On irregular stars in Islamic geometric patterns

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On irregular stars in Islamic geometric patterns

Peter R. Cromwell

[email protected]

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

We present a new method for constructing a family of Islamic star patterns containingmore than one kind of star. Besides producing the standard star combinations, it alsohandles the geometry-defying patterns created in the Seljuk period whose star motifs areincompatible with each other and with the standard grids used to lay out a design. Themethod is based on sound design principles rather than mathematical properties. Eachpattern is constructed in situ in a holistic way rather than as an assembly of preformedmotifs. In some cases this leads to irregular stars but the results are consistent with

surviving examples, traditional workshop geometry, and evidence from medieval scrolls.

1 Introduction

Islamic geometric patterns present an archaeological puzzle: while we have many exam-ples of the finished product, we do not have any contemporary documents describing thetraditional methods that were used to construct them. Recovering the lost techniques isalso complicated by the fact that there is no universal method. Subtle differences in thedetail of otherwise very similar patterns need not be due to variation in style or quality of draftsmanship, but can be indicators of entirely different methods of construction.

The geometric detail is important. Methods reverse engineered from traditional patterns

should generate patterns that match the originals very closely, and also explain the con-struction lines found in contemporary pattern books such as the Topkapı Scroll. Althoughthis seems obvious, some proposed constructions do not pass this test — while the topologyis reproduced correctly, the geometric features do not agree.

In this paper we present a new method for constructing patterns that contain starswhose geometry makes them difficult to use. The method is applied in two ways. Thefirst application produces star patterns in which the primary star motifs are separatedby smaller satellite stars nestled between them. In particular, we study examples fromthe Seljuk period, mostly from Anatolia. These patterns are distinctive in two ways: (1)the secondary stars are visually very star-like, and (2) they include patterns with unusualcombinations of incompatible primary stars. In later periods, attention is focussed on the

‘rosette’ properties of such patterns — the interstitial spaces lose their star-like qualityand the star combinations are limited to those that fit naturally on standard lattices. Thesecond application of the method is more general and we study examples from a wider rangeof periods and places.

In the analysis of geometric ornament, a pattern is overlaid with additional lines —often networks of circles or polygons. While this overlay can highlight relationships or un-derlying structure in the pattern, it need not correspond to a means of construction — itmerely replaces the original problem with two more: how to construct the overlaid structure,and how to derive the pattern from it. When full constructions are given, they are oftenperformed with straight edge and compass — the tools of Euclidean geometry. These con-structions can be quite lengthy and intricate, even for patterns that appear visually simple.

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Furthermore, while it is clear that lines and circles are key elements of medieval methods,not all Islamic patterns are constructible in this sense. Craftsmen also had other devices(enabling them to divide angles into more than two parts, for example) and were pragmaticabout using approximate constructions when necessary. Some proposed constructions, likemuch Euclidean mathematics, are synthetic, one-off solutions, which provide neither generalprinciples that can be reused in a new context nor any insight into the inspiration behinda pattern.

In discussions of Islamic patterns there is a tendency to focus on geometrical propertiesand to over-mathematise the design process. This can lead us to make implicit assumptionsabout the symmetry of motifs and forms of arrangement. In this paper we replace the math-ematical property of regularity as the guiding principle with some fundamental principlesof good design based on alignment. Although we cannot limit ourselves to Euclidean tools,the technique does not rely on anything unfamiliar to medieval craftsmen. Some examplesrequire angle trisection (a non-Euclidean process) to layout the initial framework of a de-sign, but once this is done, the heart of the construction uses nothing more complicatedthan angle bisection.

Another non-standard feature of our method is that we focus on the inner corners of stars, not the tips of their spikes. More generally, we do not focus on line crossings, buton the corners or anchor points where straight line segments terminate — the pattern isformed by connecting these anchors and the crossings are a by-product of drawing theline segments. As a consequence the crossings may not end up exactly where we mightexpect based on our experiences with simpler patterns. For example, they may not lie oncircumcircles of stars or on the lines connecting star centres, but the discrepancy betweenactual and expected is often small and is unnoticeable to the eye.

2 Terminology

Let us begin with stars. The mathematician’s method for constructing a star is to take nequally spaced points on a circle and connect all pairs of points that are d steps apart bystraight lines. The resulting regular star is denoted by {n/d}. Figure 1(a) shows {10/4}.These star polygons are a generalisation of the ordinary regular polygons: when d = 1 thestar reduces to the regular convex polygon with n sides. For large d the method producesa complex cell structure in the star interior, most of which is discarded in decorative appli-cations. Typically, only the outer boundary (a 2n-sided non-convex polygon) or the outerlayer of cells (usually kite shaped) is kept — see Figure 1(b) and (c). Motifs based on these

regular stars can be found in traditional Islamic patterns. The common ones include {5/2},{6/2}, {8/2}, {8/3}, {10/3}, {10/4}, {12/3}, {12/4} and {12/5}.Another way to draw stars is to construct a radial grid with two concentric circles

and lines radiating from the centre to form spokes. The star is constructed by connectingpoints on the inner and outer circles alternately, using the intersections with the spokes tomark the endpoints, as in Figure 1(d). We call the spokes that meet the star at the outercircle outward spokes and the ones in between inward spokes. While there are very fewmathematical stars with a given number of points, with this method the shape of the starcan be varied continuously by changing the size of the inner circle. We can think of theratio of the radii of the inner and outer circles as a measure of the pressure inside the star.When the pressure is near 1 the star is quite inflated and has broad, flattish spikes, whereas

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(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 1: Various constructions of stars.

a low-pressure star has sharp spikes. We shall use this ratio frequently and will denote itby λ throughout. In Figure 1(f) the angle between the spokes is not constant but increasesfrom the top to the bottom of the star. When the spokes are evenly spaced we say thestar is equiangular ; when the interspoke angles are not equal the star is called irregular .The regular stars are a subset of the equiangular stars. Figure 1(e) is equiangular but notregular.

Figure 1(g) and (h) show a different generalisation of the mathematician’s star. Startingwith a polygon, we place a point at the midpoint of each edge and draw pairs of linesemanating from all the points into the interior of the polygon. The incidence angle betweenthe edge of the polygon and the growing lines is constant; the lines terminate when theyintersect each other. Here the process is performed on a regular polygon and an equiangular

star is produced. However, it can be done for arbitrary convex and some non-convexpolygons — this is the basis of the ‘polygons in contact’ (PIC) method for constructingpatterns from a tiling, first described by Hankin [12]. In many cases the motifs producedfrom irregular polygons are ‘star shaped’ in the technical sense (there is a viewpoint in theinterior from which one can see the whole motif) but they can be quite uneven and lop-sidedand not visually star-like at all.

We now move on to consider the pattern shown in Figure 2, a pattern that is widespreadacross the Islamic world and very common. The figure is divided into four sections. Theleft half shows both the linear and filled forms of the pattern. In the following figuresthat illustrate our constructions the finished pattern will be shown in linear form and the

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Figure 2: Some structural properties of a very common Islamic pattern.

construction lines will be overlaid on a grey filled form.In the third section of Figure 2 the pattern has been overlaid with a tiling of regular

pentagons, regular decagons and irregular convex hexagons. We shall use this tiling toidentify various elements of the pattern. Each tile is decorated with a grey motif: the motif on the pentagons is a regular {5/2} star; the motif on the decagons is a regular {10/4} starin cellular form with the outer kite-shaped cells filled in; the motif on the hexagons is twoarrowheads pointing at each other. This viewpoint focusses on the stars in the pattern —the primary 10-pointed stars have constellations of smaller satellite or secondary stars. Itis this interpretation we shall use later.

In the fourth section of the figure the pattern has been overlaid with a tiling of regulardecagons and non-convex hexagons in the shape of bow-ties. A

{10/4

} star motif sits in

the centre of each decagon and the ten white irregular convex hexagons that surround itare petals ; together with the central star they form a rose . Notice that, in this case, thewhite hexagons on the bow-ties are congruent to the petals. In these roses the outer edgesof adjacent petals are subsets of the lines connecting the midpoints of the edges of thecircumscribing decagon, and two edges of each petal are parallel. Roses can be constructedas preformed motifs and arranged to form a pattern in the same way as stars. When wewant to project this interpretation onto a pattern, we shall call it a rose pattern rather thata star pattern.

Rather like a Platonic solid, the pattern in Figure 2 is an archetypal form that has manynice mathematical properties:

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(a) (b)

Figure 3: Forms of topological equivalence: these two patterns are homeomeric but not

diffeomeric.

• the primary stars are regular• the secondary or satellite stars are regular• sides in neighbouring primary stars are collinear• sides in some pairs of secondary stars are collinear, making the arrow heads meet at

their tips rather than overlap or fall short

• the outer edges of adjacent petals are collinear

• the petals have parallel sides (hence so do the primary stars)

• some interstitial shapes are congruent to the petals.We can make various families of patterns that share some of these characteristics. One of the key choices is whether to focus on the properties of stars or roses — the black or whiteshapes in the second section of Figure 2. Lee discusses how to make rose patterns in whichthe interstitial spaces are congruent to the petals [14, pp. 111–118]. Petals with parallelsides can be extended to arbitrary length; examples of roses in the zellij tradition found inMorocco have long-rayed petals and can accommodate a large number of spikes. Differentmethods used in different times and places preserve or distort different sets of features. Weshall define the properties of interest to us in the next section.

Finally in this section we comment on the equivalence of patterns. One sometimes sees

the term ‘topologically equivalent’ applied to patterns. This is a very weak statement, es-pecially when applied to patterns with a high degree of symmetry. Let h be an invertiblemapping of the plane that carries one pattern onto another. If h and its inverse are con-tinuous then the transformation is a homeomorphism and the two patterns are said to behomeomorphic (this is what is usually meant by topological equivalence). If, in addition, hpreserves the symmetry elements (mapping rotation centres to rotation centres of the sameorder) then Grünbaum and Shephard call the transformation a homeomerism . This is stillnot adequate to capture our intuitive notion of equivalence of Islamic patterns. Figure 3shows two patterns that are homeomeric, but which we do not regard as variants of eachother — some polygons related by the mapping have different numbers of corners. If we

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add the requirement that h and its inverse be differentiable then the transformation pre-serves corners in the pattern. Grünbaum and Shephard call patterns related in this waydiffeomeric [11, p. 347]. In general, proposed constructions of traditional Islamic patternsdo reproduce something diffeomeric to the original, even if the geometry is incorrect.

3 Design Principles

In this section we introduce our design principles. Even though we refer to them as rules,we shall interpret them in the spirit of guidelines. They cannot be treated as axioms for,although they seem simple enough, in some patterns it is not possible to satisfy them allsimultaneously.

Rule 1 (stars). Each star in a pattern is constructed on a radial grid (as described inthe previous section): the geometry of the star is fixed by its inner and outer circles, and

inward and outward spokes. This applies to primary and secondary stars.

Rule 2 (crossings). At each 4-valent vertex in a pattern, opposite angles are equal. Thisproperty ensures that each crossing is the intersection of two straight line segments.

Rule 3 (alignment).(a) When the outer circles of two stars are tangent, the line connecting the star centresshould coincide with an outward spoke in each star.(b) When the petal tips of two roses meet, the line connecting the rose centres shouldcoincide with an inward spoke in each associated star.

Rule 4 (irregularity). Any star, primary or secondary, may be irregular.

The radial grid definition of a star (Rule 1) restricts the kind of irregularity that starscan exhibit. While we cannot produce stars in which some spikes are much longer or shorterthan others, we do allow variation in the angles between the spikes. Uneven angles are lessnoticeable than uneven lengths. In the visual interpretation of a 3-dimensional scene, angleis one of the cues used for depth perception. Unlike distance and alignment, angle is notone of the primary 2-dimensional geometric concepts abstracted by the brain and we arepoor at comparing angles for equality.

There is experimental evidence to support this thesis. In the 1980s Cleeveland andMcGill investigated the effectiveness of various graphical methods used for presenting sta-tistical data [7]. They performed experiments using elementary visual tasks to compare theaccuracy of judgements drawn from representations based on properties such as position,

length, direction and angle. One of their conclusions is that people are poor at comparingthe angles in a pie chart; graphics based on length or position lead to more accurate judge-ments. Comparing the angles between the spikes of a star is a similar task to comparingthe angles in a pie chart; we cannot compare the angles with any confidence and smallvariations in the inter-spike angles are not perceived.

Notice that the rules do not discriminate between primary and secondary stars. Weshall see that forcing the secondary stars to conform to our definition of star (Rule 1) leadsto conflicts between Rule 2 and Rule 3. In all but one of the patterns constructed laterthe conflict is resolved by favouring Rule 2; however, in these cases, the deviation from thealignment required by Rule 3 is less than 1◦.

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(a) (b)

Figure 4: The gestalt principle of closure applied to a star.

A useful consequence of Rule 2 is that we do not need to determine the locations of crossings — we only need to locate the corner points in a pattern and connect them bystraight line segments; the crossings are produced indirectly. This means we have no control

over the angles between the lines at crossings. Note that a star meets its outer circle atcrossings and its inner circle at corners. The conventional view is that the outer circle of astar is one of its key parameters; here the focus of attention has shifted to the inner circle,a parameter that is often omitted or ignored.

The secondary stars in rose patterns are almost always irregular. In §7 we shall con-struct patterns using combinations of apparently incompatible stars. In these cases, inorder to comply with Rule 3 (proper alignment of spikes) we also need to allow primarystars to be irregular (Rule 4). Although it would be mathematically elegant to restrictourselves to using equiangular stars, it is not visually necessary. Indeed, as misalignmentis more noticeable than uneven spacing, we could argue that using symmetric stars inap-propriately degrades the aesthetic qualities of a pattern. We shall see that it is possible to

create balanced, harmonious patterns with irregular stars and maintain the illusion of localsymmetry.Our rules are also consistent with the gestalt theory of perception, often used as the

basis for design principles. Rule 2 ensures continuity of line, guiding the eye smoothlywithout deviation as it moves from one motif to another. Aligning the spikes of nearbystars (Rule 3) creates a subliminal connection between separated motifs. Misalignment cangive the impression of carelessness in the placement of motifs.

Closure is the tendency of the mind to supply information missing in the visual stimulusto bridge gaps and construct continuous figures from disjoint line segments. Figure 4 showsthat constructing the secondary stars on a radial grid provides the right conditions forthis to happen: in the stimulus on the left the gaps are small and the line segments are

sufficiently well aligned that the mind can construct the figure on the right. By assemblingsegments and bridging gaps in this way, the mind can form quite long curves that sweepthrough a pattern. Look out for this effect in the patterns we construct later.

The gestalt theory proposes that the mind perceives something other than a collectionof parts. Continuation, alignment and closure are some of the organising principles em-ployed by the visual system as it constructs whole figures from given parts and seeks outrelationships between them.

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(a) (b) (c)

Figure 6: Stars constructed from incircles in triangles.

the lines from the incentre to the corners of the polygon are the inward spokes, and theperpendiculars to the edges of the polygon are the outward spokes. Note that the outwardspokes do not necessarily meet the edges of the polygon at their midpoints. The inter-spokeangles either side of an inward spoke are equal, but those either side of an outward spokemay differ: the ‘dents’ of the star have mirror symmetry but the spikes may not.

In the following constructions we only use the 5-pointed stars. Assume that the radiusof the incircle is 1. Then the sides of the circumscribed pentagon are:

PQ = 2 t an(α)

QR = QD + DR = tan(α) + tan(β )

RS = 2 t an(β )

SC = SE + EC = tan(β ) + tan(2γ )

CP = CF + FP = tan(2γ ) + tan(α).

Suppose the star is equiangular so that α = β = 2γ . Summing the angles around M gives4α + 4β + 4γ = 360◦ so α = 36◦. Hence triangle ABC has angles 36◦–36◦–108◦. In all otherobtuse triangles, the star is irregular.

Figure 6(c) shows how the star is connected to other motifs in the completed pattern.It is derived from Figure 6(b) as follows. The edges of the star that pass through X havebeen extended outside the incircle until they meet the sides of triangle ABC . The twointersection points are equidistant from A, as indicated by the circular arc in the figure.The shaded area will form the spike of a primary star motif centred at A and the arc willbe part of its inner circle; the primary star’s outer circle is centred at A and passes through

X . An analogous construction produces the shaded area meeting B .Note that this process proceeds in a totally different order from many attempts to

reconstruct traditional Islamic patterns, which start with complete primary stars and thenconstruct a connecting matrix. Here, all that is required to initiate the construction are thecentres and orientations of two primary stars; the radii of the inner and outer circles (andhence all the other geometric properties) of the primary stars are outputs of the process.

More specifically, triangle ABC is chosen so that A and B are the centres of the primarystars S A and S B, respectively; sides AB and AC coincide with adjacent inward spokes of S A and sides BA and BC coincide with adjacent inward spokes of S B . The incircle of thetriangle is the outer circle of the secondary star and also determines the outer circles of theprimary stars (they are tangent to it). The radius of the inner circle of the secondary star

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is the only free parameter in the construction. Once it is chosen, the inner circles of theprimary stars are determined as shown in Figure 6(c). From this viewpoint, the secondarystars are an integral part of the construction, not secondary in nature. We shall see manyapplications of this method in the remainder of the paper.

5 First steps in pattern construction

Figure 7 shows two patterns that have been constructed from tilings of congruent trianglesby applying the method of the previous section to each triangle. The patterns are overlaidwith selected subsets of the construction lines from Figure 5 to highlight different structuralfeatures of the patterns.

The left third of each pattern shows the basic triangles ABC and their incircles. In(a) the triangle is isosceles with angles 30◦–30◦–120◦ and the construction produces an

equiangular 12-pointed star at each acute angle. The inner and outer circles of the secondarystar are in the ratio 1/2(3−√ 5), as in the regular {5/2} star. In (b) the triangles have angles30◦–60◦–90◦ and we get an equiangular 12-pointed star at the 30◦ corner and an equiangular6-pointed star at the 60◦ corner. Here the inner circle of the secondary star is chosen sothat the white region of the pattern bounded by four secondary stars is a regular octagon.

In the middle third of the figure, the radii MD , ME and MF (perpendicular to the sidesof triangle ABC ) are drawn to form an equilateral tiling. In these examples, the tilingsare Archimedean: (a) is of type 3.12.12 composed of triangles and dodecagons, and (b) isof type 4.6.12 having squares, hexagons and dodecagons. The circles are centred on thevertices of these tilings. In §8 we shall use this process in reverse and construct patternsbased on non-Archimedean tilings. The key requirement is that the tilings be equilateral

so that adding a circle centred on each vertex of the tiling with diameter equal to the edgelength produces a set of tangent circles that can be used as the outer circles of stars.

The final third of the figure shows a tiling formed by outlining the pentagons PQRSC .It is often possible to overlay such a polygonal network on a star pattern. The questionis whether the network has anything to do with the construction of the pattern or is a

just convenient means of visualising its structure after it has been created. Although theoverlay in the figure looks superficially like an application of the PIC method, many of thecrossings do not lie at the midpoints of the edges of the polygons. In this case the tiling isan intermediate by-product of the construction, not a starting point.

Although the patterns in Figure 7 seem simple enough, we have already broken therules. Figure 8(a) shows two copies of the triangle from Figure 6(c) reflected in line AB .

Our technique produces the crossings in a pattern by connecting anchor points with straightlines. In this case, the crossing on the mirror line is the intersection of two lines connectingpoints on the inner circle in the upper triangle to points on the inner circle in the lowertriangle. Our intuition developed from experience with simple patterns leads us to expectthat the crossing will lie on the line connecting the star centres. However, this is not thecase — we cannot satisfy Rules 2 and 3 simultaneously.

In Figure 8(b) we have enlarged the area between the stars and changed the anglesα = ∡DMQ and β = ∡DMR to exaggerate the problem and make it more visible. Thelabels Q, D, R and M are copied from Figure 5. The spokes MQ and MR intersect the innercircle at points V and W , respectively. The points M ′, V ′ and W ′ are the reflections of thepoints M , V and W in the line QR. The lines of the pattern are produced by connecting

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(a)

(b)

Figure 7: The three overlays highlight different structural properties in each pattern andsuggest three different methods of construction.

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(a) (b)

Q D R

M

V

W

M ′V ′

W ′

Figure 8: A conflict between Rules 2 and 3.

the anchor points by straight lines. In this case the lines VW ′ and V ′W form the crossingY between the two secondary stars centred at M and M ′. Because α = β this crossing doesnot coincide with D. The point Y lies just outside the outer circle so the spike boundedby lines VY and WY is slightly longer than it should be. The additional line in the upperhalf of Figure 8(b) is MY . In this example the angle ∡DMY is just over 3◦. This is quiteextreme — in real applications of the method the values of α and β are closer. When theratio of the inner and outer circles of the secondary stars is in the useful range, say 0.2–0.7,this discrepancy is always less than 1◦ so it is not noticeable.

Things can get even worse! In some of the patterns we shall construct later, two trianglesthat share an edge are not congruent. When the construction of

§4 is applied to both

triangles the feet of the two lines MD perpendicular to the shared side do not coincideand the tips of the two secondary stars do not meet. However, we resolve the problem byemploying the same philosophy as before — we connect anchor points on the inner circlesin the correct topology.

The original Seljuk examples of the patterns we shall study later are executed in linearform using interlaced ribbons. This helps to hide the discrepancy between Rule 2 and Rule 3as the difference in the two possible locations of the crossing is usually less than the widthof the ribbon; the line connecting the star centres passes through the area where the tworibbons overlap (the crossing), but it does so off-centre.

6 Patterns with equiangular primary starsFor our first example, we shall use the 45◦–45◦–90◦ triangular template to make a family of patterns containing 8-pointed and 12-pointed stars. The templates are shown in Figure 9and the resulting patterns are shown in Figure 10.

Consider Figure 9(a). The 12-pointed star will be centred at the top-right corner of thetemplate; the 8-pointed star will be centred at the bottom-left corner of the template; thesecorrespond to points A and B , respectively, in Figure 5. Point C lies where the extension of one of the spokes of the 12-pointed star meets the bottom edge of the template. The processdescribed in §4 determines all the other construction lines shown in the figure except forthe inner circles of the stars. The four parts of Figure 9 show different ways to choose the

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parameter.In Figure 9(a) we choose the 12-pointed star to be a regular {12/5} star. This determines

the locations of the two solid dots in the figure: the point on the outer circle of the secondarystar is fixed by §4 and sets the scale of the star, which in turn fixes the point on thehypotenuse of the template. The line passing through the two solid dots meets an inwardspoke of the secondary star at the open dot. This determines the radius of the inner circleof the secondary star, and so also the inner circle of the 8-pointed star. The extension of the edge of the secondary star (shown as a broken line in the figure) meets the right edge of the template. When the template is replicated, this leads to two disconnected arrowheadspointing at each other. To increase the connectivity of the design we adapt the pattern asshown.

In Figure 9(b) we want the extension of an edge of the secondary star to pass through thecorner of the template (this will make the arrowheads touch). This time our two fixed points(solid dots in the figure) are a point on the outer circle of the secondary star and the cornerof the template; the line passing through them meets an inward spoke of the secondary starat the open dot. This determines the radius of the inner circle of the secondary star, andso also the inner circles of the two primary stars.

In Figure 9(c) we choose the ratio of the inner and outer circles of the secondary star tobe 1/2(3 −

√ 5), as in the regular {5/2}. We also force the edge extension of the secondary

star to pass through the corner of the template. The open dot in the figure shows wherethe line crosses the outward spoke of the star — it is noticeably inside the outer circle of the star. This example is just to illustrate the method; in the next case we shall use it toproduce a more subtle design.

In Figure 9(d) we choose the 8-pointed star to be a regular {8/3} star. As in case (a)this determines inner circle of the secondary star. Using this geometry, the edge extensionof the secondary star passes very close to the corner of the template. As in case (c), weforce the line to pass through the corner, resulting in a small relocation of one of the outerpoints of the secondary star.

Notice that we have only one free parameter — we cannot have {8/3} and {12/5} starssimultaneously. Patterns in which this occurs are constructed using a different method andthe secondary shapes will not be stars as defined here.

Analogous experiments with the 30◦–60◦–90◦ triangular template yield the patterns of 9-pointed and 12-pointed stars shown in Figure 11. In (a) the geometry is determined bychoosing the 12-pointed star to be the regular star {12/5}; in (b) the 9-pointed stars haveparallel sides. In (b) the edge extension of the secondary star meets the boundary of thetemplate on the bottom edge, leading to overlapping arrowheads in the pattern.

All the geometry performed in Figure 9 can be done with straight edge and compass,making these patterns constructible in the Euclidean sense. In Figure 11 there is one stepthat cannot be accomplished with Euclidean tools: trisecting the 60◦ angle in the top-rightcorner of the template to produce the spokes of the 9-pointed star. Medieval craftsmenin the Islamic world and in Europe had good approximations for trisecting angles andconstructing approximate 9-sided regular polygons [15].

Bodner’s paper [3] includes five photographs of rose patterns containing both 9-pointedand 12-pointed stars. In all cases the star centres are arranged on the vertices of thestandard triangular grid, as in Figure 11. It is clear that her examples exhibit evidence fora variety of different construction methods. For example, in the pattern from the Alhambra

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(a) λ = 0.326 (b) λ = 0.442

(c) λ = 0.382 (d) λ = 0.460

Figure 9: Templates for a family of patterns containing equiangular 8-pointed and 12-pointed stars.

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(a) λ = 0.326 (b) λ = 0.442

(c) λ = 0.382 (d) λ = 0.460

Figure 10: Patterns generated from the templates in Figure 9.

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(a) λ = 0.326 (b) λ = 0.417

(c) λ = 0.326 (d) λ = 0.417

Figure 11: Two patterns containing equiangular 9-pointed and 12-pointed stars.

full rosettes have been preformed and arranged on the grid; the secondary stars have a longspike directed towards the 12-pointed stars. She focusses her discussion on two patterns:one from a ceiling design in the Mausoleum of Oljeitu and one from a template in theTashkent Scroll; these are diffeomeric to patterns (c) and (d) in Figure 11, respectively.

Like the templates in the Topkapı Scroll, the Tashkent Scroll contains uninked con-struction lines inscribed in the paper. Bodner observes that the stars are constructed usinginscribed and circumscribed circles. The scroll shows the inner circles of the secondary stars,but they do not appear in her proposed construction. She follows the conventional practiceof constructing the primary stars and inserting connecting lines in an ad hoc manner.

7 Patterns with irregular primary stars

The patterns analysed in this section are some of the most challenging Islamic patterns toexplain, and the methods presented in this paper were developed while trying to understandtheir geometry. In each case, primary stars whose symmetries are incompatible with eachother and with the standard underlying lattices are combined to produce a pattern thatappears to be composed of equiangular stars with aligned spikes. One pattern containsfour different kinds of star. We shall demonstrate how this deception can be achieved byallowing at least one of the primary stars to be irregular.

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(a) (b)

Figure 12: Locating star centres.

7.1 Turkey: 12-point, 10-point and other stars

Our first two examples can be seen as parallel explorations of the two standard triangulartemplates. Figure 12 illustrates the starting points of the constructions. In both caseswe place an equiangular 10-pointed star centred at the right angle of the template and anequiangular 12-pointed star at the other end of the baseline — centred at the 45◦ angleand the 30◦ angle in the respective templates. At this point, we do not know the size of the stars and are interested only in the angles between their spokes. This information isindicated by the ‘wheels’ in the figure — the wheel of an n-pointed star has 2n segments inthe ring, corresponding to the 2n spokes of the star. We extend a spoke from the 10-pointedstar until it meets the hypotenuse of the template, as shown. In each case, we shall place

another primary star centred at the point of intersection so that the three line segmentsthat meet there coincide with some of its spokes.In Figure 12(a) the angles in the corners of the triangle determined by the three star

centres are 45◦, 54◦ and 81◦. In an equiangular 9-pointed star the spokes are 20◦ apart. Thethird wheel in the figure shows how the spokes of an irregular 9-pointed star can be chosento satisfy the alignment criterion (Rule 3): in the area shown in grey the angle between thespokes is increased to 20.25◦, and it is reduced to 19.8◦ in the white area to compensate.

In Figure 12(b) the angles in the triangle determined by the star centres are 30◦, 36◦

and 114◦. In an equiangular 11-pointed star the spokes are 16.36◦ apart. The third wheel inthe figure corresponds to an irregular 11-pointed star — in the grey area the angle betweenthe spokes is increased to 16.5◦ and in the white area it is reduced to 16.28◦.

In each case, the use of an irregular star means that the spokes of the three stars satisfyRule 3. However, the irregular star is very close to an equiangular one — the inter-spokeangle differs from the equiangular case by less than 0.25◦. Note that although we havedistributed the distortion evenly so that the grey sectors in the wheel are equally dividedand the white sectors are equally divided, this is just to motivate the development of thedesign and need not be the case in the final construction. We shall now see how eachtemplate is developed into a finished design.

In Figure 13(a) the template ABC is divided into triangles as follows. The point Dis the point on BC such that angle CBD is 36◦ — we extend a spoke of the 10-pointedstar centred at B until it meets line AC . Similarly, the point E is determined by extendingspokes until they intersect: the 12-pointed star is equiangular so angle ∡EAB = 15◦ and the

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A B

C

D

E

F

G

(a) (b)

(c) Alay Han, Akseray–Kayseri road, Turkey, 1192

Figure 13: Pattern containing equiangular 10-pointed and 12-pointed stars and irregular

9-pointed stars.18


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10-pointed star is equiangular so angle ∡EBA = 18◦. Connect E to A, B and D . The lineED does not bisect angle ∡ADB but it is very close: ∡ADE = 40.88◦ and ∡EDB = 40.12◦.Reflect E in the line BD to get F on BC ; by symmetry angles ∡BDF and ∡BDE are equal.To obtain the remaining spokes of the 9-pointed star we trisect angle ∡ADF ; this gives Gon BC with angle ∡FDG = 39.25◦. (This is the only step in this construction that cannotbe performed with Euclidean tools.)

Having constructed the spokes of the primary stars, we apply the method of §4 totriangles ADE , BDE and BDF to obtain the outer circles of the secondary stars. We usetriangle BDE to obtain the outer circles of the 10-pointed and 9-pointed stars, and triangleADE obtain the outer circle of the 12-pointed star. Because angles ∡ADE and ∡EDB arenot equal, the incircle of triangle ADE is not tangent to the outer circle of the 9-pointedstar. However, the gap is less than the line width in the figure. The centre of the semicirclein the top-right of the template is the intersection of the line DF reflected off line BC withthe hypotenuse AC . The semicircle is tangent to the outer circle of the 9-pointed star, butdoes not meet the line BC .

In Figure 13(b) the construction in the template is completed. The inner circles of thesecondary stars have ratio λ = 0.381 to their outer circles; the same ratio is used in thetop-right semicircle. The method of §4 is applied to produce all the stars. The choiceof λ means that the 10-pointed star is the regular {10/4} star, the sides of the 9-pointedstar diverge, and the sides of the 12-pointed star converge. The arrowheads are formed byconnecting anchor points to anchors in adjacent copies of the template in a way that satisfiesRule 2. This produces a rather lop-sided arrowhead in the upper-right of the template; thealternative is to have a more symmetric arrowhead which does not point directly at itscounterpart and violates Rule 2. Figure 13(c) shows the result of replicating the template.Photograph TUR 0526 in Wade’s collection [16] shows this pattern on the Alay Han on theAkseray–Kayseri road, Turkey.

We now turn to the second template and follow a similar procedure. Figure 14(a) showsthe template ABC divided into triangles. The point D is the point on BC such that angleCBD is 54◦ — as before we extend a spoke of the 10-pointed star centred at B until itmeets line AC . The point E is determined as the intersection of two spokes. The line EB is an extension of a spoke of the 10-pointed star and angle ∡EBC = 18◦. A first attemptat locating E is to extend a spoke of the 11-pointed star; such a spoke can be producedby bisecting angle ∡BDC . This strategy makes angle ∡DCE 40.17◦. As this is so close to40◦, we place an equiangular 9-pointed star centred at C and recompute the location of E as the intersection of spokes radiating from B and C . As the 9-pointed star is equiangular,

angle ∡ECB = 20◦

(an angle that cannot be constructed with Euclidean tools). This slightadjustment means that angle ∡BDE = 33.22◦ and angle ∡EDC = 32.76◦. Point F isproduced by reflecting E in line BD so ∡BDF = ∡BDE . To obtain the remaining spokes of the 11-pointed star we divide the angle ∡ADF into five equal parts (another non-Euclideanstep).

To obtain the outer circles of the secondary stars we apply the method of §4 to trianglesCDE , BDE and BDF . We use triangle BDE to obtain the outer circles of the 10-pointedand 11-pointed stars, and triangle CDE to obtain the outer circle of the 9-pointed star. Takethe perpendicular to AB that passes through D and use it as a radius to draw a circular arccentred at D; the arc meets the hypotenuse AC at G. Since angle ∡BAD = 30◦, G bisectsAD . Unlike the other stars, the size of the 12-pointed star is not fixed by the geometry of

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A B

C

DE

F

G

(a) (b)

(c) Sari Han, Akseray–Kayseri road, Turkey, 1249

Figure 14: Pattern containing equiangular 9-pointed, 10-pointed and 12-pointed stars andirregular 11-pointed stars.

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the template’s triangular subdivision. For balance we choose the radius of its outer circleto be the same as that of the 11-pointed star.

The construction of the pattern is continued in Figure 14(b). As in the 45◦ template, weset the ratio of the inner circles to the outer circles of the secondary stars to λ = 0.381 so thatthe 10-pointed star is the regular {10/4} star. The right side of the template is completedusing the now familiar method of §4. To locate the remaining corners of the petals in the11-pointed rose, we draw an arc centred at D through a corner we have already constructedin triangle BDF and extend sides of the star until they meet this arc. Connecting anchorpoints to anchors in adjacent copies of the template produces the lop-sided arrowhead in thebottom-centre of the template; as above we could break Rule 2 to obtain a more symmetricshape. Figure 14(c) shows the result of replicating the template. Even the rather extremeasymmetry of the arrowhead does not look out of place and does not attract attention inthe way misaligned arrows would. Wade’s photograph TUR 0609 [16] shows the pattern onthe Sari Han on the Akseray–Kayseri road, Turkey.

These two examples come from nearby hans — the staging posts along the trade routesconnecting Asia to the Mediterranean, known outside Turkey as caravanserai. A typicalhan is a large stone building with high walls and a single entrance, large enough to permitheavily laden camels to pass. The only decoration is carved into the stone around the portaland frames the entrance. The star patterns are realised as interlaced ribbons, which createsa degree of ambiguity in the location of the crossings. Furthermore, after erosion of thestone, the ribbons do not have constant width or straight edges. Given these limitations, itmay be difficult to establish whether the details predicted by this new method match thearchaeological remains.

7.2 Iraq: 14-pointed and 11-pointed stars

Figure 15(d) shows a pattern reconstructed from the Mudhafaria Minaret at Erbil in north-east Iraq, near the border with Iran and Turkey. The minaret is a brick tower that haspartially collapsed and is badly eroded. Despite the damage, traces of its decoration arestill visible and it is possible to see that one of its patterns is an arrangement of 11-pointedand 14-pointed stars — the area shown in grey in the figure corresponds to the panel in theminaret.

In our previous examples, the centres of some of the primary stars have also been globalcentres of high rotational symmetry for the whole pattern. This is not possible with thestars in the Iraqi pattern. If the centre of a star coincides with a centre of n-fold rotational

symmetry of a pattern then the number of spikes in the star must be a multiple of n. Sincethe only possible rotation centres are 2-fold, 3-fold, 4-fold and 6-fold, the centre of an 11-pointed star cannot be a global rotation centre and, if the centre of a 14-pointed star is arotation centre, it must be 2-fold. Furthermore, when a star centre lies on a mirror line, thespokes must respect the reflection symmetry. Since inward and outward spokes alternatearound the star, the mirror line must contain two of the spokes.

The key building block of this pattern is a right triangle with one angle spanned bytwo spikes of an 11-pointed star and the other angle spanned by one spike of a 14-pointedstar. The angle between the spikes of an equiangular 11-pointed star is 32.73◦ and the anglebetween the spikes of an equiangular 14-pointed star is 25.71◦ so, if we used equiangularstars, we would get an angle sum of 181.17◦ for the triangle. The small excess can be

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eliminated if at least one of the stars is irregular and the angle between the spokes isreduced. We shall see one way to determine the angles later. The first problem is to takethe observation about the triangle and convert it into a template.

First, arrange four copies of the right triangle so that their right angles meet at a point.The triangles form a rhombus with an 11-pointed star at each end of the short axis and a14-pointed star at each end of the long axis. The edges of the rhombus must coincide withspokes of the stars; the spokes that lie inside the rhombus are closer together than theywould be in equiangular stars.

The rhombus itself cannot be used as a template to fill the plane so we embed it in arectangle, as shown in Figure 15(a). The corner of a rectangular template is a 2-fold rotationcentre in the complete pattern so it cannot be the centre of a star with an odd number of points. The aspect ratio of the rectangle depends on how the irregularity is distributedbetween the 11-pointed stars and the 14-pointed stars. In the figure the aspect ratio isdetermined by requiring the three circles drawn in dashed lines to be mutually tangent.This means that both kinds of star are irregular. If the 11-point star is equiangular, therectangle is slightly taller and narrower; if the 14-point star is equiangular, it is shorter andwider.

Subdividing the template into triangles is straightforward, as shown in Figure 15(b).The spokes that lie outside the rhombus are obtained by angle trisection. This leads tointerspoke angles of 16.15◦ and 16.93◦ for the 11-pointed star, and 12.69◦ and 13.07◦ forthe 14-pointed star — in each case the smaller angle is for the spokes inside the rhombus.

The inner circles for the secondary stars are chosen so that sides of the 11-pointed starand the 14-pointed star are virtually collinear. This means that the sides of the 11-pointedstars diverge slightly, and the sides of the 14-pointed stars converge. It also results in asmall value of λ = 0.32 and sharp narrow spikes on the secondary stars — see Figure 15(c).

The paper by Ajlouni and Justa [1] contains photographs of the minaret and some detailsof its recent conservation. The authors also examined the eroded panels and managed toidentify some of the patterns. Their analysis of the panel containing 11- and 14-pointed starsproceeds as follows. First, they construct complete 11-fold and 14-fold rosettes inscribedin regular polygons with equiangular stars with parallel sides in the centres — the 14-pointed stars are regular {14/6} stars. These preformed rose motifs are then positionedand the pattern is completed by extending their peripheral lines to form interconnections.The rosettes are arranged to cover the panel but not with respect to any template, so thepattern does not have a natural continuation beyond the panel. The result does not seemto match the original in the bottom corners and the secondary stars have the wrong shape.

7.3 Topkapı Scroll: 11-pointed and 9-pointed stars

We now consider the case of two different primary stars, both having an odd number of points. Our example comes from a template in the Topkapı Scroll — the pattern it generates,shown in Figure 16(d), is not known from an architectural source.

As in the Iraqi construction, the building block for this pattern is a right triangle; oneangle is spanned by a spike of a 9-pointed star and the other angle is spanned by 1 1/2 spikesof an 11-pointed star. With equiangular stars, the triangle would have angles of 40◦ and49.091◦ so the angle sum would be in deficit by just less than 1◦. As before, four of theseright triangles are assembled to form a rhombus — two 11-pointed stars span the short axis

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(a) (b) (c)

(d) Mudhafaria Minaret, Erbil, Iraq, 1190–1232.

Figure 15: Pattern containing irregular 11-pointed and 14-pointed stars.23


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and two 9-pointed stars span the long axis.The rhombus is placed in a rectangular template, as shown in Figure 16(a). Measure-

ment of the scroll indicates that the 11-pointed star is equiangular. This determines theorientation of the rhombic axes and the size of the rectangle. As both kinds of star have anodd number of points, neither can be located at the corners of the rectangle. The trianglein the top-left corner of the template is congruent to the four in the rhombus — it has thesame angles and a side in common. The remaining spokes of the 9-pointed star are obtainedby trisecting the angle outside the rhombus; the spokes in the grey sector of the wheel are20.45◦ apart and the spokes in the white sector are 19.09◦ apart.

Once this structure is in place, it is straightforward to produce the framework of trianglesshown in Figure 16(b). The 9-pointed stars in the scroll have parallel sides and this fixesthe ratio of the inner and outer circles of the secondary stars to λ = 0.417. The resultingtemplate is shown in Figure 16(c). This example illustrates that it is sometimes necessaryto allow aesthetic qualities to override a strictly formulaic application of the algorithm in

§4.

Notice that the incircle in top right triangle of Figure 16(b) overlaps the outer circle of the11-pointed star and is distant from the outer circle of the 9-pointed star. The associatedinner circle used for locating the anchor points does not have the same centre but is movedalong the inward spoke of the 9-pointed star to balance the design. The centres of bothcircles are marked in Figure 16(c).

Bodner has also studied this pattern [2]. She describes an approximate construction of an 11-sided regular polygon which is so close that, for all practical purposes, it generatesa wheel with eleven equally spaced spokes. Taking this as her starting point, she uses thespokes to define the geometry of the template rectangle and the location of the 9-pointedstar. She then constructs both primary stars with parallel sides and also so that sides inthe 9-pointed and 11-pointed stars are collinear. In the scroll the sides of the 11-pointedstar converge. Early on in the paper she notes that ‘all of the star polygons (the “nearlyregular” nine- and eleven-stars as well as the irregularly shaped pentagonal ones) . . . areconstructed using inscribed and circumscribed circles’ but she does not do this and so failsto explain the construction lines visible on the scroll.

My earlier attempt to understand this pattern [8] used the PIC technique. First, atiling is constructed which contains regular 9-sided and 11-sided polygons embedded in aconnecting matrix of irregular pentagons and hexagons. Then the pattern is derived fromthe tiling by placing a crossing at the midpoint of each edge and extending the lines untilthey meet; the incidence angle of the lines to the tile edges is constant. This processgenerates equiangular stars in the regular polygons. The method does not explain thefunction of the inner circles on the scroll. Indeed, the result does not agree with the scroll— the secondary stars do not have the required inscribed circle.

Deriving the pattern from the tiling in this algorithmic way makes the process straight-forward to computerise. Yet performing it by hand is more difficult than the point-joiningmethod proposed here. Moreover, the geometry of rose patterns produced by the strictapplication of the PIC method does not agree with the traditional examples. The resultscan be improved by varying the incidence angle and moving crossings away from the edgemidpoints. Kaplan [13] explored when and how these ad hoc adjustments should be made.He observed that a rose pattern can be derived from two different tilings (essentially, thetilings in the central and right hand sections of the overlays in Figure 7) and defined therosette transform to convert one tiling into the other. The crossings are located at the

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(a) (b) (c)

(d) From the template in Panel 42 of the Topkapı Scroll.

Figure 16: Pattern containing regular 11-pointed stars and irregular 9-pointed stars.

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intersection points of these two dual tilings.In the conclusion of my earlier discussion of the Topkapı Scroll pattern I noted the

discrepancy between the PIC result and the original, and commented that medieval artistsseemed to value alignment more than regularity. The method presented in this paper followsthis observation through to its natural conclusion. It produces results that are more pleasingto the eye, more in agreement with medieval examples, and are easier to perform by hand.

8 Patterns from equilateral tilings

We now turn to the second method of producing star patterns, which was suggested by thecentral sections of the overlays in Figure 7 — stars centred at the vertices of an equilateraltiling. Both tilings in that figure are Archimedean so the vertices, and also the stars centredon them, are all the same (technically, they are transitive under the action of the symmetry

group of the pattern).Figure 17 shows four more equilateral tilings, none of which is Archimedean. The tiling

in (a) is composed of regular polygons and has three vertex species (3.12.12, 3.4.3.12 and3.3.4.12) so it is a 3-uniform tiling. The tiling in (b) also uses the triangle, square anddodecagon tiles and adds an irregular convex hexagon, sometimes called a shield . Theshield tile can be seen as an equilateral triangle with an isosceles right triangle attached toeach edge. It fits naturally with the set of regular polygons with 3, 4, 6 and 12 sides toform tilings. The tiling in (c) is based on the 2-uniform tiling with vertex species 3.12.12and 3.4.3.12 — the edges of the square tiles are not drawn. The tiling of regular pentagonsand decagons and irregular hexagons in (d) is one of the structural frameworks that can beoverlaid on Figure 2. In that case the stars are inscribed in the regular tiles; here we want

to place stars on the vertices.Wheels have been placed at the vertices of each tiling in the figure. In the top two tilings

all the wheels have five spokes, and in the others the wheels have seven spokes. Many of thespokes coincide with the edges of the tilings; where this is not the case they divide the anglewithin a tile into equal sectors. None of the wheels is equiangular — as in the earlier figuresthe grey sectors are larger than normal and the white sectors are smaller than normal. Weshall now use these arrangements of wheels to form star patterns. Each wheel defines theouter circle of a star and the spokes in the wheel become the outward spokes of the star.

Figure 18(a) is derived from the tiling in Figure 17(a). A regular {12/5} star has beeninscribed in each dodecagon. This determines the radius of the inner circles of the secondarystars — the ratio of inner to outer radii is λ = 0.326. Bisecting the angles between the

outward spokes produces the inward spokes. The inner circles and inward spokes locateall the anchor points and the pattern is produced by connecting them with straight lines.This pattern is similar to a Mamluk design on a wooden minbar in the al-Ashraf Barsbaycomplex in Cairo, Egypt, dating from about 1425 — see photograph EGY 1030 in Wade’scollection [16].

The pattern in Figure 18(b) is taken from sketches in the collection of Ernst Herzfeld’spapers at the Freer Gallery of Art and Arthur M. Sackler Gallery Archives. DrawingsD372, D373 and D389 show a doorway from the Friday Mosque in Isfahan, Iran, and areannotated ‘North-East Portal (no longer in use)’. The pattern can be derived from the tilingin Figure 17(b) by setting the inner circles of the stars. Here, we set the ratio of inner toouter radii of the secondary stars to λ = 0.381. The 12-pointed stars are inscribed in regular

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(a) (b)

(c) (d)

Figure 17: The wheels centred on the vertices of these equilateral tilings provide a basis forthe construction of irregular stars.

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(a) al-Ashraf Barsbay complex, Cairo (b) Friday Mosque, Isfahan

Figure 18: Two patterns with irregular 5-pointed stars.

dodecagons so they are equiangular; their geometry is determined by the secondary stars.In some of the secondary stars two sides are omitted and the spikes opposite these ‘gaps’point freely into open space and not attached to other elements of the design. Even thoughthe stars are incomplete, the underlying construction process is the same. It is interestingto note that Herzfeld also draws inner and outer circles for all the stars in drawing D389,which shows his construction for the pattern.

The patterns in Figure 19 are derived from the bottom two tilings in Figure 17. Althoughthe buildings they come from are separated by quite large distances in space and time, theyhave striking similarities. Both have irregular 7-pointed stars that approximate the regular

{7/2} star and, in both cases, the primary star motif does not have the typical cell structureof kites but has a corona of alternating bone-shaped octagons and irregular hexagons formed

by linking the 7-pointed stars.The example in Figure 19(a) violates Rule 2 in the groups of four 7-pointed stars sur-

rounding a cross; at the end of each arm of the cross where the lines pass from one starto the next they deviate as they go through the vertex. (This arrangement also occurs inPanel 81a of the Topkapı Scroll.) This is a consequence of our choice of wheels in the tilingand the fact that spokes in neighbouring wheels are not aligned. In Figure 17(c) we chosenot to add the edges of the squares to the tiling. In the wheels the angle between the spokesin the white sectors is 50◦ and in the grey sectors is 52.5 degrees; in an equiangular wheelthey would be spaced at 51.428◦. This small discrepancy means the 7-pointed stars areeasily recognised. If we had included the edges of the squares and forced the spokes to alignwith these edges, the interspoke angles would be 45◦, 50◦ and 60◦.

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(a) Mausoleum of Oljeitu at Sultaniya, Iran, 1313.

(b) Tower of Masud III at Ghazni, Afghanistan, 1099–1115.

Figure 19: Two patterns containing irregular 7-pointed stars that approximate the regular

{7/2}.

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The original version of this pattern is painted onto a ceiling in the Mausoleum of Oljeituat Sultaniya, Iran. The design is realised in wide, interlaced ribbons in a fluid, apparentlyfreehand style without strict geometry or straight edges. This informality and lack of precision helps to disguise the misaligned crossings, and the use of wide ribbons createslarge hidden segments where they pass over and under one another, leading to ambiguityover the exact locations of the crossings.

9 Conclusions

In this paper we have demonstrated one method that medieval craftsmen could have used toconstruct Islamic geometric patterns. In the case of simple patterns it is possible to proposemany different methods of construction and it is difficult to determine which might havebeen used traditionally as there is little historical evidence to help choose between them.

Indeed, the same pattern may have been produced using different methods in differentplaces and times. Here we have analysed complex patterns that push the boundaries of what is geometrically possible. For these examples the number of satisfying explanations ismuch more limited. Whatever method was used to produce the original patterns, it seemsreasonable to assume that it was a variation or extension of a method used to produce thesimpler contemporary patterns.

We have shown how to construct complex arrangements of stars whose symmetries areincompatible with each other and with the standard lattices used to arrange repeating mo-tifs. The method is developed from simple rules based on good design principles ratherthan the mathematics of symmetry and involves only simple geometric constructions (anglebisection) and approximations (angle trisection, construction of regular polygons). Fortu-

nately, we do have some knowledge of traditional workshop geometry from the tenth-centuryBook on those Geometric Constructions which are Necessary for Craftsmen written by Abu’lWafa. Raynaud [15] lists all its propositions and traces their connections with constructionsknown in Classical Greece and Renaissance Europe. The following are of particular interestto us: Chapter 2 on basic constructions includes trisection of an acute angle, Chapter 3 con-structs regular polygons from three to nine sides with a given side, and Chapter 4 inscribesregular polygons from three to ten sides in a given circle.

A key ingredient of the new method is the inner circles of the stars. The Topkapı andTashkent Scrolls show radial grids for constructing stars. They also show inner circles forsecondary stars — other proposed construction methods do not explain the presence orpurpose of these circles. Curiously, the outer circles of the secondary stars are not shown

in the scrolls, although something like the incircle argument of §4 must have been used toestablish the relative proportions of all the stars. We have seen in the example from theTopkapı Scroll that the method is robust and tolerates small variations in the placement of the inner circles.

The new method differs from other constructions that I am aware of in its disregardfor symmetry in the primary motifs. Each of the following statements has been violated atleast once in the constructions presented here:

• the spikes of a star have equal length• the spikes of a star are equally spaced

• the spikes of a star have bilateral symmetry

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• the petals of a rosette have equal length• the petals of a rosette have equal width• the petals of a rosette have bilateral symmetry.

All these properties are consequences of the desire to maximise local symmetry. In contrast,the method presented in this paper is holistic: a pattern is constructed as a whole, not asa collection of independent preformed motifs. The ‘asymmetry’ is distributed around thepattern, and not concentrated in the secondary stars where it is noticeable.

It is instructive to compare the patterns constructed in this paper with those producedby modular design systems. Here, the patterns are constructed as linear forms by point

joining; the shapes that appear in the resulting partition of the plane are not the primaryfocus of the method. Indeed, shapes that play equivalent roles in different patterns, andeven within a single pattern, (such as the petals of a 10-fold rose) are not necessarily

congruent. The linear nature of the patterns is emphasised by their implementation asinterlaced ribbons.

By contrast, modular systems are shape-based: patterns are produced by assemblingidentical copies of polygonal modules taken from a small standardised set. Modular systembecame very popular in the Timurid and Safavid periods when coloured tiles became avail-able. In an early modular system based on the Star and Cross pattern [10] the modulesthemselves appear as shapes in the finished patterns. In later systems [9] the modules aredecorated with motifs in a manner consistent with the PIC methodology; only the motifsappear in the finished pattern — the polygons bounding the modules are merely a substrateaiding composition and are not visible. The tiles in the third section of Figure 2 are a sim-ple example of this technique. Decorated modular systems account for a large number of

Islamic patterns, which gives a misleading impression of the effectiveness of PIC; what weare seeing is the same few polygons combined and reused in a large number of examples,not essentially different PIC constructions.

The PIC method is promoted by Bonner under the name polygonal subgrids as a generaltechnique that underlies most patterns, and he applies it to create compound star patterns of the kind discussed here [5]. He claims [4] ‘The polygonal method is the only technique thatallows for the creation of both simple geometric patterns and the most complex compounds,often made up of combinations of seemingly irreconcilable symmetries.’ It is true that PICcan produce patterns that are diffeomeric to the Seljuk originals studied here. However, aswe noted earlier, a strict application of PIC does not work well and ad hoc adjustmentsare necessary to make the geometric details agree. The polygonal networks constructed

as an intermediate step in our method look superficially like the starting point of a PICconstruction, but their function is very different: in our method it is the corners of thepolygons that are important (they determine inward spokes), not the midpoints of theedges as in PIC.

Each pattern that we constructed in §8 has irregular stars centred on the vertices of an equilateral tiling with some of their outward spokes aligned with edges of the tiling.In these patterns, the irregular stars are secondary motifs that form an interconnectingmatrix between the primary motifs; the primary motifs are usually centred in large regularpolygons of the tiling. Islamic patterns of this form are quite varied and are distributed intime and place.

This is very different from the kind of pattern constructed in §7: these patterns, with

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their odd combinations of stars, are concentrated in the Seljuk period in Anatolia and itsnear neighbours. In our reconstructions of these patterns some of the primary stars areirregular. Examples of the more standard star combinations are constructed in

§6 using

the same method; both sets of patterns contain the distinctive secondary stars. Diffeomericversions of the standard patterns appear in other periods and places, and may exhibitdifferent geometric details that cannot be recreated using the technique presented here.In these cases stars or roses are used as preformed symmetric motifs and the pattern iscompleted by extension of their peripheral edges into the interstitial spaces.

The Seljuk period was a time of great experimentation and creativity in many fields.The examples we have studied here show that this was also true in the development of starpatterns. The discovery of unusual combinations of stars that can be made to fit togethermust have proceeded by trial and error. We admire the resulting patterns in part for theingenuity required to create them and, for this reason, we place them among the outstandingtechnical achievements in this art form. But it is unclear whether they were regarded asspecial when they were created — they are not displayed proudly in prominent positionsbut are ‘just another panel’ alongside simpler patterns.

It is also unclear whether the patterns were widely reproduced. This is partly becausemany Seljuk buildings have not survived, being destroyed naturally through erosion andearthquakes, or deliberately in the Mongol invasions of the thirteenth century. Figure 13(d)can be found in several places in Turkey, and Bourgoin [6, Plate V] recorded its use in ascreen (since destroyed by fire) in the Great Mosque in Damascus, Syria. Today, however,these patterns are rare — there may be ten different designs and most are known from asingle source.

Patterns containing incompatible stars disappeared with the Seljuks. Why were thesepatterns discontinued? It seems unlikely that they would have been abandoned for aes-thetic reasons for they are attractive and, when well executed, the transitions between thedifferent elements appear effortless and do not disturb the eye as incongruous or clumsyarrangements. It is not apparent to the viewer that these patterns are any more difficultto construct than the common pattern of Figure 2. Even the novelty of the different starcombinations will not be appreciated by many.

The method for constructing the patterns would have been a closely protected secretand we can see from recent attempts to understand these patterns that, if such a methodwere lost, it is not something that can be reinvented easily. However, the method was knownto the compilers of the Topkapı Scroll some 200 years after the Seljuks.

One possibility is that the Seljuk experiments provided a step in the evolution of therose motif. Once the rose archetype was abstracted, it was used as a preformed buildingblock and assembled to form patterns in the same ways that stars had been. Togetherwith the increased use of modular design, such a move could be seen as a trend towardsthe standardisation of techniques and motifs. Variable and imperfect stars would becomeunacceptable. Rose geometry does not have the flexibility to accommodate combinationsof incompatible stars. Further work is required to explore this art-historical question.

References

[1] R. Ajlouni and P. Justa, ‘Reconstruction of eroded and visually complicated archae-ological geometric patterns: Minaret Choli, Iraq’, J. Geoinformatics CTU FCE 6

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(2011) 18–24. Online version available at Proc. 23rd CIPA Symposium, (Prague, 2011)

http://cipa.icomos.org/fileadmin/template/doc/PRAGUE/080.pdf

[2] B. L. Bodner, ‘The eleven-pointed star polygon design of the Topkapı Scroll’, Proc.Bridges: Mathematics, Music, Art, Architecture, Culture , (Banff, 2009), eds. C. S.Kaplan and R. Sarhangi, Tarquin Books, 2009, pp. 147–154.

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Documents

On irregular stars in Islamic geometric patterns