Islamic Star Patterns from Polygons in Contact

Craig S. Kaplan

School of Computer ScienceUniversity of Waterloo

Abstract

We present a simple method for rendering Islamic starpatterns based on Hankin’s “polygons-in-contact” tech-nique. The method builds star patterns from a tiling ofthe plane and a small number of intuitive parameters.We show how this method can be adapted to constructIslamic designs reminiscent of Huff’s parquet deforma-tions. Finally, we introduce a geometric transformationon tilings that expands the range of patterns accessibleusing our method. This transformation simplifies con-struction techniques given in previous work, and clari-fies previously unexplained relationships between certainclasses of star patterns.

Key words: Islamic star patterns, Islamic art, Tilings,patterns

1 Introduction

Islamic star patterns represent one of the world’s greatornamental design traditions [2, 6, 7]. Star patterns area harmonious fusion of mathematics, art, and spirituality,and expressions of symmetry, balance, and ingenuity.

Star patterns also embody an enduring mathematicalmystery. Most of the original design techniques are lostto history, and we are forced to probe the minds of an-cient artisans and mathematicians via the patterns theyleft behind. Many scholars and hobbyists have discov-ered or rediscovered techniques that produce Islamic pat-terns [1, 9, 11, 20].

This paper presents a simple technique based on Han-kin’s “polygons-in-contact” method [12]. Given a tilingof the plane, Hankin’s method produces an Islamic starpattern based on that tiling (Section 3). By modifyingthe construction slightly, we are able to construct designsin the style of Huff’s parquet deformations [16, Chap-ter 10] (Section 3.1). Finally, we show how an operationon tilings called the “rosette transform” can expand therange of patterns available using Hankin’s method (Sec-tion 4). The rosette transform demonstrates the power ofHankin’s method, and formalizes previously unexplainedrelationships between certain classes of star patterns.

2 Previous work

Our approach follows most directly from recent work byKaplan and Salesin [18]. They construct designs by fill-ing tiles in a tiling with fragments of star patterns. Fortiles that are regular polygons, they give a parameterizedspace of “design elements” drawn from historical exam-ples. They then complete the pattern by filling irregularpolygons using an “inference algorithm”. Because theirconstruction is independent of Euclid’s parallel axiom,they can draw star patterns seamlessly in Euclidean andnon-Euclidean geometry. We simplify their approach byeliminating design elements and applying an inference al-gorithm uniformly to all tiles. We can still produce com-plex elements such as rosettes by moving more informa-tion into the underlying tiling. There is a large overlap inthe designs produced by the two systems – given suitabletilings, Najm could reproduce all the star patterns pre-sented here. But this paper offers insights into the tilingsthat underlie star patterns, and provides a technique thatis simpler and easier to control.

Jay Bonner is an architect who has studied Islamic starpatterns extensively. In an unpublished manuscript [3],Bonner gives a systematic presentation of star patternsdeveloped over a vast space of tilings (which he calls“polygonal sub-grids”). Some of the techniques from thebook also appear in a recent paper [4]. Bonner’s work isintended as a resource for designers, and not specificationfor software writers. He draws patterns manually using aCAD tool. This paper is in part an attempt to formalizethe algorithms that underlie his technique, and to expressthose algorithms in software for architects, designers, andartists.

3 The polygons-in-contact method

A tiling-based approach to Islamic star patterns seemsfirst to have been articulated in the west by E.H. Han-kin in the early part of the twentieth century. In a seriesof papers [12, 13, 14, 15], he explains his discoveries andgives many examples of how the technique can be used.Hankin’s description of his technique provides an excel-lent starting point for an algorithmic approach (and helpsdrive contemporary work by Bonner).

In making such patterns, it is first necessaryto cover the surface to be decorated with a net-work consisting of polygons in contact. Thenthrough the centre of each side of each poly-gon two lines are drawn. These lines cross eachother like a letter X and are continued till theymeet other lines of similar origin. This com-pletes the pattern [12, Page 4].

Since that time, scholars such as Lee [19] andCritchlow [8] have referred to Hankin’s “polygons-in-contact” technique. This method immediately suggestsan algorithm for turning a tiling into an Islamic star pat-tern. Given a tiling of the plane by polygons (Hankin’s“network”), we identify the midpoints of the edges of thetiling as “contact points” where the design will be born.We place a small X at every contact point and “grow” thearms of the X until they encounter lines growing fromother contact points. There is one obvious degree of free-dom in this process: the angle formed by the arms of theX with the edge from which they emanate. We call thisangle thecontact angleof the pattern. An illustration isgiven in Figure 1(a).

We can regard this process as growing a small arrange-ment of lines for each unique tile shape in a tiling. Wegrow a pair of rays, forming half an X, inward from themidpoints of the tile’s edges. We call the arrangementof lines associated with a single tile itsmotif. An imple-mentation of this construction technique should accept atiling and a contact angle as input, build a motif for eachtile shape, and assemble the motifs into a pattern that canthen be decorated.

Given ann-sided polygonal tile and a contact angle,we must develop a motif from the2n rays entering thattile through its edge midpoints. A successful motif willpartition the rays into pairs, where each matched pair rep-resents a distinct path through the tile. The best possiblemotif will be the pairing of rays that optimizes a chosenaesthetic goal. We choose the simple goal of minimiz-ing the sum of the lengths of all of the line segments inthe motif. This goal reflects the sense of economy andinevitability in Islamic design, and is justified throughmany historical examples.

Ideally, then, we would iterate over all possible pair-ings of rays, and find the one that minimizes total length.Unfortunately, this algorithm is not practical – there are

n!(n/2)!2n ways to partition2n rays into pairs, or over halfa billion possibilities for a region with 10 sides.

Instead we use a greedy approach, based on a sim-plified version of Kaplan and Salesin’sinference algo-rithm [18]. We consider all possible pairs of rays. If tworays

−−→AB and

−−→CD intersect at a pointP , we store that pair

in a collection together with a cost equal to the sum of the

(a) (b)

Figure 1: In the first step of Hankin’s method, a pair ofrays is associated with every contact position on everytile. In (a), a single contact position gets its two rays,each of which forms the contact angle θ with the edge.In (b), we separate the ray origins by distance δ.

Figure 2: A demonstration of Hankin’s method. On theleft, contact points sprout an X-shaped arrangement ofrays that grow until they meet other rays. When the orig-inal tiling is removed, the result is the pattern on the right.

lengthsAP andPD. If the rays are collinear and pointtowards each other, we store the pair together with thelengthAD. We can then sort the collection by cost andwalk over it in order. For each pair of rays, we incorpo-rate that pair’s path into the motif provided neither of therays has yet been used.

In practice, this algorithm performs well on a widevariety of polygons. It certainly performs perfectly onregular polygons, where it constructs star-shaped motifs.It sometimes produces motifs with unmatched rays, andsometimes paths that venture too far from the underlyingtile. In the cases where it fails, it usually does so not be-cause it is greedy, but because the pairing technique is notwell-suited to the tile shape in question. In some cases,the inferred motif can be improved by moving the contactpoints away from the edge midpoints. This adjustment isdiscussed in greater detail in Section 4.

Figure 2 illustrates the process of growing rays fromcontact positions. Figure 3 shows some typical designsthat can result from using our implementation of Han-kin’s method.

(4.82) θ = 22.5◦ θ = 45◦ θ = 67.5◦

(4.6.12) θ = 45◦ θ = 60◦ θ = 75◦

(3.4.3.12; 3.122) θ = 35◦ θ = 60◦ θ = 75◦

Altair θ = 30◦ θ = 45◦ θ = 72.5◦

Figure 3: Examples of star patterns constructed using Hankin’s method. Each row shows a tiling together with threedesigns that can be derived from it using three different contact angles. The bottom row features an amusing tiling bynearly regular polygons. It is reproduced from Grunbaum and Shephard [10, Page 64], where it serves as a reminderof the danger of over-reliance on figures. A related design also appears in Bourgoin [6, Plate 163].

(a) (b)

(c) (d)

Figure 4: A demonstration of two cases where an ex-tension to the inference algorithm can produce a slightlymore attractive motif. In (a), a star pattern is shown withlarge unfilled areas that were the centers of regular do-decagons in the original tiling. Adding a layer of inferredgeometry to the inside of the motif produces the improveddesign in (b). The process is repeated with a differenttiling in (c) and (d).

There are some cases where simple modifications tothe basic inference algorithm can improve the generatedmotif. Consider, for example, the star pattern given inFigure 4(a). This pattern contains large regions, derivedfrom regular dodecagons, that are left unfilled. A moreattractive motif can be constructed using a second pass ofthe inference algorithm, building inward from the pointswhere the rays from the first pass meet. The resultingdesign, shown in Figure 4(b), is more consistent with tra-dition. In the inference algorithm, it is easy to recognizewhen the provided tile shape is a regular polygon and torun the second round of inference when specified by theartist. Kaplan and Salesin [18] solve this problem by pro-viding a more explicit parameterization of the range ofmotifs that can be used to fill regular polygons.

A further enhancement is to allow the contact positionto split in two, as shown in Figure 1(b). The split can beaccomplished by providing the inference algorithm witha second real-valued parameterδ that specifies the dis-tance between the new starting points of the rays. Theparameterδ can vary from zero (giving the original con-struction) up to the length of the shortest tile edge in the

Figure 5: Examples of two-point star patterns constructedusing Hankin’s method. Each row shows a templatetiling, a star pattern with δ = 0, and a related two-pointpattern with non-zero δ. The structure of the tiling in thebottom row will be explained in Section 4.

tiling. This modification gives what Bonner calls “two-point patterns,” a set of designs that are historically im-portant in Islamic art [3]. Examples of two-point patternsconstructed using theδ parameter are shown in Figure 5.The designs corresponding to two-point patterns tend tobe made up of very short closed strands, each one forminga loop around a single tiling vertex in the original tiling.The contact angle is typically chosen to be45◦, formingsquares around the midpoints of the tiling’s edges.

3.1 Islamic parquet deformationsParquet deformations are a style of ornamental designcreated by William Huff, and later popularized by Dou-glas Hofstadter [16, Chapter 10]. They are a kind of “spa-tial animation,” a geometric drawing that makes a smoothtransition in space rather than time. Parquet deforma-tions are closely related to M.C. Escher’sMetamorphosisprints [5, Page 280], though unlike Escher’s work theyare purely abstract, geometric compositions.

Figure 6: Islamic parquet deformations based on Hankin’s method. The top diagram shows the effect of continuouslyvarying the contact angle of a ray depending on the horizontal position of the ray’s starting point. When the process iscarried to all other tiles, the design in the middle emerges. In this design, the contact angle varies from a minimum of22.5◦ at the sides up to 67.5◦ in the middle. In the design on the bottom, the contact angle varies from 36◦ at the sidesup to 72◦ in the middle.

By exploiting Hankin’s method, we can introduce anew style of Islamic design that we call “Islamic parquetdeformations”. We simply modify the inference algo-rithm so that the contact angle varies along a line. Thecontact angle for a ray is chosen according to a functionof a horizontal position of that ray’s start position. Inthis way, the four rays leaving a given contact positionstill form an X, even though the contact angle may varywithin a single tile.

Smooth variation of the contact angle results in a gen-tly changing geometric design that is still recognizablyIslamic (see Figure 6). We believe that these parquet de-formations occupy an interesting place in the world of Is-lamic geometric art. The structure is recognizably in theIslamic tradition, but they would not have been producedhistorically because very little repetition is involved. Theeffort of working out the constantly changing shapes byhand and then executing them would have tested the pa-tience of any erstwhile artisan.

4 The rosette transform

A curious property of Hankin’s polygons-in-contactmethod is that different tilings may give rise to the same

star pattern under suitable choices of contact angle. Thepattern in the top row of Figure 7 is one example. It canbe produced from the tiling on the left using a contact an-gle of 54◦, or from the tiling on the right using a contactangle of36◦. We might therefore suspect a relationshipbetween the two tilings.

Further evidence for this relationship can be found inthe star patterns produced by the Najm system of Kaplanand Salesin [18]. The bottom row of Figure 7 shows atiling on the left with “rosette” motifs placed in regulardecagons. Kaplan and salesin provide an explicit param-eterization of these rosettes. But the right hand side ofthe figure demonstrates that the same pattern can arisevia inference alone from the ostensibly related tiling.

Our experience with Hankin’s method suggests thatthere are many pairs of tilings that are related in thisway. Referring to Figure 7, we call the tiling on the left a“Najm tiling” and the tiling on the right a “Hankin tiling”.The two have similar structure, except that in the Hankintiling the large regular polygons are separated by rings ofpotentially irregular polygons, usually pentagons.

In his manuscript [3], Bonner uses both kinds of tilingsto create star patterns, and he too observes that a single

Figure 7: Examples of distinct tilings that can produce the same Islamic design. In each case, the tilings on the left isfilled in using a combination of design elements and inference, and the tiling on the right uses inference alone. Theymeet in the shared design in the center.

design may originate from multiple tilings. In this sec-tion we introduce a transformation on tilings called the“rosette transform” that explains the connection betweenNajm tilings and Hankin tilings. We also compare ourmethod to Najm in terms of the patterns each approachcan produce.

The algorithm for the rosette transform is reminiscentof the inference algorithm used in Section 3. Given atiling, it constructs a planar map for each distinct tileshape. The planar maps are then assembled, this time intoa new tiling rather than a final design. It is also a kind ofdualization: the most common operation is to erect per-pendicular bisectors to edges in the original tiling. Themap for each tile shape is constructed in one of two ways.

Regular polygons. If the tile is a regularn-gonP ofradiusr with five or more sides, then the map is con-structed as in Figure 8. We build a new regularn-gonP ′

with radiusr′ < r and place it concentric with the origi-nal polygon but rotated byπ/n relative to it. We then addline segments connecting the vertices ofP ′ to the edgemidpoints ofP. The inner radiusr′ is chosen so that thelength of each of these new segments is exactly half of theside length ofP ′. Some trigonometry shows that givennandr, the correct value ofr′ is given by

r′ = r

(cos

π

n− sin

π

ntan

(π(n− 2)

4n

))The map returned isP ′ together with the segments

joining it to the edge midpoints ofP.Irregular polygons. If the tile is a polygonP that does

not satisfy the conditions above, we extend perpendicularbisectors of the sides ofP towards its interior, as shownin Figure 9. The bisectors are truncated where they meeteach other. The result becomes the map for this tile.

This step is similar in spirit to running the inference

algorithm with a contact angle of90◦ and is subject to thesame pitfalls. We do not expect it to return a meaningfulanswer for every possible polygon, but in the cases ofpolygons that occur in Najm tilings, the map it discoversis usually appropriate.

Some heuristics also work here that do not apply in theinference algorithm. One moderately successful heuristicis to consider the intersection points of all pairs of raysand to cluster those points that lie inside the tile. Eachcluster can then be averaged down to a single point thatall rays contributing to that cluster can use as an endpoint.This adjustment may move some rays away from perpen-dicularity, introducing “kinks” in the middles of edges ofthe transformed tiling. We can correct this problem laterby detecting the kinks and replacing them with straightline segments.

When this algorithm is run on Najm tilings, it tendsto produce Hankin tilings. For instance, the two tiles inFigures 8 and 9 correspond to the Najm tiling on the lefthand side of Figure 7. The rosette transform produces theHankin tiling on the right hand side of the figure. Twoadditional examples of tilings and their rosette transformsare given in Figure 10.

The rosette transform takes the intelligence out of Ka-plan and Salesin’s “design elements” and embeds it in thetiling. Suppose that in a given tiling, Najm is used to filla regular polygonP with a rosette. In the rosette trans-form of that tiling, a scaled-down copyP ′ of P will besurrounded by a ring of irregular pentagons. With ourmethod, these pentagons will conspire to form the hexag-onal arms of a rosette around a central star constructedinsideP ′. The rosette transform is motivated by (andnamed after) the goal of making the pentagons as closeas possible to regular, producing rosettes that are nearlyideal in the sense given by Lee [19].

Figure 8: The rosette transform applied to a regular poly-gon. Here, a regular 10-gon of radius r is filled witha smaller regular 10-gon of radius r′ together with seg-ments that join the vertices of the inner polygon to theedge midpoints of the outer one. The inner radius is cho-sen so that the marked edges have the same length.

Figure 9: The rosette transform applied to an irregularpolygon. On the left, a perpendicular bisector is drawnfor every tile edge as a ray pointing to the interior of thetile. The rays are cut off when they meet each other, aswith the inference algorithm.

In some cases, Hankin’s method generates an unsat-isfactory pattern when applied to a rosette-transformedtiling. Figure 11 shows an example where Hankin’smethod discovers the topology of the correct pattern, butproduces uneven rosettes. Bonner discusses how to cor-rect this situation by adjusting the contact positions on thepentagons away from the centers when necessary. How-ever, he gives no indication of when or how to carry outthis adjustment.

We can appeal to the relationship between tilings andtheir rosette transforms to resolve this mystery. The de-sign of Figure 11(a) was constructed using the Najmmethod. We choose rosettes to fill the regular octago-nal tiles. By design, the hexagonal arms of the rosettes(two of which are shown shaded in the diagram) contactthe edges of their surrounding octagons.

In Figure 11(b), the rosette arms are formed from thering of pentagons introduced in the rosette transform. Theconstruction of one such pentagon is shown in Figure 12.The problem is that segmentsAE andBE have differ-ent lengths. If we simply place contact positions at edgemidpoints, then the contact positions along edgesABandCD will not be equidistant from the center of theiradjacent octagon, producing an uneven arrangement of

Figure 10: Two demonstrations of the rosette transform.The transformed tiling is shown superimposed in boldover the original.

(a) (b)Figure 11: An example where Hankin’s method can pro-duce imperfect rosettes. In (a), Najm is used to placeperfect rosettes inside regular octagons. When Hankin’smethod is used on the rosette transform in (b), the rosettehexagons (shown shaded) are of two different sizes.

Figure 12: An illustration of the adjustment to contact po-sitions that recovers perfect rosettes in Hankin’s method.The horizontal line indicates the midpoint of the edge ofthe shaded tile in the rosette transform. But the correctlocation for the contact point is the intersection of thatedge with the edge of the original tiling.

rosette arms. Although the shaded pentagon is topologi-cally dual to the vertex it contains, it is not dually situated(its edges are not bisected by the original tiling). We cor-rect the discrepancy by recording the intersection pointsof the rosette transform with the original tiling, and usingthem as contact positions.

Figure 13: A decorated star pattern produced using thepolygons-in-contact method.

Thus there a deep connection between these two kindsof tilings, which can expose the logic behind what mayhave seemed like an arbitrary but essential adjustment.Furthermore, these revised contact positions can be cal-culated easily while the rosette transform is computed.

5 Implementation

We have implemented the approach described in this pa-per as a standalone Java application. The application’sinterface shows a tiling with a corresponding Islamic starpattern superimposed on it. The user is able to selectthe tiling to work with and modify the contact angleθand distanceδ used to produce two-point patterns. Theycan also choose whether to incorporate a second pass ofthe inference algorithm in large, regular polygons, andwhether to adjust the contact positions away from theedge midpoints as described at the end of Section 4. Allof these changes are reflected in the design interactively,making it easy and enjoyable to browse a wide range ofstar patterns. When the user has decided upon a design,they can render it using the decoration styles first devel-oped for Taprats [17]. A decorated pattern is shown inFigure 13.

Our current software handles only periodic tilings(though there is no such limitation in the underlying tech-nique). A tiling is represented by two translation vectorsand a collection of untransformed polygons. Each poly-gon holds a list of transformations that map it to its oc-curences in a single translational unit. This informationis sufficient to cover any region of the plane with a subsetof the tiling. In the interactive designer, planar maps rep-resenting motifs are associated with the tile shapes anddrawn with them. A standalone Java application can beused to construct periodic tilings by hand. The user cancreate regular polygonal tiles and snap them together, fill

Figure 14: A classic Islamic star pattern that cannot eas-ily be expressed using a combination of Hankin’s methodand the rosette transform.

Figure 15: An unusual star pattern, reproduced from Bon-ner’s manuscript, featuring 11- and 13-pointed stars.

in irregular holes with new tiles, and specify translationvectors. Another program accepts a periodic tiling as in-put, and computes its rosette transform.

The construction of Islamic parquet deformations re-quires many separate invocations of the inference algo-rithm, and is currently too slow to run interactively. It isimplemented as a command-line application that builds adesign from parameters specifying the bounding box andcontact angle range.

6 Future work

Despite the power of the Hankin’s method combined withthe rosette transform, some important historical patternsare still out of reach. One example is given in Fig-ure 14. This pattern is easy to construct from a tilingof dodecagons and triangles using Kaplan and Salesin’sNajm. Their technique provides an explicit parameteri-

zation of the motifs in the dodecagons, which they call“extended rosettes”. It is tempting to assume that we canarrive at the same pattern via two iterations of the rosettetransform on this tiling. Closer examination reveals thatwhere lines cross, they form angles of both30◦ and60◦.Any inference-based representation of this pattern wouldneed either to allow multiple different contact angles, ornon-greedy choices in the inference algorithm. Eitherway, more work is necessary to discover the principlesthat govern these choices.

Hankin tilings are better suited than Najm tilings forconstructing some patterns. Bonner exhibits several re-markable designs with unusual combinations of motifs;Figure 15 shows a pattern with 11-pointed and 13-pointedrosettes. These remarkable designs are possible becausethe extra layer of irregular tiles can absorb the error whenreconciling the incompatible angles of the regular 11- and13-gons. Note that the tiling that produces this design isnot the rosette transform of any tiling. Hankin tilings cantherefore be considered “primitive” in some cases. An-other primitive Hankin tiling is the “Altair” tiling in thebottom row of Figure 3. It would be interesting to exam-ine what other unusual combinations of regular polygonscould be accommodated in a single pattern in this way,and how to generate the associated Hankin tilings auto-matically.

One other unexplored direction in this work is itsextension to non-Euclidean geometry, as demonstratedby Kaplan and Salesin [18]. This extension would bestraightforward. The only change would be a general-ization of the formula for scaling regular polygons in therosette transform. It should be possible to express a gen-eral formula using the “absolute trigonometry” given inthe appendix of their paper.

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