Reverse engineering of Islamic geometric patterns: a scientific approach to art history Peter R. Cromwell This article is based on a presentation I gave at the International Workshop on Geometric Patterns in Islamic Art held at the Istanbul Design Centre, 22–29 September 2014. 1 Introduction One of the problems in art history is to chart the evolution of form and style. Developments may be interpreted as responses to the introduction of new ideas and technologies, the influence of patronage, or contact with different cultures. In the case of geometric ornament, some of the elements of form (the artist’s use of line and shape and the arrangement of motifs) are also regulated by the techniques used to devise the patterns. Unfortunately, in the case of Islamic patterns, we do not have any contemporary documents describing the traditional methods that were used to create new designs, or the principles for applying designs as ornament in architectural or other settings. We do, however, have many examples of the finished products and, through careful examination and analysis, it is possible to recover some of the lost techniques for composition. Reverse engineering is the process of applying the scientific method not to a natural phenomenon but to a man-made system or device. It is performed when the system itself is available for experiment and analysis, but knowledge about the original design, production, or use of the system has been lost, destroyed or withheld. The system is analysed from different viewpoints to identify its components and their relationships, and to provide a range of descriptions and representations of the system. Ultimately, the aim is to understand and document the system to a level where we are able to replicate its behaviour. This includes how it works and also its limitations: what it does, what it cannot do, and how it performs when used in unexpected ways. The scientific method is a key element of this process. It is based on a cycle of revision and refinement consisting of the following stages: • make observations of the subject matter • make hypotheses or a model to explain the observations • make predictions — logical deductions from the hypotheses • design and perform experiments to test the predictions. The tests should check that all observed behaviour can be reproduced very closely, and that any predicted behaviour is observed. We shall illustrate how this approach can be applied to the study of Islamic geometric patterns. First we shall develop our main hypothesis and its consequences, and explore how it interacts with other techniques. We shall then test it against alternative explanations. 1 2 Creating patterns The subject matter for our study is the collection of traditional Islamic geometric pat- terns. While we have many examples of the end product, we have no documentation about how they were conceived. Competition between suppliers means that methods are closely guarded and information is tightly held. Medieval craftsmen did produce pattern books, but these are just catalogues of designs, not how-to manuals. A few such manuscripts have survived but, even though they are a valuable primary source, they may be not known or not available to researchers. The initial phase of the investigation involves repeated observations of the source mate- rials, looking for common motifs and configurations, and abstracting properties that can be used to group patterns into families. This may be as simple as sorting patterns by the type of star they contain, or it may require some understanding of the method of construction. We shall not discuss this part of the process further. For the purposes of this study we shall focus on a family defined as the output of a modular design system. Our hypothesis is that modular design was one of the methods used traditionally to create new patterns and, in particular, that the set of modules we shall use has a historical basis. 2.1 Modular design Modular design is a method for creating patterns or other structures that is both versatile and easy to use. A modular design system is a small set of simple elements (modules) that can be assembled in a large variety of ways. The modules can be grouped into a unit that is then repeated, or used in a more free-form and playful way to produce more ‘organic’ compositions. rhombus pentagon barrel bobbin bow-tie decagon Figure 1: Elements of a modular design system. Figure 1 shows a modular design system with six modules. The modules are outlined by equilateral polygons, shown in red. Each module is decorated with a motif that meets each side of the polygon at its midpoint. The decagon module carries a regular 10-pointed star motif composed of ten small kites arranged in a ring. These kites are congruent to the two kites forming the motif on the bow-tie module. I have named the modules for ease of reference. Some have the standard geometrical name for the boundary polygon: rhombus, pentagon, decagon. The pentagon and decagon are regular polygons. The three other polygons have six sides so they are all (irregular) hexagons. The bow-tie and the barrel are named after the shape of the polygon; the bobbin module is named after the shape of the motif (which is also called a spindle or a bottle). The bobbin motif is very distinctive and its presence in a pattern is a strong indicator that it may be possible to generate the pattern using this modular system. 2 Patterns are produced by assembling the modules to form a tessellation. Patterns in the family generated by these modules are widespread in Turkey, Iran and Central Asia. Figure 2 shows some traditional examples. The patterns in (a)–(d) use only the bow-tie, bobbin and decagon modules. Patterns that can be generated from these three modules alone are so common in Iran that this subsystem could be considered to be a modular system in its own right. The remaining patterns exhibit the other modules: (e) uses only the rhombus and decagon modules, (f) adds the pentagon, (g) adds the bobbin, and the barrel appears in (h). In the figure the red lines of the modules’ boundaries are drawn to reveal the underlying modular structure of the patterns, but they are not included in the finished ornament. The modules play the role of an invisible substrate that is used to organise the motifs. Figure 3 shows modular decompositions of two early, common star patterns. These simple patterns can be constructed in many ways and, on their own, do not contribute much support for the modular hypothesis. However, they would have provided familiarity with some of the basic motifs of the modular system before it was abstracted. The first examples of more complex patterns that can be generated with this modular system appear on Seljuk architecture of the late twelfth century. Figure 4 shows some geometric ornament from Turkey in a variety of media: • (a) is Seljuk tile work from the Sircali Madrasa in Konya. • (b) is a carved stone from the Seljuk period in the Ince Minaret Madrasa in Konya. • (c) is a panel from the Alaeddin Mosque in Konya. It is one of a sequence of ‘Turkish triangles’ that form the transition from a square base to the 20-sided polygon that carries a dome. • (d) is a wooden door panel from the Selimiye Mosque in Edirne. These examples can all be generated with the modular system. Notice that the stars in (a) and (d) have additional internal structure. 2.2 Emergent properties Although the red lines of the modules have been left visible in Figures 2 and 3, they are present only to mark the underlying structure of the design. In the finished product, the red lines are deleted and the white regions in the corners of the modules are fused together to form larger units. We can think of the shaded motifs as foreground and the (fused) white areas as back- ground. The shapes that appear in the background are an emergent property of the design system — they are a consequence of the choice of motifs and the rules for assembling the modules. People often overlook the fact that not all observed behaviour has to be explicitly built in to a system. How many background shapes are there? To enumerate the possibilities we list all the ways that modules can be arranged around a point. Because the modules are equilateral, the shape of the white area in a corner of a module depends only on the internal angle in the corner. The angles in the corners of the modules are all 2/10, 3/10, 4/10 or 6/10 of a whole turn. Therefore, the possibilities for surrounding a point correspond to the partitions of 10 using 2, 3, 4 and 6. These are shown in Figure 5 — we see that there are only four 3 (a) (b) (c) (d) Figure 2: Traditional patterns that can be produced with the modular system of Figure 1. 4 (e) (f) (g) (h) Figure 2: (continued). 5 (a) (b) Figure 3: Modular decompositions of two common star patterns. different background shapes. The three combinations in the top row all produce a regular pentagon; the two combinations in the middle row both produce an irregular convex hexagon shaped like a tooth. The partition into two 2s and two 3s is the only case where the order matters and it generates two different shapes: one is the tooth and the other is the hexagon on the bottom row. The final shape is a bone-shaped octagon that arises from packing modules round a bow-tie.