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Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
Abstract
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.
... In addition to these studies, Graünbaum and Shephard (1986) attempted to define and visualize the concept of tessellation [5]. The studies on tessellation has continued in the twenty-first century as well [6][7][8]. It is possible to see impressive 2 examples of the tessellation in contemporary architecture not only as a static building element but also as kinetic ones. ...
... This set consists of a regular decagon, a rhombus, a regular pentagon and a bowtie-like shape ( Figure 10). The sides of each polygon are equal, and the lines emerging from the midpoints of both sides intersect with each other at 72 and 108 degrees [7]. ...
... The date when Girih pattern was first used in architecture is still unknown [7], but it is known that this pattern was used in the buildings of Seljuk and Ilkhanians. It was also used in decorative arts in the 14th century. ...
Tessellation, which has examples of use in art and architecture, is the covering of a surface using one or more geometric shapes without overlapping or gaps. Based on Roman mosaics, the tessellation has an important place in architecture since the ancient times. Through the history, different patterns have been used by many cultures for various applications ranging from decorative covering elements to multi-functional latticework screens. The tessellation has still been used in contemporary architecture since it not only allows creating the geometrical surface in an order but also provides multi-functionality to the surface when applied as shading elements. The tessellation can be reviewed under three categories such as regular, semi-regular and demi-regular tessellations. Two- and three-dimensional examples of the tessellations can be seen in contemporary architecture either as façade elements or patterns used for structural elements. Because the tessellation plays a significant role in architecture in terms of geometrical or structural design, the interest on this topic has been increased in recent years. Due to their great potentials, more studies should be conducted on the tessellations. For this reason, within the scope of this paper, the applied examples of the tessellations in buildings from past to present are examined which include both static and kinetic ones. In this paper, the geometric design principles, combination methods and iteration processes of the examples are also presented. As well as providing a deeper understanding of such tessellation methods, this study will serve as a basis of reference for future studies in this field.
... Discrete shapes were generated, maintaining the logic of the pattern. This was done by referencing one of the methods in which Zellige masters traditionally generated patterns, subdivision (Lu and Steinhardt, 2007). In this method, the craftsman began by identifying the intended symmetry, in this case, a fivefold symmetry. ...
... In this method, the craftsman began by identifying the intended symmetry, in this case, a fivefold symmetry. Next, they took a basic fivefold pattern, often a rosette, subdividing it repeatedly until revealing the desired level of detail (Lu and Steinhardt, 2007). In this research, the symmetry of the pattern and its generation process are revealed and reinterpreted through modern means, highlighting the craftsmen's impressive undertaking. ...
The intricate details of Zellige tile work in Islamic architecture have high cultural significance and important sacred associations. The quasi-crystalline geometry found in el Attarine Madrassa Zellige patterns are a splendid example of the geometric rigour and exceptional craftsmanship of these sacred decorations. Modern methods have been used to reproduce these complex patterns, but these methods are mostly focused on duplication, rather than contributing to the analysis and re-contextualization of these motifs. Where recent studies have investigated the tectonics and structural limitations of ceramic additive manufacturing, few have explored its unique capabilities of re-interpretation and abstraction. This paper investigates the Zellige tiles of el Attarine Madrassa in Fes through formal and tectonic prototyping via clay 3D printing. Adjustments to the pattern by twisting, shifting, and shrinking were evaluated to observe what new interpretations additive manufactured ceramics can provide to the reading of the pattern at an architecture scale. In parallel, custom tool path design strategies were developed to address the unique intersection and overlapping issues that resulted from the intricate linework of the mosaic's geometric pattern. Several full-scale physical prototypes were developed to assess the success and design opportunities of each strategy. Design iterations were also conducted to develop a key stone assembly sequence while also testing assembly tolerances and bonding between component units. Lastly, a dome mosaic assembly was designed for an arbour structure to evaluate the architectural implications of the developed tile system. The objective is for this 3D printed investigation to simultaneously honour and innovate upon the rich continuum of craft tradition, culture, embodied knowledge, and spirit within these geometries.
... The discovery of historical Islamic patterns exhibiting non-periodic geometry has triggered an interest in understanding the traditional principles that generated these complicated formations (Makovicky, 1992;Lu & Steinhardt, 2007;Al Ajlouni, 2012, 2013Ajlouni, 2017). To date, three types of rotational symmetries have been documented. ...
... To date, three types of rotational symmetries have been documented. These include 10-fold quasi-periodic patterns (Makovicky, 1992;Makovicky et al., 1998;Lu & Steinhardt, 2007;Al Ajlouni, 2012), 8-fold quasi-periodic patterns (Emil Makovicky & Fenoll Hach-Alı, 1996;Al Ajlouni, 2013), and 12-fold non-periodic patterns (Makovicky & Makovicky, 2011). Figure 1 shows the central cartwheel formations of three historical examples; the pattern on Darb-i Imam shrine in Esfahan (1453) (Figure 1a), the pattern on the walls of the Hall of the Ambassadors in Alhambra, Spain (1354) (Figure 1d) and the dodecagonal pattern found in the tympanum of the entrance of the Zaouia Moulay Idriss II in Fez, Morocco (1308) (Figure 1g). ...
To cite this article: Rima Ajlouni (2023): Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries, Journal of Mathematics and the Arts,
... Penrose studied the nonperiodic patterns and tried to cover the surface in a nonperiodic method with a tile group consisting of two tiles to cover the surface. Lu and Steinhardt evidently proposed that Girih tiles can create complex 15th-century aperiodic patterns (Lu and Steinhardt, 2007). Their recent investigations show these medieval tiles contain symmetries like those found in aperiodic Penrose tiling first investigated in the West in the 1970s. ...
... The generating process of the form starts with shapes based on the construction of 2D Girih patterns, which are the parameters used to provide a corresponding way to construct 3D Girih patterns; these processes can tessellate through periodic and aperiodic fractal space (Bonner 2017). Their digital Girih shading device was built based on the research of Lu and Steinhardt (Lu and Steinhardt, 2007). Specifically, the design develops Islamic geometric patterns from twodimensional coordinates to three dimensions through their repeat units in contemporary utilization. ...
https://doi.org/10.55554/2785-9649.1012
... In the world, traditional unreinforced brick architecture is no longer used in new construction. However, masonry's architectural history must be conserved; this necessitates structural study, because masonry has been the primary construction material in architecture from antiquity to the present [7]. In contrast, its ability to maintain its position under horizontal loading is referred to as its stability. ...
... In the world, traditional unreinforced brick architecture is no longer used in new construction. However, masonry's architectural history must be conserved; this necessitates structural study, because masonry has been the primary construction material in architecture from antiquity to the present [7]. ...
The process of combining various parts to create a structure is called building. The most effective and significant component of any construction is masonry. The Colosseum, buildings from ancient Greece and Rome, Central American buildings, and Mycenaean structures all used this material as one of their primary building elements. The oldest form is dry masonry of irregularly shaped stones. The ecological qualities of masonry, as a restorative material with a low impact on the environment, as well as the environmental control capacity of the massive wall, bring masonry back to attention as a suitable material for sustainable building in the context of current concerns for sustainable architecture. This article takes the form of a review of the journey of masonry as the primary construction material—from prehistoric structures to modern-day edifices. This article will go through the fundamentals of masonry construction to support its usage in structures throughout history and in many architectural styles, as a crucial representation of human construction in architectural history. This article aims to create a historical review, presenting masonry as an essential building material and assessing its role in the history of building materials.
... Duarte analysed design patterns of Alvar Siza's patio houses at Malagueira using shape grammars (Duarte, 2005). Paio, along with five other authors, generated a shape grammar based computational tool for developing a sustainable and integrated urban design (Paio et al., 2011). Eilouti had used shape grammar as a reverse engineering method for the generation of architectural façade design (Eilouti, 2019). ...
... Eilouti had used shape grammar as a reverse engineering method for the generation of architectural façade design (Eilouti, 2019). Cromwell, Lu and Steinhardt used shape grammar rules to derive Islamic ornamental patterns (Cromwell, 2009) (Lu and Steinhardt, 2007). Li's article uses shape grammar as a tool to teach Yingzao L. FATHIMA, C. K. ...
... Gazneli ve Karahanlı mimar ve sanatçılar pişmiş toprak, stükko ve mermer malzeme üzerinden monokrom çokgensel örüntü repertuarlarını geliştirirken bu tekniğe dair yeni geometrik olasılıkları sağlayabilecek prosedürler 12. yüzyılda Selçuklulara ve sonrasında sırasıyla Mısır, Fas ve İspanya'ya ulaşmıştır (Bonner ve Kaplan, 2017, s.27-30). Simetri Kavramı ve Periyodik Yüzey Tasarımları Üzerine Geçmişten Günümüze Bir Araştırma A Research from the Past to Present On the Concept of Symmetry and Periodic Surface DesignGirih adı verilen bu teknikte örüntü bir dizi simetrik dönüşüme imkan verebilen birim hücrelerden oluşur ve birbirleriyle ayrıt-ayrıt, köşe-köşe ilişkisi kurabilen farklı geometrik karakteristiklerin türetilmesine yardımcı olabilmektedir ve bu düzende komşu çokgenlerin ayrıtları kavşak (''+ birleşimi'') şeklinde birbirini takip eder(Lu ve Steinhardt, 2007, s.1106. Bu örüntüler çoğunlukla sınırlı sayıdaki biçimin birbirleriyle hata1sız ilişki kurabilecek düzenlerdeki tekrarları sonucu oluşur. ...
Simetri çeşitli uğraşı alanları için birleştirici bir kavram, nitel ve nicel değerlere bağlı disiplinlerarası bir bağlantı yoludur. Simetri tasarlama eylemine binlerce yıldır dahildir. Geçmişte tasarımcılar simetri kullanımında astronomi, fizik, biyoloji gibi doğa bilimlerinden, matematiksel keşiflerden, felsefi metinlerden ve teolojik unsurlardan sıklıkla beslenmiştir. Bugün de pek çok çalışma alanının kesişiminde bulunan simetri kavramı, çeşitli bilim alanlarında birbirlerine paralel fakat anlam olarak özgün tanımlamalara sahiptir. Çalışmanın amacı tanımlar, teknik ifadeler ve bulguları kronolojik sırada takip ederek simetrinin periyodik yüzey tasarımı için günümüzdeki teknolojik araçlarla kullanım sınırlarını değerlendirmektir. Çalışmada simetriye ve periyodik yüzeylere dair nitel ve nicel bilgiler sunmuş bilim insanları ve düşünürlerin fikirleri değerlendirilmiş, bu düşüncelere paralel olarak mimarlık ve sanat tarihi açısından önem atfedilen döşeme örüntülerine ve simetrik yüzey konusunda örnek teşkil edebilecek eserlere yer verilmiştir. Simetrinin ve periyodikliğin tasarım kurgusunda ve üretim aşamalarında sağladığı olanaklar nedeniyle tasarruflu bir yol olduğu söylenebilir. Bu verim etkeni, algoritma destekli tasarım sürecinde simetrik dönüşümlerin hazır yöntem paketleri halinde kaydedilmesi sürecinde de görünür olmaktadır. Günümüzde geçerli olan ve matematiksel doğruluğu yakın bir dönemde ispatlanmış iki boyutlu simetri grup ) çalışmada bağlayıcı bir öğe olmuştur. Kesin kuramsal sınırlar belirtmesi, hem yeni örüntülerin üretilmesi sürecinde, hem de mevcut periyodik örüntüleri kimliklendirme fırsatları nedeniyle mekânsal anlamda faydalı görülmelidir. Bu kuram bilgisayar destekli tasarım ve üretken tasarım araçlarıyla kullanıma oldukça uygundur. Sayısal tasarım yoluyla birden fazla sonuç üretecek grup algoritmaları yapılandırılabilir, tarihsel simetrik döşeme örüntüleri sınıflandırılabilir ve yeni sonuç verecek farklı yöntem dizinleri geliştirilebilir.
... Examples of few references in this direction are the following: Refs. [31][32][33][34] generate Islamic geometric patterns, Refs. [35][36][37] Indian Kolam patterns, and Refs. ...
Due to their practicality and convenient parametrization, fractals derived from iterated function systems (IFSs) constitute powerful tools widely used to model natural and synthetic shapes. An IFS can generate sets other than fractals, extending its application field. Some of such sets arise from IFS fractals by adding minimal modifications to their defining rule. In this work, we propose two modifications to a fractal recently introduced by the authors: the so-called √ 2-ball fractal dust, which consists of a set of balls diminishing in size along an iterative process and delimited by an enclosing square. The proposed modifications are (a) adding a resizer parameter to introduce an interaction between the generator and generated ball elements and (b) a new fractal embedded into the √ 2-ball fractal dust, having the characteristic of filling zones not covered by the previous one. We study some numerical properties of both modified resulting sets to gain insights into their general properties. The resulting sets are geometrical forms with potential applications. Notably, the first modification generates an algorithm capable of producing geometric structures similar to those in mandalas and succulent plants; the second modification produces shapes similar to those found in nature, such as bubbles, sponges, and soil. Then, although a direct application of our findings is beyond the scope of this research, we discuss some clues of possible uses and extensions among which we can remark two connections: the first one between the parametrization we propose and the mandala patterns, and the second one between the embedded fractal and the grain size distribution of rocks, which is useful in percolation modeling.
... His tiles are decorated in the style of Girih tiles, like the decorative Islamic geometric patterns used in architecture. Some researchers have indeed investigated how such art could be designed long before we knew about quasiperiodic structures that look alike [LS07,AA12]. ...
Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons ($H$), boats ($B$) and stars ($S$) were soon exhibited and many physicists published on what they later called $HBS$ tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. Yet this set yields exactly the strict $HBS$ tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings for instance. Another new tileset called Gemstones is also presented, derived from the Star tileset.
... However, the discovery of Islamic historical patterns with similar geometric properties is providing new insight into resolving these symmetries; eliminating a major roadblock for many researchers outside the field of crystallography and material science. New methods are emerging; providing new research into understanding the generating principles behind these complicated patterns (Makovicky 1992, 2007, Lu and Steinhardt 2007, Ajlouni 2012, 2013, 2017. Examples of these patterns can be found on the walls of Gunbad-I Kabud tomb tower in Maragha, Iran (1197 C.E.), the walls of Darb-i Imam shrine and the Friday Mosque in Isfahan, Iran (1453 C.E.), the walls of the Patio of Virgins, at the Royal Alcazar in Seville (1364-1366 C.E.), the walls of the Hall of the Two Sisters, Alhambra (1354-1359 C.E.), and in the medieval Topkapi Scrolls (15th or 16th centuries) in the library of the Topkapi Palace, Istanbul. ...
The prevalence of using computational methods for design exploration is profoundly transforming the way in which we conceive, craft and experience our built environment. By imposing a high degree of imbedded conformity to the digital logic, our physical, virtual and perceptual realities are gradually changing. The ability to use algorithms to manipulate form has enabled designers to achieve more complex geometric schemes. In particular, the use of generative-based approaches have rapidly became widespread among many design professionals. This design logic encodes design intent within sequences of processes that are structured to evolve independently beyond the designer's initial input. However, this design approach often incorporates randomness as a self-governed system for generating design response. Unfortunately by incorporating chance into the design process, this irrational design approach often steers the design response towards expressions of disorganized complexities and chaos. In this context, it is critical to emphasize that randomness in the built environment can be harmful to humans. Instead, schemes of organized complexities should be utilized. These organized structured schemes can satisfy the need for complexity while maintaining a global order. Such organized schemes are often derived from natural systems, which can be understood instinctively by humans. To address this need, this paper is proposing a method for generating different schemes of organized complexity by utilizing quasi-crystalline symmetries. The goal is to introduce a methodology for generating complicated geometric schemes with al long-range order that can be used to inform architectural design. The research employs case studies as its main methodology. The general population of the study includes 8-fold, 10-fold and 12-fold quasi-crystalline symmetries.
... These monuments stand at the beginnings of a rich development that resulted in such achievements as the quasi-periodic patterns of the Gunbad-e-Kabud (Blue Tower) at Maragha, W Iran (cf. Makovicky 1992Makovicky , 2008Makovicky , 2016Lu and Steinhardt 2007). In parallel to the results and conclusions concerning the geometric character of early Islamic art, this study offers insight into symmetry concepts developed (or those still absent) at this stage of art and architecture development. ...
Ornamental adornment of the Kharraqan tomb towers, the most outstanding funeral monuments of the Seljuk era in NW Iran, and those of four best-preserved Seljuq brick minarets in northern Iran, documents the artistic canon of the pre-glaze stage of Iranian Islamic architecture. Despite some later interruptions, these monuments and their plain-brick ornaments, as well as the ‘virtually interlaced’ brick ornaments, stand at the beginnings of a rich development that led to the Safavid architecture of Iran. Besides documentation and study of the geometric character of early Islamic art, which was based on limited technical resources, this study offers insight into symmetry concepts developed at this stage of art and architecture development. This is the last and most complete study of the Kharraqan towers performed before their overwhelming destruction in the 2002 earthquake.
... The golden ratio is the division of a line segment into two parts in such a way that the ratio of the length of the segment is larger. The length of the entire line segment is equal to the length of the smaller segment to the larger segment and its decimal equivalent is about 1.618 (Lu et al, 2007(Lu et al, : 1108. The golden ratio is also called the Phi number (Putz, 1995: 275). ...
The ability to recognize and compare shapes and understand the
concept of proportions is one of the first human findings from environmental
geometric data. And certainly, the formation of geometry as a science of
recognizing the arrangement of lines, shapes, levels, and proportions leads
to the definition of golden proportions in the mathematical sciences as a
new way of organizing shapes and forms. Golden and geometric proportions
are effective as a determining element in the scale of spatial forms, and the
formation of spatial forms by not using or misuse of geometric proportions
is due to functional and psychological interactions on human perception
and cognition. Which causes the construction and formation of scales and
proportions of the environment and the emergence of psychological effects
due to human activities in the environment. And then the abnormal effects
and environmental dysfunction on the saccade system of the human eye and
as a result the feeling of fear and lack of belonging to the place interacts.
On the other hand, nature is the raw material of creation, and the principal
and mother of man, which causes him to model the proportions in nature.
Understanding the laws governing the relationships between elements and
spatial shapes lead to the formation of geometric and golden proportions in
the spending of space. Because knowing the symbol of shapes, proportions,
and scales in the world of meaning and kingdom are reminiscent of
the divine order in all dimensions of the world (even in astronomy).
Therefore, evaluating the values of spatial proportions in nature as the
the bedrock of human life and the pattern of psychological perceptions leads
... Grunbaum and Shephard (1987) defined the concept of tessellation in their comprehensive book, explained the basic rules, and visualized the patterns that can be formed with many geometries. Many other studies on tessellation have been conducted in the 21st century (Beatini, 2015(Beatini, , 2017Chang, 2018;Goodman-Strauss, 2016;Greco, 2014;Lu and Steinhardt, 2007;Pesenti et al., 2015). As well as scientific research, various examples of tessellation have been applied in contemporary architecture as both fixed and kinetic building elements. ...
Responsive facades can reduce building energy consumption and control daylight and natural ventilation to improve user comfort. This study aims to develop alternative responsive facade systems based on semi-regular and demi-regular tessellations. For this purpose, first, the tessellation method used to generate responsive facades is introduced. Then, the geometric and parametric design principles and the movement capabilities of the proposed facade systems are presented. Finally, a set of analyses are performed to test and compare the performances of the facade systems based on daylight metrics and indoor glare comfort. This study contributes to the literature with the proposed facade systems that can adapt to changing environmental conditions, provide flexibility in shape control and simplicity in mechanism design, and improve building performance. The analysis results show that all the proposed facade systems provide the desired visual comfort and daylight levels at different configurations.
... The latter paper was widely covered in the press around the world, however some of the coverage exaggerated and misinterpreted the real findings. (Lu and Steinhardt 2007) Even though Islamic motifs have local ten-fold symmetry, they are frequently used as part of a flat periodic pattern with no overall five or ten-fold symmetry. This work incorporates these motifs into patterns based on Penrose tilings. ...
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.
... On the other hand, some studies show that IGPs have all the conceptual elements necessary to produce quasicrystalline girih, shah-girih, and muqarnas as aperiodic symmetry [48][49][50]. They include a five-old and seven-fold symmetry system (Fig. 7, down). ...
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.
... ġekil 14: Motiflerde kullanılan iç içe geçmiĢ bobin 2.1.c. Bow-tie olarak isimlendirilen geometrik Ģekil Battalgazi Ulu Camii"ndeki desenlerde net bir Ģekilde görülmektedir (Lu, 2007(Lu, : 1108. ...
Malatya ili Battalgazi ilçesinde 1224 yılında I. Alaeddin Keykubad tarafından yaptırılan Ulu Camii’nin mimarı Yakub Bin Ebubekir’dir. Battalgazi Ulu Camii içerisinde yer alan geometrik süslemeler, düzgün yıldız çokgenler ve çizim kuralları, çinilerin renk ve dokusundaki ahenk, simetri ile oluşturulan uyum, bordürlerdeki ayetler, tuğla ve taş işçiliğindeki ustalığın sanata dönüşümü dikkate şayan bir güzelliktedir. Yine merkezde düzgün yıldız çokgenlerin oluşumu ve sonrasında yayılarak başka düzgün yıldız çokgenler ve geometrik motiflerin oluşturulduğu desenlerin, doğadaki minerallerin kristalografik görüntüleri ile benzerlikleri dikkat çekicidir. Matematik, geometri, kimya, sanat tarihi gibi disiplinler arası işbirliğinin estetik uyumu Battalgazi Ulu Camii’nde en iyi şekilde yansıtılmıştır. Battalgazi Ulu Camii, İran’daki Büyük Selçuklu İmparatorluğu camii mimari geleneğinin Anadolu'da temsil eden en önemli ve tek temsilcisidir. Çalışmamızda bu kadar özellikli ve eski bir dini yapı olan Battalgazi Ulu Camii, tarihsel ve yapısal özellikleri ile birlikte iç ve dış süslemelerdeki geometrik motifleri görsel zenginlikler katılarak ele alınmaya çalışılmıştır.
... Many techniques have been proposed that use substitution tilings or quasiperiodicity to guide the placement of Islamic motifs [4,27,8,9,29]. Some researchers have even credited ancient designers with an explicit understanding of quasiperiodicity [1,26], though such claims are controversial [14]. Non-periodic patterns with long-range organization occupy an aesthetic sweet spot: they advertise global structure, but the precise nature of that structure is not trivially unravelled by the eye. ...
Islamic geometric patterns are a rich and venerable ornamental tradition. Many classic designs feature periodic arrangements of rosettes: star shapes surrounded by rings of hexagonal petals. We present a new technique for generating 'freeform' compositions of rosettes: finite designs that freely mix rosettes of unusual sizes while retaining the aesthetics of traditional patterns. We use a circle packing as a scaffolding for developing a patch of polygons and fill each polygon with a motif based on established constructions from Islamic art.
... Surprisingly, eight centuries before their discovery in modern science, ancient craftsmen had created patterns with quasi-periodic formations (Makovicky1992; Makovicky, Rull P´erez, and Fenoll Hach-Al 1998;Makovicky and Makovicky 2011;Rigby 2005;Lu, and Steinhardt, 2007;Saltzman, 2008). By using the most primitive tools; "a compass and a straightedge", a creative multi-level proportional method was used to construct a wide range of ancient quasiperiodic ornamental patterns. ...
The discovery of quasi-crystalline atomic order in the solid-state physics has challenged decades of foundational knowledge in crystallography. The atoms in these novel quasi-crystalline structures are not arranged according to regularly spaced intervals similar to traditional crystals, instead they exhibit a long-range translational order that is not periodic. Three decades after their initial discovery, hundreds of quasicrystals have been reported; exposing a wealth of untapped potentials. Because of their unique isotropic, self-similar and hierarchical order, quasi-crystalline structures offer unique opportunities for addressing questions related to their acoustical behavior. In 2018, Ajlouni demonstrated that the quasi-periodic formations have the ability to diffuse and orchestrate the flow of sound energy; eliminating a major limitation with the repeating logic of traditional periodic diffusers. A major limitation with periodically arranged diffusers, is that they create repetitive energy loops that significantly reduce their ability to uniformly disperse sound energy. The goal of this paper is to introduce a generalized structural method for designing surfaces with quasi-periodic geometry for architectural acoustics. The paper also explores two methods for the fabrication of these surfaces using ceramic casting and vacuum forming processes. By utilizing the qualities of quasi-periodic structures, this research hopes to inspire a new wave of acoustical surface diffusers that allow designers to encode a wide range of acoustical behavioral properties without scarifying the aesthetic qualities.
... While older technologies are usually not entirely lost they become infrequent and exotic. (There are obviously instances of loss from time to time; for a spectacular example see Lu and Steinhardt, 2007.) That is one of the most fundamental characters of technological (and cultural, in general) evolution; there is no conceivable reason to expect that this will not be valid for extraterrestrial cultures. ...
In a recent refreshing paper, Cowie (2022, The Philosophical Quarterly) analyses the hypothesis of artificial origin of the mysterious interstellar object 1I/2017 U1 ʻOumuamua, as well as the wider question of justification of the artefactual origin explanation for anomalous astronomical phenomena. This highly commendable philosophical project should be further developed in order to establish more general methodology for dealing with traces and manifestations of extraterrestrial intentional actors. In the present note, I demonstrate a couple of weaknesses of the standard account, mainly dealing with philosophy of technology, and suggest ways of improving upon it.
... Among them are the iconic diffraction pattern of the quasicrystal with 10-fold rotational symmetry (Figure 1 Interest in these aperiodically ordered patterns emerged in the arts long before they were studied by scientists. Aperiodically ordered structures appear in medieval Islamic architecture [LS07], [AA12], [ATR14]. German renaissance artist Albrecht Dürer also experimented with aperiodic tilings [Lüc00]. ...
The Thue-Morse sequence is an aperiodically ordered infinite binary sequence. It is used as a one-dimensional way to model the structure of a quasicrystal. For example, taking autocorrelations of these sequences (roughly, measuring how similar a Thue-Morse sequence is to translates of itself) we can gain understanding of the diffraction patterns of quasicrystals. We generate abstract art images from these Thue-Morse autocorrelation functions, that capture the aperiodic structure of the Thue-Morse sequence in a compelling way.
... Plants, fungi and animals can establish tissue boundary patterns through developmental programmes using independently evolved yet similar genetic toolkits, organizers and morphogen fields 1,3,4,[17][18][19] . Mathematically, such interface (or boundary) patterns can often be described as tilings or tessellations 20,21 , which elegantly combine generic algorithmic design principles with functional structure and aesthetic appeal, as exemplified by neuronal self-avoidance tilings 22,23 , DNA tiling self-assembly 24 , Islamic architecture 25 and the Tetris video game 26 . Synthetic biology aims to engineer patterned multicellular systems to facilitate applications including programmable biomaterials, artificial tissues and metabolic consortia, and to provide a build-to-understand methodology for natural systems [5][6][7][8] . ...
Multicellular systems, from bacterial biofilms to human organs, form interfaces (or boundaries) between different cell collectives to spatially organize versatile functions 1,2 . The evolution of sufficiently descriptive genetic toolkits probably triggered the explosion of complex multicellular life and patterning 3,4 . Synthetic biology aims to engineer multicellular systems for practical applications and to serve as a build-to-understand methodology for natural systems 5–8 . However, our ability to engineer multicellular interface patterns 2,9 is still very limited, as synthetic cell–cell adhesion toolkits and suitable patterning algorithms are underdeveloped 5,7,10–13 . Here we introduce a synthetic cell–cell adhesin logic with swarming bacteria and establish the precise engineering, predictive modelling and algorithmic programming of multicellular interface patterns. We demonstrate interface generation through a swarming adhesion mechanism, quantitative control over interface geometry and adhesion-mediated analogues of developmental organizers and morphogen fields. Using tiling and four-colour-mapping concepts, we identify algorithms for creating universal target patterns. This synthetic 4-bit adhesion logic advances practical applications such as human-readable molecular diagnostics, spatial fluid control on biological surfaces and programmable self-growing materials 5–8,14 . Notably, a minimal set of just four adhesins represents 4 bits of information that suffice to program universal tessellation patterns, implying a low critical threshold for the evolution and engineering of complex multicellular systems 3,5 .
... Recently, the discovery of ancient ornaments in Islamic architecture with similar quasicrystalline symmetries has triggered significant investigations into understanding their mathematical laws and generating principles (Makovicky 1992;Makovicky et al. 1998;Bonner 2003;Lu and Steinhardt 2007;Makovicky 2008;Makovicky and Makovicky 2011). Astonishingly, eight centuries before their discovery in modern times, ancient artists had constructed patterns with perfect tenfold quasicrystalline formations (Al Ajlouni 2012). ...
... This allowed for precise patterns to be developed over large surfaces using aperiodic Penrose patterning. (Lu 2007) Using the five Penrose patterning geometries, we attempted to make extruded clay forms which could be stacked to construct screen walls for arid climates. Similar to traditional techniques, these walls would permit shading and light filtration; however, because the units feature an interstitial space, water could be introduced between the units to enhance passive cooling effects. ...
... In addition, a great debate was opened about the quasi-periodicity of some Islamic geometric patterns, such as the geometric pattern of the Darb-i-Imam in Iran. As a result, several papers have been studying the quasi-periodic structure of Islamic geometric patterns (Hargittai and Makovicky, 1992;Makovicky, Pérez and Hach-Alí, 1998;Lu and Steinhardt, 2007;Cromwell, 2009;2015;Al Ajlouni, 2011;Makovicky and Makovicky, 2011;Bonner, Pelletier et al., 2012;Al Ajlouni, 2013;Khamjane and Benslimane, 2018b;Khamjane et al., 2019;Khamjane, Benslimane and Ouazene, 2020). ...
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.
... Bridging the 'neoplatonic' and spiritualist approaches to meaning, recognition of the mathematics of Islamic geometries during the 1980s and 1990s enabled a more thorough understanding of the intellectual sophistication of patterns, as well as modes of their intellectual transmission, previously understood as manifestations of non-intellectual artisanry (Chorbachi, 1989;Bier, 2012;Lu & Steinhardt, 2007;Özdural, 1995, 1996. Neither formalist nor spiritualist, it suggests mathematics as producing meaning through technical aptitude rather than through articulating experience beyond the quiddity of form itself. Scholarship engaging with tessellated calligraphy enables this association without addressing the semiotic agency of geometry (Ahuja & Loeb, 1995). ...
Discussions of surface pattern in Islamic art resonate within broader tensions about the role of figural representation in communicating meaning. The question of whether geometric pattern communicates—whether it functions as a language without a code—reflects broader tensions about the relationship between secular and spiritual communication. Poised between discussions of modernism and Islam, the attribution of linguistic capacity to geometry serves as a measure for the possibility of abstracting pure reason from the religious roots of representationalism. This paper explores this question in four parts. First, it examines the articulation of geometric meaning in Islamic discourses. It then explores European discussions of geometry between language and ornament poised between a search for universal reason and universal spiritualism, both in modernist painting and in modern exhibitions. The final section the expressive capacity of artistic responses to historical Islamic isometric geometries in the adaptive reuse of historical Iranian arts as contemporary abstraction in the sculptures of Monir Shahroudy Farmanfarmaian (1928–1919) and in the popular transformation prints of M. C. Escher (1898–1972). It argues that the frequent recognition of intrinsic meaning in geometry, expressed in both premodern Islamic and modern secular interpretations, undermines art historical expectations of theorization as a necessary intermediary for communication. Geometry thereby functions as a language without a code.
... If we express the angle of rotation as n 2Л, only when we use 2, 3, 4 and 6 for n, point group symmetry and translational symmetry work in harmony. When there are 5, 7, 9 and 11, the harmony is broken, and these symmetries are expressed as forbidden (Lu &Steinhardt 2007, 1106. ...
Our study is an introduction to the underlying reasons why the concept of crystal (crystallographic) pattern, which emerged from the union of mathematics and crystallography in the 20th century, is a frequently used ornamental element in Turkish-Islamic architectural works dating back about 9 centuries. Our aim is the geometric ornaments, which are generally handled by art historians; the aim is to underline that it is an area that needs attention for historians of mathematics, crystallography and architecture. We will try to reveal the emergence of crystal patterns, one of the patterns used by Turkish-Islamic architecture for centuries to cover many surfaces, in the fields of mathematics and crystallography, with the perspective of the history of science, in the context of two basic theories. The first of these theories is symmetry, and the second is group theory, which provides the basis for the mathematical explanation of symmetry. The examples of crystal structures, which are the subject of the science of crystallography, which developed with the help of new technologies discovered after the mathematical possibilities provided by symmetry and group theory, were applied centuries ago in Turkish-Islamic architecture will be emphasized. In this way, it will be noticed that the geometrical ornaments used in the artifacts from the Turks and the formation principles of the crystals are the same. These patterns, which were produced in a period when symmetry groups or crystallography were not known at all, were used as a kind of cultural element, especially by the Karakhanids, Seljuks, Anatolian Seljuks…etc. It is surprising that it is seen in all geographies that Muslim Turks have reached. Considering the history of mathematics and crystallography, it is understood that the crystal patterns used in architecture are a manifestation of the knowledge of Turkish-Islamic civilization in many sciences, especially geometry.
... 2 A review of basic methods is given by Dabbour (2012), without any reference to architectural proportions. Pentagonal symmetry and the golden section based 2D patterns in particular are studied in articles by Lu and Steinhardt (2007), Makovicky (1992), Fleurent (1992), Chorbachi and Loeb (1992) and Sarhangi (2012). Multilayered patterns are studied by Bier (2012). ...
This research constitutes an alternative to proportional composition studies of the original Great Mosque and its four extensions in Córdoba, based on diagonals of a square and rectangles in ratio 1:√2 and 1:√3 (Fernández-Puertas, 2000, 2008). The method for this alternative research consists of graphic analyses by iteration of hypothetical products of the golden section in AutoCAD 2D software conducted on architectural drawings of the original Great Mosque and its four extensions, in reconstruction, according to measurements from the relevant literature. The alternative method insists on geometric harmonization derived from only one starting length in all drawings of same scale. It resulted in the production of a single harmonization pattern based on the golden section, with an additional sequence of a√2/Φn, successively developed for the original monument and its four extensions. It also includes otherwise excluded basic composition elements (minarets) and reveals otherwise hidden proportional qualities. The alternative approach enabled a deduction of algebraic expressions having only one variable for all drawings of the same scale. Their arithmetic values and deviations from real dimensions are calculated. Geometric harmonization by golden section with another starting length is applied to the drawing of the elevation at a different scale.
... Thus, symmetry analysis IS design structure analysis. This structural approach to design is applicable on patterned designs worldwide (e.g., Hann, 2012, Hardonk, 1999, Hargittai & Laurent, 2002, Lu & Steinhardt, 2007, Sarri, 2020 Content courtesy of Springer Nature, terms of use apply. Rights reserved. ...
The analysis of pattern design on ceramics, textiles, tile and other plane surfaces remains dominated by typological, design element and stylistic approaches. In this paper, I compare these approaches with a classification system that describes patterns by the geometric symmetries that combine and repeat the pattern parts. I argue that these standard plane pattern symmetry classes enable systematic, objective descriptions of decorated material. I use an analysis of the symmetric structures and colors of designs on whole and partial Chihuahuan polychrome vessels from the Pueblo IV period site of Paquimé (Casas Grandes), Chihuahua, Mexico, to illustrate how a standardized classification system that focuses on the structure of pattern parts rather than on the pattern parts themselves reveals a number of new insights on a body of well-studied material.
Do geometric patterns characterizing many surface treatments in Islamic cultures represent? If so, can pattern function as an image? The question of whether geometric pattern communicates reflects tensions between secular and spiritual understandings of communication. Poised between discussions of modernism and Islam, the attribution of linguistic capacity to geometry serves as a measure for the possibility of abstracting pure reason from the religious roots of representationalism. This paper explores this question in four parts. First, it examines the articulation of geometric meaning in Islamic discourses. It then explores European discussions of geometry between language and ornament poised between a search for universal reason and universal spiritualism, both in modernist painting and in modern exhibitions. The final section the expressive capacity of artistic responses to historical Islamic isometric geometries in the adaptive reuse of historical Iranian arts as contemporary abstraction in the sculptures of Monir Shahroudy Farmanfarmaian (1928–1919) and in the popular transformation prints of M. C. Escher (1898–1972). It argues that the frequent recognition of intrinsic meaning in geometry, expressed in both premodern Islamic and modern secular interpretations, undermines art historical expectations of theorization as a necessary intermediary for communication. Geometry thereby functions as a language without a code.
Too often, Western encounters with the Islamic world commence with stereotypes and end with a renewed distance. Drawing from decades of experience studying the Muslim world, Lawrence Rosen challenges these narrow understandings. Adopting an interdisciplinary approach, Rosen shows the wide-ranging significance of Muslim art, culture, and law around the world. Exploring political, economic, and social encounters within and with the Muslim world across the eras, he considers a wide range of contexts – from fifteenth-century mosaics in Central Asia that reveal a complex understanding of mathematics, to the political choices available to the youth of modern-day Morocco and Cairo. With in-depth analyses of art, law, and religion, and how they informed one another, Rosen develops a vibrant, nuanced portrait of the Islamic world. Drawing linkages across time, regions, and cultures, this is a significant anthropological study of the Islamic world from a seasoned scholar.
The Thue–Morse sequence is an aperiodically ordered infinite binary sequence. It is used as a one-dimensional way to model the structure of a quasicrystal. For example, taking autocorrelations of these sequences (roughly, measuring how similar a Thue–Morse sequence is to translates of itself), we can gain understanding of the diffraction patterns of quasicrystals. We generate abstract art images from these Thue–Morse autocorrelation functions that capture the aperiodic structure of the Thue–Morse sequence in a compelling way.
(This paper was approved for the 2023 Bridges Conference, but it was not published as the author broke his ankle and couldn't attend to present the paper.)
The subject is modular 5-fold patterns. Breaking boundaries can sometimes enable discovery of new insights. Here I break the edge rules to be able to create tile motifs that are more common among historical 5-fold patterns. I introduce two approaches to a concept I call Pentalining. This concept came to me from an Islamic geometric pattern in a Moroccan restaurant. It looked very traditional, but its appearance was deceptive. I found several uncommon shapes. To be able to tile it, I had to break the edge rule.
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
Quasiperiodic ornamental patterns represent only a small percentage of patterns when compared to the entire body of periodic patterns. Decagonal pattern is known since twelfth century Iran and fourteenth century western Islam (Andalusia and Morocco). A rich spectrum of octagonal patterns exists at the latter localities (fourteenth century and later), whereas a sole example of a dodecagonal pattern comes from Morocco. Later copies exist in all these regions. My most recent studies were concentrated upon the Andalusian and Moroccan regions, in which the fourteenth century (and later) wall mosaics occur as uninterrupted coatings of entire walls so that the motif of individual panels had to be adjusted to secure continuity of their underlying bar-and-band structure. In Andalusia, the tetragonal structure of the panels and their complexes were locally adjusted to become octagonal quasiperiodic. Only two geometric types of such octagrids were derived in Andalusia, in agreement with the rarity of quasiperiodic ornaments in general. In Morocco, before the panel substructure became heavily masked by an overflow of rosettes of several sizes, the mosaic panel was based on an octagonal quasiperiodic grid and ornamental rosettes were placed in it, disposed in the form of concentric octagons. As a prominent example, the octagonal motif of the Nejjarine Fountain and its plaster encasement will be discussed.
Recent developments in archeological research extend diverse technological methods for the geometric deciphering and cultural understanding of various historical building components. One of the emerging methods in this field is the development of generative algorithms to develop computational models for the comparative study of variation among different structures belonging to a common era, style, or region. In this study, we present a novel approach for the computational analysis and parametric modeling of muqarnas found among Anatolian Seljuk architecture in Kayseri and Sivas built in the 13th century. Using four different octagonal muqarnas structures, we outline common generative rules showing recursive stacking of geometric layers, fractal patterns and hierarchical branching of the axis of symmetry. A recursive algorithm is developed that can offer a generative study of muqarnas structures using proportions based on the ‘silver ratio.’ The development of the algorithm is presented through rules and variations that can offer a novel perspective for the geometric understanding and categorization of muqarnas in the region.
p>Diplomatic gifts, war prizes, or library treasures of royal and princely libraries—handwritten Qur ʾ ans have also been endowed to mosques, tombs, and other religious complexes to perpetuate and transmit their baraka (divine blessing). Artistic, historic, and religious contexts and materiality of Qur ʾ ans are investigated, from use of costly materials such as gold and parchment to development of special scripts, intricate illuminated designs, and meticulously tooled bindings. This edited collection resulted from a 2016 symposium at the Smithsonian’s Arthur M. Sacker Gallery in conjunction with the exhibition The Art of the Qurʾan: Treasures from the Museum of Turkish and Islamic Arts . </p
In this book, Patricia Blessing explores the emergence of Ottoman architecture in the fifteenth century and its connection with broader geographical contexts. Analyzing how transregional exchange shaped building practices, she examines how workers from Anatolia, the Mediterranean, the Balkans, and Iran and Central Asia participated in key construction projects. She also demonstrates how drawn, scalable models on paper served as templates for architectural decorations and supplemented collaborations that involved the mobility of workers. Blessing reveals how the creation of centralized workshops led to the emergence of a clearly defined imperial Ottoman style by 1500, when the flexibility and experimentation of the preceding century was levelled. Her book radically transforms our understanding of Ottoman architecture by exposing the diverse and fluid nature of its formative period. It also provides the reader with an understanding of design, planning, and construction processes of a major empire of the Islamic world.
Sir Roger Penrose was awarded the 2020 Physics Nobel Prize at the age of 90 for his research on the black holes of the universe, which he had initiated many decades earlier. Penrose’s insights and creativity span many fields. Among his many interests are Escher’s works and quasi crystals. The meeting between Penrose and Escher was inevitable. It took place at the world congress of mathematics in Amsterdam in 1954. At the time, Penrose was a student of mathematics. The conference was the site of the first major exhibition dedicated to the Dutch graphic designer. I met Penrose while shooting my Escher film.
Decagonal motifs found in the traditional tile designs of Islamic art and architecture can, with a little refinement, be arranged according to the fundamental quasiperiodic structure of the Penrose tiling. A simple tessellation is applied to six traditional motifs to create analogues to the pentagonal Penrose tiles. Substitution rules guiding hierarchical assembly of these Penrose tiling arrangements are then applied, mediated by the tessellation tiles. I then explore how the artist may include hints to the viewer about the mathematical characteristics of Penrose tilings that these designs represent, including their correct arrangement, substitution rules, quasiperiodicity, and other features.
In this study, an alternate method to evaluate structural systems, especially those with historical importance has introduced by applying Fourier Transform (FT) to the damage ratio of time history outcome in the frequency domain. The concept of damage content (DC) regarding the mechanical characteristics of the used material, including plastic strain, failure plane, and ultimate load-bearing capacity, along with drift value, record selection criteria, and architectural aspects, have employed. Due to its valuable aesthetic and architectural view, Timche Haj-Mohammad-Qoli of Tabriz Historic Bazaar, one of the traditional covered spaces with the complicated configuration of spatial masonry intersecting arches, selected for further assessment in the current research program. The required experimental samples for obtaining the mechanical properties and relevant geometrical measurements to prepare the numerical model of the structure obtained. The strong ground motions according to seismological and geological characteristics of the construction site selected. The records with different durations were merged by Fourier Transform (FT) and Damage Content (DC) analysis. According to the outcomes, the damage state of the structure due to the imposed strong motion at every stage of the lateral loading from the failure initiation to the final collapse was traceable. As a straightforward outcome, the vulnerability of the near-field earthquakes is more tangible than that of the far-field earthquakes. The provided methodology could have proper use in future similar studies for evaluating the performance of the structures.
Le dialogue avec la tradition est toujours un enjeu majeur dans la société iranienne d’aujourd’hui. Différents mouvements architecturaux mêlant l’architecture traditionnelle et l’architecture contemporaine se sont succédé ces dernières décennies en Iran. Par ailleurs, nous assistons au développement de plus en plus rapide du « digital » dans tous les aspects de la discipline architecturale, ce qui a comme conséquence le rapprochement de la conception et de la fabrication. L’émergence des outils numérique en architecture, notamment les outils paramétriques et algorithmiques, facilite et encourage le recours à la géométrie (complexe).L’un des fondements de l’architecture traditionnelle iranienne est la géométrie, dont la présence est fort visible au sein des éléments structurels et décoratifs. C’est donc la réémergence de la géométrie grâce au numérique qui nous a amenés à nous pencher sur la réinterprétation du Muqarnas pour réfléchir à sa transposition dans le contexte contemporain iranien. Le Muqarnas est un élément à caractère spatial et géométriquement complexe de l’architecture iranienne en période islamique, à la fois structure et ornement. Dans cette démarche de génération de nouveaux Muqarnas, nous ferons appel aux outils paramétriques et algorithmiques qui ouvrent de nouvelles perspectives. C'est l'objet premier de ce travail de recherche que de savoir comment et avec quelle approche réinterpréter le Muqarnas à l’aide des outils numériques. En prenant connaissance des études antérieures sur les Muqarnas, nous avons acquis la conviction qu'une étude historique de l’évolution des Muqarnas nous permettrait de structurer notre démarche de réinterprétation avec des outils numériques.Il s'agit ainsi, dans un premier temps, de réinvestir l’histoire des Muqarnas en étudiant leur évolution, en particulier dans sa dimension géométrique. Cette étude nous permettra de déduire des « principes géométriques », essentiels pour la réinterprétation numérique des Muqarnas. Les principes géométriques issus de l’étude historique sont, dans un deuxième temps, intégrés dans plusieurs approches génératives numériques. Chaque approche présente une logique algorithmique et paramétrique de génération des Muqarnas dont on évalue les avantages et les limites.Enfin, une troisième partie est consacrée à la mise en œuvre des modèles numériques issus des différentes approches génératives et à la réalisation de prototypes à différentes échelles
In this study, an alternate method of evaluating structural systems, especially for the sensitive structures with historical importance by applying Fourier Transform (FT) to the damage ratio of time history outcome in the frequency domain has been introduced. The concept of damage content (DC) regarding the mechanical characteristics of the used material, including plastic strain, failure plane, and ultimate load-bearing capacity, along with drift value, record selection criteria, and architectural aspects, have been employed. Due to its valuable aesthetic and architectural view, Timche Haj-Mohammad-Qoli of Tabriz Historic Bazaar, one of the traditional covered spaces with the complicated configuration of spatial masonry intersecting arches, was selected for the assessment in the current study. The required experimental samples for obtaining the mechanical properties and relevant geometrical measurements to prepare the numerical model of the structure obtained. The strong ground motions according to seismological and geological characteristics of the construction site selected. The records with different durations were merged by Fourier Transform (FT) and Damage Content (DC) analysis. According to the outcomes, the damage state of the structure due to the imposed strong motion at every stage of the lateral loading from the failure initiation to the final collapse was observable. As a direct outcome of this study, the vulnerability concerning the near-field earthquakes is more tangible than far-field earthquakes. The provided methodology could have proper use in future similar studies for evaluating the performance of the structures.
We present a unified method to meshing surfaces with unconventional patterns, both periodic and aperiodic. These patterns, which have so far been studied on the plane, are patterns comprising a small number of tiles, that do not necessarily exhibit translational periodicity. Our method generalizes the de Bruijn multigrid method to the discrete setting, and thus reduces the problem to the computation of N -Directional fields on triangle meshes. We work with all cases of directional symmetries that have been little studied, including odd and high N. We address the properties of such patterns on surfaces and the challenges in their construction, including order-preservation, seamlessness, duality, and singularities. We show how our method allows for the design of original and unconventional meshes that can be applied to architectural, industrial, and recreational design.
Sir—A number of reports suggest that some occupational exposures of males may alter the gender ratio of their offspring. Reduction in male offspring has been observed after exposures to hyperbaric chambers 1 and among professional drivers, 2 applicators of dibromochloropropane, 3 carbon setters 4 and aluminum smelter workers. 5 Weijin and Olsen 6 have suggested that offspring gender ratio, because of its correlation with reduced fertility, may be a possible epidemiological indicator of reproductive hazards. We have conducted a study to evaluate the possible role of exposure to metal fumes (particularly nickel and chromium) on the reproductive health of male workers in an Italian mint. The study was designed to evaluate specifically fertility problems, and the observed reduction in fertility is reported separately. We also noted, however, an interesting alteration of the gender ratio with reduced male births which we feel is worth reporting. The workers were divided into administrative staff (nonexposed to metal fumes), technical and maintenance staff (with minimal exposure), coin stampers (exposed to metal fumes) and founders (highly exposed to metal fumes). Air sampling and biological monitoring of workers yielded concentrations of metals within the TLV-ACGIH. However, the blood nickel levels were very near the biological reference value of 0.5 μg/dl, especially among founders. The proportion of male children among all offspring born to these men while working in the four job categories is shown in Table 1. Founders had a significantly reduced proportion of male children when compared to both the administrative staff and to the general population, where the proportion of males is 51%. We focused more specifically on the most recent birth of these workers, because information on exposure was more detailed for the last pregnancy of the couple. Again we note a reduction in the births of males among founders, which is statistically significant. The limited number of cases included in our study do not allow a conclusion, but these data add to the evidence that occupational exposure might affect the gender ratio in favour of female births. The mechanism of this effect is under debate. James 7 hypothesized that gender ratio is affected by hormone concentration. High levels of gonadotropin and low levels of testosterone, favour the production of daughters. However, among welders Bonde 8 has observed hormonal alterations, poor semen quality and reduced fertility but no alteration in the gender ratio. The other possible mechanism may be related to a greater susceptibility of the fetuses, to early undetected pregnancy loss. This hypothesis is compatible with alterations of the sex ratio observed in Danish female physiotherapists. 9
The present note is an attempt to supplement previous argument on the hypothesized parental hormonal antecedents of mammalian
offspring sex ratios. It is intended that a reader who has read both this paper and my previous paper should be aware of all
the data relating to the hypothesis [James WH (1996), J Theor Biol 180, 271–286].
There have been serious doubts regarding the ability of petroleum production to meet demand. Crude oil production in the continental US peaked in 1970 and has declined continuously since that time. Production in the North Sea reached a plateau in the late 1990s and has recently dropped sharply. The non-OPEC production of conventional oil would peak in the near future probably by about 2010. OPEC producers have adequate reserves and would meet market demand far into the future.
There is good evidence that paternal (and maternal) hormone levels at the time of conception are associated with offspring sex ratios (proportions male) at birth. The mechanisms underlying this association (pre- or postzygotic) are not of primary relevance here. When people are exposed to endocrine-disrupting agents, these agents may have different hormonal effects on men and women. So, if endocrine disruption is to be revealed by offspring sex ratios, it is necessary to categorize the sexes of subsequent offspring by the four possible parental mating classes, viz. exposed/unexposed mothers/fathers. In general, substantially altered sex ratios may reveal endocrine disruption, but the tiny (admittedly significant) secular meanderings of national live birth sex ratios across the 20th Century (and before) are not now readily interpretable.
A quasi-crystal is the natural extension of the notion of a crystal to structures with quasi-periodic, rather than periodic, translational order. Two and three-dimensional quasi-crystals are here classified by their symmetry under rotation, and it is shown that many disallowed crystals symmetries are allowed quasi-crystal symmetries. The diffraction pattern of an ideal quasi-crystal is analytically computed, and it is shown that the recently observed electron-diffraction pattern of an Al-Mn alloy is closely related to that of an icosahedral quasi-crystal.
Part I. General Concepts: Hamiltonian dynamics; Stability and chaos;
Part II. Dymanic Order and Choas: The stochastic layer; Stochastic
layer - stochastic sea transition; The stochastic web; Uniform web;
Part III. Spatial Patterns: Two-dimensional patterns with
quasi-symmetry; Two-dimensional hydrodynamic patterns with symmetry and
quasi-symmetry; Chaos and streamlines; Part IV. Miscellanea: Patterns in
art and nature; References; Index.
Two mathematical sources, On the Geometric Constructions Necessary for the Artisan, by Abu'l-Wafā' (ca. 940–998), and the anonymous work, On Interlocks of Similar or Corresponding Figures (ca. 1300), provide us with insight into the collaboration between mathematicians and artisans in the Islamic world. In this paper I present a series of quotations from these two sources, which show that mathematicians taught geometry to artisans by means of cut-and-paste methods and of geometrical figures that had the potential of being used for ornamental purposes. Copyright 2000 Academic Press.Matematikle ilgili iki kaynak bize İslam dünyasında matematikçiler ile sanatkarlar arasındaki işbirliği konusunda aydınlatıcı bilgiler sunuyor. Bu kaynaklardan biri Abu'l-Vefa (ca. 940–998) tarafından yazılan “Sanatkarın ihtiyaç duyduğu geometrik çizimler,” diğeri anonim bir yazarın kaleme aldığı “İçiçe geçen benzer veya karşılıklı şekiller” (ca. 1300). Bu iki kaynaktan derlediğim bir dizi alıntıya yer verdiğim bu makalede görüyoruz ki matematikçiler sanatkarlara kes-ve-yapıştır yöntemiyle geometri öğretirken, aynı zamanda önerdikleri geometrik şekillerin bezeme sanatlarında kullanılabilir olmasına özen gösteriyorlardı. Copyright 2000 Academic Press.MSC subject classifications: 01A30, 01A20.
A personal account of an interdisciplinary inquiry into the study of Islamic geometric design and architectural decoration touching on the fields of History, History of Science, Scientific Theory of Symmetry and History of Art. The study stresses the necessity of the use of a common scientific language of Symmetry Notation in order to discuss and communicate in a precise manner about Islamic geometric pattern. To understand Islamic geometric design, it is necessary to move beyond the symmetry issues, to the step-by-step process of design. This is based on primary sources of scientific manuscripts of practical geometry written specifically for the Muslim artisans. The research demonstrates not only a direct meeting but a collaborative work between science and art in Islamic civilization.
The current study summarizes an 8-year reassessment of testicular function and reproductive performance in 15 workers with dibromochloropropane (DBCP)-induced azoospermia and oligozoospermia. Recovery of spermatogenesis was observed in four oligozoospermic and three azoospermic men whose plasma follicle-stimulating hormone (FSH) concentration was normal during the whole period. A marked increase in FSH and luteinizing hormone concentrations above the upper limit of normal was found in the azoospermic workers who did not recover. No significant changes in FSH concentrations were detected in both recovered and nonrecovered oligozoospermic men. Testosterone levels of all patients were normal at all times. Paternal exposure to DBCP was not associated with increased risk of fetal malformations or spontaneous abortion.
Observations of body weight, testis size, antler status, plasma testosterone and prolactin were made on 12 red deer stags during their first 2 years of life. Six of the stags were fed to appetite throughout the study (Group A) and 6 were fed a 70% restricted diet during each winter (Group B). In addition 6 of the stags , 3 from each group, were studied in more detail; LH and testosterone were measured either after a single injection of LH-RH or in samples taken at frequent intervals over a period of 8 or 24 h. During the study the stags became sexually mature, developed first their pedicles and then antlers and showed at least one complete cycle of casting and regrowth of the antlers . The stags in Group A developed their testes and pedicles about 2 months earlier than did those in Group B. Pedicle initiation was associated with increasing plasma testosterone levels in response to changes in LH secretion, and antler development occurred when testosterone levels were low or decreasing. Cleaning of the velvet was associated with high levels of plasma testosterone. Antler casting occurred when plasma testosterone concentrations were low or undetectable and prolactin levels were high or increasing. The relationship between LH and testosterone varied during the study; in spring when the testes and antlers were growing, relatively high levels of LH were associated with only small peaks of testosterone, yet in summer, when antler growth was complete and the antlers were clean of velvet, low LH concentrations were associated with large peaks of testosterone.
A series of six studies were carried out in red deer stags to test hypotheses concerning the importance of steroid control of velvet antler growth and to investigate mechanisms by which these hormones exert their effects. Medroxyprogesterone acetate (MPA) an LH inhibitor administered to stags during hard antler caused premature antler casting, reduced subsequent antler weight and caused a reduction in the LH and testosterone responses to GnRH. In two separate studies blockade of testosterone receptors with cyproterone acetate (CPA) administered to stags, either during early velvet antler growth or during the hard antler stage, significantly reduced LH and testosterone responses to GnRH. In both studies antler length, but not weight, was increased by CPA treatment. In another study testosterone implants were used to prevent the gradual decline in plasma testosterone levels normally observed during winter. Implants were removed 3 weeks before the anticipated date of antler casting. The implants significantly increased plasma testosterone levels and subsequent antler growth (expressed as a proportional increase compared with the previous year) compared with untreated controls. To determine whether the annual cycle of plasma testosterone response following GnRH stimulation was due simply to a lack of LH stimulation, ovine LH was injected on six occasions at defined stages of the antler cycle to red deer stags and the testosterone response measured. The testosterone responses were low at antler casting and during velvet antler growth compared with antler cleaning and peak rut. It appears low testosterone levels are due, in part, to a loss of responsiveness by the testes to LH as well as a low level of secretion of LH during the antler growing season. Finally synthetic ACTH was injected at the same defined stages of antler growth as in the previous study to determine whether cortisol and adrenal androgen production altered with the stage of the antler cycle. No significant differences were found in the dehydroepiandrosterone (DHEA) response, but cortisol responses were higher from late velvet antler growth to peak rut, compared with the times of antler casting and early velvet growth. Overall it was concluded that velvet antler growth can occur without testosterone stimulation during the period of velvet growth, but the data reinforce the concept that the timing of antler growth is linked to the annual cycle of testosterone.
Evidence is adduced for the hypothesis that mammalian (including human) sex ratios at birth are partially controlled by the hormone levels of both parents at the time of conception. Sex ratio variation is considered under three main headings viz that identified by Clutton-Brock & Iason (1986); that identified since that paper; and other variation mainly relating to human beings. Some of the evidence is strong.
In a recent paper, we introduced the concept of quasicrystals [Phys. Rev. Lett. 53, 2477 (1984)], a new class of ordered atomic structures. Quasicrystals have long-range quasiperiodic translational order and long-range orientational order. In the present paper and the following one, we discuss the details of our analysis of the mathematical and structural properties of quasicrystals. We begin with a general overview of our analysis. We then discuss our computation of the diffraction pattern of a quasilattice, using as an example the case of icosahedral orientational symmetry. We demonstrate that two quasilattices with the same orientational symmetry and quasiperiodicity which are not locally isomorphic will have diffraction patterns with different peak intensities. Finally, we describe some examples of computer modeling of atomic quasicrystals.
Lu and Steinhardt (Reports, 23 February 2007, p. 1106) claimed the discovery of a large, potentially quasi-crystalline Islamic tiling in the Darb-i Imam shrine but regard the earlier Maragha tiling, previously described as quasiperiodic, as a small isolated motif. We demonstrate that the Darb-i Imam pattern is periodic and that the quasi-crystalline discs superimposed on its lattice are derivatives of the Maragha pattern.
This paper presents the OPEC Secretariat's latest outlook to 2025 for oil supply and demand. The results have been developed using the OPEC World Energy Model, OWEM.
The next two decades are expected to see increases in energy demand met predominantly by fossil fuels, with oil set to continue to maintain its major role. There is also a clear expectation that the oil resource base is sufficiently abundant to satisfy this demand growth. Global oil demand rises in the reference case by 12 million barrels per day to 89 mb/d from 2002 to 2010, an average annual growth rate of 1.5 mb/d, or 1.8 per cent per annum, over the period. In the following decade, demand grows by a further 17 mb/d to 106 mb/d by 2020, and then by another 9 mb/d to 115 mb/d by 2025. Almost three-quarters of the increase in demand over the period 2002–25 comes from developing countries.
In the short-to-medium term, overall non-OPEC supply is expected to continue to increase—rising to a plateau of 55–57 mb/d in the post-2010 period. The key sources for the increase in non-OPEC supply will be Latin America, Africa, Russia and the Caspian. In the longer term, OPEC will increasingly be called upon to supply the incremental barrel.
Uncertainties over future economic growth, government policies and the rate of development and diffusion of newer technologies raise questions over the future scale of investment that will be required. These uncertainties, coupled with long lead times, inevitably complicate the task of maintaining market stability. Medium-term prospects suggest that there is a need to ensure that spare capacity is not too high and that it is consistent with sustained market stability. There are genuine risks of downward pressure on oil prices, and this could sow the seeds of instability.
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