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The Whirling Kites of Isfahan:

Geometric Variations on a Theme

Peter R. Cromwell and Elisabetta Beltrami

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

Introduction

In medieval times the city of Isfahan was a major centre of culture, trade and scholarship.It became the capital of Persia in the Safavid era (1617th centuries) when the creationof Islamic geometric ornament was at its height. Many of the most complex and intricate

designs we know adorn her buildings, including multi-level designs in which patterns ofdifferent scales are combined to complement and enrich each other. In this article we studyfive 2-level designs from Isfahan built around a common motif. They illustrate a variety oftechniques and the analysis exposes some of the ingenuity and subtle deceptions needed toreconcile incompatible geometries and symmetries, and produce satisfying works of art.

Theme

Kites are a characteristic design element in Islamic geometric art. They can be arrangedas motifs in their own right or used to provide a structural framework for other elements.Figure 1 shows two patterns created by arranging kites with squares. Part (a) shows a

cheiral arrangement of four kites chasing around a central square in a finite composition.For want of a name we shall refer to this as the Whirling Kitespattern. We shall also saythat the pattern with this orientation is the clockwise variant, and that its mirror-imageis counter-clockwise. Part (b) shows a repeating pattern that can be extended to fill theplane. It contains the Whirling Kites pattern in both its mirror-image forms.

There are three canons of Islamic ornament: calligraphy, arabesque and geometric. Allhave been applied to the Whirling Kites figure as a secondary form of decoration. InFigure 2(a) the compartments are decorated with stylised Kufic calligraphy; the design istaken from a small tiled panel in the al-Hakim Mosque (Masjid Hakim), Isfahan; photographIRA 1017 in Wades collection [17] shows the original. The website [14] is a useful resourceon Kufic calligraphy and gives translations of many inscriptions. Figure 2(b) shows an

arabesque design carved in relief on a wooden door panel in the Great Mosque of Uqbain Kairouan, Tunisia. Another floral example from the Tilla-Kari Madrasa in Samarkand,Uzbekistan, can be seen in photograph TRA 0732 in [17] or in [15, p. 236]. It is of silvergilded plasterwork and has simple floral trails along the bands. Two much larger andmore elaborate floral examples are placed on either side of one of the great iwans in theImam Mosque (Masjid-i Imam) in Isfahan, formerly known as the Royal Mosque (Masjid-iShah). They form a mirror-image pair of Whirling Kites motifs and are executed in paintedpolychrome tiles. Photograph IRA 0225 in [17] and [15, p. 260] show an overall view. Weshall see geometric examples later.

While it is the finite figure in Figure 1(a) that is the focus of this article, we shallalso cite a few examples of the repeating pattern of Figure 1(b). Wades archive contains

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(a) (b)

Figure 1: Finite and unbounded Whirling Kites patterns.

(a) Kufic calligraphy (b) Floral arabesque

Figure 2: Examples of simple decoration applied to Whirling Kites patterns.

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Photograph reproduced courtesy of Paul Rudkin.

Figure 3: The west iwan of the Friday Mosque, Isfahan.

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A B

CD

E

F

G

Figure 4: Construction of a kite in a square.

photographs of carved stone reliefs from the Fort at Agra (IND 0404 and IND 0407), and alatticework screen in the Maharajahs Palace in Jaipur (IND 1019). Examples in woodendoor panels and brickwork from the Khan Mosque, Isfahan, can be seen in [1].

As these examples show, the Whirling Kites figure is widespread in the Islamic world.Besides being a motif in its own right, it also provides a useful device to organise a largercomposition and, consequently, a range of styles and techniques have been applied to buildcomplex designs on this simple form.

Figure 3 shows the west iwanof the Friday Mosque (Masjid-i Jami), Isfahan. An iwanis an open, high, vaulted porch that provides a large facade for decoration. This exampleis of interest as it has five Whirling Kites panels of three different designs: there are twoin each of the tall narrow panels that run the full height of the front face either side of thearch, and another on the north side of the inner wall. We shall examine the constructionsof these designs plus two others.

The geometry of the Whirling Kites figure

A kite is a convex quadrilateral having two pairs of adjacent equal-length sides. We shallassume that a kite is not equilateral so that it has two short sides of length s and two longsides of length t. In all the examples used here, the two angles where sides of differentlengths meet are right angles. If is the acute angle between the two long sides then theobtuse angle between the two short sides is 180 . Note = 2 tan1(s/t).

The geometry of Whirling Kites patterns is straightforward. Call the four lines makingthe outer square the frameand the four lines bounding the inner square and radiating fromit the rotor. Letxbe the length of a side of the frame and y be the length of a side of thesmall square in the rotor. Thenx= t +s and y =t s. In fact, any pair ofx, y, s and tdetermine the other two. If we ignore the scale, the whole figure is determined by .

Figure 4 shows one way to lay out a Whirling Kites pattern. First take a square ABCDwith side length s + t. Mark each side with a point that divides it into segments of lengthss and t so that the long and short segments alternate around the square. In the figure,two such points are marked EandF. Scribe a circular arc centred at Eof radius EA, andanother centred atFof radius FA. The two arcs intersect at G, and AEGFis the requiredkite.

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(a) (b)

Figure 5: Possible sources of inspiration for the Whirling Kites motif. In (a) t : s = 2 : 1and in (b) t : s = 4 : 3.

The Whirling Kites figure is simple to construct, but this property is not sufficient toexplain its origin as an ornamental motif. It is possible that mathematical diagrams pro-

vided the inspiration. The 10th century Persian mathematician and astronomer Abul WafawroteOn the Geometric Constructions Necessary for the Artisan, which includes referencesto meetings between geometers and craftsmen at which theoretical constructions were pre-sented and practical applications discussed [13]. In Chapter 10 cut-and-paste arguments areused to construct squares of given area. For example, to construct a square of area 5, placetwo unit squares so they share an edge and cut the resulting rectangle along a diagonal; twosets of these pieces plus another unit square can be arranged to form a square of area 5 Figure 5(a). Removing the dashed segments produces a a template for the periodic patternin Figure 1(b); the template is repeated by reflection in the sides of the bounding square.

A similar figure occurs in one of the many proofs of the Pythagorean Theorem. Theancient Chinese text The Arithmetical Classic of the Gnomon and the Circular Paths of

Heaven contains a discussion of the theorem using the 3-4-5 triangle as an example Figure 5(b). The oldest surviving manuscript is a 13th century copy in Shanghai library,but much of the content predates Islam by hundreds of years. There is Chinese influence inIslamic art so it is possible that this proof was known to medieval Islamic scholars, too.

Ozdural suggests [13] that figures such as those in Figure 5 may have inspired the artisticimagination of the craftsmen. Once the periodic pattern is known, it is a simple matterto extract the Whirling Kites motif. He cites the Whirling Kites pattern on the inner wallof the west iwan of the Friday Mosque as a possible application. Cheiral motifs such asthe swastika are widespread and found in many cultures; the Whirling Kites motif seemsto be unique to Islamic ornament. Perhaps some mathematical input was required for itsdiscovery.

Variation 1

Figure 6(c) shows a geometric design based on the upper panels in the front face of the westiwan of the Friday Mosque. Both panels are counter-clockwise. Here the kites are decoratedwith a section of a periodic pattern constructed on a triangular grid. Figure 6(a) shows ahexagonal repeat unit for the pattern. The black motif is related to a common square Kuficrepresentation of the name Ali. Here the text is truncated and reflected; a well-formedhexagonal treatment of the text appears in the centre of panel 91 of the Topkap Scroll [12].Figure 6(b) shows how hexagons can be used to fill a kite whose small angle is 60 . WhileFigure 6(c) is not a true reproduction of the panel on the mosque (the mosaic is not laid

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These two techniques (filling and outlining) correspond to Type A and Type B, respectively,in the classification of 2-level designs introduced by Bonner [3].

In the best examples of 2-level designs, the large and small scale patterns are comple-mentary in the sense that prominent features of one are highlighted or supported by theother. This is not achieved in Figure 6(c): the hexagonal subdivision of the kite providesa good basis for the construction of a small scale pattern, and two of the directions withinthe small scale pattern are aligned with the long sides of the kite, but the focal points ofthe pattern are not strong enough to add emphasis where it is needed. Also, the kites aretreated independently rather than as parts of composite figure, so there is no continuityacross their boundaries. Other examples we shall analyse reveal that finding a small scalepattern that is compatible with the features of the Whirling Kites figure is a challengingproblem.

Variation 2

Figure 7 shows a Type A 2-level Whirling Kites design from the Madar-i Shah Madrasa(Mother of the Shah or Royal Theological College), also known as the Chahar Bagh Madrasa.Each corner of the large central courtyard is canted with an arch leading to a small octagonalcourtyard giving access to the rooms of the college. See [15, p. 293] for a general view. TheWhirling Kites design is repeated just below roof level around the small courtyards. Thedesign is used in both mirror-image forms, and the composition of the small scale patternvaries.

The mosaic is made using the cut tile technique: large ceramic tiles with a single colourglaze are cut into small tesserae, which are then assembled to make the mosaic panel. Here,

the yellow star-shaped tesserae mark out the shapes of the compartments and manifest the2 : 1 ratio of the long and short sides of the kites. The kites are filled with a seeminglyrandom arrangement of black and turquoise tesserae. This small scale pattern is based ona modular design system that underlies many Islamic patterns [6, 7, 10, 11]. The basicsystem comprises the three equilateral tiles shown in Figure 8: a regular decagon decoratedwith ten small kites arranged to form a{10/3} star motif, a hexagon shaped like a bow-tie decorated with two kites congruent to those on the decagon, and a convex hexagonwith a bobbin-shaped motif. The boundaries of these underlying tiles are not apparent inthe finished mosaic but they can be recovered from the design: the black tesserae are theforeground motifs on the tiles, the yellow tesserae are the centres of the decagons, and theturquoise tesserae are formed by fusing the background regions at the edges of the tiles.

The arrangement of the tiles is a typical application of the modular system to this styleof 2-level pattern: decagons are placed so their centres coincide with prominent features ofthe large scale pattern, other tiles are placed so that their edges or mirror lines are alignedwith the outlines of the kites see Figure 9. The interiors of the compartments are theninfilled with more tiles. In this case the decagons are centred on the corners and junctionsof the lines in the large scale pattern and also divide the long sides of the kites. The centresof the decagons on the frame divide each side into three equal parts.

If the long and short sides of a kite are in the ratio 2 : 1 then , the small angle ina kite, is about 53.13. In a mosaic context, this angle cannot be distinguished from 54

an angle compatible with the 10-fold geometry of the modular system. However, it isnot compatible with the 4-fold symmetry of the Whirling Kites pattern. Observe that all

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Photograph reproduced courtesy of Brian McMorrow.

Figure 7: A Type A 2-level design from the Madar-i Shah Madrasa, Isfahan.

decagon

bow-tie

bobbin

Figure 8: Elements in a common modular design system.

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Figure 9: First stage analysis of Figure 7.

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(a) (b) (c)

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Figure 11: Properties of the tiles in Figure 8.

tiles in terms of the parameters and shown in Figure 11(a). Recall that the lengths ofa diagonal and an edge of a regular pentagon are in the golden ratio, . We have

=

5 + 1

2 , = cos(72) =

5 1

4 and = sin(72) =

5 +

5

8 .

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 times its edge length.

(v3) Figure 11(b) shows that the distance across the waist of the bow-tie is 1 2.(v4) Figure 11(c) with relation (v2) shows the long diagonal of the bobbin is 2( ).

Now we consider some horizontal distances in Figure 11.

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

(h2) Recalling that the lengths of the two red lines in Figure 11(b) are in ratio, we candeduce that the apothem (centre to edge mid-point) of the decagon is .

(h3) Using Figure 11(c) with relations (h1) and (h2) we can deduce that the length of theshort mirror line of the bobbin is 2( 1).

The vertical lines of the square in Figure 9 must be covered by rational combinationsof the distances (v1)(v4). These are parametrised byand so the length of the side ofthe square must belong to Q[

5]. The horizontal lines of the square must be covered by

rational combinations of the distances (h1)(h3); these are parametrised by and . Thedouble radical is irreducible in Q[

5]. Therefore the vertical and horizontal distances

covered by the tiles are incommensurable.

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Variation 3

Figure 12 shows one of the lower pair of Type A 2-level Whirling Kites designs from the

front face of the west iwan of the Friday Mosque. As with the upper pair (Variation 1), bothpanels are counter-clockwise. The mosaic is predominantly in black and gold with the kitesoutlined in white. The problems in the Madrasa design (Variation 2) arising from the useof 10-pointed stars are avoided here by using 12-pointed stars. These stars are compatiblewith the 4-fold symmetry of the whole design and its 90 angles at the corners of the frameand the inner square.

Figure 13 shows the underlying structure of the design. The 12-pointed stars are repre-sented by circles. Using the distance between adjacent centres as the unit, we see that thelong and short sides of the kites are in the ratio 4 : 2 so 53.13. The panel is subdividedinto 20 unit squares, and 8 small kites with sides in the ratio 2 : 1. To form the mosaic eachsmall square is filled with a standard star pattern that has the centres of 12-pointed stars

at the corners and an 8-pointed star in the centre. This pattern covers over half the panel.The decoration in the small kites is based on tiles analogous to the bow-tie and bobbin tilesof Figure 10, but adapted to the angles of a dodecagonal tiling scheme. The 12-pointedstars are not compatible with the local geometry in the shaded circles (if the spikes areequally spaced, they cannot align with the sides of the kite), but this does not intrude onthe eye.

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

Variation 4Figure 14 shows the famous 2-level Whirling Kites design from the inner wall of the westiwan 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 by fragments of 10-pointed stars.Connecting the centres of these stars divides the bands into strips of approximately squarecells, as shown in Figure 15(a). Using the side of a square as a unit, and measuring alongthe centre-line of the band, we see that the frame is 15 units along each side, and the centralsquare is 5 units. Therefore, the sides of each kite are (again) in the ratio 2 : 1.

The small scale design is created by filling each square cell with a pattern based onthe template shown in Figure 15(b). This pattern is constructed using another modularsystem, this time having four decorated tiles: a regular decagon with a {10/4}star motif, aregular pentagon with a {5/2}star (or pentagram) motif, an isosceles triangle with sides inthe golden ratio decorated with a kite, and a trapezium decorated with an arrowhead. Thetemplate can be repeated to form periodic star patterns see photograph IND 0705 in [17]for an example. Applying the template to the square cells of Figure 15(a) is problematicas the template itself is not square (the height is about 95% of the width). Hence, some

juggling of the tesserae is required to make things fit. The pentagrams are most affectedby the deformation they are noticeably irregular in the mosaic.

Even though the large scale pattern has 4-fold symmetry, and is decomposed intosquares, the design on the template has only 2-fold symmetry. In Figure 15(a) the ori-

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Photograph reproduced courtesy of Daniel Sanderson.

Figure 14: A Type B 2-level design from the Friday Mosque, Isfahan.

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(a) (b)

Figure 15: Analysis of Figure 14.

entation of the template is indicated with the double arrow motif from the centre of thetemplate; the copies around the frame are vertically aligned and those in the rotor are

aligned top-right to bottom-left.The angle between the bands in the rotor and those in the frame is approximately 54

so it is compatible with the 10-fold geometry underlying the template. This means that it ispossible for the stars and other motifs in the small scale pattern to be aligned consistentlythroughout the design (as in Variation 2). However, if the craftsmen who made the mosaicrecognised this, either they did not consider it important or they have made a mistake inlaying out the design. In the mosaic, the stars in the frame have a vertical spike while thestars in the rotor have a horizontal spike. If the rotor were rotated by 90, all the starswould have the same alignment, and the small scale designs would be compatible at the

junctions where the rotor meets the frame. In the mosaic this is not the case and furtherjuggling is required to disguise it.

Star Placement

When the length parameters x, y, s and t are integers, discrete motifs such as flowers orstars can be placed on the Whirling Kites figure so their centres lie on the figure, somecoincide with the corners and intersections of the lines, and they are equally spaced alongall its lines.

Figure 16(a) shows a template for a Whirling Kites design with x = 11 and y = 3.This implies 59.49, an angle 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

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(a)

(b) (c)

(d) Design Copyright P. R. Cromwell 2010.

Figure 16: A new Type B 2-level design with 12-pointed star motifs.

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Figure 17: A band network with integer boundaries.

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Variation 5

Figure 18 shows another Type B 2-level design from the Friday Mosque. Other views are

shown in photographs IRA 0721 and IRA 0722 in [17]. In this example, placing the stars atprominent points has taken precedence over equal spacing. The underlying structure of thesmall scale pattern is shown in Figure 19. The lines dividing the band into cells connectthe star centres. Left to right along the bottom of the frame we find four squares, fourrectangles, and a final square. The width of each rectangle is determined by the equilateraltriangle it contains. This arrangement is repeated around the other sides of the frame. Theratio of the lengths is AB :BC =

3 : 1 so= 60.

To construct the rotor, erect a line from A making an angle of 60 with the bottom ofthe frame. Repeat on each side of the frame and extend the four lines until they meet. Forexample, the line starting at A meets the line starting at D in the point E. These fourlines bound a square in the middle of the figure, which is subdivided into a 3

3 array of

congruent squares. These squares are smaller than those in the frame: EFis about 94% ofCD. The kite in the lower right is completed with the line CF. Note that CF and DE arenot parallel but diverge away from the frame. Label the midpoint ofCF as G.

This cellular structure provides a framework for laying out the small scale pattern. Theprincipal star motifs have 12 points and so are compatible with the 90 and 60 angles atthe corners of the band. The 16 stars in the central array are aligned so that their spikeslie on the cell boundaries; the tips of spikes of adjacent stars touch. The 12-pointed stars inthe frame are aligned so that the cell boundaries pass between the spikes this differencemay help to disguise the fact that they are further apart than the others. The star at Gmarks the transition between the two orientations and has 13 points. The square cells inthe frame contain 8-pointed stars at their centres. Triangle CDGis almost equilateral (the

angle at C is about 62.19) and this is close enough for the decoration used in the othertriangles to be applied.

Each kind of cell has its own filling, and these are consistently applied. The patternhas no awkward juxtapositions or abrupt changes, it is a masterly display of apparentlyeffortless transitions between a progression of patterns.

Conclusion

The examples discussed above have highlighted some of the problems encountered in tryingto design and fabricate 2-level Whirling Kites designs. The mathematics required to create

designs with discrete motifs such as flowers evenly spaced along the centre-lines of theband in a Type B pattern is straightforward and could have been understood by medievalcraftsmen. Working with star patterns is more difficult, but it is possible to discover byexperiment some configurations of the Whirling Kites figure whose angles are compatiblewith the geometry of stars. Even so, applying a star pattern to cover the bands presentstheoretical as well as practical challenges and the medieval artists produced ingenious andattractive solutions.

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Photograph reproduced courtesy of Steven Achord.

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

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A B

C

D

E

F

G

Figure 19: Analysis of Figure 18.

Acknowledgements

We are grateful to everyone who has given us permission to reproduce their images, and toMamoun Sakkal for explaining the Kufic basis of the decoration in Figure 6.

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