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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.
... 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 Fashi's architectural style (Li, 2001). ...
Every Architectural style consists of an Architectural language with vocabulary, syntax, and semantics. The compositional principles of a particular style can be defined over as a set of rules. These rules can be reformed and converted using mathematical computational techniques using Shape Grammar (A systematic method used for interpreting spatial design and activities). Researchers across the world used shape grammar to analyse design patterns of traditional architectural styles, master architects' works, etc. These rule-based methods can be adopted into computer languages to produce new designs. Traditional Architecture of a region portrays culture integrated with all aspects of human life. The proposed paper is to study the potentials of shape grammar to use as a tool in the research of traditional architectural styles by analysing case studies. The research methodology reviews the previous shape grammar studies conducted in various conventional styles and comparative analysis of the approaches of authors in shape grammar generation. The research by Lambe and Dongre on the formulation of shape grammar of Pol houses of Ahmadabad and Cagdas's work on traditional Turkish houses is an example of this. T Knight had formulated shape grammar of Japanese tea houses, and Yousefniapasha and Teeling developed a grammar of vernacular houses facing rice fields of Mazandaran, Iran. Similarly, many researchers used shape grammars as a tool to analyse traditional architecture. So the study will compare the different traditional shape grammar generations and formulate a sample shape grammar of a traditional prototype to conclude the scope of further research in the domain.
... Most architectural images appeared to be somewhat periodic in the horizontal and vertical directions ( Figure 1A). Because periodicity brought aesthetic value to images (Lu & Steinhardt, 2007), we assessed whether façades were more periodic than paintings. To do so, we performed a power spectral density estimate, using the Cooley-Tukey algorithm for the Fast Fourier Transform (Cooley & Tukey, 1965). ...
... When one looks at façades such as those in Figures 1A and 3A, one sees that the windows and mansards tend to form periodic patterns both vertically and horizontally. Spatial periodicity is another aesthetic variable because it supplies order to the image (Lu & Steinhardt, 2007). Interestingly, however, such a periodicity is rarely plain in artistic paintings. ...
In visual perception, aesthetic values arise from image variables like color, balance, symmetry, harmony, complexity, and positive/negative spaces. Are such variables weighed similarly for different domains of aesthetic activity, like paintings, sculptures, or façades of buildings? Computational theories for the learning of aesthetic values have not considered them to be domain-specific. Here, we test this predicted domain independence by comparing the statistics of several aesthetic variables in artistic paintings and the architecture of building façades. We perform the study for paintings and façades realized during the Hausmann’s Renovation of Paris. The focus on that time and place is due to the relative aesthetic freedom afforded to painters and architects. Our results reveal significant differences in aesthetic values between these two domains of activities. We see more balance, symmetry, harmony, complexity, and rhythmicity in building façades than in artistic paintings. In contrast, artistic paintings give up some of these aesthetic values to obtain more drama, dynamism, and focus on the main subjects. We conclude that theories for the learning of aesthetic values must include a separation in domains of activity. Such a domain specificity may be needed because of their different demands, such as giving the impression of both balance and continuous urban identity in buildings but not paintings. Importantly, domain-specific value systems are compatible with ideas of contingent information from evolutionary-psychology research.
... 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,
... An aperiodic example is found in Girih tiles, show-ing the rich articulation and symmetry properties of geometric tiles (Kaplan, 2002;Khamjane and Benslimane, 2018). Pentagonal and decagonal symmetry is found in most Islamic ornaments that coincide with mathematical definitions of crystal forms observed in nature (Dabbour, 2012;Lu and Steinhardt, 2007). In various Islamic elements such as zellij and muqarnas, the planimetric drawings and analysis led to the definition of common tile types and symmetry groups that show crystalline behavior (Castera, 1999). ...
... As a stacked dome structure, muqarnas are composed of layers that are formed through triangular or quadrilateral pieces that can be compared through trigonometric angles, ratios and tile types. A fundamental characteristic of muqarnas is determined through the symmetry of the overall muqarnas that can follow octagonal, decagonal or dodecagonal patterns-all of which can form quasicrystal behavior (Lu and Steinhardt 2007). While the cities of Kayseri and Sivas present examples of all the radial types (Gökmen et al. 2022), in this paper the focus will be on the identification of ...
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.
... 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
... From an architectural engineering perspective, it might be more pragmatic to start the base tiling by repeating geometric polygon shapes in regular tiling patterns, e.g. as in the Penrose tiling pattern [20]. This approach is traditionally known as Girih-Chini in Persian Architecture [21], [22], [23]. ...
Building one-off free-form or form-found vault shapes requires various custom non-reusable structural elements (or formworks) and an expert workforce. The aim of this paper is to propose an alternative direction which is about improving the constructability of the vaults and the versatility of the shapes that can be approximated rather than optimizing one-off free-form shapes for ideal vaults. This research introduces a topological design process for shaping muqarnas-like vault pieces that can be mass-produced for approximating suitable vault forms (a.k.a. funicular forms). The idea is to derive a small set of shapes with which one can combinatorically create a large plethora of approximated vaults, akin to muqarnas vaults in Middle-Eastern and North African architecture traditions. The gist of the proposed process is to start from vault or dome tessellations (a.k.a. tilings), inspired by Gothic and Persian vaulting patterns; mark the apex points and/or the top ridges of the vault; find load-transfer paths from the faces (tiles) incident to the top of the vault to the faces incident with the supporting walls on a dual graph corresponding to the base tiling through graph traversal methods; and then convert the pairs of consequent faces corresponding to the links in the topological load-paths to the base meshes for creating the shapes of the stackable corbel-vault pieces; and later carve out cornices (the Greek namesake of muqarnas vault pieces).
... Importantly, quasicrystalline order can be found in various systems, spanning from solid-state materials to soft matter [9][10][11][12][13][14][15]. Moreover, QCs are intimately related to mathematical tiling concepts explored well before their discovery in actual materials [16,17] and emerge in more exotic systems such as vibrating (macroscopic) granular materials [18] and quantum phase transitions [19]. ...
We present a mesoscale field theory unifying the modeling of growth, elasticity, and dislocations in quasicrystals. The theory is based on the amplitudes entering their density-wave representation. We introduce a free energy functional for complex amplitudes and assume nonconserved dissipative dynamics to describe their evolution. Elasticity, including phononic and phasonic deformations, along with defect nucleation and motion, emerges self-consistently by prescribing only the symmetry of quasicrystals. Predictions on the formation of semicoherent interfaces and dislocation kinematics are given.
Published by the American Physical Society 2024
... In one of the earliest applications of non-periodic tilings in computer graphics, Penrose tilings [12,14,44] were employed by Rangel-Mondragon and Abas [42] in the design of decorative patterns inspired by Islamic art. They had effectively reinvented the medieval trade secrets of the craftsmen of fifteenth century Islamic mosques [28] who created by hand highly intricate mosaics closely resembling quasicrystal tilings only discovered by modern science in the late twentieth century. Wang tilings [12,14] were first introduced by Stam [48] in order to enable wave texture patches to cover water surfaces of arbitrary size without the appearance of regularly repeating artifacts. ...
We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.
... We define an order parameter which quantifies these variations and provides a systematic comparison between supramolecular arrays prepared using similar methodologies. Furthermore, we show that the observed variations may be rationalised within a complex phase diagram which can be related to the adsorption of dimers on a periodic lattice 7,8 , a paradigmatical problem in statistical mechanics which has been studied for several decades, and is a well-known example of a physical system which can be mapped onto a non-periodic tiling of the plane [9][10][11][12][13][14][15][16] . In particular, using numerical simulations, we relate the order parameter to an orientational dependence of the intermolecular interactions which stabilise the network. ...
The degree of randomness, or partial order, present in two-dimensional supramolecular arrays of isophthalate tetracarboxylic acids is shown to vary due to subtle chemical changes such as the choice of solvent or small differences in molecular dimensions. This variation may be quantified using an order parameter and reveals a novel phase behaviour including random tiling with varying critical properties as well as ordered phases dominated by either parallel or non-parallel alignment of neighbouring molecules, consistent with long-standing theoretical studies. The balance between order and randomness is driven by small differences in the intermolecular interaction energies, which we show, using numerical simulations, can be related to the measured order parameter. Significant variations occur even when the energy difference is much less than the thermal energy highlighting the delicate balance between entropic and energetic effects in complex self-assembly processes.
... It relies on the regular decagon and two hexagons and has individual dihedral symmetry D 10 . It was published in [LS07a,LS07b]. However, the complete set of substitution rules can be found in [Ten08,Cro08]. ...
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.
... Quasicrystals (QCs) are solids, typically ternary metallic alloys, that lack translational symmetry but still exhibit long range order through a deterministic nonperiodic tiling structure, leading to unusual geometric properties such as 5-, 8-, 10-, or 12-fold rotational symmetry and scale invariance [1,2]. Such non-periodic tilings have been studied by mathematicians, most notably in the 1970's by Penrose [3], and have existed for centuries in art and architecture [4], but material examples were unknown until the serendipitous synthesis and discovery of QCs in an AlMn alloy by Shechtman was announced in 1984 [5], for which he was awarded the 2011 Nobel Prize in Chemistry. QCs have also been found naturally occurring in meteorites [6], and form spontaneously in some organic polymer systems [7]. ...
Quasicrystals are nonperiodic structures having no translational symmetry but nonetheless possessing long-range order. The material properties of quasicrystals, particularly their low-temperature behavior, defy easy description. We present a compact optical setup for creating quasicrystal optical potentials with 5-fold symmetry using interference of nearly co-propagating beams for use in ultracold atom quantum simulation experiments. We verify the optical design through numerical simulations and demonstrate a prototype system. We also discuss generating phason excitations and quantized transport in the quasicrystal through phase modulation of the beams.
... Along this path, we take inspiration from geometric motifs in ancient architectures and present a concept for the design of planar bistable auxetics that is generic and readily applicable to other patterns while can be manufactured through simple cuts at various length-scale. We should note that this is not the first time that geometric art has had an impact on materials science; in fact, it was used, for example, to study quasicrystal patterns [45]. We recognize that other geometric patterns predate Islamic era and can provide equivalent source of inspiration. ...
Auxetic materials become thicker rather than thinner when stretched, exhibiting an unusual negative Poisson's ratio well suited for designing shape transforming metamaterials. Current auxetic designs, however, are often monostable and cannot maintain the transformed shape upon load removal. Here, inspired by ancient geometric motifs arranged in square and triangular grids, we introduce a class of switchable architected materials exhibiting simultaneous auxeticity and structural bistability. The material concept is experimentally realized by perforating various cut motifs into a sheet of rubber, thus creating a network of rotating units connected with compliant hinges. The metamaterial performance is assessed through mechanical testing and accurately predicted by a coherent set of finite element simulations. A discussion on a rich set of mechanical phenomena follows to shed light on the main design principles governing bistable auxetics.
... A significant breakthrough in this domain was presented by Lu and Steinhardt. They identified that tessellations employing Girih tiles, when adorned with lines, facilitated the formation of complex periodic Girih patterns [8]. These patterns can be observed across various historical Islamic sites and artifacts. ...
... Mimarlık-matematik ilişkisine sibernetik bir zihnin bütün alanların tevhidine duyacağı inancı üzerinden bakarsak, iki disiplin arasındaki içkinliğin kimya kanunlarının kanıtlanmasına kadar yayılageldiğini iddia edebiliriz. Nobelli kimyager Dan Shechtman, quasicrystals [kuazi kristaller] keşfini açıklayabilmek için tarihi yapıların mukarnas ve çini kaplamalarında birbirini tekrar etmeyen, fakat kurallı dizilere sahip geometrik desenlerden yararlanmıştır (Lu and Steinhardt, 2007). Mukarnaslar, parametrik mimarlık denince akla ilk gelen şeylerden biridir. ...
Bu makale, mimarlık ve matematik arasındaki içkinlik hakkındadır. Dijital tasarım araçlarının kullanımının zirvesinde, herkes tarafından artık bilindiği düşünülen “parametrik mimar” tanımını açıklamaya çalışmak bu yazının hedefidir. Parametrik mimarlık tanımının, mimarlığın matematikle kurduğu ilişkide arandığı bu çalışmada ilk olarak, matematiğin platonik boyutlarından akıp mimarlığın mükemmellik arayışına; hassasiyet (precision) ve doğruluk (accuracy) ihtiyacına; geometrik ve grafik sezgilerin temsile dönüşümüne kadar farklı konseptler üzerinden “parametrik mimar” kavramının türeyişini anlatan tarihsel bir sunuşa yer verilmiştir. Sonrasında Zaha Hadid ve Antoni Gaudí gibi zaman ve mekandan bağımsız virtüöz mimarların sunduğu temsiller üzerinden “parametrik mimarlar”ın virtüellik zemini tartışılmaktadır. Bunu da mimari manipülasyonların materyalist taleplerini karşılamak için çalışan “parametrik mimar”a gerekli kılınan matematiksel dönüşümü bir kompütasyon düşüncesi (computational thinking) üretim alanı olarak tartışmaya açmaktadır. Star mimar ve Tanrı’nın mimarı gibi erk üreten pozisyonlara sıkıştırmak yerine, bu yazıda altı çizilmek istenen virtüalite tartışması, her iki virtüöz mimar ile zaman ve mekandan bağımsız dahi olsalar, Patrik Schumacher ve Mark Burry’nin üzerinden “parametrik mimar”ın yeniden yaratımına yer verilmektedir. Sonuç olarak, bu araştırmadan elde edilen iç görüler, tasarı geometrinin sınırlarına sıkışan geleneksel mimarlık pratiğini aşmanın bir yolu olarak görünen “parametrik mimarlık” tanımını matematik ile kurulan yaratıcılık odaklı ilişki üzerinden yeni ve gelişen virtüel zemin kayması/birleştirmesi olarak yapmaktadır. Böylece mimarlık pratiği ve teorisi için çıkarımlar sunmaktadır.
... For example, photonic crystals show a fractal hierarchy of band gaps [13] among other phenomena [14][15][16][17], and acoustic metamaterials can reach isotropic effective speeds of sound and isotropic acoustical activity [18]. Characteristic quasicrystalline phenomena and structures can be seen in numerous other platforms, including ultra-cold atomic lattices [19], decorated metallic surfaces [9], polaritonic systems [20] microwave networks [21], acoustic [18] and mechanical [22] metamaterials, botanic [23] and even architecture [24]. ...
Finding optimal wave sampling methods has far-reaching implications in wave physics, such as seismology, acoustics, and telecommunications. A key challenge is surpassing the Whittaker-Nyquist-Shannon (WNS) aliasing limit, establishing a frequency below which the signal cannot be faithfully reconstructed. However, the WNS limit applies only to periodic sampling, opening the door to bypass aliasing by aperiodic sampling. In this work, we investigate the efficiency of a recently discovered family of aperiodic monotile tilings, the Hat family, in overcoming the aliasing limit when spatially sampling a wavefield. By analyzing their spectral properties, we show that monotile aperiodic seismic (MAS) arrays, based on a subset of the Hat tiling family, are efficient in surpassing the WNS sampling limit. Our investigation leads us to propose MAS arrays as a novel design principle for seismic arrays. We show that MAS arrays can outperform regular and other aperiodic arrays in realistic beamforming scenarios using single and distributed sources, including station-position noise. While current seismic arrays optimize beamforming or imaging applications using spiral or regular arrays, MAS arrays can accommodate both, as they share properties with both periodic and aperiodic arrays. More generally, our work suggests that aperiodic monotiles can be an efficient design principle in various fields requiring wave sampling.
... Importantly, quasicrystalline order can be found in various systems, spanning from solid-state materials to soft matter [9][10][11][12][13][14][15]. Moreover, QCs are intimately related to mathematical tiling concepts explored well before their discovery in actual materials [16,17] and emerge in more exotic systems such as vibrating (macroscopic) granular materials [18] and quantum phase transitions [19]. ...
We present a mesoscale field theory unifying the modeling of growth, elasticity, and dislocations in quasicrystals. The theory is based on the amplitudes entering their density-wave representation. We introduce a free energy functional for complex amplitudes and assume non-conserved dissipative dynamics to describe their evolution. Elasticity, including phononic and phasonic deformation, defect nucleation and motion, emerges self-consistently by prescribing only the symmetry of quasicrystals. Predictions on the formation of semi-coherent interfaces and dislocation kinematics are given.
... 5), the vital tradition in Islamic architecture was very clear in the formal patterns of the drawings on the windows (parasols) and the roundabouts of the mosques (the brides of the sky). And Salsabil (Shaduran) came on plant patterns inspired by nature, [10],[11] ...
... Разнообразные кирпичные узоры, украшенные голубой глазурью, можно видеть на мавзолеях Юсуфа ибн Кусейра (1162) и Момине Хатун (1186), возведённых Аджами ибн Абубакром в Нахичевани (Cromwell 2018). К этой же западно-иранской школе относятся мавзолеи Гунбад-и Сурх (1148) и Гунбад-и Кабуд (1197) в Мараге, последний -с обвивающим всю стену исключительно сложным орнаментом, на который обратили внимание авторы статьи, связавшей искусство гириха с математической теорией непериодических замощений плоскости (Lu & Steinhardt 2007, см. также Cromwell 2008, Bier 2015, Makovicky 2023a. ...
This article deals with the art of geometric ornaments, widely spread in the whole Islamic world. This art appeared at the beginning of the 11th century in Khorasan and Transoxania, rapidly developed in the next two centuries until the Mongol invasion, transferred from here to Damascus, Cairo and further to the Maghreb countries, and then flourished again in the Timurid Empire, when multi-color solutions were added to complicated geometry of star polygons. We consider various principles for constructing these patterns, with special attention to so called “polygonal technique”.
... 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
... 5), the vital tradition in Islamic architecture was very clear in the formal patterns of the drawings on the windows (parasols) and the roundabouts of the mosques (the brides of the sky). And Salsabil (Shaduran) came on plant patterns inspired by nature, [10],[11] ...
... 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.
Quasi-periodic structures of quasicrystals yield novel effects in diverse systems. However, there is little investigation on employing quasi-periodic structures in the morphology control. Here, we show the use of quasi-periodic surface structures in controlling the transition of liquid droplets. Although surface structures seem random-like, we find that on these surfaces, droplets spread to well-defined 5-fold symmetric shapes and the symmetry of droplet shapes spontaneously restore during spreading, hitherto unreported in the morphology control of droplets. To obtain physical insights into these symmetry transitions, we conduct energy analysis and perform systematic experiments by varying properties of both liquid droplet and patterned surface. The results show the dominant factors in determining droplet shapes to be surface topography and the self-similarity of the surface structure. Our findings significantly advance the control capability of the droplet morphology. Such a quasi-periodic patterning strategy can offer a new method to achieve complex patterns.
Girih patterns, prominent in Islamic art and architecture, exhibit complexity and adherence to specific mathematical rules. Understanding and preserving these patterns requires knowledge of their design processes and underlying rules which require transferable knowledge of computation. This paper aims to generate three prominent Girih patterns in Islamic architecture, namely ‘ Hasht-Chahar-Lengeh’, ‘ Hasht-Panj’, and ‘ Hasht-Bazoubandi’, gaining insights from graph theory definitions and algorithmic design within the Grasshopper program. Employing the graph theory, it pioneers a novel approach to pattern generation. The approach facilitates the faithful reproduction of existing patterns and enables the generation of entirely new designs with a comprehensive comparison of both computational methodologies.
Skyrmion structures play critical roles in solid-state systems involving electric, magnetic and optical fields. Previous approaches to the study of skyrmions have involved specific structures in magnetic materials, liquid crystals and polymers in addition to two-dimensional arrays used for electrical control. These methods have encountered limitations and constraints on both the microscopic and macroscopic scales related to the physical properties of materials. The present work demonstrates an origami-based skyrmion engineering strategy that suggests a new approach to topological control. This technique utilizes the unique properties of orientational origami, combining polarization techniques with rotationally symmetric, periodically folded designs. This strategy enables the transformation of flat sheets into three-dimensional structures with associated changes in optical topology, similar to the characteristics of proteins. Topological defects such as misalignments and dislocations in folded molecularly oriented sheets lead to the creation of skyrmion clusters at boundaries having different orientational orders. The strategy reported herein involves the construction of unique metamaterial platforms that could provide new applications for twistronics in graphene and photonic crystals.
In numerous instances of Islamic architecture, the geometric arrangement of muqarnas was established using square-rhombus tessellations in their flat projections. Similarly, in Türkiye, square-rhombus projection schemes have been employed in muqarnas designs since the era of the Anatolian Seljuks. This study introduces a computer-aided design (CAD) tool called “Anatolian Muqarnas,” which aims to recreate muqarnas schemes with square-rhombus projections. The tool is built upon a novel geometric analysis approach specifically tailored for Anatolian Seljuks muqarnas, drawing inspiration from Ammann-Beenker tiling and supertiles. Users of Anatolian Muqarnas can easily simulate existing or new designs by following predetermined rules. This contribution is expected to enhance discussions regarding the geometric analysis of muqarnas and the interplay between computing and cultural heritage at large.
This paper details a qualitative study examining the experience and motivation of participants, facilitators, and organizers of the Math Circus, an event hosted by the University of Arkansas Honors College in Fayetteville, Arkansas, as part of the university’s 150th anniversary celebrations. The aim of the Math Circus was to entice participants to explore advanced mathematical concepts through collective mathematical art creations and dramatic storytelling of comic tales around the history behind mathematics. The study was carried out in two phases; an autoethnographic case study during the event; and a series of deep, semi-structured interviews with participants, facilitators, and organizers. The study examined pedagogical motivations behind the event’s activities and their mathematical content, as well as the experiences of the interviewees. Findings indicate that an interactive and festive event like the Math Circus is likely to encourage confidence to explore mathematical concepts further
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.
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
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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