Reverse Engineering of Islamic Geometric Patterns: a Scientiﬁc Approach to Art History

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 patterns. 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 materials, 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 background. 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)

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. The four background shapes can all be found in the example patterns shown in Figure 2; the hexagon derived from partition 2.3.2.3 is not common, but it appears in the centres of (f) and (g).

2.3 Where do modules come from? The set of modules presented in Figure 1 generates patterns that are balanced and interesting to look at. There are enough different shapes that the eye does not get bored, yet not so many that the result seems cluttered. The detail is visible — the shapes are not too small and the angles are not too sharp. None of the shapes is so large that it overwhelms the others and dominates the pattern. The modules are very versatile and can be assembled in many configurations. A set with such desirable properties is difficult to contrive and does not spring into existence fully formed. It is likely that it evolved through a process of trial and error. Figure 6 illustrates two mechanisms that can produce candidates for modules, in this case by experimenting with decagon modules. Regular decagons will not tile the plane on their own — they either leave gaps or they overlap. In the first case, the pattern lines on the decagons can be extended into the void until they meet each other; the ‘decorated gap’ can then be abstracted and used as a design element in its own right. In Figure 6(a) this process produces the bow-tie module. The intersection of the two decagons in Figure 6(b)

6 (a) (b) (c)

(d)

Figure 4: Photographs reproduced courtesy of Mirek Majewski.

7 2.2.2.2.2 2.2.2.4 2.4.4

2.2.3.3 3.3.4

2.3.2.3 2.2.6 and 4.6

Figure 5: The different background regions that emerge by placing modules around a point. has the shape of a bobbin module, and the pattern lines it contains outline the bobbin motif. The following example may be the result of this kind of experiment. The photograph in Figure 7(a) shows the lower panel of a wooden door from the Suley- maniye Mosque in Istanbul. It is redrawn in Figure 8, both with and without its modular structure. The tessellation uses the pentagon, barrel and decagon modules — they are assembled in a way that leaves a rhombus-shaped void in the centre of the panel that cannot be filled with the modules of Figure 1. The pattern is extended into the void by continuing existing lines until they meet each other. The resulting filler shares some attributes with the other modules — it is an equilateral polygon and it is decorated with a motif that aligns properly with its neighbours. Should we add the thin rhombus to our set of six modules? All of the modules shown in Figure 1 have widespread use and are found in many traditional patterns. However, Figure 7(a) is the only pattern I know of where this thin rhombus module appears. The panel seems to be an experiment whose outcome was not regarded as successful and the rhombus module was not adopted into the standard vocabulary. In other areas of art history we may have records such as sketchbooks that show prelim- inary workings and ideas that did not produce satisfactory results. In the study of Islamic ornament spurious examples like Figure 8 are important because they can also provide insight into the creative process.

(a) (b)

Figure 6: Arranging decagon modules with gaps and overlaps.

8 (a) (b)

Figure 7: Photograph (a) reproduced courtesy of Mirek Majewski, photographs (b) and (c) reproduced courtesy of David Wade.

9 (a) (b)

Figure 8: Construction of the pattern in Figure 7(a).

2.4 Rectangular templates Many traditional Islamic patterns are based on a rectangular template. The template is usually replicated by reflection in the sides of the rectangle. Note that this does not require left-hand and right-hand templates — in practice, it is achieved by turning the template over. Figure 9(a) shows a rectangular template outlined in blue and decorated with an asymmetric motif. In (b) the basic template has been reflected vertically and horizontally, and these images have been reflected again to produce the standard quartering arrangement often used in design; part (c) shows a more extended panel produced by repeated reflection of the template.

(a) (b) (c)

Figure 9: Repetition of a rectangular template by reﬂection.

We shall combine this common replication procedure with our modular design system. To complete a template we cannot simply fill a rectangle by packing it with modules because the angles in the corners of the modules are not compatible with the right angles of the rectangle — we cannot pack modules into corners of the template. However, we do not need to fill the template exactly — we only need to cover it. Any modules that extend beyond the boundary of the template can be truncated and cut to fit. Figure 10 illustrates the process. The blue rectangle in (a) marks the boundary of the

10 (a) (b)

Figure 10: Construction of the pattern in Figure 7(b).

template; it is covered by a tessellation of modules. In (b) the parts of the modules that lie outside the template have been removed. In (c) the template is replicated by reflection, and in (d) the underlying framework has been deleted to leave the finished design. This pattern (rotated by 90◦) appears on the stone inlay panel shown in Figure 7(b) taken from the Tilla Kari Madrasa, one of the three madrasas in the Registan complex in Samarqand. Notice that the replication of the template creates complete modules across the whole of the design in Figure 10(c). Why did this happen? All the modules in Figure 1 have two perpendicular lines of mirror symmetry. This means that they can be divided into halves or quarters and reconstructed by applying reflections. This is very convenient for filling a rectangle because the quarter-modules contain a right angle. Notice that quarter- modules are placed in the bottom-left, top-left and bottom-right corners of the template in Figure 10(b). In the top-right corner two half-modules fill the space. When the boundary of the template meets a module, it coincides with an edge or a mirror line of the module. This ensures that a whole module will be regenerated when the template is replicated.

2.5 Patterns with anomalies What happens when this rule is broken? Suppose that we are not so careful about the alignment of the modules with the boundary of the template. Figure 11 shows the same steps as Figure 10. In this case the template is almost square (the width is about 97.5% of the height). The arrangement of the modules covering the template in Figure 11(a) has 2-fold rotational symmetry so we shall only describe the right half of the template. A decagon is placed in the top-right corner and aligned so that the

11 sides of the template coincide with two of its mirror lines. The bow-tie module below it also has a mirror line lying over the side of the template. The other bow-tie modules are set diagonally; they do intersect the boundary of the template but not in the required manner. This unusual arrangement of the modules does not cause any problems for the replication process. We can still crop the overhanging modules and reflect the resulting template — see Figure 11(b) and (c). A pattern line that meets the boundary of the template will never be left as a loose end — it always joins up with its reflection to produce a continuous line. What does go wrong is that the modular structure breaks down and we spawn anomalous shapes in the foreground and background of the resulting pattern. For example, the shaded arrowhead motifs on the horizontal centre-line of Figure 11(d) do not appear in Figure 1. Notice that the alignment of the stars in this pattern is also unusual. In most traditional patterns stars are positioned so that a line segment connecting the centres of neighbouring stars passes along the centre-line of the spikes of the stars or midway between the spikes. This property does not hold in this example. The pattern shown in Figure 7(c) is from the carved stone minbar in the Mausoleum of Barquq in Cairo. It resembles the design in Figure 11(d), but close inspection shows that the two are not identical. The corners have a cuspidal form — this is most apparent in the central star, and also the narrow waists of the bone-shaped octagons. The standard patterns that can be made with this modular system are rare in Egypt, and we only find simple examples. Although our method can explain the structure of this design, it may have been created in another way. Patterns that contain anomalies are not common, but there are other examples. Fig- ure 7(d) shows part of the ceiling vault in the Karatay Madrasa in Konya. The template for this design is a large rectangle than contains more than 20 modules. Except for two modules in one corner of the template, all the modules are either contained inside the template or meet the boundary of the template in an edge or a mirror line. The two misplaced modules lead to an anomaly that is in the centre of the photograph. The details are shown in Figure 12. The bottom-right corner of the template is shown in (a). Although the bobbin module on the bottom has a mirror line that coincides with the horizontal side of the template, the module extends beyond the right side of the template; the vertical side of the template meets the module in a line that is not in a mirror line of the module. The results of replicating the template, with and without the underlying structure, are shown in (b) and (c). The white background shape in the centre and the shaded arrowhead motifs above and below it are shapes that are foreign to the modular system used to generate the rest of the pattern.

3 Testing the hypothesis

We have demonstrated a method for generating geometrical patterns. However, what we are really interested in is the art historical question of whether this is a traditional method for creating Islamic patterns. We need to consider the following questions:

• is our hypothesis consistent with the data? • is there another explanation? • is there other evidence?

12 (a) (b)

Figure 11: Construction of the pattern in Figure 7(c).

(a) (b) (c)

Figure 12: Construction of the pattern in Figure 7(d).

13 At this point we should remark that care is required when reading the literature on Islamic patterns: some published papers fail even the first test. In these cases, the proposed construction produces a figure that has some features in common with the source pattern it is trying to explain, but which is not an accurate reproduction. The difference may be due to the stylistic treatment of the design by the artist, or because some of the salient features have not been recognised in the analysis or are not recreated by the method. In some cases the authors do not comment on the discrepancies. When they do, the errors may be explained away as poor quality workmanship on the part of the original builders or later restorers of the pattern — that is, they are seen as mistakes in the data, not in the analysis. Let us return to our hypothesis. We have seen that the modular system of Figure 1 can reproduce a large family of traditional Islamic patterns. Although we have not shown examples here, the family includes patterns from Iran that exhibit a more playful and free- form expression of modular design. This is seen most clearly in the production of 2-level patterns [1, 6]. We have also seen that when we use our method in unexpected ways, it generates faults in the modular structure along the edges and corners of the templates. In this way we can predict and explain both the existence and location of anomalies in some patterns. As an explanatory tool, our proposed method works well and captures many properties of the data. Let us now consider whether there are alternative explanations.

3.1 Non-modular construction Modular design is not the only method for creating patterns. The construction shown in Figure 13 uses some standard motifs from the Islamic vocabulary and a technique called symmetry breaking to generate the template. It starts with a 10-pointed star (a) and attaches a ring of regular pentagons in the spaces between the spikes (b). The blue rectangle outlined in (c) passes through the top and bottom points of the star and the corners of some of the pentagons. In (d) the design is cropped to fit the rectangle to form a template; the 10-fold symmetry of the rosette in (b) is reduced to 2-fold symmetry. Replicating the template produces the pattern shown in (e); the bobbin motif in the centre of the pattern is an emergent feature of the construction that arises from the corners of the template. Figure 13(f) shows that the same pattern can also be created by assembling the bow-tie, bobbin and decagon modules, so it belongs to the family of patterns that can be generated by the modular system shown in Figure 1. In this case, the modular design hypothesis is not needed to explain its construction, and indeed, many of the simpler patterns in the family may have been made with non-modular techniques. However, the existence of a large number of patterns composed of the same elements in different arrangements suggests that some kind of modules were in use, even if they were not those shown in Figure 1 (some alternatives are discussed below). Furthermore, some of the more complex designs have a low density of stars; in these cases the modular approach provides a simple mechanism to construct a visually effective interconnecting matrix, something that is difficult to achieve with other means.

14 (a) (b) (c) (d)

(e) (f)

Figure 13: Generating a pattern by symmetry breaking.

3.2 Undecorated modules In our figures illustrating modular constructions, the patterns are coloured in two colours like a chessboard so that any region of one colour is surrounded by regions of the other colour. The two roles are created automatically by the decoration on the modules in Figure 1: motif and corners, dark and light, foreground and background. The chessboard colouring is a prominent feature of many Iranian cut tile mosaics — the foreground motifs are usually black and a variety of pale colours are used for the background. Sometimes, several background colours are used in the same panel to highlight different elements of the composition. However, this division is not made apparent in any of the artworks shown in the photographs in Figures 4 and 7. In these examples the regions are not separated into background and foreground, but are all treated equally to provide a uniform ground for the linear designs. We can use the foreground and background regions themselves as modules. The ten different shapes are shown in Figure 14. With this modular system, the shapes you see

Figure 14: A design system with undecorated modules.

15 in the finished product are the modules used in its conception. This what-you-see-is-what- you-get approach removes the need to fuse corner regions, and we no longer need to imagine an invisible substrate that is used as a framework. Simplifying the composition process in this way creates a corresponding increase in complexity elsewhere: there are more modules and they are not so easy to assemble (there are three different edge lengths). We also lose the chessboard separation of the two roles that is automatically enforced by the decorated modules: there is nothing a priori to prevent modules that play the same role from being placed next to each other. Rigby and Wichmann [8] explored patterns that can be generated using four of these shapes and created over 60 designs. The tiles they chose and the assembly rules they used mean that all their examples can also be generated with the bow-tie and bobbin modules of Figure 1. Figure 15 shows two patterns I have constructed from the undecorated modules — in both cases some modules that play the same role are placed next to one another. These patterns look very different from the others we have been discussing. In the other patterns the vertices are crossings — they can be seen as the intersection of two straight lines that pass through each other. In Figure 15(a) only three edges meet at each vertex, and in (b), although four edges meet at each vertex, the vertices are still not crossings because opposite edges are not aligned to produce a straight path through the vertex. Let us add a requirement that the undecorated modules must be assembled to form a pattern with crossings. The following arguments work through the consequences of this restriction.

1. The crossings are formed when the external corners (those on the convex hull) of modules come together. Re-entrant angles cannot form crossings.

2. The angles in the external corners of the background shapes are all larger than 90◦ so we cannot ﬁt four background shapes around a vertex to form a crossing. Therefore, the pattern must contain some foreground shapes.

3. The kite can only be assembled in two local conﬁgurations. The two short sides of the kite match only two other shapes — the star and the bone-shaped octagon — so the kites must be arranged in rings of ten or in opposing pairs (as on the decagon and bow-tie modules in Figure 1). Therefore, the obtuse angle in a kite is never part of a crossing — it is always just a corner in the pattern.

4. Except for the obtuse angle in the kite, the angles in the external corners of the foreground shapes are all 72◦.

5. Opposite angles at a crossing are equal. In this system it happens that none of the angles in the background shapes is 72◦ so, wherever foreground shapes appear in the pattern, they must be placed opposite each other at crossings. This is equivalent to arranging the motifs of Figure 1 and we will generate exactly the same set of patterns.

3.3 Modules based on background shapes The four patterns shown in Figure 16 are the same as those in Figure 3(a), Figure 2(d), (g) and (h). In the new ﬁgure they have been overlaid with red lines that decompose the

16 (a) (b)

Figure 15: Tessellations made from the undecorated modules by the author. patterns into modules based on the white shapes. The same set of modules is used for all the patterns. It consists of four modules decorated with the background shapes we found in Figure 5, and one decorated with a {10/2} star. To these we need to add three fillers — a thin rhombus, a small pentagon and a long hexagon. These filler modules are not decorated and are just used to plug the gaps between the others. We now have three different representations of the data: the pattern itself, a decomposition into modules based on the foreground motifs, and a decomposition into modules based on the background shapes. These three representations are equivalent and can be converted into one another. Which one is ‘best’ will depend on the situation. Economy and internal consistency of the solution may also influence the choice. The modular system based on background shapes does not have the simplicity of that in Figure 1: the polygons are not equilateral, the modules come in two kinds (decorated and undecorated) and there are more of them.

3.4 Modules with new motifs Something interesting happens when we apply the idea of modules based on the background shapes to the patterns we created in §2.5: recall that we broke the rule about how to arrange modules in a template and, as a result, we created anomalous patterns with foreign motifs and in which the modular structures have defects. Figure 17(a) shows the pattern from Figure 11; in this case the template has been replicated so that the anomalous shapes are displayed clearly in the centre of the pattern, not disguised by being truncated at the borders of the panel. The red overlay is the decomposition that results from using modules based on the white regions. We see that the anomalous white shape in the centre now sits inside an equilateral hexagonal module. In fact, the decomposition uses four modules with the same outlines as the pentagon, barrel, bobbin and decagon modules of Figure 1 — they are just decorated with diﬀerent motifs. From this viewpoint, the central element is just another module, consistent in design with the others, and there is no discontinuity in the modular structure. The Karatay pattern from Figure 12 is redrawn in Figure 17(b). It is also decomposed into modules based on the background regions. This time, the foreign motif in the centre of the pattern sits inside a rhombus module. Again, the discontinuity in the modular structure has disappeared, although we may regard the thin rhombus ﬁller as an impurity. In this

17 (a) (b)

Figure 16: Modular decomposition based on the background shapes.

18 (a) (b)

Figure 17: Background decomposition of anomalous patterns.

case the modular decomposition does not extend across the rest of the pattern in a natural manner. We can take the five new modules and add a bow-tie module, as shown in Figure 18. The motif on the bow-tie is constructed so that the incidence angles where the motifs meet the polygons are all equal. (This ensures continuity of line when the modules are assembled.) This is an example of Bonner’s approach to pattern design based on polygonal grids [1] in which the same grids are re-used with different decorations. He calls the systems in our Figures 1 and 18 the ‘middle’ and ‘obtuse’ 5-fold systems, respectively. The obtuse modular system has the same set of boundary polygons as Figure 1. There- fore, it is capable of producing the same rich variety of patterns as the middle modular system because the modules can be assembled to form the same tessellations. If the obtuse system had been used traditionally to create Islamic patterns, we would expect to find a large variety of examples, and possibly some free-form compositions such as an application to filling compartments in a 2-level pattern. However, the examples we have are sporadic, and evidence of such playful experimentation with this system is missing from the historical record.

3.5 Other forms of evidence At this point in an investigation we have usually exhausted our primary sources, we have the results of our analysis, and we are left to apply judgement and interpretation to reach conclusions. In this case, we are very fortunate because we also have documentary evidence

Figure 18: A modular design system with the same polygons and diﬀerent motifs.

19 to consider. The Topkapı Scroll is an important source for the study of Islamic geometric ornament [3, 4, 7]. It contains a series of geometric figures drawn on individual sheets, which are glued end to end to form a continuous roll about 33 cm high and almost 30 m long. It is in mint condition with no signs of use, and is believed to be about 500 years old. The scroll is not a how-to manual as there is no text, but it is more than a pattern book as the designs are annotated with additional lines. Three of the panels from the scroll have been redrawn in Figure 19. They show rectangular templates containing the design drawn in black solid lines and some supporting lines drawn in red dotted lines (the figure imitates the marking on the original). These diagrams are very similar to the one in Figure 10(b). The dotted lines outline rhombus, bobbin, bow-tie and decagon modules. Panels 28 and 52 of the scroll contain more complex templates that are annotated in the same way; panel 28 contains pentagon modules. Notice that the barrel module in panel 53 carries a more intricate motif than the one we used in Figure 1, and that it is divided into two trapezia. This motif also appears at the top and bottom edges of the panel in Figure 7(a). There is a variant of the modular system we have been using that has two additional modules — this trapezium and a kite. Patterns that can be produced with this augmented system are found almost exclusively in Turkey [2]. The scroll shows that tessellations (the red dotted lines) were certainly associated with patterns. However, we do not know whether the tessellation comes first and the pattern is derived from it, or whether the tessellation is added after the pattern has been constructed in order to highlight particular properties or relationships. The scroll also contains indentations — lines that are scored in the paper with a stylus but not inked. In the bottom-left corner of panel 49 these lines include outlines of rhombus, pentagon, bobbin, bow-tie and decagon modules. However, the scored lines usually reveal the construction lines for laying out a pattern and radial grids for constructing stars. We may be seeing evidence of two distinct processes here: the red dotted lines record the modular tessellation used during the creation of the pattern, and the invisible lines are the construction used during its reproduction to fit in a given space. While modular design is a useful tool to stimulate creativity, it does not help transfer the composition to its place of use.

4 Conclusions

We have studied one example in detail to illustrate how the principles of reverse engineering can be applied to the analysis of geometric ornament. We took modular design (a mechanism for creating new patterns) and reflection of rectangular templates (a standard means of replication) and explored how they interact when used together to create a new composition. This demonstrated that the proposed method can reproduce a family of traditional patterns, and can also explain the anomalous shapes found in a few patterns. The model can explain the observed behaviour and we can find examples of the predicted defects. We also explored some alternative techniques for generating the patterns: composition with motifs rather than modules (§3.1), a modular system based on the shapes visible in the pattern (§3.2), and a decorated modular system based on the background shapes (§3.3). Table 1 summarises the pros (X) and cons (×) of the different methods. Relying on a hidden

20 (a) Panel 50 (b) Panel 53 (c) Panel 62

Figure 19: Templates from the Topkapı Scroll.

Method Properties foreground X modules have simple shapes that are easy to assemble modules (Fig. 1) X can explain some anomalous patterns X the Topkapı Scroll shows modular tessellations − modular structure is invisible in the final product non-modular × only work for simple patterns methods (§3.1) × cannot explain the repeated use of the same shapes in many patterns undecorated X no need to postulate an invisible substrate modules (§3.2) X no fusing of module corners to produce background shapes × uses many modules × modules have complex shapes that are difficult to assemble × does not enforce the separation of foreground and background shapes unless we add assembly rules background X sometimes removes anomalies from the tessellations modules (§3.3) − modular structure is invisible in the final product × modules are convex but not equilateral × two kinds of module — 5 decorated plus 3 fillers

Table 1: Summary of analysis.

21 structure may be regarded as convoluted or valuable (the artist’s method is concealed); this property is marked −. Part of the comparative analysis concerns the relative complexity of the various systems. The elegance and internal consistency of the system in Figure 1 is one of the features in favour; the simplicity of its polygons is unlikely to arise by chance as an emergent property of a different method. We have remarked several times that a proposed technique can generate patterns with particular properties, but that no examples of such patterns are known in the historical record: predicted behaviour is not observed. There are legitimate reasons why evidence may not exist. Material sources can be destroyed through natural processes such as weathering or earthquakes, or more actively by invaders; patterns are replaced in response to changes in fashion. It may also be that the patterns generated by the technique were not seen as very effective or attractive, so fewer examples were made. Even so, such absences are worrying and weaken an argument. I believe that the arguments I have put forward demonstrate that, before the time of the Topkapı Scroll, modular design was being used to create some Islamic geometric patterns, and that the modular system in Figure 1 is one example of a traditional Islamic design system. The analysis also provides a plausible line of development. The bobbin motif, a key element of our system, first appears on twelfth-century Seljuk architecture. As we saw in §3.1, this motif can be generated without using the modular system. The family of patterns that can be created with Figure 1 also contains other simple, common, early patterns — examples that can be generated in non-modular ways. Such patterns provided the raw materials in which craftsmen could identify shared elements — elements that were then extracted, re-used and combined in new ways. The familiarity of the motifs meant that the new patterns would be seen as a continuation and extension of an existing tradition. Initially, the elements may have been used ‘raw’, like the undecorated modules of §3.2. However, there are some techniques [5] for constructing patterns that naturally produce a polygonal network as an intermediate step in the process; in this context, the tessellation is a side-effect of the method and it would be just as easy to abstract the polygons as to abstract the motifs. The crucial step, however it was made, is the discovery that arranging a small standardised set of decorated polygons to form a tessellation is a good method for creating new patterns. Once this basic process had been identified and adopted, simple experiments would lead to an effective set of modules, as we saw in §2.3. I hope I have persuaded you that applying scientific methodology is sometimes appro- priate and that it can make a valuable contribution that enriches our understanding of the history of ornament.

References

[1] J. Bonner, ‘Three traditions of self-similarity in fourteenth and ﬁfteenth century Islamic geometric ornament’, Proc. ISAMA/Bridges: Mathematical Connections in Art, Music and Science, (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1–12.

[2] P. R. Cromwell, ‘Hybrid 1-point and 2-point constructions for some Islamic geometric designs’, J. Math. and the Arts 4 (2010) 21–28.

22 [3] P. R. Cromwell, ‘Islamic geometric designs from the Topkapı Scroll I: unusual arrangements of stars’, J. Math. and the Arts 4 (2010) 73–85.

[4] P. R. Cromwell, ‘Islamic geometric designs from the Topkapı Scroll II: a modular design system’, J. Math. and the Arts 4 (2010) 119–136.

[5] P. R. Cromwell, ‘On irregular stars in Islamic geometric patterns’, preprint 2013. http://www.liv.ac.uk/~spmr02/islamic/.

[6] P. R. Cromwell, ‘Modularity and hierarchy in Persian geometric ornament’, preprint 2013. http://www.liv.ac.uk/~spmr02/islamic/.

[7] G. Necipo˘glu, The Topkapı Scroll: Geometry and Ornament in Islamic Architecture, Getty Center Publication, Santa Monica, 1995.

[8] J. F. Rigby and B. Wichmann, ‘Some patterns using speciﬁc tiles’, Visual Mathematics, 2006. http://www.mi.sanu.ac.rs/vismath/wichmann/joint3.html