed in the desired reference plane. Then, in the front view, these polygons were repeated to generate the 3⁶ (A) and 8².4 (B) tessellations, as seen in Figure 10.

Based on the tessellation drawn to create new geometric patterns, regular polygons should be assigned a pattern. By placing these patterns side by side, as repeatable units (module), the final shape and geometric pattern of South Iwan can be formed. Each regular polygon is a separate family that can be customized within its environment. By saving them with new names and reloading them into the environment where the Tessellations 3⁶ (A) and 8².4 (B) in Figure 10 were created, you can select the available regular polygons from the type selector to turn them into the final polygons designed as repeatable units. To model the geometric pattern in Figure 4, an octagon was initially designed using the sweep tool, as shown in Figure 11.

After designing the octagon pattern, its family was saved with a custom name, and then it was imported into the generic model containing the 8².4 tessellation. Upon importing the pattern, all the existing octagons were selected using the “select similar” command and entered into the type selector. By choosing the name of the new octagon pattern, all the octagons were transformed into the imported pattern, creating a new geometric pattern, as shown in Figure 12.

As shown in Figure 12, using the mosque tessellation and modifying its repeatable unit, a geometric pattern can be realized. In this pattern, the squares are kept empty, and an eight-sided pattern is drawn with perpendicular lines from the center of each side, resulting in a final geometric pattern.

To use this family in the Revit project environment, the decision to use it as a window family was made. Initially, a window creation environment was opened, and the designed geometric pattern was inserted. To control the visibility of the tessellation lines in the project environment, the “visible” parameter was assigned to these lines. After inserting the pattern into the window creation environment, a void was required to align the geometric pattern with the window frame. This void was created using the generic model face base environment and designed with width, length, and height parameters to cut out the geometric pattern model’s additional parts (Figure 13).

After designing the void, it was imported into the window environment, and its parameters were linked to the existing opening wall parameters so that changes in the window’s openness would also affect the void. Subsequently, a frame for the window was modeled using the sweep tool, and material parameters were assigned to both the window frame and the geometric pattern model. This allows for easy adjustments in the project environment, as seen in Figure 14, where one of the latticework patterns with an Islamic geometric pattern has been parametrically adjusted for length and width (Figure 15).

4. Results

Following the steps mentioned earlier, described above in detail, another geometric pattern located on the eastern side of the southern Iwan of Varamin Grand Mosque has been modeled. Both geometric patterns present in the Iwan, as introduced in Figure 4 and Figure 7, have been modeled and described in

Table 1. Geometric patterns with tessellation 8².4 and 3⁶ (authors).

Number	Repeating Unit (Module)	2D Latticework with Shading in Revit	3D Latticework with Shading in Revit
1
2

.

Both of these geometric patterns have been modeled using the software, reviewed, and are capable of being utilized in contemporary projects. Given the mentioned approach, designers can generate a more diverse range of geometric patterns by modifying the repeating units, as demonstrated in these tessellations, and creating various geometric latticework. In addition to decorative purposes, the possibility of utilizing these patterns as architectural elements in urban areas, such as fences or semi-open space enclosures, offers numerous other benefits, such as limiting visibility to ensure the privacy of interior spaces of residential buildings and enhancing urban aesthetics.

5. Conclusions

Gereh in geometric patterns exhibits significant diversity, each governed by specific rules and drawing methods. Precision and adherence to the drawing rules for each pattern play a crucial role in visual appeal and coherence. Uniformity and harmony in these patterns are essential for visual attractiveness and coordination in these artworks. Particularly in patterns made from consecutive combinations and continuous transitions, maintaining precision in the exact implementation of drawing rules and paying attention to symmetry details will produce endless beauty through combinations and symmetries in geometric patterns.

In this article, two Islamic geometric patterns of the south Iwan of Varamin Grand Mosque were examined and modeled with a new method. Creating various geometric patterns with these tessellations using this method is feasible, and there are no limitations to the design and creation of these lattice works. Based on its specific project needs and objectives, each design can be used to create novel geometric patterns by modifying the original patterns (module) and utilizing the 8².4 and 3⁶ tessellations. This suggested method also applies to the infill architectural design on the buffer zone of historic buildings. While these designs are not direct copies of historical motifs, they are harmonious due to their similar context. This method, with adjustable parameters for latticework, can easily accommodate a variety of materials envisioned by designers for different projects. This article aims to facilitate and simplify the creation of new geometric patterns based on the region’s cultural and artistic values in line with contemporary requirements.