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A computerized method for generating Islamic star patterns
Abstract
In this paper, we will present a method based on the symmetry group theory to construct Islamic star patterns. The method builds star patterns using one or multiple kinds of stars/rosettes from a fundamental region of a given symmetry group and a few intuitive parameters. This paper explains the procedure used in the arrangement and construction of the stars/rosettes within the fundamental region. Given a symmetry group, the method begins by constructing, within the fundamental region, the minimal essential information (template motif) needed to generate a particular symmetry type. The constructed template motif, on which we operate a set of transformations that depend on the symmetry group, gives birth to the unit motif (content of the unit cell). Finally, the entire pattern is obtained by replicating the unit motif in a suitable net. The paper gives a set of generated patterns for the most encountered symmetry groups in Islamic geometric art.
... IGPs are commonly categorized based on the number of "points" the base geometry consists of. A study by Khamjane and Benslimane [10] with a primarily mathematical approach aimed to construct, within the fundamental region, the minimal key information needed to generate a particular symmetry type of IGPs (Fig. 5). The constructed template motif depends on the symmetry group, creating the unit motif. ...
... After conducting a thorough SWOT analysis, it is highlighted that further research is needed in regards to patterns' role in defining the image of a city and should be analysed in the form of a comparative analysis between the perception of architects and inhabitants. Figure 5: Twelve ray star parameters [10]. Figure 6: Doha high-rise office tower [11]. ...
... One such paragon of this artistic brilliance graces the board of the Madrassa Attarine in Fez, Morocco. Recognized among the most intricate geometric configurations in Islamic art, this design has garnered significant scholarly interest [20][21][22]. A comprehensive analysis of this pattern unveils its origin and underlying structures, amplifying our understanding of its aesthetic and mathematical constructs. ...
... 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 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.
... Aljamali et al (2009) emphasize unit pattern attributes, motif components, and geometric properties, introducing the IGP Designer tool. Conversely, Khamjane and Benslimane (2018) employed symmetry groups theory to automate periodic star and rosette patterns, introducing novel variations. The template motifs are constructed based on continuous strands. ...
... Some researchers have utilized symmetry group theory to categorize, examine, and produce Islamic geometric patterns. Computerized methods based on symmetry group theory are also accessible for generating these patterns [18][19][20]. Furthermore, there are comparable methods capable of constructing Islamic self-similar and quasiperiodic patterns [21][22][23]. ...
6-point girihs (6-point Islamic geometric patterns) are among the earliest and most popular geometric patterns created by Muslim artists in Iran. Some of them have distinctive behavior compared to the majority of Islamic geometric patterns. This paper examines the motifs used in 6-point girihs and categorizes them into basic and secondary motifs. We propose a method for constructing 6-point girihs based on symmetry group theories. The method involves a step-by-step process starting from constructing the minimal essential information (template motif) required to generate a particular symmetry type within the fundamental region, to replicating the unit girih in a two-dimensional plane using a suitable net of 6-point girihs. Finally, we present a notation system consisting of six parameters to construct the basic motifs of p6m 6-point girihs.
... Sayed et al. (2015) employed parameterised shape grammar to generate Islamic motifs, while Alani (2018) examined the metamorphosis of these motifs using a parametric model. Other studies involved the production and evaluation of motifs using CNC milling machines (Hamzaoğlu and Özkar 2018), the analysis of mosaics based on symmetry groups (Albert et al. 2015), and the reconstruction of Islamic star motifs utilising symmetry groups (Khamjane and Benslimane 2018). The parameterised shape grammar method has been used to create 3D Islamic motifs (Sayed et al. 2015), and symmetry groups have been employed to generate the sevenfold pattern (Naserabad and Ghanbaran 2023). ...
This study investigates the traditional religious architecture through the examination of Armenian Khachkar motifs at Varagavank Monastery in Van, Turkey. “Khachkar” is pivotal within Armenian religious architecture, denoting a minor architectural style and symbolising “cross-stone.” The study employs a parametric identification method to extract the fundamental unit of Khachkar motifs and develop a model to explore various variations. By utilising a mathematical analysis, the research showcases the practicality of this approach in facilitating diverse transformations within intricate designs. The study offers a comprehensive analysis of the built heritage found in Armenian religious architecture, enhancing the understanding and appreciation of these cultural artefacts. The findings of this study on Khachkars make valuable contributions to architectural research and deepen our comprehension of their significance in Armenian religious art.
... Düzgün çokgenler gibidirler, çünkü uç noktalarında kesişen düzlemdeki bölümlerin birleşiminden oluşurlar, ancak bitiş noktalarından başka noktalarda da kesişirler [1]. Ayrıca düzgün yıldız çokgenlerin geometrik süsleme sanatında geniş bir uygulama alanına sahip olduğu bilinmektedir [8], [9], [10], [11]. ...
In this study, the subject of regular star polygons, which has been studied for hundreds of years, is discussed. Regular star polygons, which were studied in the set of positive integers in the literature, were defined and analyzed for the first time in the set of rational numbers. The two variables we use to express regular star polygons are defined in rational numbers. This situation has been investigated according to whether regular star polygons have a spiral structure or not. For regular star polygons formed by the overlapping of their simplest forms at an equal angle, the relation giving this angle was obtained and proved by the proof without words technique. This angle is also the angle between the vertices in regular star polygons in spiral structure. Again, depending on whether the regular star polygon is spiral or not, two kinds of general number sequences are formed with these numbers by placing consecutive integers, starting with the number 1, at the corner points in the counterclockwise direction and following the formation lines. In addition, on the traces of the formation lines of regular star polygons, successive integers, again starting with 1, were placed at the corner points in the counterclockwise direction, and number sequences were obtained with the numbers at the corner points, respectively. With both methods, number sequences can be obtained without drawing a regular star polygon shape. With the study, the definition range for regular star polygons has been increased to the set of rational numbers.
... In Islamic patterns, stars are formed by intertwining lines. Many such methods are based on the symmetry group theory, which is used to construct Islamic star patterns by repeating the decorative shape within a suitable grid (Art & Shephard, 2015;Khamjane & Benslimane, 2018). The Penrose tilings are notable for being non-periodic (i.e., there is no translational symmetry) but well-organized. ...
Islamic floral patterns warrant further research and analysis as they are an important aspect of the cultural heritage of Islamic patterns. These floral patterns are aesthetically inspired by flowers, leaves, vines, and stems and feature characteristics such as symmetry, interlacing, and pattern repetition. This study analysed a five-pointed rose pattern (peony flower) and its elements, such as the curved lines that make up the leaves and flowers. A new floral pattern featuring a botanical motif and curved lines was designed and distributed using kite and dart tiling. The floral pattern was designed using the pentagram reflection of the Penrose tiling method to suit modern design requirements of looking like a Shamsah. The results of the floral ornament and newly designed patterns were then reviewed in order to facilitate the generation of new patterns accurately and quickly through computer design software. Thus, the problem of time and effort in designing Islamic floral patterns was solved. This study also provides suggestions for future studies on Islamic floral patterns.
... Aljamali [112] proposed a new method to classify and design star/rosette IGPs using computer software and the implementation of symmetry groups. Khamjane and Benslimane [113] presented a computerized method based on the symmetry groups theory to create periodic Islamic star and rosette patterns. The number of stars/rosettes and their parameters enabled them to develop novel ones. ...
Currently, there is a tendency to use Islamic Geometric Patterns (IGPs) as important identities and cultural elements of building design in the Middle East. Despite high demand, lack of information about the potential of IGPs principles have led to formal inspiration in the design of existing buildings. Many research studies have been carried out on the principles of IGPs. However, comprehensive studies relating to new possibilities, such as structure-based, sustainable-based, and aesthetic-based purposes, developed by computer science and related technologies, are relatively rare. This article reviews the state-of-the-art knowledge of IGPs, provides a survey of the main principles, presents the status quo, and identifies gaps in recent research directions. Finally, future prospects are discussed by focussing on different aspects of the principles in accordance with collected evidence obtained during the review process.
... Jowers (et al., 2010) proposed a generative method based on the symmetry group approach and shape grammar formalism. In their paper, Khamjane and Benslimane (2018a) proposed a computerized and general method for constructing Islamic star patterns based on the symmetry group theory. ...
Thanks to its beauty and its mathematical rigor, Islamic art has attracted the
attention of crystallographers, mathematicians, designers, artists, and architects. Among
scientific literature, various works have been performed to analyze the mathematical
structures of its ornaments and investigate their construction techniques. This paper will
first analyze the traditional Tastir style to design the sixteen-fold rosettes frequently used
in Moroccan and Andalusian geometric art. Based on this analysis, we will then propose
a computerized method to construct patterns with a sixteen-fold rosette.
... Other works provide mathematical tools for analyzing and constructing Islamic geometric patterns [9][10][11][12]. On the other hand, several works have focused on the pattern generation process [13], the symmetry group theory [14], and the quasi periodicity [15][16][17]. This work aims to present the relationship between the eight-point star and the other derived tiles. ...
This article concerns the traditional geometric patterns from Morocco and Andalusia. It shows the relationship between the 8-pointed star and other geometric tiles according to the traditional method called “Tastir”. It also presents the symmetry groups characterizing these geometric patterns.
... The second step is to construct a template motif within the right triangle of each golden triangle (Fig. 9b). The algorithm of the construction of the template motif within a right triangle is described in detail in (Khamjane and Benslimane 2018b). To obtain the content of each golden triangle, we apply mirror reflection to the content of its associated righted angle triangle. ...
Because of its unequal beauty and mathematical sophistication, Islamic art has received a great attention from several scientists. Hence, several works have been done to investigate its mathematical structure, and to discover its principle of construction. Up to now, no method of constructing new self-similar patterns were proposed. In this paper, we will present a method for constructing new self-similar patterns. The proposed method is based on successive subdivisions of the golden mean triangles.
... Other researchers have investigated Islamic patterns to reveal the principle beyond the construction of these patterns. Most of these methods are tiling based approaches [8]- [13], while other methods are based on the symmetry group theory [14], [15]. In addition, several scientists have analyzed several patterns to prove their quasi-periodicity [16]- [22], while, others have provided generation methods for producing quasiperiodic patterns [13], [23]- [25]. ...
We present a simple method for constructing patterns of Islamic geometric art. The method allows creating a parameterized set of motifs within a fundamental region of a periodic pattern. The constructed template motif will undergo the inner symmetries of a chosen symmetry group to obtain the content of the unit cell. The whole pattern is obtained by applying several linearly independent translations to the generated unit motif. We show how this method can be adapted to construct Islamic quasi-periodic patterns.
... The second step is to construct a template motif within the right triangle of each golden triangle (Fig. 9b). The algorithm of the construction of the template motif within a right triangle is described in detail in (Khamjane and Benslimane 2018b). To obtain the content of each golden triangle, we apply mirror reflection to the content of its associated righted angle triangle. ...
Because of its unequal beauty and mathematical sophistication, Islamic art has received a great attention from several scientists. Hence, several works have been done to investigate its mathematical structure, and to discover its principle of construction. Up to now, no method of constructing new self-similar patterns were proposed. In this paper, we will present a method for constructing new self-similar patterns. The proposed method is based on successive subdivisions of the golden mean triangles.
... Yi utilized symmetry group theory to analyze 17 different patterns in the design of a building facade [10]. Aziz Khamjane proposed a method to generate Islamic star patterns based on symmetry group theory [5]. José Pedro Sousa utilized digital technology to further explore the application of symmetry in architectural design and manufacturing in the context of teaching experiments [8]. ...
Islamic geometric patterns known as “girihs” are typically seen as 2D designs. However, in traditional Iranian architecture, girihs have been used to decorate curved and 3D surfaces. While there have been many studies on 2D girihs and methods for creating them, there has been less focus on 3D girihs. Additionally, there isn’t a comprehensive digital technique for constructing 3D girihs. This paper aims to fill this gap by proposing an automated approach for constructing girihs on curved surfaces to create 3D girihs. We use the UV coordinate relative to the NURBS (Non-Uniform Rational B-Spline) surfaces to identify the corresponding generative points of 2D girihs on 3D surfaces, then connect the points to construct 3D girihs. With this method, any geometric girih can be mapped onto a 3D surface, and under the right conditions, any 2D designs and motifs can also be mapped. We implement the proposed method in Grasshopper for Rhino 8 for Win, controlling the density and mapping quality of 3D girihs through parameters NMU and DL. By integrating this algorithm with a 2D girih generation algorithm, we provide the possibility of real-time customization and interactive construction of 3D girihs.
Hybrid girihs refer to Islamic geometric patterns that include various stars/rosettes in their final pattern. In this paper, we first identified historical hybrid girihs and then categorized them based on symmetry groups and the number of stars/rosettes folds. In the next step, we analyzed the existing hybrid girihs to identify the generative parameters and present a method for generating historical and novel systematic and non-systematic hybrid girihs. The proposed method of this paper is a computational and parametric approach based on the symmetry groups theory. Its general steps include generating the minimal essential information (template motif) within the fundamental region, applying appropriate symmetry operations on the content of the fundamental region to create the content of the unit girih, and replicating the content of the unit girih in a suitable network according to the symmetry group to create the whole pattern. Our method is used to generate hybrid girih for adorning surfaces in digital spaces and for constructing facade modules (adorned with Islamic geometric patterns) and interior decorative partitions and furniture in physical spaces according to the aesthetic judgment of users.
This paper presents a morphological study of Islamic geometric patterns (IGP) and their role in application of formal grammar in computational modeling of IGPs. Through a comprehensive literature review and data collection, we analyze the morphological properties of these patterns using techniques such as geometric transformations, pattern classification, and symmetry analysis. Based on our findings, we explore how these properties can be used in constructing a formal grammar through string rewriting system for a CAD application. Building on a study on the potential of a string rewriting system for modeling IGPs, the current research suggests an update to the previous system and introduces a new morphological structure for IGPs. The new method has an expanded sample set and is tested on a class of 5-fold star patterns with 12 members and demonstrates successful development. The results are implemented in a Grasshopper add-on, providing a flexible platform to generate the patterns through strings and to control their parameters. This tool opens up new possibilities to bridge traditional patterns with contemporary technologies and make them more accessible. Furthermore, it contributes to the preservation of IGPs as a significant cultural and architectural heritage, while also advancing the evolution of these patterns to a new and contemporary generation.
Bu çalışmada, üzerinde yüzlerce yıldır çalışılan düzgün yıldız çokgenler konusu ele alınmıştır. Literatürde pozitif tam sayılar kümesinde çalışılmış olan düzgün yıldız çokgenler ilk kez rasyonel sayılar kümesinde tanımlanmış ve analiz edilmiştir. Düzgün yıldız çokgenleri ifade ederken kullandığımız iki değişken rasyonel sayılarda tanımlanmıştır. Bu durum düzgün yıldız çokgenlerin sarmal yapıda olup olmamasına göre incelenmiştir. En sade formlarının eşit bir açıyla üst üste gelmesiyle oluşan düzgün yıldız çokgenler için, bu açıyı veren bağıntı elde edilmiş ve sözsüz ispat tekniği ile ispatlanmıştır. Bu açı sarmal yapıdaki düzgün yıldız çokgenlerde de köşeler arası açıdır. Yine düzgün yıldız çokgenin sarmal yapıda olup olmamasına göre köşe noktalarına saat yönünün tersi yönde birden başlayarak ardışık tam sayıları yerleştirip oluşum çizgilerini takip ederek bu sayılarla iki çeşit genel sayı dizisi oluşturulmuştur. Ayrıca düzgün yıldız çokgenlerin oluşum çizgilerinin izinde köşe noktalarına saat yönünün tersi yönde birden başlayarak ardışık tam sayıları yerleştirip birden itibaren sırasıyla köşe noktalarındaki sayılarla sayı dizileri elde edilmiştir. İki yöntemle de düzgün yıldız çokgen şekli çizmeden sayı dizilerimizi elde edeceğimiz metotlar ortaya çıkartılmıştır. Yaptığımız çalışma ile düzgün yıldız çokgenler için tanım aralığı rasyonel sayılar kümesine yükseltilmiştir.
The construction of the tiling is a crucial step for the construction of Islamic geometric patterns by the polygonal method. However, the majority of the works based on the polygonal method ignore this step. Hence, the researchers try to extract tessellations from existing geometric patterns. Therefore, it would be useful to develop a method to build a wide variety of tilings. This broadens the spectrum of constructed geometric patterns based on the Polygons In Contact (PIC) method. The goal of this paper is to provide guidelines for generating periodic tilings of the symmetry groups pmm and cmm. The proposed method is based on the symmetry group theory.KeywordsIslamic artgeometric artstarrosettetilingpolygonsymmetry group
Girihs (Islamic geometric patterns) are an example of graphic designs in architectural spaces. Some wonderful Islamic geometric patterns are produced by 7-fold polygonal sub-grid elements. In this paper, after examining 7-fold patterns, we will present a method based on the symmetry groups theory to construct 7N-point girihs. This method is based on a step-by-step process from constructing the minimal essential information (template motif) needed to generate a particular symmetry type within the fundamental region to replicating the unit girih in a two-dimensional plane based on a suitable net. Finally, we will present a notation system for the method of this paper consisting of generative parameters to construct girihs of the 7-fold system and other Bonner’s systems classified into four families: acute, middle, obtuse, and two-point.
In this study, the subject of regular star polygons, which is one of the important topics of geometry and which has taken
place in many works since the fourteenth century, is discussed. A significant number of new definitions and features were
added at the end of the process to the known definitions and features related to the subject. Based on the basic logic of
regular star polygons, a new drawing technique has been developed, which is different from the known drawing technique
using regular polygons. Equivalent parts of regular star polygons connecting the corner points are called formation lines
and the study is dealt with around this scope. The lines of formation are classified as acute and obtuse according to the
angles they form with the horizontal axis and important results are obtained. The angular arrangement of the layout of
acute and obtuse formation lines on regular star polygons and the number of acute and obtuse formation lines are
generalized according to mode 4. The circles connecting the points formed by the intersection of the formation lines on
any regular star polygon are called the circle layers and the intersection points of the formation lines are called star
coordinates and their properties are examined. A method in which we can determine the position of the star coordinates
in any regular star polygon in four steps without drawing shapes has been developed and supported by applications. The
determination of the position of the star coordinates for all star polygons without the need for drawing will prepare the
ground for new studies. Conceptually, the fact that all regular polygons are also regular star polygons puts a special
emphasis on the subject.
Geometric patterns are one of the most important features of Islamic monuments that are formed based on Islamic codes and concepts. The use of geometry in arrays and works of art and Islamic architecture has been a gateway to the knowledge of unknown universal codes, the manifestation of which is derived from the concepts of God. In Islamic arts, every motif has a secret that the correct interpretation of its concepts is achieved when the researcher can gain a deep understanding of the cultural components and religious beliefs of that society. Soltan Mohammad Khodabandeh (Al-Jaito), Ghazan Khan's brother, was born to an Uzbek father and an Armenian mother. During his lifetime, Al-Jaito chose different religions until he was finally influenced by the Shiite religion and became very devoted to the Shiite dynasty. So much so that he decided to build a great tomb with various sacred decorations of Islamic art in Zanjan (the capital of Iran during the Ilkhanate) and move the bodies of Imam Ali (AS) and Imam Hussein (AS) from Najaf and Karbala to that place. However, after completing the construction, he faced strong opposition from the imitators of that time and could not achieve his long-held dream, and at the end of his life, he turned to the Sunnis again. After him, during the reign of his son, who had a special devotion to the Sunnis, the interior decorations were tailored to his beliefs, which had the color and scent of the Sunni concepts. The construction of the historic city of Soltaniyeh began during the reign of Arghun Khan, the Mongol ruler, and developed during the reign of Soltan Mohammad Khodabandeh as one of the great Islamic cities. By the order of Al-Jaito, the dome of Soltaniyeh located in Zanjan city was introduced in 704 AH, in the city of Soltaniyeh, the capital of the Ilkhans at that period, and in 712 AH the tomb of Al-Jaito was erected in the central core of the city of Soltaniyeh by his order. It is known as the tallest building and the largest brick dome in the world and one of the most beautiful historical buildings in the world, which is the manifestation of the glory and art of Islamic architecture and the objective manifestation of the evolution of Iranian architecture. The height of the dome is 48.50 meters, the diameter of the opening is 25 meters with 1.60 meters thickness, constructed by the double-shell technique. The purpose of this study is to interpret the concepts and geometric codes hidden in the decoration and architecture of Soltaniyeh Dome based on an analysis of Islamic concepts from the perspective of Quranology, Hadithology, Numerology, Abjad, Al-Jafr, Asma al-Hassani along with symbolism of geometric shapes and color in decoration and architecture. Also in the analysis of golden proportions of the building, 2020 PhiMatrix 1.618 Pro, Atrise Golden Section 2018, 2018 Golden Ratio and PhiMatrix Golden Ratio Design 2020 software have been used. For this purpose, the main research question is: In what geometric forms are the meanings and concepts used in the decoration and architecture of Soltanieh dome reflected in the arrays of this monument with the aid of numbers? What is the role of geometry in creating spiritual connections between material, mystical worlds and creating a single message in the decorative arrays of Soltaniyeh dome and the effectiveness of the arrays can be classified into how many levels? The research method was historical-interpretive data collection and comparative-analytical data analysis. The results show that the concepts and meanings are reflected in the arrays in physical forms (in the form of buildings and the hierarchy of formation of spaces), religious symbols (such as Shamseh, celestial bodies), linear shapes (divine names and verses), colors (to create sense of space). The use of sacred numbers, symbols and religious forms in the spatial and physical structures of the building has caused a spiritual connection between the earth (material) and light (spiritual) world. Monument arrays are also categorized into 5 levels: geometric shapes (square, octagonal, and circle), numbers (4, 5, 6, 8, 10, 12, 16, 23, 36, 40, 42, 60, and ... 110), holy names (Allah, Mohammad (PBUH), Ali (AS)) and Sunni religious concepts, use of spaces, Quranic and hadiths verses in the form of Bannai script. Also, the results of the research show that all the components of the Soltaniyeh Dome are based on Islamic concepts that are appropriate to its use, which was a place for the burial of Imams. In fact, the master artists and architects of Soltaniyeh Dome with full knowledge of various sciences such as geometry (complete knowledge of formal and mystical concepts of geometric shapes), mathematics (recognizing sacred numbers and their use in structural elements), occult sciences, .
Domes constitute a significant part of Iranian traditional architecture. Iranian architects not only created hemisphere shape domes, they also created conical shape domes. Documents about conical shape domes of Iran are scarce; among conical shapes, data about Ourchin domes is particularly limited due to the rarity of remaining monuments. Most of these domes can be found in the south and southwest of Iran, which suffers from severe weather conditions, dust storms, earthquakes, floods, high and dry temperatures, and even war hazard. Therefore, conservation and documentation of these monumental domes are essential and must be done immediately. This research attempts to document the geometry and construction process as well as discussing the typology of Ourchin domes.
Any stepped dome in which the layers are self-similar and the whole structure could fit in a cone or a paraboloid is an Ourchin dome. Ourchin dome geometrical design process modeling is a critical task, in order to conserve, protect, and restore these particular domes. Required data to preserve the geometry of Ourchin domes is presented through text and illustrations. This modeling divides Ourchin domes into three distinct types, which are: polygonal, star, and circular shapes.
Results indicate that although Ourchin domes have complex 3D geometry, the building process was actually simple and practical. In the design process, four key factors (plan shape, Hanjar shape, number of regular polygon sides, and ratio between Sha'hang and radius of dome base) define elongation of the dome. Due to the fact that the number of layers is theoretically infinite, the construction process proceeded by the designer's preference. Other circumstances, such as material cuts, fragility of smaller brick pieces, and inability of constructing, all had a profound impact on this choice as well.
Geometry is one of the most prominent aspects of Muslim art that has dominated all the works of influential Islamic artists. In other words, geometry in the thought of Islamic art is the door to the spiritual word from the material world. If we want to point out the importance and the role of geometry in art and architecture, we can state that geometry can be considered as an asset of the work, and its various creations can be easily understood by its definition and limitation. In other words, with the dominance of geometry on any structure, from the structures of a simple atom to the components of a work of art, architecture, and urban, we can see a single order in all dimensions. The order, which has always been disrupted by geo-disciplinary principles due to the neglect of many urban architects and designers, and has caused many problems at many levels of life. It is undoubtedly one of the most successful examples in which geometric principles are of great interest at all levels of art and architecture. The historic village of Abyaneh, located in the central part of Natanz, Isfahan province, has become nationally recorded as a valuable and historic monument. The level of utilization of the geometric principles in this village is such that if we want to look at all the structures of art in a superficial way, we can see the same footprint of homogeneous geometric designs used in all their structures and architecture; so, in this research, we are confronted with the question of which geometric patterns exist in the Abyaneh historical village? Are the same geometric patterns repeated throughout the artistic, architectural, decorative, apparel, musical, and texture structures? According to the content mentioned in the present article, to meet the main questions and achieve the research purpose, the data collection by the historical-interpretive method with the phenomenological approach and the analysis of information will be analytic and adaptive. Therefore, the main objective of this paper is to recognize the geometric language in Abyaneh architecture by understanding the geometric structures that exist in all the arts of Abyaneh village. The most important results of this research are the use of geometric patterns, golden proportions, golden rectangles, golden spirals, golden polygons, the series of Fibonacci numbers and Pimon, which is a quarry used in all artistic structures such as architecture, handicrafts, rugs, carpets, textiles, traditional music, rural structures, and the way buildings are erected in the context of the Abyaneh village. In other words, it can be pointed out that geometric patterns and golden proportions are the common language among all the arts in the Abyaneh historical village, which are used continuously in all architectural structures. In other words, the use of geometry in this historical village creates the same beautiful and coherent structures in small artistic scales such as artistic, musical, ornamentation, textiles, carpets, cloths or on larger scale such as the general form of buildings and the way buildings are erected in Abyaneh village.
One of the large jali screens adorning the mausoleum of Muhammad Ghaus in Gwalior (N India), built in 1565, contains panels composed of disordered composite octagons and Salomon stars. These elements show a rotational disorder with some interdependence. Analysis of these partially disordered patterns with rotatable configurations of the above elements suggested that they may be approximants of a quasiperiodic octagonal tiling based on a new type of composite tiles. Comparisons with the Amman’s quasiperiodic tiling were made. Instances of similar or related periodic ornamental patterns at other northern Indian localities are analyzed as well.
The tympanum of the entrance of the Zaoua Moulay Idriss II in Fez contains the only known example of a dodecagonal cartwheel quasiperiodic pattern in Islamic art, dating possibly from the Merinid epoch. This pattern, carved in a marble plate, is based on a type of Ammann quasilattice known also from modern mathematical literature. The central portions of this pattern were used as elements in a periodic pattern on the walls of the Saadian mausoleum in Marrakech. © 2011 International Union of Crystallography Printed in Singapore - all rights reserved.
Many works report the classification and analysis of geometric patterns, particularly those found in the Alhambra, Spain, but few authors have been interested in Moroccan motifs, especially those made on wood. Studies and analyses made on nearly a thousand Moroccan patterns constructed on wood and belonging to different periods between the 14th and 19th centuries show that, despite their great diversity, only five plane groups are present. Groups p4mm and c2mm are predominant, p6mm and p2mm are less frequent, while p4gm is rare. In this work, it is shown that it is possible to obtain the 17 plane symmetry groups by using a master craftsmen’s method called Hasba. The set of patterns are generated from n-fold rosettes, considered as the basic motif, by the Hasba method. The combination and the overlap between these basic elements generate other basic elements. By repeating these basic elements, it is possible to construct patterns having various symmetry groups. In this article, only uncoloured patterns are considered and the interlace patterns are disregarded.
The Topkapı Scroll is an important documentary source for the study of Islamic geometric ornament. Here we give a mathematical analysis of some its exemplary star patterns that illustrate a variety of methods of construction. Highlights include a design combining 9- and 11-point stars, and another with 13- and 16-point stars. We show that the practice of producing a design by replicating a template using reflections in its sides restricts the range of symmetry types produced. In particular, the lack of 3-fold symmetry is related to the exclusive use of rectangular templates. By transferring the distributions of stars traditionally seen in a square template to their equivalent arrangements in an equilateral triangle, we produce new 3-fold designs in the Islamic style.
This article presents mathematical tools for computer-generated ornamental patterns, with a particular attention payed to
Islamic patterns. The article shows how, starting from a photo or a sketch of an ornamental figure, the designer analyzes
its structure and produces the analytical representation of the pattern. This analytical representation in turn is used to
produce a drawing which is integrated into a computer-generated virtual scene. The mathematical tools for analysis of ornamental
patterns consist of a subset of tools usually used in the mathematical theory of tilings such as planar symmetry groups and
Cayley diagrams. A simple and intuitive step-by-step guide is provided.
Repetitive interface patterns are one of the hallmarks of Islamic and Moorish art. Through the study of various collections of such patterns, it is easy to verify that, despite the considerable complexity of the designs, most of the interlaces are formed by strands of a small number of shapes – often just a single shape stretching over many repeats of the design. This observation is described and documented by the authors, who present a simple explanation for this phenomenon.
Tilings and patterns have been made and enjoyed for thousands of years. Their mathematical treatment was begun by n J. Kepleren but was then forgotten until the nineteenth-century development of crystallography. In this unique book, with its abundant illustrations, the authors explain exactly what one means by "tiling" and "pattern", restricting the treatment to two dimensions. There are many surprises; for instance, Figure 1.2.2 shows the 24 "heptiamonds", with the remark that only one of them cannot be repeated by congruent copies to fill and cover the whole plane. Chapter 2, on "Tilings by regular polygons", includes n A. J. W. Duijvestijn'sen "squared square", in which 21 different squares, with sides , are fitted together to fill a square of side 112. Any solution to the slightly simpler problem of "squaring a rectangle" can be extended to a tiling of the whole plane by infinitely many squares of different sizes. Chapter 3, on "Well-behaved tilings", tells us precisely when a tiling can be called "normal". One counterexample is the remarkable Figure 1.0.1 (repeated as a cover design) which is monohedral (all tiles congruent) but is abnormal in that some pairs of tiles share a disconnected set of boundary points. Euler's theorem is used to prove that, if every tile of a normal tiling has k vertices, where the valences are , then . Figure 3.8.6 illustrates a nice paradox: it shows a particular pentagon which is entirely surrounded by seven congruent replicas although the arrangement cannot be extended to a monohedral tiling of the whole plane. Chapter 4 describes the transition from metrical to topological tilings. Chapter 5 introduces the subject of patterns, beginning with a fascinating example (Figure 5.0.1) based on a maze. Except in some of the exercises, a discrete pattern means a planar family of mutually disjoint congruent copies of a motif with the property that for each pair of copies, say and , there is an isometry of the plane that maps the whole pattern onto itself and onto . According to this strict definition, the abnormal Figure 1.0.1 is not a pattern: every two of its infinitely many tiles are related by an isometry that maps one onto the other, but in no case is this isometry a symmetry of the whole pattern! The authors have undertaken the almost incredibly difficult task of classifying patterns so that one can say in what sense any two given patterns are of different types. Table 5.2.1 lists the 3 types of finite patterns, each type-symbol involving an integer n which is the smallest period of a rotatory symmetry; Table 5.2.2 lists the 15 types of frieze patterns; Table 5.2.3 lists the 52 types of discrete periodic patterns. The number of types becomes smaller when the arbitrary motif is replaced by a dot or other symmetrical shape. Further complications arise when the motif is allowed to be infinite, or when copies of the motif overlap. Chapter 6 combines the two topics (tilings and patterns) by making even subtler distinctions: it can happen that several tilings are "really different" even though they have the same topological type and the same pattern type. There is a historical account of the classification of tilings. Attempts by some highly respected crystallographers, such as n A. V. Shubnikoven and n V. A. Koptsiken ref[ Symmetry in science and art, English translation, Plenum, New York, 1974], "led to an almost unbelievable number of errors". A different method of classification is developed in Chapter 7. Chapter 8 describes the complications that arise when tiles are distinguished by being variously colored. Chapter 9 deals with tilings by polygons, not necessarily regular; for instance, there are 24 types of tilings by congruent pentagons. The most exciting developments are reserved for Chapters 10 and 11, on "Aperiodic tilings". Before 1966, nobody could imagine the existence of a set of prototiles which would admit infinitely many tilings of the plane although no such tiling is periodic. Obviously, even such a simple prototile as a domino admits a nonperiodic tiling; but the exciting new idea, embodied in the term "aperiodic", is a set of n prototiles which cannot possibly be arranged in a periodic fashion. n R. Bergeren ref[Mem. Amer. Math. Soc. No. 66 (1966); MR0216954 (36 #49)] discovered the first aperiodic tiling, with n=20426. Berger himself soon reduced this fantastic number to 104, n D. E. Knuthen to 92, n H. Läuchlien to 40, n R. M. Robinsonen to 35, n R. Penroseen to 34, n Robinsonen (again) to 32, and later to 24, n R. Ammannen to 16, and later to 6, then Penrose (again) to 5, and ultimately to 2! Many of the amazing ramifications of this theory, including some by n J. H. Conwayen, are published here for the first time. Chapter 11 deals with "Wang tiles": square tiles having colored edges which must match with their neighbors, only translations being allowed. These aperiodic tilings are relevant to questions of logic and computing, because it is possible to find sets of 16 Wang tiles which mimic the behavior of any Turing machine. Finally, Chapter 12 relaxes the restriction that the tiles should be topological disks, or that the tiling should cover the plane only once. The book ends appropriately with a 42-page bibliography and a 6-page index. Reviewed by H. S. M. Coxeter
This note considers the frequency of the 17 planar symmetry groups in Islamic Geometric Art. The collection used for this analysis contains over 600 patterns and is thought to the largest available.
We present a simple method for rendering Islamic star patterns based on Hankin's "polygons-in-contact" technique. The method builds star patterns from a tiling of the plane and a small number of intuitive parameters. We show how this method can be adapted to construct Islamic designs reminiscent of Huff's parquet deformations. Finally, we introduce a geometric transformation on tilings that expands the range of patterns accessible using our method. This transformation simplifies construction techniques given in previous work, and clarifies previously unexplained relationships between certain classes of star patterns.
In this article, we propose a general computational model for the extraction of symmetry features of Islamic geometrical patterns' (IGP) images. We describe IGP images using the discrete symmetry groups theory. Our model contains the three following steps. (1) By noting that these patterns fall into three major categories, we begin our indexation process by classifying every pattern into one of these categories. The first pattern category describes all the patterns generated by translation along one direction. Every pattern of this category can be classified into one of the seven Frieze groups. The second type of pattern contains translational symmetries in two independent directions. Patterns of this category can be classified into one of the seventeen Wallpaper groups. The last type, called rosettes, describes patterns which begin at a central point and grow radially outward. We use rosette symmetry groups to classify patterns of this latter category. (2) For every pattern, we extract the symmetry features, namely, the symmetry group and the fundamental region, which is a representative region in the image from which the whole image can be regenerated. But for rosette groups, we can also compute the number of folds. (3) Finally, we describe the fundamental region by a simple color histogram and build the feature vector which is a combination of the symmetry feature (defined in the second step) and histogram information. Experiments show promising results for either IGP images' classification or indexing. Efforts for the subsequent task of classifying Islamic geometrical patterns' images can be significantly reduced.
We present Najm, a set of tools built on the axioms of absolute geometry for exploring the design space of Islamic star patterns. Our approach makes use of a novel family of tilings, called "inflation tilings," which are particularly well suited as guides for creating star patterns. We describe a method for creating a parameterized set of motifs that can be used to fill the many regular polygons that comprise these tilings, as well as an algorithm to infer geometry for any irregular polygons that remain. Erasing the underlying tiling and joining together the inferred motifs produces the star patterns. By choice, Najm is build upon the subset of geometry that makes no assumption about the behavior of parallel lines. As a consequence, star patterns created by Najm can be designed equally well to fit the Euclidean plane, the hyperbolic plane, or the surface of a sphere.
This paper presents a method of coding and generating infinite classes of comlex periodic geometric pattern found in various cultures. The method uses a pattern code for the known symmetry groups and is based on the infinite classes of subdivisions of the fundamental regions of each symmetry group. The subdivisions provide a skeletal grid from which selected point, line or polygonal sequences can be extracted to produce an infinite variety of geometric patternswithin a symmetry group. Surprisingly, many complex architectural patterns from different periods and cultures, notably those from historic Islamic buildings, and Chinese window lattices, can be systematically obtained by this method. The pattern-codes provide a basis for generation of simple and complex geometric patterns, and their transformation from one to another. Some directions for systematic transformations based on meta-structural principles are outlined.
Thesis (Ph. D.)--University of Washington, 2002 Throughout history, geometric patterns have formed an important part of art and ornamental design. Today we have unprecedented ability to understand ornamental styles of the past, to recreate traditional designs, and to innovate with new interpretations of old styles and with new styles altogether.The power to further the study and practice of ornament stems from three sources. We have new mathematical tools: a modern conception of geometry that enables us to describe with precision what designers of the past could only hint at. We have new algorithmic tools: computers and the abstract mathematical processing they enable allow us to perform calculations that were intractable in previous generations. Finally, we have technological tools: manufacturing devices that can turn a synthetic description provided by a computer into a real-world artifact. Taken together, these three sets of tools provide new opportunities for the application of computers to the analysis and creation of ornament.In this dissertation, I present my research in the area of computer-generated geometric art and ornament. I focus on two projects in particular. First I develop a collection of tools and methods for producing traditional Islamic star patterns. Then I examine the tesselations of M. C. Escher, developing an "Escherization" algorithm that can derive novel Escher-like tesselations of the plane from arbitrary user-supplied shapes. Throughout, I show how modern mathematics, algorithms, and technology can be applied to the study of these ornamental styles.
The question which of the seventeen wallpaper groups are represented in the fabled ornamentation of the Alhambra has been raised and discussed quite often, with widely diverging answers. Some of the arguments from these discussions will be presented in detail. This leads to the more general problem about the validity and meaning of the answers to such questions. The second part of the paper deals with the approaches to symmetry in ornaments of various cultures that should replace the mechanical counting of the wallpaper groups that occur. A more reasonable investigation would deal with symmetries as they may be considered and understood by the people of the societies in question. Such an approach to the remarkable orderly decorations of ancient Peruvian fabrics is presented.
The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture
were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge
and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations
of a special set of equilateral polygons (“girih tiles”) decorated with lines. These tiles enabled the creation of increasingly
complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations
to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West.
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