islamic star patterns

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A . . J . LEE

ISLAMIC STAR PATTERNSThe geometrical ornament of Islam has fascinatedWestern observers for well over a century, and con-tinues to provide material for numerous studies at-tempting to explain its nature in esthetic, mathemat-ical, mystical, or even cosmological terms. The "geo-metrical" content of this ornament is obvious, but inspite of the fact that many works have been publishedon the construction and analysis of Islamic patterns, 1none has delved very deeply into the subject, and nocomprehensive work has yet appeared dealing in acomparative and systematic manner with the wholerange of patterns, both geographically and historically. 2There is clearly a need for a more scientifically basedaccount of these patterns, although it is not so much thetheoretical knowledge of the professional mathemati-cian that is required for such a study, as the precisemathematical language and rigor which he might bring'to bear on the subject. There is still no generallyaccepted terminology for the many different kinds ofmotifs used in Islamic geometrical ornament, nor forthe methods of forming repeating patterns from them.In the absence of a definitive work along these lines, Iwill attempt here to show ways in which some of thesimpler patterns might be developed from a number ofelementary principles.

Opinions have differed as to the part played bymathematics in the genesis, development, and con-struction of the more complicated Islamic patterns."Some have seen a gradual, convergent evolution frommany different types of pre-Islamic ornament,culminating in the often bewildering complexities of thelater, fully differentiated Islamic patterns. According tothis view it was presumably the widening practicalexperience of skilled craftsmen which alone chose thepaths taken at each level of increasing elaboration.Others, on the contrary, have seen clear evidence forthe intervention of the professional mathematician inthe design and invention of these patterns, or at leasthave ascribed to the artisans themselves a considerableknowledge of theoretical geometry and an ability toapply this knowledge to the development of new kindsof geometrical ornament.

We shall perhaps never have a final answer to thiscontroversial question, + yet it seems that Westerncritics have seriously underestimated the ability ofnative craftsmen to retain large amounts of empiricalknowledge on pattern design and construction in theabsence of any understanding of the theoretical back-ground which a professional mathematician mightbring to bear on these problems." It may be an advan-tage for a modern author to develop a systematicanalysis of Islamic patterns in purely mathematicalterms, but a knowledge of pure mathematics orgeometry is unnecessary for those who wish merely todraw Islamic patterns or invent new ones. A theoreticalbackground will often allow the artist to see a numberof combinatorial possibilities more quickly than the useof trial-and-error methods, but it forms no substitutefor true creativity. It is perhaps significant that a genu-ine application of mathematical insight to a systematicanalysis of Islamic geometrical patterns reveals a fargreater range of possibilities than were ever discoveredby the Muslims themselves."

A great deal of this ornament is immediately andunmistakably recognizable as "Islamic," and yet in itsentirety it does not form such an easily distinguishablebody of geometrical ornament. In fact the whole rangeof Islamic patterns represents an amalgam of many dif-ferent styles, some simply adapted and absorbed fromclassical sources and from various cultures with whichIslam came into contact during its early expansion.Since it is not possible to cover all types of Islamic pat-terns here, r will limit myself to an examination of cer-tain of the more interesting and typical of them.

There is one class of geometrical patterns whichIslam has made its own. This group comprises whatone might term the" star patterns," since they includestar-like motifs, linked or oriented according to certainprecise rules to produce endlessly repeating two-dimensional patterns. The star patterns are unques-tionably the most beautiful and intricate of all Islamicpatterns, and they owe their beauty in no smallmeasure to a high degree of symmetry at all levels.Indeed, the star motifs themselves invariably possess n-


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ISLAMIC STAR PATTERNS 183fold rotational symmetry, n representing a range ofwhole numbers from 3 to almost 100. Although anypattern which repeats in two directions must ofnecessity pertain to one or another of 17 fundamentallydistinct arrangements, 7 a classification by means ofsuch symmetry groups is of little use in a detailedanalysis of Islamic geometrical patterns. A givenIslamic pattern will frequently employ a small numberof precisely determined shapes, some of which becomerepeated within the pattern in many ways not allowedfor under the four elementary operations, or isometries,of classical plane symmetry; 8 each of the latter must acton the whole two-dimensional plane, not merely locallyon small parts of it. Similarly, although many motifsthemselves possess higher than 6-fold rotational sym-metry, they cannot form repeated centers of similarrotational symmetry in the plane as a whole, since thepermitted centers can have no more than 2-, 3-, 4-, or6-fold symmetry (such "rotocenters" are termedrespectively diads, triads, tetrads, and hexads). On thesurface of the sphere the restrictions are somewhat dif-ferent, whereas in the hyperbolic plane virtuallyanything is possible."

In their simplest form all Islamic geometrical pat-terns are examples of periodic tilings (or tessellations)of the t~o-dimensional plane, consisting of polygonalareas or cells of various shapes abutting on neighboringcells at lines termed the edges of the tiling, and withthree or more cells meeting at points termed the ver-tices, or nodes, of the tiling. In general, the cells are notrequired to be convex polygons, nor the edges to bestraight lines, and there is no restriction on the numberof edges or cells meeting at each vertex."?

From the earliest times Islamic ornament adoptedthe widespread interlacing band form of linear decora-tion, whereby the original lines of the pattern arerepresented by straps or bands, executed in such a wayas to give the impression of weaving alternately overand under one another. This style of ornament had itsorigins in antiquity, and must ultimately derive byimitation from various types of weaving, plaiting, orbasketwork. As an artistic device it serves to give cohe-sion to a whole design. We may note that an inter-lacing-band style can only be achieved when theoriginal tiling consists entirely of 4-way nodes, that is,four edges (and therefore cells) meet at every node. 11Ideally, opposite angles at each node should be equal,which means that the four edges meeting at that nodebecome a pair of lines intersecting at a crossover point.Patterns consisting entirely of 4-way nodes may be

referred to as true interlacing patterns, whether or notthey are drawn as interlacing bands.

Interlacing patterns have another important prop-erty: in their linear, as opposed to interlacing-band,form the cells of any pattern may be colored alternatelyin two modes-say, black and white-so that no twocells of the same mode meet at a shared edge. Achessboard is a familiar example. Strictly speaking, thepossibility of a two-mode coloring is inherent in anytiling in which an even number of cells or edges meetsat every node. Non-interlacing patterns contain at leastsome nodes which are not 4-way. If some of these areodd-numbered, then a two-mode coloring is no longerpossible. The star patterns in general include examplesfrom both interlacing and non-interlacing categories.

Islamic star motifs owe their beauty and regularity toa feature which they share with the regular convexpolygons: all derive from sets of points equallydistributed around the circumference of a circle. Whenpairs of adjacent points in any such set are joined bystraight lines until a single circuit is completed, theresult is a regular convex polygon. However, it is possi-ble to continue joining up every other point, or everythird point, and so on, to produce many beautiful star-like figures. In a loose sense these may all be termed"star polygons," although a star polygon, properlyspeaking, results only when all the lines so producedform a single circuit surrounding the center of thefigure more than once. The set of points on the initialcircumference comprises the vertices of the star polygon,but the sides of a star polygon intersect one another atvarious additional points, which are not counted as ver-tices, although they divide each side into a number ofsegments.

The earliest Islamic star motifs were based on a starpolygonal construction, but complete star polygonswere rarely used as ornamental motifs (usually in isola-tion, as medallions). Initially such a construction pro-duces a space at the center of the figure, in the shapeof a regular polygon (fig. 1). In authentic Islamic orna-ment this central space is usually transformed into astar-shaped area by the omission of one or more of themiddle segments on all sides of the star polygon. Usu-ally all but the last two segments at each end of a sideare omitted, thus producing the typical Islamic starmotif (fig. 2). This figure therefore consists of an innercell, or central star, and a number of outer cells in theshape of kites. Occasionally all but the outermost seg-ment are omitted (fig. 3), and we thereby arrive at thesimplest form of a regular geometrical star.


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184Star motifs of these types can be distinguished by a

concise notation, giving data on three quantities: thenumber of initial vertices; the method of joining up thevertices to produce the original star polygon (i.e., twoby two, three by three, and so on); and the number ofend segments remaining at each end of the sides of thestar polygon. These three quantities can be representedby n, d, and s, respectively, and the complete symbolfor the basic Islamic star as (nld)s. Thus, the star shownin fig. 2 can be designated an (8/3)2 (this may be readas an "eight over three, two-segment star"). Obviouslythis notation, which is derived from the mathematicalnotation for star polygons, is applicable to any Islamicstar constructed by drawing straight lines between pairsof points on a circle. Indeed, it may even be adaptedto certain other types of construction, if we allow non-integral values for d.

Motifs based directly on star polygons (sens. lat.) areeasily constructed using a single circle and a number ofpoints equally distributed on the circumference of thatcircle. No other initial construction is necessary. Manylater Islamic star motifs, however, are not derived inthis way, although they may still consist of a central starand surrounding kites. In these later star motifs theslope of the lines forming the star is such that they can-not be constructed simply by joining pairs of points onthe circumscribing circle of the motif. In these cases,one or more additional concentric circles are needed todetermine the inner points of the star motif and henceto complete the lines (fig. 4). Islamic star motifs are ofmany different types, and the main varieties will beindicated below, but all include a simple star as a cen-tral cell. The precise metrical properties of any starmotif may sometimes be arbitrarily chosen, but fre-quently depend on geometrical considerations concern-ing the relation of the motif to other elements in apattern.

The symmetrical properties of any regular n-pointedstar motif or n-sided regular polygon (n being anywhole number greater than two) can be represented asa system of 2n radii diverging from the geometricalcenter of that motif. This figure is conveniently termeda star-center; it consists of n principal radii, through themain, outer points of the motif, and n secondary radii,alternating with them. The continuing invention offundamentally new star patterns entails a search for allsuitable arrangements of star-centers in the two-dimen-sional plane, but the number of possibilities is limitedby the precise way in which each star-center must beoriented in relation to its nearest neighbors. In the

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3 4Figs. 1-4. The formation of star-motifs from star polygons.

method of linking star-centers that is preferred aboveall others in Islamic patterns, one radius from eachstar-center of a neighboring pair is coincident with thestraight line joining their centers. In the case of a pairof star motifs, their centers and shared point of contactlie on a single straight line (fig. 5). This is thereforetermed a co ll in ea r l in k .

Another acceptable method of linking nearby star-centers is by having their nearest radii parallel, ratherthan collinear. In this case the straight line joining theircenters no longer coincides with a radius of either star;such a relation may be termed a p ara ll el lin k (fig. 6).Parallel links between star motifs are principally foundin certain derivative patterns (described below) whichare usually far more difficult to construct with a rulerand compass than those using collinear links. (It isprobable that most of these derivatives were originallycomposed through rearrangements of certain elemen-tary mosaic shapes-for example, cut tiles or pieces ofwood inlay.)

Rules such as these for linking adjacent star motifswere never explicitly stated by Muslim artists, but wereapplied, probably unconsciously, in large numbers ofvaried patterns throughout Islam. Indeed, the employ-ment of collinear links to join star- or flower-like motifs


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ISLAMIC STAR PATTERNS 185

Fig. 5. A collinear link between two 9-pointed stars.

Fig. 6. A parallel link between two IO-pointed stars.

antedates their use in Islamic patterns by many cen-turies, and has probably always been felt instinctivelyto represent a more elegant method of pattern com-position.

Many early forms of repeating patterns were basedon simple fractional divisions of underlying grids ofequilateral triangles or squares, both of which requirethe ruling of sets of parallel lines right across the pat-terned area. It was to such grids that the first designersof Islamic patterns turned to produce the earliest starpatterns. The simplest procedure is to center a starmotif on every vertex of the grid, with a radius equalto half the edge length of the grid polygons. The motifsare then oriented so that certain of their principal radiicoincide with grid lines, and each pair of adjacentmotifs meets at the midpoint of an edge of the grid.Thus, collinear links are automatically established.Since the triangular and square grids themselves haverespectively 6- and 4-way vertices, the simplest motifswhich can be used in this way will have 6 of 4 points.In fact, it should be obvious that motifs with anynumbers of points which are integral multiples of thesevalues can be similarly placed on the grid vcrtices.!"

It is not easy to establish the exact date at which sim-ple star patterns of this kind were first used in Islamicarchitectural decoration. Some attempts have beenmade to explain the historical and developmentalorigins of the first rectilinear star patterns.":' but sincecurvilinear versions of many of these occurred evenearlier, perhaps one should really try to explain theorigins of the latter. In any discussion of historicalorigins, we must bear in mind that although it is stilltheoretically possible to locate the earliest survivingversion of every distinct pattern, it cannot necessarilybe assumed that anyone of these really represents thefirst historical occurrence of that particular pattern. Itis more than likely that most early examples of the arthave long ago crumbled to dust. Nevertheless, suffi-cient material has survived from the eighth to the tenthcenturies to give a tantalizing glimpse of what musthave been an extremely rich fund of very early geo-metrical ornament. Recognizable precursors of the firststar patterns started to appear in the Middle East asearly as the beginning of the eighth century, in the formof open work window grilles in the Great Mosque atDamascus (715),14 the palace of Qasr al-Hayr al-Gharbi in Syria (727),15and the palace of Khirbat al-Mafjar in Jordan (743).16 Although some of thesedesigns contain star-like elements, it is not always clearwhether they represented distinct motifs in the artist'soriginal conception, or whether they arose merely asresidual spaces between groups of overlapping circles,circular arcs, or other shapes. In some cases, however(for example, that shown in fig. 7, of which almost iden-tical versions survive from Qasr al-Hayr al-Gharbi andKhirbat al-Mafjar) , we are probably correct in inter-

Fig. 7. A pattern from the eighth century with curvilinear 8-pointedstars.


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186preting the pattern as consisting essentially of an arrayof star motifs (in this case, of curvilinear 8-pointedstars) in contact.

Two of the earliest repeating patterns using simplerectilinear stars consisted of 6- and 8-pointed stars,respectively (figs. 8 and 9), both of which had theirorigins in classical antiquity."? In an Islamic context thestar-and-cross pattern (fig. 9) first occurs in ninth-century Samarra.!" a version of the pattern with6-pointed stars occurs in the mosque of Ibn Tulun atCairo (876-79);19while both can be seen in stucco fromthe palace of Madinat al-Zahra (936-76) in Spain.!"and in al-Hakim's mosque in Cairo (1003).21However,the star-and-cross pattern probably appeared even ear-lier, since a rectilinear version occurs on one of the win-dows from Qasr al-Hayr al-Charbi.P in which all linesegments are extended as straight lines running ininterlaced form right through the pattern. Anotherexample of the pattern with 6-pointed stars occurs in anearly-eleventh-century house excavated in Siraf. 23

Thus by the end of the tenth century at the latest,patterns containing 6- and 8-pointed rectilinear starswere fairly widespread, and it probably required littleintellectual effort on the part of the original artists toconceive the idea of incorporating simple stars withgreater numbers of points in repeating patterns.v" It isnot until the second half of the eleventh century thatexamples survive, however. A pattern with 12-pointedstars superimposed on a triangular grid occurs on theearlier of the two Kharraqan tomb towers (1067-68)25in northwestern Iran (fig. 10). A related pattern, usingthe same (12/4)2 stars, but on the square grid (fig. 11),might be expected to have been discovered at about thesame period, but no early examples appear to have sur-vived. These patterns were later among the most wide-spread of all star patterns; they are found fromMorocco to Central Asia, indicating in both cases veryearly discovery and dissemination. Similarly a patternof (8/3)2 stars (fig. 12) had probably been discovered atthis time, since the basic star itself was already inexistence.s" but again no early examples appear to havesurvived, though what could be regarded as a cur-vilinear version from the eighth century is illustrated infig. 7.

The patterns developed so far immediately illustratean important feature in the execution of nearly allIslamic star patterns. When the number of points in thestars is greater than the number of lines radiating fromeach node of the grid on which they are superimposed,there remains the problem of what to do with the

A. J. LEE

Figs. 89. Two of the earliest rectilinear star patterns.

Fig. 10. 12-pointed stars on a triangular grid.

Fig. 11. 12-pointed stars on a grid of squares.


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ISLAMIC STAR PATTERNS 187"free" vertices of each star, i.e., those not in directcontact with neighboring stars. The simplest and mostelegant solution is to continue the edges of the starsthrough each free vertex until they meet similarly pro-duced lines from neighboring stars. In the case of thesquare array of 8-pointed stars (fig. 12) this results ina small interstitial 4-pointed star, whereas in the case ofthe triangular array of 12-pointed stars (fig. 10) a smallequilateral triangle is produced. In these two patternseach star contributes only a single free vertex in eachgrid polygon, and the total number of free vertices ineach polygon is obviously equal to the number of sidesin that polygon. The majority of star patterns obtain agreater measure of continuity between their constituentstar motifs by bridging the space between free verticesin this way, although the artist is always at liberty notto do so if he wishes. The additional lines thus pro-duced by joining up such free vertices may be referredto as constituting the interstitial pattern, since they occurin the residual space between groups of three or moreneighboring stars. The greater the number of free ver-tices available in this space, the more complex is theinterstitial pattern. The nature and symmetry of theelements in the interstitial pattern are clearly related tothe symmetry of the pattern as a whole, as well as to thetypes of constituent star motifs and their precise con-struction.

Patterns using 10-pointed stars occur in the northdome chamber of the Masjid-i Jamie at Isfahan(1088).21 These "decagonal" patterns represent adeparture from traditional methods of geometrical pat-tern design, although they result inevitably from alogical generalization of the principles developed so far.

Fig. 12. 8-pointed stars on a grid of squares.

It is not immediately obvious how to form repeatingpatterns with 10-pointed stars, since neither of the gridsconsidered so far is suitable. If the stars are required toremain in contact, using collinear links, however, it isfound that there is only one simple arrangement possi-ble, and this could easily have been obtained after alittle trial and error (fig. 13). This pattern and the newgrid which underlies it require a more detailed analysis(which will be given below) that leads to the formaldiscovery of many kinds of patterns. These includesimultaneously two kinds of star motifs, including someof the most widespread of all Islamic star patterns. It isby no means suggested, however, that the originaldiscovery of such patterns followed from the methods ofanalysis outlined here. Rather more elaborate patternswith lO-foldmotifs occur on the later, so-called VictoryTower of Masud III at Ghazni in Afghanistan (prob-ably dating from around 1100).28This structure alsopresents what appears to be the first occurrence inIslam of geometrical patterns incorporating 7-foldmotifs (in fact these same patterns also include 20- and14-pointed stars, respectively).

These "heptagonal" patterns, in which the propor-tions ofthe basic repeat are determined by angles whichare multiples of 18017, are extremely difficult to workwith. This difficulty is reflected in the paucity ofexamples throughout the whole range of Islamic geo-metrical ornament. About 6 percent of Bourgoin's col-lection'" consists of true heptagonal patterns, and inthis respect it is probably representative of Islamic pat-terns as a whole. Indeed, patterns using many otherkinds of odd-numbered star-motifs are often very dif-ficult to incorporate in repeating patterns, and were

Fig. 13. 10-pointed stars of Type I on a grid of 72/108 rhombs.


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188therefore seldom or never used. This is particularly soin the case of those motifs in which the numbers ofpoints are prime numbers-say, 11, 13, 17, 19, and soon.

Before analyzing some of these new kinds of pat-terns, wemust first consider the discovery and develop-ment of what is perhaps the most typically "Islamic"of all star motifs: the geometrical rose tte (fig. 14). The pro-totype for the general n-rayed rosette almost certainlyconsisted of a 6-pointed star surrounded by six regularhexagons (fig. 15). This configuration becomes auto-matically incorporated in the pattern of 6-pointed starsdescribed above (fig. 8), but the prototypical 6-rayedrosette seems to have been used for the first time as adistinct motif on the Arab-Ata mausoleum (978) atTim, in Uzbekistan"? (fig. 16), and the same patternreappears, with slightly different proportions, on theearlier of the two Kharraqan tomb towers."! A methodof constructing the isolated 6-rayed rosette can bederived from the construction used to produce the

Figs. 14-15. 12- and G-rayed geometrical rosettes.

Fig. 16. The earliest occurrence of a 6-rayed rosette as a discretemotif (late 10th century onwards).

A. J. LEEoriginal pattern of 6-pointed stars (fig. 17). The centersof the (six) surrounding per ipheral s ta rs form the verticesA of a l im i ti n g po lygon of the rosette, and circles centeredon these vertices, with a radius AB, equal to half theedge length of the limiting polygon, determine the posi-tions of points such as C, and hence the radius CD ofthe interior star of the rosette. The hexagons may betermed the o uter cells of the rosette; the terminalsegments a of these outer cells are obtained by drawingstraight lines between such points as Band E, while thesides b of the outer cells are obtained by drawing linesthrough points such as C, parallel to a principal radiusBD of the rosette. In such a case the outer cells of therosette may be said to possess collinear terminalsegments, and parallel sides. Alternatively the rosetteitself can be described as a parallel-sided rosette withcollinear terminal segments.

The construction just given has purposely made noreference to 6-fold symmetry or to absolute angle sizes,so it is capable of immediate generalization to includegeometrical rosettes with any number of rays (fig. 18),provided only that the limiting polygon is a regularpolygon. A geometrical rosette of this kind will auto-matically be produced, with collinear terminal seg-ments and parallel sides. In addition, although it is notimmediately obvious, segments a and b are a lw ay s e qu al(this can be proved quite easily). The outer cells of thegeneral n-rayed rosette are always symmetrical hex"agons, but only when n = 6 are they regular.

When n is greater than 6, the geometrical rosette isnearly always constructed so as to include a 2-segmentstar, as defined above, in the circle with radius CD (fig.18). This may be termed the outer s tar of the rosette, andthis in its turn contains the inner or c en t ra l s ta r. Withregard to constituent polygonal areas, the n-rayedrosette now consists of n hexagonal outer cells, n kite-shaped midd le c el ls (or midcells), and an n-pointed cen-tral star. Since the angle BAC is greater when n isgreater than 6 (cf. figs. 17, 18) it is obvious that exactlyregular G-pointed peripheral stars are only possiblewhen n = 6. For higher values there is not enoughroom for a perfectly regular 6-pointed peripheral star,but a regular 5-pointed peripheral star only becomespossible when n = 10. Furthermore, since the anglebetween a pair of adjacent sides of any regular polygonis never 1800, it is obvious that regular 4-pointedperipheral stars are impossible, with the propertiesgiven above. Thus, 5-pointed peripheral starsinevitably corne to be included in many patterns con-taining geometrical rosettes, and although they can be


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ISLAMIC STAR PATTERNS 189.......... . . . .

1 9Figs. 17-19. The formal construction of the geometrical rosette, gen-

eralized from the 6-rayed example in fig. 17.

regularly formed in relatively few cases, ideally they aremade as regular as is possible without reducing thesymmetry of the main rosettes. The highest degree ofregularity is achieved as follows: the outer points of aperipheral star must lie on a circle, the center of whichis a vertex of the limiting polygon of the parent rosette;the angles at its outer points should be equal; the sidesor segments bordering all outer points should be equal;and the peripheral star should form collinear links withthe outer star of its parent rosette(s), and with neigh-boring peripheral stars. Furthermore, in this most reg-ular form, the bisectors of its outer angles meet at asingle point, which may be termed the center of theperipheral star, since it coincides with the center of thecircumscribing circle.If a pair of peripheral stars is shared between two

equal rosettes which are joined by a collinear link (fig.19), then it is obvious that not only do the rosettes sharean outer point B, but their limiting polygons share anedge AA'; in other words the edge length of the twolimiting polygons is the same. In fact, a pair of linked

rosettes could be constructed on the basis of a pair ofregular polygons sharing an edge, and indeed the con-struction is easily generalized to include different kindsof regular polygons, and therefore rosettes of differentsizes, but in this case only one of the rosettes can beconstructed strictly according to the principles we haveestablished so far. The slopes of the various line seg-ments of the other rosette are then largely determinedby those in the first rosette, but the sides and terminalsegments of the outer cells of the second rosette can stillbe constructed as equal lengths, if required, by notingthat the shoulder, or subterminal point (G, fig. 18)should always lie on the bisector of angle BAC. In orderto produce repeating patterns with geometrical rosettesone might then search for tessellations or open arrange-ments of regular polygons in which pairs of adjacentpolygons share an edge. These are the essentials ofwhat might be termed the "polygons in contact" (PIC)method of constructing certain Islamic star patternsthat was first enunciated as a general principle by E. H.Hankin.V although it was occasionally used earlier byBourgoin.!" However, in most cases it is not necessaryto draw a complete arrangement of limiting polygonssince the construction of just one shared edge is suffi-cient to determine the relative circumradii of a pair ofadjacent rosettes, and subsequently the radii of theirouter stars. In fact there are other exact constructionspossible which achieve the same result without usingthe limiting polygons at all.

The earliest patterns with geometrical rosettes weremostly constructed with the properties, if not themethods, outlined above, i.e., with collinear terminalsegments and parallel sides. Apart from the case whenn 6, rosettes are completely absent from theextremely rich ornamentation of the two Kharraqantomb towers (1067-68 and 1093, respectively), butgeometrical rosettes with 10 rays occur in the northdome chamber of the Masjid-i jami", Isfahan (1088).34Rosettes with 8, 11, 12 and 16 rays occur in variouspatterns round the mihrab of the mosque at Barsian ,Isfahan (1134)35-and, incidentally, include the earliestexample of a remarkable pattern with 4-, 5-, 6-, 7-, and8-pointed stars. 36 Patterns with 8- and l.U-rayed rosettesoccur on wooden minbars from the mosque of Ala al-Din at Konya (1155),37 and from the Aqsa Mosque atJerusalem (1168).38It seems possible, however, that the6-fold case was generalized to include 8-, 10-, and per-haps 12-rayed rosettes well before the end of the: elev-enth century, and that the majority of early exampleshave simply not survived. A more thorough search of


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ISLAMIC STAR PATTERNS 191Unfortunately it is not possible to enter into con-

structional and other details here, so further patterntypes on the same basis can only be briefly describedand illustrated. Types IV (fig. 23) and V (fig. 24) aresimply elaborations of type I patterns and are confinedto Central Asia. Type VI (fig. 25) is also derived froma type I, this time by expanding each kite-shaped celluntil it overlaps its neighbors in small rhombs. Initiallythe size of the expanded cells can be arbitrarily chosen,but in certain derivatives of this type the relative sizesof the pattern cells become rigidly determined. A fre-quent addition to this pattern is the incorporation oftype II rosettes at the center of each type VI motif (seefig. 25, right side). Types VII (fig. 26) and VIII (fig.27) are also Central Asian and are ultimately derivablefrom elements of types I and III and their later elabora-tions. In type VIII all star-centers are surrounded byregular decagons, but only alternate centers of type VIIare so surrounded. The interiors of the decagons arevariably treated in authentic sources. The fact that typeVII patterns simultaneously contain two kinds ofcenters means that either kind can be given prominencein a small pattern area, creating quite different effects.The same remarks apply to types IX (fig. 28) and X(fig. 29), alternate centers of which employ regulardecagons and 20-gons, respectively. These last two,and many related patterns, are almost exclusively usedin wooden lattice work. Types XI (fig. 30) and XII (fig.31) again form a related pair of patterns, with similarperipheral elements. Other varieties are possible, butthese twelve types are the most common variants.

In addition there are many derivative variations,some of which are described below. There is somejustification in giving distinct designations to thesevariations, since analogous treatments were applied, orcan be applied, to many other arrangements using starmotifs of different sizes. Similar variations may there-fore be given the same type designation, irrespective ofdifferences in the numbers of points in the constituentstar motifs.

The pattern of 10-fold star-centers which underliesall the foregoing pattern types is formally related to agroup which shares the same structural basis as the72/108 rhombus, with respect to the pattern of prin-cipal and secondary radii within the rhombus. The twoaxes or diagonals of a rhombus divide it into four equalright-angled triangles (fig. 21) and in the present caseeach has, in addition to the right angle, interior anglesof 54 and 36. Radii from the star-centers on thesetwo vertices divide their respective angles into three

Fig. 23. A pattern with lO-fold motifs of Type IV.

Fig. 24. A pattern with IO-fold motifs of Type V.

Fig. 25. A pattern with lO-fold motifs of Type VI. On the right aType II rosette is inserted.


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Fig. 26. A pattern with lO-fold motifs of Type VII. Motifs are of twokinds, one of which is surrounded by complete decagons. Authentic

treatment of the latter is variable, as shown above.

Fig. 27. A pattern with lO-fold motifs of Type VIII. All motifs aresurrounded by decagons.

and two divisions (fig. 32), each division of 180/10, or18. Let us label the vertex with three divisions the m-center, and that with two divisions the n-center. Thesetwo symbols represent the number of principal radii ineach complete star-center, so that initially m =n =10.This number may be termed the star number (or rosettenumber). Let p then represent the number of inter-radial divisions contributed by the m-center, and q thenumber of divisions at the n-center, so that p =3 andq =2. These quantities, however, may be regarded asspecific values of a general relationship between thefour variables m, n, p . and q. Ifwe express the anglesat m and n as fractions of 180 they become respectively

A. J. LEE

Fig. 28. A pattern with lO-fold motifs of Type IX. This and thefollowing pattern are usually executed in wooden lattice.

Fig. 29. A pattern with lO-fold motifs of Type X. As in fig. 28, motifsare of two kinds. Here, one kind is surrounded by complete 20-gons.

plm and qln; the right angle is obviously 1/2. As frac-tions of 180 these three angles will therefore sum tounity, i.e. plm + ql n + 112=1, from which we obtain therelationships m =2npl(n - 2q) and n =2mql(m - 2P).

We are interested in cases where rhombs might occurwhich are structurally similar to the decagonal examplepreviously considered, i.e., in which p =3, q =2. Sub-stituting these values in the two expressions above weobtain two equations which can be solved for possiblepairs of integral values of m and n, one of which mustobviously be m =n =10. In fact the number of possiblepairs is finite, and there are only eight integral solu-tions. These are 30,5; 18,6; 14,7; 12,8; 10,10; 9,12;


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ISLAMIC STAR PATTERNS 193

Fig. 30. A pattern with l Ovfold motifs of Type XI.

Fig. 31. A pattern with 10-fold motifs of Type XII.

p = = 3 divisions~1\\1\\ m

n~'I= 2 divisions

Fig. 32. Division of the 72 1108 rhomb into equal angles of 18,producing a (3 x 2) rhomb.

8,16; and 7,28 (in each case the m-center is the first ofthe pair of values). 41

These solutions represent in effect a series of rhombsin which m-fold and n-fold centers occur in oppositepairs on alternate vertices of each rhombus. We cannotimmediately build up repeating patterns with theserhombs, however, since none of them will form a sim-ple grid like the pattern of 10-fold star-centers withwhich we began. But patterns incorporating all exceptthe first and last solutions have been used in authenticornament, although only four of them-18,6, 12,8,10,10 and 9,12-can be used by themselves to formrepeating patterns; the others require additionalshapes. The beauty of this series is that similar varia-tions, based on the decagonal types I-XII that we havebriefly described above, can be adapted to eachrhombus in the series. Theoretically, one might there-fore expect at least 96 distinct patterns from this seriesalone, but in cases where the resulting star-motifs aretoo dissimilar in size the patterns are often not verysatisfactory.

Using the same symbols as before, we may refer toseries of (p x q) rhombs, and the specific solutions ofsuch a series may be expressed in the form (p x q )m , n .Thus, the rhombs in the grid shown in fig. 21 become(3 x 2)10,10 rhombs. The general notation for par-ticular pattern types in the (3 x 2) rhomb series may bewritten as (3 x 2)m,nIT, where T stands for one of thetwelve distinct types given above. Each rhombus in thepattern of fig. 20 thus becomes (3 x 2)10,10/II. Strictlyspeaking, of course, this notation ought to refer to theright-angled triangle which constitutes one quarter of arhombus, but it is natural to extend it to include notonly the complete rhombus, but also if necessary thetwo kinds of isosceles triangles which can be producedfrom two such identical right triangles. The context willmake it clear to which we are referring at any particulartime. It must be pointed out that the notationdeveloped so far does not designate a repeating pattern,but merely a potential elementary unit for one or morerepeating patterns. Additional symbols are required toindicate ways in which such elementary units are incor-porated in repeating patterns, but this is unfortunatelya question which cannot be pursued here.

As we have remarked above, the decagonal types 1-II I form special cases of an infinitely variable series, in

,which the two interstitial cells are congruent to theouter cells of the star motifs in each case. A similarproperty is characteristic of other (3 x 2) rhombuseswith dissimilar star numbers, but in these cases the


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194interstitial cells can be made congruent to the outercells of only one of the two motifs, namely the m-motifs(although this congruence was rarely achieved inauthentic patterns). This is an inescapable consequenceofthe underlying geometry of this series ofrhombs, andit is capable of rigorous geometrical proof (a similargeometry is exhibited by the (2 xl) rhomb series). Fun-damental distinctions of this nature between the m- andn-centers of the (3 x 2) and other rhomb series wereseldom understood by the original artists, and have cer-tainly not been appreciated by Western authors.Among a number of lines of evidence leading to thisconclusion, we may cite the many differently con-structed versions of patterns using the (3 x 2)12,8/IIrhombus in existence (the "rhombus" being in fact asquare), some of which indicate that the artists con-cerned had no idea of the "correct" method of con-struction.P The PIC method will certainly allow thecorrect relative sizes of different star motifs to beachieved.P and this method is essential for type I pat-terns with dissimilar stars, but it is not appropriate forall twelve variations dealt with above, and there areeven certain rosette patterns in which it cannot be used.

Among other rhombs in the (3 x 2) series whichbecome incorporated in the more common patterns, wemay mention (3 x 2)12,8/11, which occurs in a varietyof versions from India to the Maghrib, as does (3 x2)9,12/II. Type I patterns using these two rhombsoccur from India to Egypt, but seem to be absent fromthe Maghrib. (3 x 2)8,16/11 is commonly incorporatedin Maghribi patterns but is absent or rare elsewhere;(3 x 2)8,16/1, on the other hand, seems to appear occa-sionally in Iran, but is entirely absent elsewhere.(3 x 2)12,8 types VII and VIII are common in CentralAsia, but do not appear in other areas. It is, however,difficult to compile an inventory of pattern typesthroughout Islam from published sources, since it isextremely rare to find a work which illustrates everypattern occurring on a single monument, let alone awhole geographical area or historical period. Of recentpublications notable exceptions are the work ofStronach and Young referred to above, and that of theErdmanns on thirteenth-century Anatolian caravan-serais.t"

Variations such as the types I-XII we have been con-sidering do not of course constitute all possible sourcesfor variety in patterns which use (3 x 2) rhombuses. An,extremely common derivative type I pattern is shownin fig. 33. Here the stars are separated by a "twinnedpentagon" shape, forming a linear sequence which also

A. J. LEE

Fig. 33. A derivative pattern of lO-pointed stars, formed by re-arrangement of elements in fig. 13.

occurs along the short axis of the original type Irhombus (fig. 13). One could continue this process,taking linear sequences along the axes of the rhombusof a basic pattern type and using the sequence obtainedas the edge of a new, larger rhombus of the same shape(or of a different shape: there are two kinds of rhombspossible with 10-fold star-centers, which may be distin-guished as (3 x 2) and (4 xl) rhombs, using the presentnotation). However, although this is a very fruitfulmethod for generating derivative patterns, and mayindeed have been used on occasion by Muslim artists,there are many similar patterns which cannot bederived in this way.

A different source of variety makes use of parallellinks (fig. 34). Only one example is illustrated, but it ispossible to classify such patterns in many differentcategories; there is no end to this kind of variation. Ofeven greater interest, it is possible to imitate many ofthese derivative decagonal patterns in other rhombs ofthe (3 x 2) series, especially in the case of (3 x 2)12,8and (3 x 2)9,12, since in these there is very little distor-tion of the pentagons, pentagrams, and other shapes ofthe decagonal patterns. A number of such derivativesexist as authentic patterns throughout Islam.

In the limited space available it has been possible togive no more than a glimpse of some of the methods bymeans of which these star patterns can be investigated.There are of course many numerical procedures whichcan be used to investigate different combinatorialaspects of star-pattern construction beyond thosebriefly mentioned above. We have dealt with onlyrhombic configurations of star-centers-and even here,


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195SLAMIC STAR PATTERNS

Fig. 34. A further derivative pattern of 10-pointed stars, in whichsome stars are joined by parallcllinks. The underlying grid is rhom-bic, but the rhomb angles arc no longer simple multiples of 18.

with only one series, the (3 x 2) series-and yet we havenot described ways in which rhombs of one or morekinds can be arranged to build up repeating patterns.This would entail a classification of rhombic tessella-tions along lines somewhat different from thoseemployed III classifying tessellations of regularpolygons.

There are also many possibilities for repeating pat-terns using almost regular star-centers, relatively few ofwhich exist as authentic Islamic ornament, and thesealso lend themselves to numerical methods of investiga-tion. Although by no means all authentic patterns arebased on rhombic tessellations, this does form a veryuseful approach since it accounts for most of the com-moner patterns, including hexagonal and square-basedarrangements as special cases. It is not suggested thatthe first designers of Islamic patterns used similar linesof reasoning to those outlined above, but theoreticalstudies of this kind are of particular value in that onecan attempt a systematic and exhaustive enumerationof all possible ways of combining authentic star motifsand their variation. Since no one can possibly study thevast profusion of existing Islamic patterns at first handin order to be able to classify all authentic varieties, atheoretical approach is clearly an advantage. It thenbecomes of interest to compare the results of suchstudies with the results actually achieved by the originalMuslim artists, since the comparison may provide cluesto the early craftsman's understanding of the deepergeometry of his patterns.

NOTES1. K. A. C. Creswell, A Bibliography oj theArchitecture, Arts and Crajts'if Islam to lsl January, 1960 (Cairo: American University at

Cairo Press, 1961), "Ornament," pp. 963-78, and Supplement(1973), "Ornament," pp. 29091; D. Wade, Pattern in IslamicArt (London, 1976); I. El-Said and A. Parman, GeometricCon-cepts in Islamic Art (London, 1976); K. Critchlow, Islamic Pat..terns: An Analytic and CosmologicalApproach (London, 1976). Fora general appraisal of the role of ornament in Islamic art, secD. Jones, "The Elements of Decoration: Surface, Pattern andLight," in George Michell, ed., Architecture oj the Islamic World(London, 1978).

2. Perhaps the intended collaboration between Gomez-Morenoand Prieto Vives was to have been such a work, but only partof it was ever published (A. Prieto Vives and M. Gomez-Moreno, EI Lazo: Decoracion geomitrica musulmana [Madrid,1921]). A great deal ofunpublished material still exists from thiscollaboration, however, and it is to be published in the nearfuture (see M. Gomez-Moreno, "Una de rnis teorias delLazo," Cuademos de laAlhambra 10-11 [Granada, 1974-75]). Fora discussion of this abortive collaboration and a list of paperspublished by Prieto Vives, see D. Cabanelas, "La Antiguapolicromia de techo de Comares en la Alhambra," Al-Andalus35 (Madrid, 1970): 4-35-37.

3. See the discussion by J. M. Rogers, "The 11th Century-ATurning Point in the Architecture of the Mashriq?," in D. S.Richards, ed., Islamic Civilisation 950-1150 (Papers on IslamicHistory 3) (Oxford, 1973), pp. 221-24.

4. Oleg Grabar (The Alhambra [London, 1978]), referring (onpp. 195-96) to an unpublished doctoral thesis, cites "evidencewhich is only now being discovered" indicating that newmethods of composing geometrical patterns after the tenth cen-tury, emphasizing polygons and stars, were "made possible bya conscious attempt on the part of professional mathematiciansand scientists to explain and to guide the work of artisans." Itisnot yet clear whether this guidance refers to a choice of motifsor their construction, or to methods of combining motifs inrepeating patterns, or whether perhaps the mathematiciansthemselves are supposed to have designed new patterns effec-tively. Even if this claim is true, the influence of the mathemati-cians cannot have been very widespread or long lasting, judgingby the many clumsily constructed patterns in existence, oftenemploying widely varying and sometimes arbitrary methods oflayout, or even drawn more or less freehand.

5. R. Orazi, WiJOdenGratings in Sa(avid Architecture (Rome, 1976),text in Italian and English, p. 104 and note.

6. My own unpublished research.7. E. H. Lockwood and R. H. Macmillan, Geometric Symmetry

(Cambridge, 1978).8. These are (1) rotation about a point, (2) displacement in a given

direction, (3) "reflection" across a straight line, and (4) a"glide-reflection," which combines (2) and (3) in oneoperation.

9. Although many examples of Islamic patterns occur on thecurved surfaces of domes, true spherical patterns are rare; theyare principally found as quarter-spheres capping the hemicylin-drical niche of many mihrabs. Hemispherical bosses sometimesoccur, decorated with geometrical patterns. A sphere is a realsurface of constant positive curvature, and Arab mathemati-cians were familiar with its geometry. The hyperbolic plane on


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196the contrary is an infinite imaginary surface of constantnegative curvature (i.e., it is everywhere saddle-shaped)discovered by European mathematicians in the early nineteenthcentury. It is possible to map the whole hyperbolic plane insidea circle, and in this form many interesting tessellations havebeen made visible. The Dutch artist Maurits Escher adaptedmany of his bizarre tessellat ions to this circular representat ion.See J. L. Locher, ed., The World of M . C. Escher (New York,1971). No one seems yet to have attempted to adapt Islamic pat-terns to the hyperbolic plane.

10. The class of tilings I have described here have been termededge-to-edge tilings. However, other types exist in which, forexample, one of two vertices limiting an edge of one tile lies partway along an edge of an adjacent ti le . Some Islamic ti le pat ternsare of this type.

11. If the interlacing bands form only closed circuits or infinitelengths without any loose ends, it can be proved that thecrossovers can always be arranged so that along any band theyrun alternately over and under other bands; two adjacentcrossovers of the same type need never occur. Thus the artisthas no need to worry that his interlacing pattern will not "workout," provided he does not start the crossovers simultaneouslyin more than one place. The interlacing principle is linked tocertain topological properties of the plane, and there are othermathematical surfaces on which it does not hold. See H. A.Thurston, "Celtic Interlacing Patterns and Topology," ScienceNews 33 (1954): 50-62.

12. In the tessellat ion of equilateral tr iangles, the vertices consti tutehexads, the centers of the triangles triads, and the midpoints ofthe edges diads. Any stars can be placed on these rotocenters inwhich the numbers of points are multiples of 6, 3, and 2 respect-ively. Similarly the tessellation of squares can accommodatenumbers that are multiples of 4,4, and 2 respectively. Most, ifnot all, of the simpler permutations of this kind are encounteredin authentic Islamic patterns.

13. J. M. Rogers, "The 11th Century" (cited above, n. 2).14. K. A. C. Creswell, A Sh ort Accou nt of Early M uslim Architecture(Harmonsworth: Penguin Books, 1958), pp. 75 ff., and pl. 16.15. M. Selim Abdul-Hak, "La reconstitution d'une partie de Kasr

al-Heir al-Gharbi au Musee de Damas" (in Arabic), L es A n n al esArcheologiques d e S yr ie 1(I)B (Damascus, 1951), pp. 5-57, pIs.15-24.

16. R. W. Hamilton, Kh ir b at a l-Maf ja r (Oxford, 1959).17. B. Pavon Maldonado, E l A rte h ispan om usu lm dn en su decoracion

geometrica (Madrid, 1975).18. E. Herzfeld, D ie A usg ra bu ng en v on S am aria I.D er W an dsch mu ck d er

B au te n von S am arra u nd se in e O rn am en tik (Berlin, 1923), fig. 234,house 3, room 16.

19. K. A. C. Creswell, "Some Newly Discovered Tulunid Orna-ment," B ur li ng to n Ma ga zi ne 35 (1919): 180-88, pl. II G.

20. Pavon Maldonado, Ar te h i spanomusu lmdn , pl. 31.21. K. A. C. Creswell, Mu slim A rc hite ctu re o f E gy pt (Oxford, 1952),vol. 1, fig. 29 and pl. 26e.22. M. S. Ahdul-Hak , "Reconstitution d'une partie de Kasr al-

Heir," pl. 19 (bottom).23. D. Whitehouse, "Excavations at Siraf: Third Interim Report,"

Iran 8 (1970): 1-18, pl. lla.24. Some authors would perhaps consider this a moot point. These

crafts tend (0 remain conservative, the artisans continuing tousc traditional formulae and resisting any sudden innovations.It is therefore conceivable that the initial idea for generalizing

A. J. LEEthese star patterns to include stars with any number of pointscame from professional mathematicians, or at least from some-one with sufficient imagination to glimpse something of thepossibilities inherent in this kind of ornament. If so, this initialpush must have occurred sometime toward the end of the tenthcentury and probably no later than the first half of the eleventh.Even granted such an external stimulus, however, the solutionssubsequently discovered follow quite logically and inevitably,and it is entirely possible that the artisans themselves could thenhave designed the definitive patterns, using trial-and-errormethods.

25. D. Stronach and T. C.Young, "Three Seljuq Tomb Towers,"Iran 4 (1966): 1-20, pl. 9a.

26. Ibid., pl. 4b.27. A. U_ Pope, "Note on the Aesthetic Character of the North

Dome of the Masjid-i J ami of Isfahan," in S tu dies in Islam ic A rtand Architecture in H onour of Prof K. A. C. Creswell (Cairo:American University in Cairo Press, 1965), pp. 179-93; fig. 7.

28. Fairly clear pictures of this ornament arc given in D. Hill andO. Grahar, I sl am ic A rc hi te ct ur e a nd I ts D e co ra ti on (London, 1964),figs. 146-48.

29. J. Bourgoin, Les Elem en ts de Ia rt a ra be : Ie tra it d es enirelacs (Paris,1879), reissued by Dover Publications as A ra b ic G e ome tr ic al P at -tern an d D esig n (New York, 1973).

30. G. A. Pugachenkova, "Mavzolei Arab-Ata," Iskusstuo ZodchikhUzbekistana IT (Tashkent, 1963), figs. 16, 17.

31. Stronach and Young, "Three Seljuq Tomb Towers," pl. 4b.32. E. H. Hankin, "On Some Discoveries of the Methods of

Design Employed in Mohammedan Art," Journal of t he S oc ie ty o fArts 53 (1905): 461-72; idem, "The Drawing of Geometric Pat-terns in Saracenic Art," M em oirs of th e A rch ae olog ical S urvey ofIndia 15 (Calcutta, 1925).

33. Bourgoin, E lim en ts de L'art arabe, pIs. 123, 124.34. U. Vogt-Goknil, Mosquees (Paris, 1975), p. 78.35. M. B. Smith, "Material for a Corpus of Early Iranian Islamic

Architecture, II: Manar and Masdjid, Barsian (Isfahan)," Ar sIslamica 4 (1937): 7-11, figs. 2 . 5 , 27.36. Ibid., fig. 27. Another design on the same basis is shown inBourgoin, E lem en ts d e L 'a rt a ra b e, pl. 163, and appears to havebeen taken from paired window grilles on the main facade of themosque of Sarghatmish in Cairo. Yet another version OCcurs onthe mausoleum of Mumineh Khatun (1186-87) at Nakhichevanin southern Azerbaijan (see M. A. Useinov, Pamya tn i ki A e er ba id -[ a ns kogo zo dch es to a [Moscow, 1951], central face of plate 5). Thelimiting polygons of the stars or rosettes of these patterns forma tessellation of almost regular polygons, whieh is the basis forE. Holiday's first series of Altair Design sheets (Altair Design[London: Longman Group, 1970]).

37. O. Aslanapa, T urk ish A rt a nd A rc hite ctu re (London, 1971), fig.20.

38. E. Herzfeld, "Damascus: Studies in Architecture, IV," Ar sIslamica 13-1+ (1948): 118-38, fig. 24.39. For a contrary view, cf. B. Pavon Maldonado, Artehispanomusulmdn , pp. 338-39, and fig. 85, drawings 5 and 6.

40. A fact known to SOme fourteenth-century artists, since types Iand II are used superimposed as decoration in Uljaytu'smausoleum at Sultaniyya, Iran. Sec A. U. Pope, cd., A S urveyof P ersian A rt (London, 1939), vol. 4, pl. 383.

41. Curves can be drawn from these equations in order to find pairsof values for other kinds of rhombs. The values which are ofinterest lie on the positive arms of hyperbolas, with asymptotes


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ISLAMIC STAR PATTERNS 197at m =2p , n = Zq . Each series has a pair of values for whichm = n = 2( P + q),

42, In a sense the construction ofmany Islamic patterns is arbitraryuntil rules have been formulated defining the degrees ofregularity one wishes to achieve in each part or in the patternas a whole, Since rules of this kind were never explicitly statedby the Muslim artist, it might be thought irrelevant to describeauthentic examples as "correctly" or "incorrectly" con-structed, Nevertheless it seems evident that the majority ofartisans were striving for the highest level of symmetry andregularity attainable in each pattern, and that the proportionsof the parts within the patterns were adjusted accordingly,Using the principle of the greatest possible symmetry andregularity as a basis for comparison, it becomes feasible todefine an "incorrectly" constructed pattern as one which fallsshort of achieving the highest degree of symmetry possible for

that particular pattern, Unfortunately one cannot appeal toauthentic patterns as the final arbiter, since there is rarely aunique and universally adopted construction of any givenpattern,

43, The relative radii of two dissimilar stars constructed accordingto the PIC method are in a precise ratio, the numerical valueof which can be calculated to any desired accuracy byrrigonometrical or other means, However, it is a curious factthat the more nearly equal are the two values m and n, the closerthe respective radii are to the ratio m : n, So close are some ofthese approximations to the theoretical values that it becomesimpossible to distinguish between the theoretical and m : n ratiosfrom measurements on authentic patterns,

44, K, Erdmann and Hanna Erdmann, Das anatolische Karauansaraydes 13, Jahrhunderts, pts. 2-3 in one vol. (Berlin, 1976), Die Orna-ment, pp, 109-205,

islamic star patterns

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