From Substitution Tilings to Geometric Patterns in Islamic Styles
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Substitution tilings are geometric structures that are orderly and systematic , but need not be periodic. We exploit such tilings to create new geometric patterns in several traditional Islamic styles found in Iran, Turkey and Central Asia. The patterns are produced by decorating the prototiles with motifs abstracted from historic examples. We describe a new substitution tiling that is well-suited for this purpose with kite, rhombus and trapezium prototiles. Some families of Islamic patterns can be generated by known modular design systems. Three systems correspond directly to the prototiles in substitution tilings, and lead naturally to plain, hierarchical , and self-similar designs. Other patterns are produced by decorating prototiles with groups of modules or using non-modular approaches. Exploring these possibilities prompts discussion of what characteristics constitute Islamic style.
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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.
We survey 40 examples of 2-level patterns in Islamic geometric ornament from Iran and Central Asia, and identify a range of methods that can be used to explain their construction. Techniques based on modular design are shown to simplify the process. Each 2-level pattern is listed in a catalogue that gives the location, construction method, pattern type of the large and small-scale patterns, and references to photographs and other literature. We also show that a tympanum pattern in the Darb-i Imam in Isfahan is constructed by filling compartments and not by subdividing modules, as has been previously suggested. None of the medieval patterns in the catalogue is quasiperiodic.
The similarity between the structure of Islamic decorative patterns and quasicrystals has aroused the interest of several crystallographers. Many of these patterns have been analysed by different approaches, including various kinds of ornamental quasiperiodic patterns encountered in Morocco and the Alhambra (Andalusia), as well as those in the eastern Islamic world. In the present work, the interest is in the quasiperiodic patterns found in several Moroccan historical buildings constructed in the 14th century. First, the
zellige
panels (fine mosaics) decorating the Madrasas (schools) Attarine and Bou Inania in Fez are described in terms of Penrose tiling, to confirm that both panels have a quasiperiodic structure. The multigrid method developed by De Bruijn [
Proc. K. Ned. Akad. Wet. Ser. A Math. Sci.
(1981),
43
, 39–66] and reformulated by Gratias [
Tangente
(2002),
85
, 34–36] to obtain a quasiperiodic paving is then used to construct known quasiperiodic patterns from periodic patterns extracted from the Madrasas Bou Inania and Ben Youssef (Marrakech). Finally, a method of construction of heptagonal, enneagonal, tetradecagonal and octadecagonal quasiperiodic patterns, not encountered in Moroccan ornamental art, is proposed. They are built from tilings (skeletons) generated by the multigrid method and decorated by motifs obtained by craftsmen.
We introduce a modular design system, which we call the Central Asian modular design system (CAMS), that is based on the pre-Islamic Star and Cross pattern. It can be used to generate a large family of traditional Islamic patterns found in Central Asia and Iran. In other examples of Islamic modular systems, the modules form a substrate that is used in construction and then deleted; in this case the modules themselves form the finished pattern. We also analyse some traditional two-level geometric patterns as hierarchical structures of CAMS modules, corroborating the principles of two-level pattern construction found in other Islamic modular systems.
We present a technique for blending the 1-point and 2-point applications of the ‘polygons in contact’ method of constructing Islamic geometric patterns. Two special tiles provide the bridge. Evidence is presented for the historical use of this method, and hybrid structures are shown to underlie some traditional Turkish patterns.
Aperiodic tilings of Euclidean space can profitably be studied from the point of view of dynamical systems theory. This study takes place via a kind of dynamical system called a tiling dynamical system.
In medieval times the city of Isfahan was a major centre of culture, trade and scholarship. It became the capital of Persia in the Safavid era (16–17th centuries) when the creation of Islamic geometric ornament was at its height. Many of the most complex and intricate designs we know adorn her buildings, including multi-level designs in which patterns of different scales are combined to complement and enrich each other. In this article we study five 2-level designs from Isfahan built around a common motif. They illustrate a variety of techniques and the analysis exposes some of the ingenuity and subtle deceptions needed to reconcile incompatible geometries and symmetries, and produce satisfying works of art.
The discovery of quasi-crystals has led to a great debate about their unusual structure. The big surprise is that these structures were found in Islamic art several centuries ago. This latest discovery drew the attention of scientists to propose several approaches for the comprehension of these structures by analyzing several quasi-periodic patterns spread around the Islamic world. In this article, we propose a systematic method for generating new quasi-periodic patterns inspired by existing Islamic historical patterns. The method builds Islamic quasi-periodic patterns based on a quasi-periodic tiling and a few intuitive parameters. Given a quasi-periodic tiling, the method divides its tiles (rhombs) into symmetric right triangles and constructs their template motifs. The construction of these template motifs is achieved by a systematic and well-organized process. The content of the tiles is obtained by applying mirror reflections to the constructed template motifs. Finally, the pattern is drawn by putting the content of the constructed tiles in the tiling. To show the effectiveness of this generative method, examples of new quasi-periodic patterns will be presented