the whirling kites of isfahan

The Whirling Kitesof Isfahan:Geometric Variationson a ThemePETER R. CROMWELL AND ELISABETTA BELTRAMI

Introduction

IIn medieval times the city of Isfahan was a major centreof culture, trade and scholarship. It became the capital ofPersia in the Safavid era (16–17th centuries) when the

creation of Islamic geometric ornament was at its height.Many of the most complex and intricate designs we knowadorn her buildings, including multi-level designs in whichpatterns of different scales are combined to complementand enrich each other. In this article we study five 2-leveldesigns from Isfahan built around a common motif. Theyillustrate a variety of techniques and the analysis exposessome of the ingenuity and subtle deceptions needed toreconcile incompatible geometries and symmetries, andproduce satisfying works of art.

ThemeKites are a characteristic design element in Islamic geometricart. They can be arranged as motifs in their own right or usedto provide a structural framework for other elements.Figure 1 shows two patterns created by arranging kites withsquares. Part (a) shows a chiral arrangement of four kiteschasing around a central square in a finite composition. Forwant of a name we shall refer to this as the Whirling Kitespattern.We shall also say that thepatternwith this orientationis the clockwise variant, and that its mirror-image is counter-clockwise. Part (b) shows a repeating pattern that can beextended to fill the plane. It contains the Whirling Kitespattern in both its mirror-image forms.

There are three canons of Islamic ornament: calligraphy,arabesque and geometric. All have been applied to theWhirling Kites figure as a secondary form of decoration. In

Figure 2(a) the compartments are decorated with stylisedKufic calligraphy; the design is taken from a small tiledpanel in the al-Hakim Mosque (Masjid Hakim), Isfahan;photograph IRA 1017 in Wade’s collection [17] shows theoriginal. The website [14] is a useful resource on Kuficcalligraphy and gives translations of many inscriptions.Figure 2(b) shows an arabesque design carved in relief ona wooden door panel in the Great Mosque of Uqba inKairouan, Tunisia. Another floral example from the Tilla-Kari Madrasa in Samarkand, Uzbekistan, can be seen inphotograph TRA 0732 in [17] or in [15, p. 236]. It is of silvergilded plasterwork and has simple floral trails along thebands. Two much larger and more elaborate floral exam-ples are placed on either side of one of the great iwans inthe Imam Mosque (Masjid-i Imam) in Isfahan, formerlyknown as the Royal Mosque (Masjid-i Shah). They form amirror-image pair of Whirling Kites motifs and are executedin painted polychrome tiles. Photograph IRA 0225 in [17]and [15, p. 260] show an overall view. We shall see geo-metric examples later.

While it is the finite figure in Figure 1(a) that is the focusof this article, we shall also cite a few examples of therepeating pattern of Figure 1(b). Wade’s archive containsphotographs of carved stone reliefs from the Fort at Agra(IND 0404 and IND 0407), and a latticework screen in theMaharajah’s Palace in Jaipur (IND 1019). Examples inwooden door panels and brickwork from the Khan Mos-que, Isfahan, can be seen in [1].

As these examples show, the Whirling Kites figure iswidespread in the Islamic world. Besides being a motif inits own right, it also provides a useful device to organise a

84 THE MATHEMATICAL INTELLIGENCER � 2011 Springer Science+Business Media, LLC

DOI 10.1007/s00283-011-9225-4

larger composition and, consequently, a range of styles andtechniques have been applied to build complex designs onthis simple form.

Figure 3 shows the west iwan of the Friday Mosque(Masjid-i Jami), Isfahan. An iwan is an open, high, vaultedporch that provides a large facade for decoration. Thisexample is of interest as it has five Whirling Kites panels ofthree different designs: there are two in each of the tallnarrow panels that run the full height of the front faceeither side of the arch, and another on the north side of theinner wall. We shall examine the constructions of thesedesigns plus two others.

The Geometry of the Whirling Kites FigureA kite is a convex quadrilateral having two pairs of adjacentequal-length sides. We shall assume that a kite is notequilateral so that it has two short sides of length s and twolong sides of length t. In all the examples used here, thetwo angles where sides of different lengths meet are rightangles. If h is the acute angle between the two long sidesthen the obtuse angle between the two short sides is180� - h. Note h ¼ 2 tan�1ðs=tÞ:

The geometry of Whirling Kites patterns is straightfor-ward. Call the four lines making the outer square the frameand the four lines bounding the inner square and radiatingfrom it the rotor. Let x be the length of a side of the frameand y be the length of a side of the small square in therotor. Then x = t + s and y = t - s. In fact, any pair ofx, y, s and t determine the other two. If we ignore the scale,the whole figure is determined by h.

Figure 4 shows one way to lay out a Whirling Kitespattern. First take a square ABCD with side length s + t.Mark each side with a point that divides it into segments oflengths s and t so that the long and short segments alternatearound the square. In the figure, two such points aremarked E and F. Scribe a circular arc centred at E of radiusEA, and another centred at F of radius FA. The two arcsintersect at G, and AEGF is the required kite.

The Whirling Kites figure is simple to construct, but thisproperty is not sufficient to explain its origin as an orna-mental motif. It is possible that mathematical diagramsprovided the inspiration. The 10th century Persian mathe-matician and astronomer Abu’l Wafa wrote On theGeometric Constructions Necessary for the Artisan, whichincludes references to meetings between geometers andcraftsmen at which theoretical constructions were pre-sented and practical applications discussed [13]. In Chapter10 cut-and-paste arguments are used to construct squaresof given area. For example, to construct a square of area 5,place two unit squares so they share an edge and cut theresulting rectangle along a diagonal; two sets of thesepieces plus another unit square can be arranged to form asquare of area 5—Figure 5(a). Removing the dashed seg-ments produces a template for the periodic pattern inFigure 1(b); the template is repeated by reflection in thesides of the bounding square.

(a) (b)

Figure 2. Examples of simple decoration applied to Whirl-

ing Kites patterns. (a) Kufic calligraphy. (b) Floral arabesque.

.........................................................................................................................................................

AU

TH

OR

S PETER R. CROMWELL graduated from

Warwick, and completed his Ph.D. at Liv-

erpool working in knot theory, is interested

in anything geometric, and has written

books on polyhedra and knot theory.

Pure Mathematics Division

Mathematical Sciences Building

University of Liverpool, Peach Street

Liverpool, L69 7ZL

Englande-mail: [email protected]

ELISABETTA BELTRAMI graduated from

Milan, and studied for her Ph.D. at Pisa and

Liverpool, also working in knot theory. She

has taught at Trinity College Dublin and Liv-erpool. She enjoys walking and cooking.

Pure Mathematics Division

Mathematical Sciences Building

University of Liverpool, Peach Street

Liverpool, L69 7ZL

England

(a) (b)

Figure 1. Finite and unbounded Whirling Kites patterns.

� 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011 85

A similar figure occurs in one of the many proofs of thePythagorean Theorem. The ancient Chinese text The Arith-metical Classic of the Gnomon and the Circular Paths ofHeaven contains a discussion of the theorem using the 3-4-5triangle as an example—Figure 5(b). The oldest survivingmanuscript is a 13th-century copy in Shanghai library, butmuch of the content predates Islam by hundreds of years.There is Chinese influence in Islamic art so it is possible thatthis proof was known to medieval Islamic scholars, too.

(a) (b)

Figure 5. Possible sources of inspiration for the Whirling

Kites motif. In (a) t : s = 2 : 1 and in (b) t : s = 4 : 3.

Figure 4. Construction of a kite in a square.

Figure 3. The west iwan of

the Friday Mosque, Isfahan.

(Photograph reproduced

courtesy of Paul Rudkin).

86 THE MATHEMATICAL INTELLIGENCER

Ozdural suggests [13] that figures such as those in Figure 5may have inspired the artistic imagination of the craftsmen.Once the periodic pattern is known, it is a simple matter toextract the Whirling Kites motif. He cites the Whirling Kitespattern on the inner wall of the west iwan of the FridayMosque as a possible application. Chiral motifs such as theswastika are widespread and found in many cultures; theWhirling Kites motif seems to be unique to Islamic ornament.Perhaps some mathematical input was required for itsdiscovery.

Variation 1Figure 6(c) shows a geometric design based on the upperpanels in the front face of the west iwan of the Friday Mos-que. Both panels are counter-clockwise. Here the kites aredecorated with a section of a periodic pattern constructed ona triangular grid. Figure 6(a) shows a hexagonal repeat unitfor thepattern. Theblackmotif is related to a common squareKufic representation of the name ‘Ali’. Here the text is trun-cated and reflected; a well-formed hexagonal treatment ofthe text appears in the centre of panel 91 of the TopkapiScroll [12]. Figure 6(b) shows how hexagons can be used tofill a kite whose small angle is 60�. While Figure 6(c) is not atrue reproduction of the panel on the mosque (the mosaic isnot laid out so accurately), this method or something similarclearly underlies its construction.

Here we pause for a few comments on terminology. Thelines in the figures drawn in red show the underlying geo-metric structure of a design but are not apparent in thefinished product. We shall use this convention throughout.We shall also refer to the shapes outlined by red lines as tiles,and to a collection of tiles as a tiling. This is to distinguishthem from the individual ceramic shapes, which we shall calltesserae, that are assembled to form a panel or mosaic.

Figure 6(c) is a simple example of a 2-level design: twogeometric patterns of different scales used in a single design.Many examples of the interplay of patterns on multiple scalescan be found in Islamic ornament. In the early works, voids inthebackgroundof a large-scalepattern are progressivelyfilledwithfloral or geometricmotifs to leave adesignwithnovacantspaces. In some of the finest examples of 2-level geometricdesign, mathematical processes such as subdivision were

applied to generate large- and small-scale patterns that areintimately related [3, 4, 6, 10].

In the patterns featured in this article, the Whirling Kitesfigure establishes the basic framework of a large-scalepattern, and it is embellished with secondary decoration inthe following ways:

• calligraphy, arabesques or geometric patterns are used tofill the interiors of the kites and central square

• bands of arabesques or geometric patterns may be usedto outline the compartments, thickening the lines of theframe and rotor.

These two techniques (filling and outlining) correspondto Type A and Type B, respectively, in the classification of2-level designs introduced by Bonner [3].

In the best examples of 2-level designs, the large- andsmall-scale patterns are complementary in the sense thatprominent features of one are highlighted or supported bythe other. This is not achieved in Figure 6(c): the hexagonalsubdivision of the kite provides a good basis for the con-struction of a small-scale pattern, and two of the directionswithin the small-scale pattern are aligned with the long sidesof the kite, but the focal points of the pattern are not strongenough to add emphasis where it is needed. Also, the kitesare treated independently rather than as parts of a compositefigure, so there is no continuity across their boundaries.Other examples we shall analyse reveal that finding a small-scale pattern that is compatible with the features of theWhirling Kites figure is a challenging problem.

Variation 2Figure 7 shows a Type A 2-level Whirling Kites design fromtheMadar-i ShahMadrasa (Mother of the Shahor Royal Theo-logical College), also known as the Chahar Bagh Madrasa.Each corner of the large central courtyard is canted with anarch leading to a small octagonal courtyard giving access tothe rooms of the college. See [15, p. 293] for a general view.The Whirling Kites design is repeated just below roof levelaround the small courtyards. The design is used in bothmirror-image forms, and the composition of the small-scalepattern varies.

The mosaic is made using the ‘cut tile’ technique: largeceramic tiles with a single colour glaze are cut into small tes-serae, which are then assembled to make the mosaic panel.Here, the yellow star-shaped tesserae mark out the shapes ofthe compartments and manifest the 2:1 ratio of the long andshort sides of the kites. The kites are filled with a seeminglyrandom arrangement of black and turquoise tesserae. Thissmall-scale pattern is based on a modular design system thatunderlies many Islamic patterns [6, 7, 10, 11]. The basic systemcomprises the three equilateral tiles shown in Figure 8: a reg-ular decagon decorated with ten small kites arranged to form a{10/3} star motif, a hexagon shaped like a bow-tie decoratedwith twokites congruent to thoseon thedecagon, andaconvexhexagon with a bobbin-shaped motif. The boundaries of theseunderlying tiles arenot apparent in thefinishedmosaicbut theycan be recovered from the design: the black tesserae are theforeground motifs on the tiles, the yellow tesserae are the

(a)

(b) (c)

Figure 6. A 2-level design from the Friday Mosque, Isfahan.


centres of the decagons, and the turquoise tesserae are formedby fusing the background regions at the edges of the tiles.

The arrangement of the tiles is a typical application of themodular system to this style of 2-level pattern: decagons areplaced so their centres coincidewithprominent featuresof thelarge-scale pattern, other tiles are placed so that their edges ormirror lines are aligned with the outlines of the kites—seeFigure 9. The interiors of the compartments are then infilledwith more tiles. In this case the decagons are centred on thecorners and junctions of the lines in the large-scalepattern andalso divide the long sides of the kites. The centres of thedecagons on the frame divide each side into three equal parts.

If the long and short sides of a kite are in the ratio 2:1then h, the small angle in a kite, is about 53.13�. In a mosaiccontext, this angle cannot be distinguished from 54�—anangle compatible with the 10-fold geometry of the modularsystem. However, it is not compatible with the 4-foldsymmetry of the Whirling Kites pattern. Observe that all the

decagons have the same orientation (vertex at the top), andthis alone reduces the symmetry of the design as a whole to2-fold rotation. The complete design is asymmetric becauseeach kite has its own irregular filling.

The lines in the large-scale pattern fall into two cate-gories according to whether they connect two vertices ortwo edges of the decagon tiles. The lines connecting twovertices are covered by the diagonals of two bobbin tilesand a tile edge; the other lines (except for the bottom-centre) are covered by various sequences of two bow-tiesand two bobbins. In fact, the two lengths produced bythese combinations of tiles are not quite equal and theconstruction illustrated in Figure 9 is a geometric fallacy.This is made clear in Figure 10 which shows the small-scale

Figure 9. First-stage analysis of Figure 7.

Figure 7. A Type A 2-level design from the Madar-i Shah

Madrasa, Isfahan. (Photograph reproduced courtesy of Brian

McMorrow).

Figure 8. Elements in a common modular design system.

Figure 10. Part of the small-scale pattern from Figure 7 when

it is not constrained to fit a kite.


pattern in the top left kite with its natural geometry. Noticethe two half bow-tie tiles at the inner end of the fracture.The gap and the misalignment of the boundary at the top-centre are small and, with minor adjustments to the sizeand shape of a few tesserae, the small-scale pattern can bemade to fit the available space without drawing attention.

Similar adjustments are made in the fillings of the otherkites—in the bottom right kite, the ‘problem’ is pushed intothe small angle of the kite where it affects the frame (asindicated by the discontinuity in Figure 9).

In fact, it is impossible to cover a square frame exactlywith the tiles using this strategy. We shall now sketch aproof of this—readers who are not interested in the tech-nicalities can skip to the next section.

In any tiling composed of the three tiles in Figure 8 all thedecagon tiles have the same orientation, and the bow-tie andbobbin tiles can both occur in five orientations aligned atmultiples of 36� to each other. We have the requirement thata tile which intersects a line in the square must do so in anedge or a mirror line of the tile.

We take the edge length of the tiles to be 1. We shallexpress some distances across the tiles in terms of theparameters d and k shown in Figure 11(a). Recall that thelengths of a diagonal and an edge of a regular pentagon arein the golden ratio, s. We have

s ¼ffiffiffi

5pþ 1

2; d ¼ cosð72�Þ ¼

ffiffiffi

5p� 1

4and

k ¼ sinð72�Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5þffiffiffi

5p

8

s

:

First we consider distances that are vertical in Figure 11.

(v1) The edge length is 1.(v2) The pentagon in Figure 11(a) shows that the radius

(centre to vertex) of the decagon is s times its edgelength.

(v3) Figure 11(b) shows that the distance across the waistof the bow-tie is 1 - 2d.

(v4) Figure 11(c) with relation (v2) shows the long diag-onal of the bobbin is 2(s - d).

Now we consider some horizontal distances in Figure 11.

(h1) Figure 11(b) shows the length of the long mirror lineof the bow-tie is 2k.

(h2) Recalling that the lengths of the two red lines inFigure 11(b) are in ratio s, we can deduce that the

apothem (centre to edge mid-point) of the decagon isks.

(h3) Using Figure 11(c) with relations (h1) and (h2) wecan deduce that the length of the short mirror line ofthe bobbin is 2k(s - 1).

The vertical lines of the square in Figure 9 must becovered by rational combinations of the distances (v1)–(v4). These are parametrised by d and s so the length of theside of the square must belong to Q½

ffiffiffi

5p�: The horizontal

lines of the square must be covered by rational combina-tions of the distances (h1)–(h3); these are parametrised byk and s. The double radical k is not in the field Q½

ffiffiffi

5p�:

Therefore the vertical and horizontal distances covered bythe tiles are incommensurable.

Variation 3Figure 12 shows one of the lower pair of Type A 2-levelWhirling Kites designs from the front face of the west iwanof the Friday Mosque. As with the upper pair (Variation 1),both panels are counter-clockwise. The mosaic is pre-dominantly in black and gold with the kites outlined inwhite. The problems in the Madrasa design (Variation 2)arising from the use of 10-point stars are avoided here byusing 12-point stars. These stars are compatible with the4-fold symmetry of the whole design and its 90� angles atthe corners of the frame and the inner square.

Figure 13 shows the underlying structure of the design.The 12-point stars are represented by circles. Using thedistance between adjacent centres as the unit, we see thatthe long and short sides of the kites are in the ratio 4:2 soh & 53.13�. The panel is subdivided into 20 unit squares,and 8 small kites with sides in the ratio 2:1. To form themosaic each small square is filled with a standard starpattern that has the centres of 12-point stars at the corners

(c)(b)(a)

Figure 11. Properties of the tiles in Figure 8.

Figure 12. A Type A 2-level design from the Friday Mosque,

Isfahan. (Reproduced from [15, p.220] courtesy of Henri and

Anne Stierlin, Geneva).


and an 8-point star in the centre. This pattern covers overhalf the panel. The decoration in the small kites is based ontiles analogous to the bow-tie and bobbin tiles of Figure 10,but adapted to the angles of a dodecagonal tiling scheme.The 12-point stars are not compatible with the local geom-etry in the shaded circles (if the spikes are equally spaced,they cannot align with the sides of the kite), but this doesnot intrude on the eye.

We have seen three examples of Type A (filling); wenow consider examples of Type B (outlining).

Variation 4Figure 14 shows the famous 2-level Whirling Kites designfrom the inner wall of the west iwan of the Friday Mosque.A wider view and some details are shown in photographsIRA 0520, IRA 0604, and IRA 0605 in [17].

The bands outlining the frame and rotor are bordered byfragments of 10-point stars. Connecting the centres of thesestars divides the bands into strips of approximately squarecells, as shown in Figure 15(a). Using the side of a squareas a unit, and measuring along the centre-line of the band,we see that the frame is 15 units along each side, and thecentral square is 5 units. Therefore, the sides of each kiteare (again) in the ratio 2:1.

The small-scale design is created by filling each squarecell with a pattern based on the template shown in Fig-ure 15(b). This pattern is constructed using another modularsystem, this time having four decorated tiles: a regulardecagon with a {10/4} star motif, a regular pentagon with a {5/2} star (or pentagram)motif, an isosceles trianglewith sides inthe golden ratio decorated with a kite, and a trapeziumdecorated with an arrowhead. The template can be repeatedto form periodic star patterns—see photograph IND 0705 in[17] for an example. Applying the template to the square cellsof Figure 15(a) is problematic as the template itself is not

square (the height is about 95% of the width). Hence, somejuggling of the tesserae is required to make things fit. Thepentagrams are most affected by the deformation—they arenoticeably irregular in the mosaic.

Even though the large-scale pattern has 4-fold symmetry,and is decomposed into squares, the design on the templatehas only 2-fold symmetry. In Figure 15(a) the orientation ofthe template is indicated with the double arrow motif fromthe centre of the template; the copies around the frame arevertically aligned and those in the rotor are aligned top-rightto bottom-left.

The angle between the bands in the rotor and those in theframe is approximately 54� so it is compatiblewith the 10-foldgeometry underlying the template. This means that it ispossible for the stars and other motifs in the small-scale pat-tern to be aligned consistently throughout the design (as in

Figure 13. Analysis of Figure 12. Figure 14. A Type B 2-level design from the Friday Mosque,

Isfahan. (Photograph reproduced courtesy of Daniel Sanderson).

(a) (b)

Figure 15. Analysis of Figure 14.


Variation 2). However, if the craftsmen who made the mosaicrecognised this, either they did not consider it importantor they have made a mistake in laying out the design. In themosaic, the stars in the frame have a vertical spike whilethe stars in the rotor have a horizontal spike. If the rotor wererotated by 90�, all the stars would have the same alignment,and the small-scale designs would be compatible at thejunctions where the rotor meets the frame. In the mosaic thisis not the case and further juggling is required to disguise it.

Star PlacementWhen the length parameters x, y, s and t are integers, dis-crete motifs such as flowers or stars can be placed on theWhirling Kites figure so their centres lie on the figure, somecoincide with the corners and intersections of the lines, andthey are equally spaced along all its lines.

Figure 16(a) shows a template for a Whirling Kitesdesign with x = 11 and y = 3. This implies h & 59.49�, anangle that is indistinguishable from 60� for practical

purposes. The figure is formed from strips of squares;where the rotor meets the frame the strips are simplyoverlaid and the two squares that meet at each junction areconcentric.

To create a Type B pattern we need to find a star patternthat has a square repeat unit and is compatible with 90� and60� junctions. We shall not explain the construction of starpatterns here—see [5, 8, 9] for information. However, it isclear that a 12-point star is a good candidate for a motif thatmeets these requirements.

Figure 16(c) shows four repeat units of a traditional pat-tern taken from Plate 94 of Bourgoin [2]. It is constructedusing another modular system of decorated tiles: regularpolygons of 3, 4 and 12 sides and a shield-shaped tile formedby erecting a right isosceles triangle on each side of anequilateral triangle [16, p. 18]. Figure 16(d) shows the resultof placing this repeat unit in each square of Figure 16(a); thejunctions between the rotor and the frame aremadeusing thesimple mitre joint shown in Figure 16(b). The result is a TypeB 2-level Whirling Kites pattern with 12-point stars (colouredyellow in the figure) equally spaced along the centre-line ofthe bands.

Even though it would have been possible for medievalartists to construct patterns like this, we are not aware of aWhirling Kites example in which the principal star motif runsalong the centre-line of the bands. The closest we have comeis the border pattern shown in photograph EGY 1609 of [17],which shows a band with 12-point stars turning a 90� corner.

In most 2-level designs where the small-scale design is astar pattern, the defining features (corners and intersections)of the large-scale design are located in the centres of stars inthe small-scale design. In Type B designs, the defining fea-tures of the large-scale design are the corners in theboundaryof the band.

In the example of Figure 14 the centres of the 10-pointstars are evenly spaced along the band edges as far aspossible. The exterior corners of the frame, the corners ofthe central square, and the 90� corners of the kites are alllocated at star centres. The obtuse and acute angles of thekites do not coincide naturally with star centres, althoughstars have been placed at the acute angles in the top andbottom sides of the frame.

Figure 17 shows that it is possible to create a WhirlingKites design in which all the band boundaries have integerlength. A 3-4-5 triangle is placed at each junction of the rotorand the frame, and the bands are 4 units wide. Measuringalong the centre-line of the bands we have x = 36 andy = 12. This means that the long and short sides of the kitesare in the ratio 2:1 and h & 53.13�.

When trying to select a star whose geometry is compatiblewith a Whirling Kites design, it is useful to find a fraction of360� that approximates h. The denominator gives an indi-cation of the number of points in a suitable star, eitherdirectly or via simple relationships. In this case 3/20 is a goodcandidate. However, it is difficult to make patterns from starswith as many as 20 points. The most natural choice for asmall-scale geometric pattern is a star pattern with 10-foldmotifs. As we have seen, this is far from easy. Motifs with 10-fold symmetry are not compatible with the 4-fold symmetryof the whole design: 10-point stars will have the same

(a)

(b) (c)

(d)

Figure 16. A new Type B 2-level design with 12-point star

motifs. Design Copyright P.R. Cromwell 2010.


orientation throughout the pattern, so some band bound-aries will pass through opposite spikes, and others will passbetween the spikes. There are also the problems of pro-ducing a square template to cover the band, and covering the3-4-5 triangles in both vertical and horizontal alignments.Even if this were done, the small-scale pattern would prob-ably appear too busy and intricate to be effective asornament—the difference in scale and apparent complexitybetween the large and small patterns is too great.

Our final example shows another approach to theproblem of star placement.

Variation 5Figure 18 shows another Type B 2-level design from theFriday Mosque. Other views are shown in photographs IRA0721 and IRA 0722 in [17]. In this example, placing the starsat prominent points has taken precedence over equalspacing. The underlying structure of the small-scale patternis shown in Figure 19. The lines dividing the band into cellsconnect the star centres. Left to right along the bottom ofthe frame we find four squares, four rectangles, and a finalsquare. The width of each rectangle is determined by theequilateral triangle it contains. This arrangement is repe-ated around the other sides of the frame. The ratio of thelengths is AB : BC ¼

ffiffiffi

3p

: 1 so h = 60�.To construct the rotor, erect a line from A making an

angle of 60� with the bottom of the frame. Repeat on eachside of the frame and extend the four lines until they meet.For example, the line starting at A meets the line starting atD in the point E. These four lines bound a square in themiddle of the figure, which is subdivided into a 3 9 3array of congruent squares. These squares are smaller thanthose in the frame: EF is about 94% of CD. The kite in thelower right is completed with the line CF. Note that CF andDE are not parallel but diverge away from the frame. Labelthe midpoint of CF as G.

This cellular structure provides a framework for layingout the small-scale pattern. The principal star motifs have12 points and so are compatible with the 90� and 60� anglesat the corners of the band. The 16 stars in the central arrayare aligned so that their spikes lie on the cell boundaries;the tips of spikes of adjacent stars touch. The 12-point starsin the frame are aligned so that the cell boundaries passbetween the spikes—this difference may help to disguise

Figure 19. Analysis of Figure 18.

Figure 18. A Type B 2-level design from the Friday Mosque,

Isfahan. (Photograph reproduced courtesy of Steven Achord).

Figure 17. A band network with integer boundaries.


the fact that they are further apart than the others. The starat G marks the transition between the two orientations andhas 13 points. The square cells in the frame contain 8-pointstars at their centres. Triangle CDG is almost equilateral (theangle at C is about 62.19�) and this is close enough for thedecoration used in the other triangles to be applied.

Each kind of cell has its own filling, and these are con-sistently applied. The pattern has no awkward juxtapositionsor abrupt changes, it is a masterly display of apparentlyeffortless transitions between a progression of patterns.

ConclusionThe examples discussed above have highlighted some of theproblems encountered in trying to design and fabricate2-level Whirling Kites designs. The mathematics required tocreate designs with discrete motifs such as flowers evenlyspaced along the centre-lines of the band in a Type B patternis straightforward and could have been understood bymedieval craftsmen. Working with star patterns is more dif-ficult, but it is possible to discover by experiment someconfigurations of the Whirling Kites figure whose angles arecompatible with the geometry of stars. Even so, applying astar pattern to cover the bands presents theoretical as well aspractical challenges and the medieval artists producedingenious and attractive solutions.

ACKNOWLEDGEMENTS

We are grateful to everyone who has given us permission

to reproduce their images, and to Mamoun Sakkal for

explaining the Kufic basis of the decoration in Figure 6.

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http://www.kufic.info/

http://www.patterninislamicart.com/

the whirling kites of isfahan

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