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Reverse engineering of Islamic geometric patterns:a scientific approach to art history

Peter R. Cromwell

[email protected]

Pure Mathematics Division, Mathematical Sciences Building,University of Liverpool, Peach Street, Liverpool L69 7ZL, England.

1 Introduction

One of the problems in art history is to chart the evolution of form and style. Developmentsmay be interpreted as responses to the introduction of new ideas and technologies, theinfluence 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 describingthe traditional methods that were used to create new designs, or the principles for applyingdesigns as ornament in architectural or other settings. We do, however, have many examplesof the finished products and, through careful examination and analysis, it is possible torecover some of the lost techniques for composition.

Reverse engineering is the process of applying the scientific method not to a naturalphenomenon but to a man-made system or device. It is performed when the system itself isavailable 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 arange of descriptions and representations of the system. Ultimately, the aim is to understandand document the system to a level where we are able to replicate its behaviour. Thisincludes how it works and also its limitations: what it does, what it cannot do, and how itperforms when used in unexpected ways.

The scientific method is a key element of this process. It is based on a cycle of revisionand 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 thatany predicted behaviour is observed.

We shall illustrate how this approach can be applied to the study of Islamic geometricpatterns. First we shall develop our main hypothesis and its consequences, and explore howit interacts with other techniques. We shall then test it against alternative explanations.

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.

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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 abouthow they were conceived. Competition between suppliers means that methods are closelyguarded 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 havesurvived but, even though they are a valuable primary source, they may be not known ornot 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 beused to group patterns into families. This may be as simple as sorting patterns by the typeof 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 thatmodular design was one of the methods used traditionally to create new patterns and, inparticular, 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 versatileand easy to use. A modular design system is a small set of simple elements (modules) thatcan be assembled in a large variety of ways. The modules can be grouped into a unit thatis 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 outlinedby 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-pointedstar motif composed of ten small kites arranged in a ring. These kites are congruent to thetwo kites forming the motif on the bow-tie module.

I have named the modules for ease of reference. Some have the standard geometricalname for the boundary polygon: rhombus, pentagon, decagon. The pentagon and decagonare 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 bobbinmodule 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 thatit may be possible to generate the pattern using this modular system.

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(a) (b)

(c) (d)

Figure 2: Traditional patterns that can be produced with the modular system of Figure 1.

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(e) (f)

(g) (h)

Figure 2: (continued).

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(a) (b)

Figure 3: Modular decompositions of two common star patterns.

different background shapes. The three combinations in the top row all produce a regularpentagon; the two combinations in the middle row both produce an irregular convex hexagonshaped like a tooth. The partition into two 2s and two 3s is the only case where the ordermatters and it generates two different shapes: one is the tooth and the other is the hexagonon the bottom row. The final shape is a bone-shaped octagon that arises from packingmodules round a bow-tie. The four background shapes can all be found in the examplepatterns shown in Figure 2; the hexagon derived from partition 2.3.2.3 is not common, butit 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 interest-ing to look at. There are enough different shapes that the eye does not get bored, yet not somany that the result seems cluttered. The detail is visible — the shapes are not too smalland the angles are not too sharp. None of the shapes is so large that it overwhelms theothers 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 doesnot spring into existence fully formed. It is likely that it evolved through a process of trialand error.

Figure 6 illustrates two mechanisms that can produce candidates for modules, in thiscase by experimenting with decagon modules. Regular decagons will not tile the plane ontheir own — they either leave gaps or they overlap. In the first case, the pattern lines onthe 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) thisprocess produces the bow-tie module. The intersection of the two decagons in Figure 6(b)

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(a) (b) (c)

(d)

Figure 4: Photographs reproduced courtesy of Mirek Majewski.

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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 bobbinmotif. 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 modularstructure. The tessellation uses the pentagon, barrel and decagon modules — they are as-sembled in a way that leaves a rhombus-shaped void in the centre of the panel that cannotbe filled with the modules of Figure 1. The pattern is extended into the void by continuingexisting lines until they meet each other. The resulting filler shares some attributes withthe 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. Thepanel seems to be an experiment whose outcome was not regarded as successful and therhombus 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 Islamicornament spurious examples like Figure 8 are important because they can also provideinsight into the creative process.

(a) (b)

Figure 6: Arranging decagon modules with gaps and overlaps.

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(a) (b)

(c) (d)

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

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(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 requireleft-hand and right-hand templates — in practice, it is achieved by turning the templateover. Figure 9(a) shows a rectangular template outlined in blue and decorated with anasymmetric motif. In (b) the basic template has been reflected vertically and horizontally,and these images have been reflected again to produce the standard quartering arrangementoften used in design; part (c) shows a more extended panel produced by repeated reflectionof 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 becausethe angles in the corners of the modules are not compatible with the right angles of therectangle — we cannot pack modules into corners of the template. However, we do not needto fill the template exactly — we only need to cover it. Any modules that extend beyondthe 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

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(a) (b)

(c) (d)

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 thatlie 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. Thispattern (rotated by 90◦) appears on the stone inlay panel shown in Figure 7(b) taken fromthe 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 wholeof the design in Figure 10(c). Why did this happen? All the modules in Figure 1 havetwo perpendicular lines of mirror symmetry. This means that they can be divided intohalves or quarters and reconstructed by applying reflections. This is very convenient forfilling 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 inFigure 10(b). In the top-right corner two half-modules fill the space. When the boundaryof 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 thealignment 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 thetemplate in Figure 11(a) has 2-fold rotational symmetry so we shall only describe the righthalf of the template. A decagon is placed in the top-right corner and aligned so that the

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sides of the template coincide with two of its mirror lines. The bow-tie module below it alsohas a mirror line lying over the side of the template. The other bow-tie modules are setdiagonally; 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 replicationprocess. 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 neverbe 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 anomalousshapes in the foreground and background of the resulting pattern. For example, the shadedarrowhead 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 traditionalpatterns stars are positioned so that a line segment connecting the centres of neighbouringstars 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 thatthe two are not identical. The corners have a cuspidal form — this is most apparent inthe central star, and also the narrow waists of the bone-shaped octagons. The standardpatterns that can be made with this modular system are rare in Egypt, and we only findsimple examples. Although our method can explain the structure of this design, it mayhave 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 templatefor this design is a large rectangle than contains more than 20 modules. Except for twomodules in one corner of the template, all the modules are either contained inside the tem-plate or meet the boundary of the template in an edge or a mirror line. The two misplacedmodules lead to an anomaly that is in the centre of the photograph. The details are shownin Figure 12. The bottom-right corner of the template is shown in (a). Although the bob-bin module on the bottom has a mirror line that coincides with the horizontal side of thetemplate, the module extends beyond the right side of the template; the vertical side of thetemplate meets the module in a line that is not in a mirror line of the module. The resultsof 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 aboveand below it are shapes that are foreign to the modular system used to generate the rest of the pattern.

3 Testing the hypothesisWe have demonstrated a method for generating geometrical patterns. However, what weare really interested in is the art historical question of whether this is a traditional methodfor 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?

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(a) (b)

(c) (d)

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

(a) (b) (c)

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

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At this point we should remark that care is required when reading the literature onIslamic patterns: some published papers fail even the first test. In these cases, the proposedconstruction 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 bedue to the stylistic treatment of the design by the artist, or because some of the salientfeatures have not been recognised in the analysis or are not recreated by the method. Insome cases the authors do not comment on the discrepancies. When they do, the errorsmay be explained away as poor quality workmanship on the part of the original builders orlater restorers of the pattern — that is, they are seen as mistakes in the data, not in theanalysis.

Let us return to our hypothesis. We have seen that the modular system of Figure 1can reproduce a large family of traditional Islamic patterns. Although we have not shownexamples 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, itgenerates faults in the modular structure along the edges and corners of the templates. Inthis way we can predict and explain both the existence and location of anomalies in somepatterns.

As an explanatory tool, our proposed method works well and captures many propertiesof 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 inFigure 13 uses some standard motifs from the Islamic vocabulary and a technique calledsymmetry 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 rectangleoutlined in (c) passes through the top and bottom points of the star and the corners of someof the pentagons. In (d) the design is cropped to fit the rectangle to form a template; the10-fold symmetry of the rosette in (b) is reduced to 2-fold symmetry. Replicating thetemplate produces the pattern shown in (e); the bobbin motif in the centre of the patternis 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 generatedby the modular system shown in Figure 1. In this case, the modular design hypothesisis not needed to explain its construction, and indeed, many of the simpler patterns in thefamily 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 thatsome kind of modules were in use, even if they were not those shown in Figure 1 (somealternatives are discussed b elow). Furthermore, some of the more complex designs have alow density of stars; in these cases the modular approach provides a simple mechanism toconstruct a visually effective interconnecting matrix, something that is difficult to achievewith other means.

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(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 colourslike 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 — theforeground 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 differentelements of the composition. However, this division is not made apparent in any of theartworks shown in the photographs in Figures 4 and 7. In these examples the regions arenot separated into background and foreground, but are all treated equally to provide auniform ground for the linear designs.

We can use the foreground and background regions themselves as modules. The tendifferent shapes are shown in Figure 14. With this modular system, the shapes you see

Figure 14: A design system with undecorated modules.

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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 imaginean 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 modulesand they are not so easy to assemble (there are three different edge lengths). We also losethe chessboard separation of the two roles that is automatically enforced by the decoratedmodules: there is nothing a priori to prevent modules that play the same role from beingplaced next to each other.

Rigby and Wichmann [8] explored patterns that can be generated using four of theseshapes and created over 60 designs. The tiles they chose and the assembly rules they usedmean that all their examples can also be generated with the bow-tie and bobbin modulesof Figure 1.

Figure 15 shows two patterns I have constructed from the undecorated modules — inboth 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 patternsthe vertices are crossings — they can be seen as the intersection of two straight lines thatpass 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 oppositeedges 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 apattern with crossings. The following arguments work through the consequences of thisrestriction.

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◦

sowe 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 — sothe kites must be arranged in rings of ten or in opposing pairs (as on the decagon andbow-tie modules in Figure 1). Therefore, the obtuse angle in a kite is never part of acrossing — 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 theforeground shapes are all 72◦.

5. Opposite angles at a crossing are equal. In this system it happens that none of theangles in the background shapes is 72◦ so, wherever foreground shapes appear in thepattern, they must be placed opposite each other at crossings. This is equivalent toarranging 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

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(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 inFigure 5, and one decorated with a {10/2} star. To these we need to add three fillers — athin rhombus, a small pentagon and a long hexagon. These filler modules are not decoratedand are just used to plug the gaps between the others.

We now have three different representations of the data: the pattern itself, a decom-position into modules based on the foreground motifs, and a decomposition into modulesbased on the background shapes. These three representations are equivalent and can beconverted into one another. Which one is ‘best’ will depend on the situation. Economyand internal consistency of the solution may also influence the choice. The modular systembased on background shapes does not have the simplicity of that in Figure 1: the polygonsare 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 backgroundshapes to the patterns we created in §2.5: recall that we broke the rule about how to arrangemodules in a template and, as a result, we created anomalous patterns with foreign motifsand in which the modular structures have defects.

Figure 17(a) shows the pattern from Figure 11; in this case the template has beenreplicated 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 thedecomposition 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. Infact, 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 withthe 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 decomposedinto 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 structurehas disappeared, although we may regard the thin rhombus filler as an impurity. In this

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(a) (b)

(c) (d)

Figure 16: Modular decomposition based on the background shapes.

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(a) (b)

Figure 17: Background decomposition of anomalous patterns.

case the modular decomposition does not extend across the rest of the pattern in a naturalmanner.

We can take the five new modules and add a bow-tie module, as shown in Figure 18. Themotif on the bow-tie is constructed so that the incidence angles where the motifs meet thepolygons 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] inwhich the same grids are re-used with different decorations. He calls the systems in ourFigures 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 modularsystem because the modules can be assembled to form the same tessellations. If the obtusesystem had been used traditionally to create Islamic patterns, we would expect to find alarge variety of examples, and possibly some free-form compositions such as an applicationto 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 historicalrecord.

3.5 Other forms of evidence

At this point in an investigation we have usually exhausted our primary sources, we havethe 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.

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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 inmint 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 bookas the designs are annotated with additional lines. Three of the panels from the scroll havebeen redrawn in Figure 19. They show rectangular templates containing the design drawnin black solid lines and some supporting lines drawn in red dotted lines (the figure imitatesthe 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 52of the scroll contain more complex templates that are annotated in the same way; panel 28contains pentagon modules. Notice that the barrel module in panel 53 carries a moreintricate 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 isa 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 arefound almost exclusively in Turkey [2].

The scroll shows that tessellations (the red dotted lines) were certainly associated withpatterns. However, we do not know whether the tessellation comes first and the pattern isderived from it, or whether the tessellation is added after the pattern has been constructedin order to highlight particular properties or relationships.

The scroll also contains indentations — lines that are scored in the paper with a stylusbut 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 revealthe construction lines for laying out a pattern and radial grids for constructing stars. Wemay be seeing evidence of two distinct processes here: the red dotted lines record themodular tessellation used during the creation of the pattern, and the invisible lines are theconstruction used during its reproduction to fit in a given space. While modular design is auseful 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 engineeringcan 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 interactwhen used together to create a new composition. This demonstrated that the proposedmethod can reproduce a family of traditional patterns, and can also explain the anomalousshapes found in a few patterns. The model can explain the observed behaviour and we canfind examples of the predicted defects.

We also explored some alternative techniques for generating the patterns: compositionwith motifs rather than modules (§3.1), a modular system based on the shapes visible inthe pattern (§3.2), and a decorated modular system based on the background shapes (§3.3).Table 1 summarises the pros () and cons (×) of the different methods. Relying on a hidden

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structure may be regarded as convoluted or valuable (the artist’s method is concealed); thisproperty 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 anemergent property of a different method.

We have remarked several times that a proposed technique can generate patterns withparticular properties, but that no examples of such patterns are known in the historicalrecord: predicted behaviour is not observed. There are legitimate reasons why evidence maynot exist. Material sources can be destroyed through natural processes such as weatheringor earthquakes, or more actively by invaders; patterns are replaced in response to changes infashion. It may also be that the patterns generated by the technique were not seen as veryeffective or attractive, so fewer examples were made. Even so, such absences are worryingand 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 designsystem.

The analysis also provides a plausible line of development. The bobbin motif, a keyelement 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 patternsthat 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 rawmaterials in which craftsmen could identify shared elements — elements that were thenextracted, re-used and combined in new ways. The familiarity of the motifs meant thatthe 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 apolygonal network as an intermediate step in the process; in this context, the tessellationis a side-effect of the method and it would be just as easy to abstract the polygons as toabstract the motifs. The crucial step, however it was made, is the discovery that arranginga small standardised set of decorated polygons to form a tessellation is a good method forcreating new patterns. Once this basic process had been identified and adopted, simpleexperiments 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 thehistory of ornament.

References

[1] J. Bonner, ‘Three traditions of self-similarity in fourteenth and fifteenth century Islamicgeometric 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 geometricdesigns’, J. Math. and the Arts 4 (2010) 21–28.

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[3] P. R. Cromwell, ‘Islamic geometric designs from the Topkapı Scroll I: unusual arrange-ments 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 designsystem’, 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’, preprint2013. http://www.liv.ac.uk/~spmr02/islamic/.

[7] G. Necipoğlu, 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

cromwell rev eng

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