Islamic Geometric Ornament from Twelfth-Century Architecture in Azerbaijan

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Islamic geometric ornament from twelfth-century architecture in Azerbaijan

Peter R. Cromwell http://girih.wordpress.com

Contents

1 Introduction 2

2 The Seljuk environment 2

3 Source material 4 3.1 Gunbad-iSurkh,Maragha(1148) ...... 5 3.2 Mausoleum of Yusif ibn Kuseyir, Nakhchivan (1161/2) ...... 5 3.3 Round Tower, Maragha (1168/9) ...... 5 3.4 Gunbad-iSeh,Orumiyeh(1184)...... 8 3.5 Mausoleum of Mu’mina Khatun, Nakhchivan (1186) ...... 8 3.6 Mudhafaria Minaret, Erbil (1190) ...... 8 3.7 Gunbad-i Qabud, Maragha (1196/7) ...... 13 3.8 Gulustan Mausoleum (about 1200) ...... 13

4 Methods of design 13 4.1 Modifyinganexistingpattern ...... 13 4.2 Creatingstarsbydoublingedges ...... 19 4.3 Designingwithstarmotifs...... 24 4.3.1 Regularstars,basicgrids ...... 24 4.3.2 Otherstars,othergrids ...... 25 4.3.3 Wheel construction for stars ...... 29 4.3.4 Decoratedtiles ...... 30 4.3.5 Constellation patterns ...... 31 4.4 Rosettemotifs ...... 35 4.5 Colouredpatterns ...... 38 4.6 Complementarypatterns...... 40 4.7 Spiralmotifs ...... 43

5 Observations 44 5.1 Patternre-use...... 44 5.2 Archimedeantilings ...... 46 5.3 Brokensymmetry...... 46 5.4 Designingwithcolour ...... 48

6 Concluding remarks 49

1 1 Introduction

In most branches of fine art or architecture, studying the work of a particular school, region, period, or other cultural group is routine and unremarkable. However, in studies of the decorative arts such as geometric ornament, this is an uncommon approach. For example, Islamic geometric patterns are often treated as a homogeneous collection and grouped according to various intrinsic geometric properties, rather than by regional or dynastic styles. With this form of presentation, patterns can become detached from any local context, and the aggregates may contain items separated by large distances in time or space. This, in turn, can obscure trends in the evolution of form and style, and could leave the misleading impression that Islamic ornament has a single, unified, universal style. While there are a few ubiquitous patterns that provide a shared foundation for artistic development, many patterns do have a style that is historically identifiable, and they can be assigned to a particular time or place with quite high confidence. In this paper we focus attention on a small region to explore the state-of-the-art at a time of rapid development and innovation. In particular, we examine forty patterns from eight twelfth-century buildings in the area around Azerbaijan. The examples do not constitute an exhaustive catalogue of each monument, but the sample is large and provides a representative snapshot of the work being produced at this time and place. These patterns provide enough material to illustrate a range of methods for composition: they feature motifs of different complexity (spirals, stars, rosettes), the motifs are organised around different structures (simple lattices, Archimedean tilings, circle packings), and the inspiration seems to come from an empirical approach to design driven by a range of experimental processes (modification, dissection, cropping, threading, colouring). This collection is supplemented with around twenty other contemporary patterns from the wider area of Iran and Turkey to provide a broader context and further examples of the methods we describe.

2 The Seljuk environment

The Seljuks originated from a Turkic nomadic tribe who lived on the steppes between the Caspian and Aral Seas. In the mid-eleventh century, they conquered lands from Afghanistan westwards through Iran to Azerbaijan and Iraq, and later expanded into Syria, Armenia and Turkey. They established a prosperous environment that sustained cultural and intellectual development in a range of fields, including poetry, science and architecture. New ideas were also introduced into geometric ornament, and the many Seljuk buildings that survive display a rich diversity of patterns. The Seljuks adopted many Persian ways and used local architects and craftsmen who, initially, developed or consolidated earlier ideas. However, by the twelfth century, it is clear that innovative approaches were being tried, and there is a level of complexity, organisation, and systematic enumeration of possibilities that is not seen before. Evidence of their freedom to explore can be found in the stucco mihrab (1105) of the Barsian Mosque. The mihrab is like a sketchbook of ideas and we shall refer to it several times. Many of the patterns are just small fragments that give glimpses of recognisable motifs, and sometimes there is insufficient information to see how a pattern develops beyond the panel. Brick was the standard building material in Iran long before the Seljuks. It was also used for decoration. There are fine examples of relief brick patterns on the eleventh-century

2 minarets in Damghan at the Tarikhaneh Mosque [14, Pl. 196] and the Friday Mosque (1058) [14, Pl. 197]. In each case, the whole surface of the cylindrical column is divided into a stack of zones, each filled with a band of ornament. Many brick patterns were made by setting standard bricks in horizontal and vertical arrangements, but bricks were also cut or carved, and sometimes specially moulded. Brick patterns were originally made through variations in the bonding pattern of structural walls; later, walls were covered with a brick revetment, decoupling the functions of structural integrity and ornament, and leading to a richer variety of patterns. The Sassanians built large brick domes over square halls, using squinches to make the transition from the square plan to the octagonal support for the base of the dome. Islamic architects continued to use this characteristic Persian form to provide dome chambers at the heart of mosques. The domed chamber archetype was also used with circular or polygonal plans in the mausoleums known as tomb towers. These mausoleums are distributed across the northern part of Iran, and Turkey where they are called türbe (tomb) or kümbet (dome). Many examples are described and illustrated in [1, §3c], and there are some comparative studies [5, 18]. Some eleventh-century tomb towers have survived near Damghan: the Gunbad-i Qabus at Gorgan (1006) [14, Pl. 176], and the mausoleum of Pir-i Alamdar (1026) [14, Pl. 198] date from the Ziyarid dynasty, and the Gunbad-i Mehmandust, also known as the mausoleum of Ma’sum Zadeh, (1096) is a Seljuk example. They are all circular and are rather plain. The Gunbad-i Qabus has ten prominent flange-like buttresses, giving it a star-like plan, but has no other decoration. Both of the other towers carry narrow bands of ornament in relief brickwork running just below the roof — two bands of meander patterns either side of a band of calligraphy. At Kharraqan, on the plains southwest of Qazvin, stand two octagonal tomb towers built in 1067/8 [14, Pl. 558] and 1093 [14, Pl. 559]. They once had double-shell domes, but the outer domes have not survived. Apart from this, they were were well preserved until recently, but suffered some earthquake damage in 2002. These towers are the earliest representatives of a new fashion for decorating mausoleums that took hold in the Azerbaijan area: they are completely covered with ornament, in this case with a revetment of brick patterns. Not everyone adopted the new style: a twelfth-century mausoleum at Hamadan known as the Ghorban Tower is a 12-sided brick tower with a pyramidal roof; each face has an arch-shaped panel recessed into wall but there is no ornament. The buildings we shall study are located in the area shown in Figure 1. Known historically as Arran in the north and Azerbaijan to the south, today it includes the modern Republic of Azerbaijan in the north, and the north-west region of Iran. The buildings were constructed in the late Seljuk period when the area was controlled by the Eldiguzids (1136–1225), also known as the Atabegs of Azerbaijan. The founder of this dynasty, Shams ad-Din Eldigüz, was appointed governor of Arran in 1136 by the Sultan of Iraq and western Iran; he established a territory extending from Nakhchivan to Hamadan and on to Isfahan. The buildings are richly decorated towers (mausoleums and a minaret) that showcase the architect’s ability in pattern design. Unlike mosques, which are often adapted or incorpo- rated into larger complexes in later periods, these freestanding towers have maintained their integrity. The buildings are dated, and each one is the work of a single designer, implying that the patterns were assembled as a coherent decorative scheme for the building. This gives us a reliable foundation for comparative analysis. Finally, a remark on the colour-coding used in the figures. The monuments include some

3 Araxes Nakhchivan Gulustan Caspian Marand Sea Tabriz Lake Orumiyeh Urmia Maragha

Soltaniyeh Qazvin Damghan Erbil Z a g r o s M o u n t a i n Kharraqan s Varamin Hamadan

Kashan Euphrates Golpayegan Baghdad Natanz Tigris Barsian Isfahan

Figure 1. Azerbaijan and surrounding region. early examples of coloured patterns. In the ﬁgures elements in blue are an integral part of the design and correspond to the use of turquoise glaze in the original artwork; elements in red are additional lines overlaid on the design to aid explanation, indicate structure, or illustrate stages in a construction.

3 Source material

In this section we present brief notes on the eight buildings that are the focus of our study. Their locations are marked in Figure 1. The seven mausoleums share the same basic construction, which is outlined here. They stand on their own, not attached to other buildings. They have central symmetry so that the footprint is a circle, square, or other regular polygon. They are divided into two chambers, one above the other. The lower chamber is built of stone and is partially below ground, forming a crypt; a brick tower is built on this base to form the upper chamber. The interior is plain, while the outside is decorated with geometric patterns in brick, terracotta or stucco. The roof may be a dome (gunbad is Persian for dome) or be conical or pyramidal, and it may have had a double skin, allowing the interior ceiling and the external roof to have diﬀerent shapes. In many cases the architect is named in the inscriptions, indicating that the profession was held in high regard. Note that I have not visited any of the places and have worked from photographs, mostly from recent high-resolution images. Early photographs that show some of the buildings before restoration can be found in [21].

4 (a) (b)

Figure 2. Patterns from the Gunbad-i Surkh, Maragha.

3.1 Gunbad-i Surkh, Maragha (1148) The Gunbad-i Surkh (red tomb) is a square tower with engaged columns at the corners, topped with an octagonal drum that supported the roof [14, Pl. 223]. Most of the ornament is formed of brick patterns; the bricks in the corner columns are specially moulded. The ornament over the entrance is made from terracotta strips, some of which have a turquoise glaze, and plaster; these patterns are shown in Figure 2. Inscriptions give the year of completion, and the names of the patron and builder; the patron is given the title ‘Guardian of Azerbaijan’ [19].

3.2 Mausoleum of Yusif ibn Kuseyir, Nakhchivan (1161/2) The mausoleum is an octagonal tower with a pyramidal roof [14, Pl. 227]. Each face is almost covered by a single panel of geometric ornament formed from brick set in plaster. Scrolls and other ﬁne-lined details are also carved into the plaster. Figure 3 shows most of the patterns; examples (a), (b), (d) and (f) are from the entrance fa¸cade. The inscription over the entrance gives the construction date and identiﬁes the occupant; the name of the architect is on a plaque inside the doorway. A Quranic inscription encircles the building just below the roof.

3.3 Round Tower, Maragha (1168/9) The mausoleum is a circular tower with a large projecting rectangular frame around the entrance called a pishtaq [14, Pl. 226]. It is dated by an inscription. The tower is plain except for the decorated portal, which contains the two patterns shown in Figure 4: (a) in the tympanum and (b) forming the frame. Both are outstanding examples in their complexity and originality.

5 (a) (b)

Figure 3. Patterns from the mausoleum of Yusif ibn Kuseyir, Nakhchivan.

6 (e) (f)

(g) (h)

Figure 3 (continued).

7 (a) (b)

Figure 4. Patterns from the Round Tower, Maragha.

3.4 Gunbad-i Seh, Orumiyeh (1184) This building has the same structure as the Round Tower at Maragha: it is a circular tower with a large decorated entrance portal. An inscription gives the date and the architect. The frame of the pishtaq has three bands or ornament: the pattern in the outer band is shown in Figure 5(c), running inside that is a band of calligraphy, then a band of pattern 5(a). Above the door is pattern 5(b) and a three-tiered muqarnas hood in an arched frame; the design in the irregular-shaped spandrel spaces on either side includes 7-pointed stars.

3.5 Mausoleum of Mu’mina Khatun, Nakhchivan (1186) The mausoleum is a decagonal tower that is about 25 metres high, much taller than similar buildings [14, Pl. 229]. It was commissioned by the second Eldiguzid ruler Jahan Pahlawan for his wife, and designed by same architect who built the mausoleum of Yusif ibn Kuseyir. A four-year project to restore the monument was completed in 2003. Every surface of the building is decorated. Around the top of the tower is a three-tiered muqarnas cornice. The band of calligraphy running below it presents Persian poetry, rather than the usual Arabic scripture [5, 20]. On most faces, a panel of geometric ornament runs the entire height of the tower; only one panel is divided and the two parts re-use patterns from the other faces. The face containing the entrance also has a tympanum over the door. Patterns from all the large panels are shown in Figure 6. They display a variety of diﬀerent genres, and mastery of some diﬃcult techniques.

3.6 Mudhafaria Minaret, Erbil (1190) The area to the west of Lake Urmia and the Zagros mountains was part of the Seljuk Empire but not under the control of the Atabegs of Azerbaijan. However, the remains of a late

8 (a)

Figure 5. Patterns from the Gunbad-i Seh, Orumiyeh.

9 (a) (b)

Figure 6. Patterns from the mausoleum of Mu’mina Khatun, Nakhchivan.

10 (e) (f)

(g) (h)

Figure 6 (continued).

11 (i)

Figure 6 (continued).

12 twelfth-century minaret in Erbil are relevant to our study. The Mudhafaria Minaret is a brick structure composed of a circular column standing on an octagonal base. The column is decorated with relief brick patterns in which the bricks are laid vertically and horizontally to imitate the warp and weft threads in woven fabric — a technique known as hazar baf. The faces of the base have two tiers of recessed arches which once carried ornament, but are now in poor condition with many panels stripped back to the bare wall. A few have fragments of glazed tiles, impressions in the plaster that held tiles, or remnants of carved stucco. Some patterns have been recovered from these remains [2] and are shown in Figure 7.

3.7 Gunbad-i Qabud, Maragha (1196/7) The Gunbad-i Qabud (blue tomb) [14, Pl. 225] stands next to the Round Tower. It is a decagonal tower, outside and inside, and once had a pyramidal roof of blue-glazed brick [5]. Like the Gunbad-i Surkh, it has engaged columns at the corners. The ornament on the lower section is not confined to neatly framed panels, but flows over the whole surface, including the corner columns. The same pattern is applied to all the faces of the tower, either as direct copies or reflected in a vertical mirror line. The top section of each face has a three-tiered muqarnas hood inside an arched frame. The spandrels of both the arch and the hood contain unusual combinations of stars (like the spandrels on the Gunbad-i Seh, noted above.) The muqarnas panels are decorated with a small selection of patterns that are re-used. On some buildings the re-use of patterns seems to be avoided as the designers show off their creative powers through variety. Here, although the number of patterns is limited, they display a high level of innovation and complexity — several examples feature the interplay of two independent patterns in the same design. Examples are shown in Figures 8 and 28.

3.8 Gulustan Mausoleum (about 1200) This mausoleum is built from red sandstone [14, Pl. 232]. Old photographs show it as a stump in the open plain, but it now has a pyramidal roof, which was added as part of restoration work completed in 2016. The upper chamber is a short tower with twelve sides. The base is square at ground level, and makes the transition to the dodecagonal drum above by facetting the corners. The base is plain, but each of twelve upper faces contains a panel of geometric ornament. Only the three patterns shown in Figure 9 appear and are re-used around the drum.

4 Methods of design

In this section we shall explain a selection of methods for creating Islamic patterns. They can all be illustrated with designs found on the buildings included in this study.

4.1 Modifying an existing pattern One of the easiest ways to create a new pattern is to use an existing one as the starting point and then modify it by removing, replacing or altering some of its elements, or adding new ones.

13 (a) (b)

Figure 7. Patterns from the Mudhafaria Minaret, Erbil.

14 (e) (f)

(g)

Figure 7 (continued). 15 Photograph reproduced courtesy of Hamid Abhari (a)

Figure 8. Patterns from the Gunbad-i Qabud, Maragha.

16 (b) (c) (d) (e)

(f)

Figure 8 (continued).

17 (a)

(b) (c)

Figure 9. Patterns from the Gulustan Mausoleum.

18 Figure 10 illustrates how this works with two patterns from the Mudhafaria Minaret (Figure 7(c) and (d)). They can be derived from the two Archimedean tilings that contain dodecagons using essentially the same process. Archimedean tilings are tessellations composed of regular polygons of more than one kind with the same arrangement around every vertex. There are eight such tilings and many of them can be found in Roman mosaics, either with the polygons filled in black and white using the chessboard colouring scheme, or as a linear framework that outlines the polygons. In the latter case, the compartments produced by the polygons may contain figurative or ornamental panels. Such mosaics, and hence the Archimedean tilings, would have been familiar in the lands around the eastern end of the Mediterranean under Roman or Byzantine control. In both parts of Figure 10 the Archimedean tiling is on the left in black. Archimedean tilings are usually labelled by listing the number of sides in the polygons that surround a vertex: we have 3.12.12 in (a) and 4.6.12 in (b). In both cases, the construction proceeds by subdividing each dodecagon into a central hexagon and a corona of alternating squares and triangles — the added lines are shown in red. (Although the resulting tilings are still composed of regular polygons, they are not Archimedean as the vertices are not all surrounded in the same way — there are two types of vertex in (a) and three types in (b).) The Islamic patterns are shown in black on the right of the figure. In each case, they are derived by placing 6-pointed stars inside hexagons then erasing some of the lines. In (a) all of the hexagons are replaced with a star. In (b) only the hexagons that belong to the initial Archimedean tiling become stars; those that are created by the subdivision remain as part of the pattern. The division of the stars into six rhombi used in (b) is also an indication of Roman influence (a more typical Islamic division would have a star and six kites). Figure 11 shows two other patterns that are related by a similar add-and-subtract process. The pattern in black on the left can be viewed as overlapping dodecagons; the shapes bounded by the lines are a 6-pointed star, an irregular hexagon sometimes called a shield, and a long, thin, bone-shaped octagon. The red lines added in the centre of the figure subdivide the shield regions. The black pattern on the right is produced by erasing line segments at the corners of the shields. The result can be viewed as overlapping barrel- shaped hexagons. This example is based on the two patterns from the Gunbad-i Seh shown in Figure 5(a) and (b). Note that the overlapping dodecagons in Figure 5(a) are slightly smaller than in Figure 11, so the bone-shaped octagons are fatter around the waist.

4.2 Creating stars by doubling edges Islamic ornament is known for patterns containing star motifs. In Figure 10 stars were directly inserted into an existing pattern: some of the hexagons in a tessellation were replaced with 6-pointed stars. In this section we shall demonstrate a simple method to generate star patterns by doubling the edges in a tessellation. Figure 12(a) shows a vertex from a tessellation with its edges radiating out like spokes around a hub. Draw lines parallel to the spokes, as shown in (b). When there are more than four spokes, this creates a star motif centred on the vertex. If we start with n spokes that are evenly spaced, and the resulting star will have n-fold symmetry. When the spokes are not evenly spaced, the star becomes irregular and may degenerate into a shape with fewer than n spikes (as we shall see later). Applying this process to all edges and vertices in a tessellation produces a simple star

19 (a)

(b)

Figure 10. Constructing patterns from Archimedean tilings.

20 Figure 11. Relationship between two patterns on the Gunbad-i Seh.

pattern. The result is usually not very effective because the underlying tessellation remains too obvious. It can be disguised by embellishing the design with additional motifs. Figure 12(c–e) show the effect of adding polygons and stars centred on the vertex. Figure 13 shows some examples of this technique. In part (a) the process is applied to a tessellation formed by overlaying the standard triangular and hexagonal grids; the triangular grid generates the 6-pointed stars. Part (b) is also derived from the triangular grid; in this case the equilateral triangles are subdivided and 12-pointed stars are produced. Both patterns are from the Ghurid Minaret at Jam, Afghanistan, which dates from the end of twelfth century. The next three examples are Seljuk patterns from Turkey. Part (c) is from the entrance to the Çifte (Twin) Madrasa in Kayseri (1205) [14, Pl. 478]. It is based on a square grid; notice that the stars have more internal structure than the basic form in Figure 12(b). Part (d) is from the entrance to the mosque in the citadel in Divrigi (1180/1) [14, Pl. 499], and was built by an architect from Maragha [14, p. 76]. The underlying tessellation is derived from the 4.8.8 Archimedean tiling composed of squares and octagons — each octagon has spokes connecting its corners to the centre. The ‘stars’ generated from the 5-valent vertices are degenerate and do not have five well-define spikes. The addition of overlapping regular octagons centred on the vertices of the underlying Archimedean tiling completes the design. Part (e) is from the mausoleum of Sultan Izzeddin Keykavus in the Sifaiye Madrasa in Sivas (1217/8), who founded the madrasa as a hospital and medical school. The mausoleum is a 10-sided tower with a pyramidal roof; it was built by an architect from Marand, and was restored in 2011. The pattern [22, Photo 29] is unusual for containing so many different stars: there are 8-, 10-, 12- and 16-pointed stars. The angles in these regular stars are not quite compatible, so there is a small bend in the channel running between the 10-pointed and 12-pointed stars. Figure 13(f) is from a Seljuk mihrab in the Masjid-i Malik (King’s Mosque, now the Imam Mosque) in Kerman, Iran [14, Pl. 532]. In this case, the underlying tessellation is

21 (a) (b) (c) (d) (e)

Figure 12. Parallels offset from edges meeting at a vertex, and various embellishments. the 3.3.4.3.4 Archimedean tiling composed of squares and triangles. The inner corners of a star and the corners of its associated supplementary polygon are chosen so that they lie on the spokes radiating from the vertex at their centre. As a consequence, the channels bounded by the pairs of parallel lines are not all the same width: the channels separating two triangles are slightly narrower than the ones separating a square and a triangle. (If the channels are made of equal width, the stars become more irregular.) In the original design on the mihrab, the eye is distracted from these weaknesses in the geometry by foliate arabesques in compartments. In our source materials patterns of this kind are included on both of the mausoleums in Nakhchivan. Of the patterns in Figure 3 from the Kuseyir mausoleum (f) is constructed on the standard triangular grid and is decorated with 6-pointed stars, (g) has 6-pointed stars placed at the midpoints of the edges of a square grid and is decorated with hexagons, and (h) has 8-pointed stars at the vertices of a square grid and is decorated with octagons. In the last case, the underlying tessellation can be seen as two dually situated copies of what is sometimes called the Cairo tiling of congruent pentagons. In Figure 6(e) from the Mu’mina Khatun mausoleum the underlying tessellation has ten spokes radiating from each vertex, and the pattern is decorated with 10-pointed stars. In all of these examples, the result of the edge-doubling process is embellished with additional polygonal or star-shaped components. With the exception of Figure 13(e), the underlying tessellations are simple and familiar, and the additional decoration helps to obscure the starting point and create a more interesting visual effect. The network underlying Figure 13(e) is neither a regular nor an Archimedean tiling — this pattern is one of several thirteenth-century examples from Turkey that are constructed from more complicated tessellations. Some of them are sufficiently intricate that they do not require much embellishment. Both Lee [16, pp. 124–130] and Bonner [8, pp. 216–7] have observed that these later patterns are derived by offsetting parallel lines from grids used to construct other star patterns. In these cases, the stars in the derived pattern are located at the same positions as the stars in the original pattern — Lee describes the process as ‘doubling stars’ because the new stars have twice as many points as the original stars. (Neither Lee nor Bonner mention that the method automatically generates star motifs from any tessellation.) Bonner [8, p. 72] also suggests that the technique (which he calls ‘extended parallel radii’) was pioneered by the Seljuks of Rum, but it could also be seen as a natural extension of the ideas shown here that were used earlier in Azerbaijan.

22 (a) (b)

Figure 13. Patterns constructed by oﬀsets from a tessellation.

23 (e) (f)

Figure 13 (continued).

4.3 Designing with star motifs In the examples of star patterns that we have seen so far, the stars have been inserted into existing patterns or they have been generated as the by-product of a construction process. How do we use stars as the primary motifs in a composition and take more control over their placement and visual eﬀect?

4.3.1 Regular stars, basic grids First, we have to create star motifs. Simple stars can be constructed by placing n points equally spaced around a circle and connecting them by straight lines. When each line connects points p steps apart, the result is what mathematicians call a regular star and is denoted by {n/p}. Stars of this kind are found in Roman mosaics. Figure 14 shows patterns formed from dense arrangements of these simple stars. The circumcircles of the stars are packed together, and the stars are aligned so that their spikes meet where the circles are tangent. This means that the pattern lines continue from one star to another without break or deviation. The voids between the circles are ﬁlled in the same manner: the lines of each star are extended outside its circumcircle until they meet lines from other stars; this process adds connections between the stars and strengthens pattern. There are Islamic examples of all the patterns in Figure 14:

(a) This simple design is pre-Islamic and can be found in Roman mosaics in France and Morocco [4, Pl. 214]. In the Islamic world it is common and widespread. The {6/2} stars ﬁt together without needing additional connecting lines.

(b) This is a variant of the Star and Cross pattern in which the collinear sides of the cross have been connected to outline a square in the centre. An early example occurs in the ninth-century al-Tariq Mosque (Masjid-i Nuh Gunbad) at Balkh, Afghanistan.

24 (c) This pattern can be found in the ruins of the thirteenth-century mosque at Farumad, north-east Iran — see [14, Pl. 576] or [15, Fig. 7]. In structural terms (8-pointed stars on a square grid), it is the same as (b), but I do not know of any earlier examples. The {8/3} motif itself does appear on earlier buildings. On the eleventh-century brick tomb tower at Demavand, north of Varamin, a single {8/3} star ﬁlls a square panel, adjacent to a panel of the Star and Cross pattern [23, Pl. 8], and on the late twelfth- century brick mausoleum of Khwaja Atabeg in Kerman {8/3} stars are repeated in a diﬀerent design [14, Pl. 534].

(d) This is one of the most common and widespread Islamic star patterns. The example in the north dome chamber of the Friday Mosque in Isfahan is elaborated with secondary geometric motifs in the compartments, so the basic form may have been used earlier. The 10-pointed stars do not ﬁt the standard square or triangular arrangements, and a rhombic grid with a compatible angle is required.

(e) The brick patterns on the early, east tower at Kharraqan include this pattern of 12-pointed stars [14, Pl. 558].

(f) This pattern uses the same {12/3} stars as (e) but they are arranged on a diﬀerent grid. It can be found in the remains of a caravanserai dating from the 1330s, just north of Marand on the road to Julfa [14, Pl. 591]; the background is ﬁlled with turquoise tiles. As with (c), I do not know of any examples from the Seljuk period.

Three of these basic patterns are used in the Seljuk dome chamber at Golpayegan (1114). The spandrels in the main mihrab contain a version of Figure 14(a) in which the lines are doubled up in the form of a wide ribbon. On the left of the mihrab is a large panel of 12- pointed stars arranged in pattern (e). Adjacent to it on the next wall, forming the southern corner of the chamber, is a large rectangular panel of pattern (d) containing 10-pointed stars [14, p. 282]; it looks rather like a hanging carpet with a fringe of calligraphy along the bottom. The pattern in Figure 5(c) from the Gunbad-i Seh is derived from Figure 14(d): the stars in the central column have been replaced with the motif shown in Figure 15(b). Apart from this adapted example, these simple star patterns are not found in our source materials. We shall see that star patterns developed in several ways: relaxing the definition of a star, exploring different ways to arrange them, and combining different types.

4.3.2 Other stars, other grids The examples in Figure 14 are based on three simple grids composed of triangles, squares or rhombi. The number of points in a star placed on a triangular grid must be a multiple of 6, and the number of points in a star placed on a square grid must be a multiple of 4. The angle in a rhombic grid is adjustable and can be selected to match any star with an even number of points. What other ways are there to create dense arrangements of stars, and do any of them allow us to use stars with an odd number of points? There are only ﬁve grids in which the vertices are all related by translation (the 2-dim- ensional Bravais lattices): they are based on equilateral triangles, squares, rhombi, rectangles, or parallelograms. Suppose we place stars centred on the vertices of such a grid. For the stars to be connected together, their circumcircles must be tangent, which means

25 (a) {6/2} on triangular grid (b) {8/2} onsquaregrid (c) {8/3} on square grid

(d) {10/3} on rhombic grid (e) {12/3} on triangular grid (f) {12/3} on square grid

Figure 14. Dense arrangements of regular stars on standard grids.

(a) (b) (c)

Figure 15. Three small motifs that have a common boundary and hence are interchangeable with one another.

26 (a) (b)

Figure 16. Seljuk variations on Figure 14(b). that the grid must be equilateral. This explains why only the first three lattices appear in Figure 14 since rectangles and parallelograms have sides of different lengths. The symmetry properties of the grid do not play any role in this analysis, so we do not need to restrict ourselves to Bravais lattices. In fact, any equilateral tiling could be used to control the placement of stars. Figure 16 shows two patterns inspired by Figure 14(b). In the original, 8-pointed stars are placed on a square grid and crosses are formed in the complementary spaces. In the new patterns, the cross motif is preserved and is blended with 7-pointed stars in (a) and 9-pointed stars in (b). Both examples are from Seljuk Iran: (a) is from the Masjid-i Malik in Kerman [25, IRA 2420], and (b) is from a border around the mihrab in the Barsian Mosque. Each pattern contains a single type of star motif, but the stars are not arranged on any of the basic grids (Bravais lattices). The solid red lines in both parts of the figure connect the centres of stars with tangent circumcircles. In (a) they form the 3.3.4.3.4 Archimedean tiling. Although (b) looks very similar, the tiling is not Archimedean as the triangles are not equilateral — the dotted lines are longer than the others, and the stars they connect cannot touch each other. To explain the constructions of these patterns, we need to reconsider our concept of a star. We have already seen two methods for making a star, and they produce different families of star motifs: a star with parallel sides (as generated in §4.2) is not regular if it has an odd number of points. We shall now explore two more methods to create stars: one is a dissection technique (sometimes called cut-and-paste), and the other uses scaffolding in the form of a wheel to control the shape. The first three parts of Figure 17 illustrate the construction of the stars in Figure 16(b). Pieces of two different stars are fused together: (a) is the regular 8-pointed star {8/2}, (b) is a 10-pointed star which has 90◦ corners where it meets its circumcircle, and (c) is assembled by interleaving two sectors from (a) with two sectors from (b) to make a 9-pointed

27 (a) (b) (c) (d) (e)

Figure 17. Constructing shapes using dissection and reassembly.

(a) (b) (c)

Figure 18. Construction of a template for Figure 16(a).

star. The resulting star has both large and small ‘teeth’. The noticeable variation in tooth size, combined with the fact that the star appears in four different orientations within the pattern, makes it difficult to compare the motifs, and leads to confusion1 as to whether the stars are all congruent. The blue components of the 3-layer design in Figure 2(b) are also produced by cut- and-paste. Figure 17(d) shows a regular octagon; in (e) three copies of the top slice of the octagon have been attached to an equilateral triangle, producing an irregular2 nonagon with 3-fold symmetry. In general, the only sources we have to help us understand traditional patterns are the surviving artefacts that carry the ornament. Sometimes, however, we can find additional evidence in manuscripts: panel 81a of the Topkapı Scroll is a template for the pattern in Figure 16(a). The scroll was composed long after the Seljuk era so we do not know whether it records a tradition that was preserved for several hundred years or is another interpretation of the same design. Figure 18(c) shows the template: the framework shown in red corresponds to lines scored into the paper with a stylus and the other lines are inked in. Figure 18 shows stages in the construction of the template. The template is square and contains four semicircles that are tangent to each other and pass through the corners of the square. Lines are drawn connecting the centres of opposite semicircles. These oblique lines divide each semicircle into large and small sectors. In the semicircle on the right-hand side of Figure 18(a) the small sector has been divided into three equal parts, and two sectors of the same size have been marked in the large sector.

1For example, Bonner [7, 8] incorrectly describes the Barsian pattern as having 7-pointed and 9-pointed stars. 2These are incorrectly shown as regular polygons in [5, Fig. 7].

28 (a) (b) (c) (d) (e)

Figure 19. Making stars with the wheel construction.

In Figure 18(b) another line has been added bisecting the remaining angle. This produces ﬁve sectors of 25◦ and two sectors of 27.5◦. We shall refer to the lines radiating from the centre of the circle as spokes. The central square marked in black is part of the pattern; its corners are located where the oblique lines meet the semicircles. Extend one of its sides until it meets one of the spokes as shown — the intersection point determines the radius of a new semicircle with the same centre. In Figure 18(c) the spokes and semicircles have been replicated in the other parts of the diagram. This framework (in red) provides all the information required to construct the pattern. One line in the pattern is highlighted in black. Notice that, apart from the end that lies in the corner of the template, all the corners of the line lie at the intersection of a spoke with an inner semicircle. These are the ‘dents’ in the stars. The ‘spikes’ of the stars occur where pattern lines cross and are not constructed explicitly.3 Notice also that the edges of the Archimedean tiling added to Figure 16(a) to indicate a structural feature of the composition are not involved in this construction. The template is for reproduction of the pattern, and this process may be diﬀerent from the original creative process.

4.3.3 Wheel construction for stars Figure 18 has introduced the wheel construction for making stars. Wheels embody the idea that a star is a zig-zag that closes round on itself. Some examples are shown in Figure 19 — the wheels are in red. A wheel is a framework consisting of two circles, one inside the other, and a set of spokes radiating out from a focal point to meet the outer circle. The star is constructed by drawing a line between adjacent spokes that bounces alternately oﬀ the inner and outer circles. Figure 19(a) shows a simple example. The method is very ﬂexible and the other parts of Figure 19 illustrate the consequences of varying some of the parameters.

• Reducing the radius of the inner circle produces a star with sharper spikes — see (b). Regular stars and stars with parallel sides have inscribed circles, so they are special cases of the wheel construction.

• In (a) and (b) the spokes come in two kinds depending on whether they deﬁne the inner or outer corners of the star. In fact, all the spokes can serve both roles, as is shown in (c). In this case, the method automatically produces the characteristic ring of kites found in Islamic star motifs. 3The crossing formed where two 7-pointed stars meet does not lie on the line connecting the star centres, but is oﬀset slightly. If the pattern were constructed using the polygons-in-contact method, the crossing would lie at the mid-point of this edge.

29 • In (d) the spokes are not evenly spaced. This kind of adjustment enables the designer to control the alignment between motifs. As we shall see later, this becomes important when combining stars of different types in a design. • In (e) the circles are not concentric and the inner circle is shifted down. The spokes are equally spaced and radiate from the centre of the inner circle. This device produces stars with larger spikes at the top and smaller, closer spikes at the bottom, and facilitates a change of scale in a design. Even though the wheel construction can generate such a wide range of motifs, all the results look star-like because the geometry of the wheel constrains the shape within visually recognisable limits. Note that not all stars can be created in this way — for example the star in Figure 17(c) does not have an incircle that touches all the ‘dents’. Wheels are a prominent feature in manuscripts that contain star patterns, although they may be present only as indentations in the paper and not immediately visible. This confirms that wheels were used traditionally for constructing final versions of a design. However, they do not seem suited to the creative process of composition.

4.3.4 Decorated tiles In this section and the next, we consider how to create designs containing more than one kind of star. One possible source of inspiration is illustrated in Figure 20. The method is based on decorated tiles. In this case, the tiles are regular polygons with 5, 6, 7, or 8 sides (shown in red), and they are decorated with star motifs that make an angle of 60◦ with the edges of the tiles. The star on the hexagon is the regular {6/2}, but the other stars are not regular. The fact that the incidence angles are all equal means that when the tiles are placed edge-to-edge, the pattern lines continue across the join without deviating. Ideas for designs can be explored by assembling the tiles in various ways. Three hexagons surround a point in the plane exactly, an arrangement that produces the regular hexagonal grid when replicated — Figure 20(a). However, the monotony of the grid quickly becomes tiresome. If we insist that the tiles should fit together with no gaps and no overlaps then there are no other ways to assemble these tiles. However, if we view the tiles as a tool to stimulate ideas then we can afford to be more relaxed about the rules and adopt a more flexible approach. Rather than place three hexagons around a point, we can try a pentagon, a hexagon and a larger regular polygon. A pentagon, a hexagon and a heptagon leave a gap of almost 3.5◦, while a pentagon, a hexagon and an octagon produces an overlap of 3◦. Figure 20(b) and (c) show two patches of tiles based on these configurations. Traditional star patterns do contain these kinds of arrangements. Hankin’s first paper on Islamic geometric patterns shows [13, Fig. 8] a pattern composed of 5-, 6- and 7-pointed stars arranged like those in Figure 20(b) that he saw on the domed ceiling of a bath house in the sixteenth-century Mughal palace of Jodha Bai at Fatehpur Sikri, India. This pattern is a good choice for a domed surface as the curvature helps to offset the angle deficit. Hankin’s figure is annotated with a network of polygons and he records how he found edges of some of these polygons scratched into the plaster. It was this observation that led him to develop the polygons-in-contact method for constructing Islamic geometric patterns, a method later adopted and promoted by Bonner [8]. Returning to our twelfth-century examples, Figure 6(a) has 5-pointed and 6-pointed stars alternating around 8-pointed stars, and

30 a small part of the same pattern appears in the mihrab of the Barsian Mosque. Figure 6(b) is similar but also includes 7-pointed stars. These two examples from the mausoleum of Mu’mina Khatun form repeating patterns that can be extended indefinitely to fill large panels. However, if the area to be decorated is small or irregular, it may be more effective to create a design that fits the available space. The other parts of Figure 20 show two examples from the Gunbad-i Qabud. In (d) the tiles cover an arch-shaped region that models a squinch. The misfitting (gaps or overlaps) of the tiles, and the resulting misalignments of the stars, do not matter for, as we saw in Figure 18(c), a pattern can be obtained by connecting the inner points of the stars. The small red dots in the figure highlight the points that determine a pattern line running from top to bottom of the panel. Applying this process throughout the panel produces the result shown in (e). Using the inner points to define the pattern means that we do not explicitly control the positions of the outer points of the stars — they materialise at the intersections where lines cross.4 This design is used several times in the bottom tier of the muqarnas hoods at the top of the tower. The same technique is applied to a spandrel in (f) and (g); it corresponds to the blue part of the 2-layer design in Figure 8(a).

4.3.5 Constellation patterns Anyone playing with combinations of regular stars soon discovers that two {5/2} stars and a {10/4} star fit exactly around a point in the plane. In fact, ten {5/2} stars can be arranged in a ring around a {10/4} star to produce very attractive motif composed entirely of regular stars. Figure 21(a) shows a design built from five of these motifs. The void in the middle has been completed by adapting the small stars nearest the centre and extending their sides beyond their circumcircles until they meet other such lines. An arched panel [7, Fig. 32] in the north dome chamber in the Friday Mosque in Isfahan is based on this arrangement. This is the earliest example (1088) I know of what I call a constellation pattern: a dense arrangement of star motifs with primary stars (usually with eight or more points) separated by 5-pointed satellite stars. Figure 21(b) shows another constellation pattern. The circle packing that determines the stars is straightforward to construct as the centres of the small circles lie at the vertices of the 3.12.12 Archimedean tiling. The primary stars are {12/5} stars, but the small stars are not regular. This pattern is common and quite widespread. Figure 22 shows an analysis of three panels from the muqarnas hood of the mihrab in the Barsian Mosque. The grey regions show the extent of the original panels, and I have developed the patterns from the fragments they contain. The arrangements in the first two are clearly related: (a) combines three 11-pointed stars and a 12-pointed star, and (b) uses four 13-pointed stars. Neither of these arrangements can be extended in an obvious way to produce a repeating pattern, but the patches provide a basis to cover the required area. It seems likely that whoever was experimenting in this way was aware of the periodic arrangement with 12-pointed stars shown in Figure 21(b), but I do not know of a surviving example of that pattern that predates the mihrab (later examples can be found in Turkey on the mid-twelfth-century mosque at Niksar and thirteenth-century buildings at Sivas and

4Note that this is the opposite of the polygons-in-contact method where the outer points are ﬁxed by the polygons and the inner points are derived.

31 (a) (b) (c)

(d) (e)

(f) (g)

Figure 20. Designing with stars that do not ﬁt together perfectly.

32 (a) (b)

Figure 21. Constellation patterns.

Divrigi). Figure 22(c) is built from 16-pointed stars. The fragment in the panel can be developed into a repeating pattern as shown, but it is not very successful because the 8-pointed stars have only 4-fold symmetry and are clearly irregular. This is not so apparent in the original panel as the central 16-pointed star dominates the design and the other motifs are only partially visible. Despite the ﬂaws present in these examples, it is clear that people were exploring ways to combine stars in constellation patterns early in the twelfth century. Designs in our source material show that, by the end of the century, they had mastered techniques for doing it and were producing well-controlled, repeating patterns that contain unusual combinations of stars. Figure 23 shows templates for two designs from the mausoleum of Mu’mina Khatun. In each case, the underlying structure is overlaid in red. The spokes of the primary stars are aligned with each other. When extended, the spokes produce a network of triangles, and the incircles of these triangles determine the size and position of the secondary 5-pointed stars. Transforming a circle packing into a star pattern is done using the wheel construction. The spokes of the wheels lie in lines connecting the centres of neighbouring circles, which ensures that the spikes of the resulting stars are nicely aligned (but also means that the spokes are not equally spaced, leading to slightly irregular stars). Each wheel has an inner and an outer circle, and the ratio of their radii determines the sharpness of the spikes. In both examples here, the ratio used for the 5-pointed stars is about 0.4. A more detailed discussion of this technique, a comparison with the polygons-in-contact method, and further examples including the pattern of Figure 7(g), can be found in [11]. The method produces visually satisfying results even when the circles do not pack together precisely. In Figure 23(b) the small circles diﬀer in size slightly, and hence are not quite tangent, and the large circles are not tangent to the small ones. Moreover, the spokes

33 (a) (b)

(c)

Figure 22. Experimental constellation patterns based on panels in the Barsian Mosque.

34 (a) (b)

Figure 23. Templates for the constellation patterns in Figure 6(g) and (i). of the wheels are not equally spaced. The circles shown in Figure 23(a) do pack properly but, in my reconstruction, the inner circles of the the wheels that define the 7-pointed stars are off-centre and are moved closer together. Although motifs may not have perfect symmetry, the distortion is distributed throughout the pattern and does not attract attention. A small misalignment between elements is easily noticed, but the mind seems to overlook quite large deviations from regularity. This bias in our visual system gave Seljuk artists the freedom and flexibility to produce constellation patterns combining stars that, in geometric terms, are incompatible with each other and with the standard ways of arranging them. The blending of 11-pointed and 13-pointed stars in Figure 6(i), and of 11-pointed and 14-pointed stars in Figure 7(g) are among the finest examples.

4.4 Rosette motifs A motif that has rotation symmetry and no translation symmetry is called a rosette. Fig- ure 24 shows rosettes with 8-fold, 10-fold and 12-fold symmetry that have a regular star in the centre. They are all constructed in the same way, using an idea inspired by the constellation patterns: tangent circles are placed around the circumcircle of the star, and lines are drawn through the points of tangency [17]. These lines divide the plane into concentric layers of cells around the star: the cells in the ﬁrst layer are convex hexagons (often called petals), then there is a layer of arrow shapes, then kites. In these examples, the stars have the form {2n/(n − 1)} so the petals have two parallel sides. To create a motif we can choose how many layers of cells to add to the star, and so produce a series of rosettes with increasing complexity. The cells in the outer layer determine the angle presented by the rosette at its circumcircle, which aﬀects how it attaches to other elements in the pattern. The three patterns on the Gulustan mausoleum shown in Figure 9 are simple arrangements of rosette motifs: (a) has 8-fold rosettes with one layer on a square grid, (b) has 10-fold rosettes with one layer on a rhombic grid, and (c) has 12-fold rosettes with two layers on a square grid. The strategy for composition is the same as for the star patterns shown in Figure 14, except that the star motifs have been replaced by something more

35 (a) (b) (c)

Figure 24. Rosette motifs.

(a) (b) (c)

Figure 25. Templates for the rosette patterns in Figures 6(c). 6(h) and 21(b).

36 complicated. All three patterns are fairly common and have twelfth-century examples. Figure 9(b) is ubiquitous in the Islamic world. The earliest surviving example is on the Ghurid arch at Bust, Afghanistan [14, Pl. 153], which dates from 1149. The arrow motifs also appear in the lower corners of the panel in the north dome chamber in the Friday Mosque in Isfahan shown in Figure 21(a) (although they are not included in the figure), so the design may be much earlier. Similar ideas are developed in Figure 6(f); notice that alternate rows of rosettes are connected in different ways. It is straightforward to extend a symmetric motif in a symmetric manner to create intricate rosette motifs. However, including large rosettes in a repeating pattern and connecting them in visually satisfying ways is more of a challenge. The patterns in Figure 14 are dense arrangements of stars, which can be produced by packing their circumcircles in basic grid formations, and then extending the lines of the stars into the spaces in between to link them together. This approach can also be used with rosettes. However, because the outer angle of a rosette is not so sharp as that of a star, large voids can remain in the pattern, and further work is then required to integrate other motifs and produce a more balanced distribution of region sizes. Another way to create repeating patterns from large rosettes is to crop them to fit inside a shape that tiles the plane (typically a rectangle or hexagon). Figure 25 shows three examples using the rosettes from Figure 24(a) and (c). In (a) the 8-fold rosette is matched with a square; the parts of the rosette that lie outside the red box are removed to leave a square template. In (b) and (c) the 12-fold rosette is cropped to fit a square and hexagonal template, respectively. The square templates generate patterns on the mausoleum of Mu’mina Khatun (Figure 6(c) and 6(h)), and the hexagonal template generates Figure 21(b). No- tice that the connecting matrix between the rosettes is generated automatically: the region fragments around the template boundaries combine with their mirror images and the result has no loose ends. In these examples, many of the induced regions have somewhat irregular shapes: octagons with 4-fold symmetry, hexagons with 3-fold symmetry, and non-convex hexagons and 5-pointed stars with bilateral symmetry. (See [10, Fig. 7] for another example where cropping generates regions with unusual shapes in a pattern.) We have used Figure 21(b) as an example of both a constellation pattern and a rosette pattern; Figure 9(b) can also be interpreted both ways. The different interpretations mark a distinct change in perspective. In a rosette pattern, the 5-pointed stars are part of the background — they are gaps between the rosettes, and are a by-product of arranging the primary motifs. In a constellation pattern, the 5-pointed stars are part of the foreground along with the larger stars; the regions we call petals in the rosette interpretation are gaps between the stars, and are part of the background. These two approaches produce different families of patterns, which overlap in the case of simple examples. Figure 19 demonstrates that our visual system is willing to tolerate quite a large variation in shape and still classify a motif as a star. This effect underlies the production of constellation patterns where both the primary and secondary stars may be irregular; it enables artists to combine stars with incompatible geometries because the small distortions required to align neighbouring stars are overlooked. On the other hand, we do not find unusual combinations of rosettes as the more complex cell structure in a rosette makes any deviations from perfect symmetry more obvious. The rosette patterns on the mausoleums in Nakhchivan and Gulustan were produced at the end of the twelfth century, shortly before the Mongol invasion. After a destructive

37 Figure 26. Design of overlapping 3-layer rosettes from the Gunbad-i Alayvian, Hamadan. and turbulent period, the production of high quality architectural ornament resumed in the fourteenth century: the mausoleum of Oljeitu at Soltaniyeh contains examples of constellation and rosette patterns, and the Gunbad-i Alayvian at Hamadan, also thought to be a mausoleum, displays the ﬁne rosette pattern shown in Figure 26 above its entrance.

4.5 Coloured patterns Some of the patterns in Figures 2–9 contain elements coloured in blue. In the original artworks these blue parts are coated with a turquoise glaze. The earliest known example of this technique in Iran is on the minaret of the Friday Mosque in Damghan, which dates from the end of the eleventh century. The colour is applied to the letters in the inscription, where it attracts the viewer’s attention. The increased contrast should also improve legibility, although Blair observes that the tightly packed letters are almost unreadable [6]. Turquoise glaze appears on several twelfth-century minarets near Isfahan. Those at Sin (1131), Sareban (1135), and Ziar all carry glazed inscriptions. The minaret at Rahravan has an inscription on a blue ground, and also a simple pattern composed of a mixture of glazed and unglazed bricks — a technique known as banna’i. Two Seljuk mosques in Qazvin also have a dash of colour: turquoise highlights can be found high inside the dome of the Friday Mosque (although these baubles are too small to be seen from the ground [26]), and the inscription at the Haydariyya Mosque (1121/2) has a turquoise frame. The Gunbad-i Surkh is the earliest building in which large periodic patterns are aug- mented with coloured glaze — see Figure 2. The 9-sided polygons in the tympanum above the entrance, and the 3-pronged arrowhead motifs in the spandrels are made from glazed strips of terracotta. A few turquoise ﬂecks can be found as small highlights in the brick patterns on the corner columns. The other towers in Maragha also make use of the blue

38 glaze. On the Round Tower it is applied to inscriptions, and on the Gunbad-i Qabud it is used for geometric patterns and calligraphy. We shall consider the latter building in more detail later. Glazed terracotta is used extensively on the mausoleum of Mu’mina Khatun. It colours the band of calligraphy under the cornice, it frames the rectangular inscription above the entrance and arch-shaped panels near the top, and also creates a few highlights. Turquoise strips are used in most of the primary panels. They perform three diﬀerent functions:

• Outline compartments in the pattern. The 7-pointed stars are outlined in Figure 6(b), and octagons are outlined in parts (a), (d), (g) and (h).

• Highlight components of the pattern. The 10-pointed stars are highlighted in Figure 6(e).

• Reveal structure in the pattern. In Figure 6(f–i) polygonal rings are added that connect the points of large stars. These rings in the constellation patterns approximate the circumcircles of the primary stars, and hint at the patterns’ underlying structure.

Shared properties like the octagonal regions and the rings around the stars link the patterns together in families, providing cohesion to the ornamental scheme of the building as a whole. Now consider the two versions of a star pattern shown in Figure 27. The one on the left is the packing of 12-pointed stars that we saw in Figure 14(e); on the right, some of the lines have been coloured blue. However, the function of colour here is new — it changes our interpretation and we see a fusion of two complementary patterns, one overlaid on the other. The black pattern can be seen as a composition of overlapping hexagons. In the monochrome pattern these large hexagons are still quite prominent, but the lines of the blue pattern are hard to isolate. The Gunbad-i Qabud is often compared to the mausoleum of Mu’mina Khatun: they are almost the same age, both are 10-sided tomb towers, and both feature turquoise glaze. They also share the same ornamental framework: a cornice of small muqarnas arches above a band of turquoise calligraphy, below which are muqarnas hoods with three layers of arches, then the main panels of ornament. However, the patterns on the two buildings are very diﬀerent. For the patterns in Figure 6, colour is used to direct the viewer’s attention onto highlights or to emphasise structural features. The blue elements do not make a connected pattern when considered on their own, and are not interesting in themselves. In Figure 27(b) each colour makes a good pattern by itself. (Indeed, both patterns can be found on their own.) The pattern in the spandrel of the Gunbad-i Qabud shown in Figure 8(a) is of the same form and, in this case, the two layers are distinguished by line style as well as by colour: the glazed lines are straight and the plain lines are curvilinear. Colour is used in many panels in the upper section of the Gunbad-i Qabud. Figure 8(b– e) shows four sketches based on some of the arched panels. Sketch (c) is a variant of Figure 13(a) with the embellishing hexagon coloured blue and the small squares replaced by swastikas. The patterns in (b) and (c) could be parts of periodic patterns as there is enough information to infer how to extend them. However, the approach taken in (d) and (e) is similar to that used in the arched panels from the Barsian Mosque in Figure 22: the motifs suﬃce to cover the panel but there is no obvious way to extend the piece shown.

39 (a) (b)

Figure 27. The impact of colour on design.

Notice the blue lines in Figure 8(d) match the red lines of the polygons in Figure 20(b). This suggests that the designer was aware of the polygonal structure, although we cannot tell whether it was used as the means of constructing the pattern.

4.6 Complementary patterns We have seen that the spandrel pattern in Figure 8(a) is composed of two different patterns that fit together such that the corners and crossings of one lie in the open spaces of the other, and vice versa. The two patterns also have the same weight and the same scale (corresponding stars are the same size). They support and complement each other. As we demonstrated in Figure 27, this example may have been discovered through serendipity, just by adding colour to an existing pattern. However, other patterns on the Gunbad-i Qabud suggest that complementary patterns were constructed deliberately. Figure 28 shows a blue curvilinear pattern threaded through the Star and Cross pattern; small pieces of this pattern are used in some of the panels in the muqarnas hoods. The lower section of the Gunbad-i Qabud is covered in a unique design of complementary patterns that flows over the walls and the engaged columns at the corners of the tower like a fabric wrapped around the building. The same panel design is applied to all the faces of the tower except for the entrance portal, with direct and mirror-image forms on alternate faces. Although this could become monotonous, the repeat unit is the size of a panel. This lack of variety in the decorative scheme of a monument is unusual for the period, so this pattern must have been recognised as something special to have been featured so prominently. The design is shown in Figure 8(f). It does not use colour; instead, the two patterns are distinguished by the depth of relief. The primary pattern (shown by the thicker lines in the figure) is set in high relief and is composed of lines of small terracotta units with a D-shaped cross-section that run across the walls like vertebrae. The other pattern is made of terracotta strips pushed into the plaster ground,

40 Figure 28. A complementary pattern for the Star and Cross pattern.

(a) (b) (c) (d) (e) (f)

Figure 29. Adding connections in polygons.

The primary pattern is composed of a small number of shapes that occur in different configurations — a property that is the signature of modular design. Figure 5(c) can also be made with the same modular system.5 The low-relief, secondary pattern is unlike any earlier example, and seems to have been specially constructed for the purpose by threading extra lines through the primary pattern. The result is a pair of related but independent patterns, each of which can stand on its own. Although the additive6 nature of this pattern has been recognised before [8], I am not aware of proposed processes for threading a complementary pattern through a given one. Figure 29 illustrates a simple idea that underlies a mechanism for deriving a secondary pattern from a given pattern. The primary pattern has polygonal compartments. In each polygon, place two points on each edge, and connect them in pairs — the connecting lines will become the complementary pattern. The figure shows two connection schemes: in (a), (c) and (e) the points are connected to a neighbour, while in (b), (d) and (f) the connecting

5There is some debate (see [9, 12]) over whether the modules that constitute the design system are the shapes visible in the ﬁnished design, or are tiles decorated with motifs analogous to the red polygons in Figure 20. 6The term additive does not have an agreed deﬁnition when applied to ornament. For Trilling [24], it describes an aggregation of motifs combined without a clear organising principle, such as a collage. In Islamic ornament Schneider [22] uses the term for patterns in which the lines do not meet in the standard way as transversal 4-valent vertices; examples include tilings that are not edge-to-edge, and the cobweb-like patterns used as grills. Bonner [8] uses the term for simple embellishment with polygonal or star components, and for more complex examples where a two or more independent but complementary patterns are intertwined.

41 (a) (b)

Figure 30. Archimedean tilings with complementary patterns. lines cross each other. The first method is useful in tight spaces to prevent a cluttered look; the second method has more interest and sometimes leads to a star-like motif in the centre. In these figures the connections are shown as smooth curves, but they are just initial sketches to provide inspiration; they can be adapted and replaced with polygonal lines to produce a composition that is visually interesting and not cluttered. Figure 30 shows two Turkish patterns that have complementary patterns of this kind added to the 3.6.3.6 and 3.4.6.4 Archimedean tilings: (a) is from the Dundar Bey Madrasa in Egirdir (1237), and (b) is from the Sifaiye Madrasa in Sivas (1217/8). We can see the origin of this idea in patterns like Figure 3(e) where octagons placed at the vertices of a grid produce such configurations of arcs in the faces. Once this device has been seen in naturally occurring contexts, it can be abstracted and introduced deliberately as a design strategy. This process evolves naturally into another method for designing patterns: the initial pattern is regarded as merely a structural framework in which to construct a secondary pattern, the framework is then deleted, leaving only the derived pattern in the finished artwork. This mechanism, with some minor changes, can also be used to produce the Gunbad- i Qabud pattern in Figure 8(f). One difference arises because the edges in the primary pattern are not all the same length, and the shorter ones have only a single point of contact with the secondary pattern. There are also some elements of the secondary pattern that do not meet the edges at all and so do not come from connections. In particular, the bobbin- shaped compartments have an additional loop component. The circles in the small bow-tie shaped hexagon compartments correspond to small studs in the original panel. The primary pattern is unusual in that it contains no stars. It does, however, contain the motif shown in Figure 15(c) in the bottom-left of the figure. This motif could be replaced by a 10-pointed star, but this may make it more difficult to generate an effective secondary pattern.

42 4.7 Spiral motifs Most of the monuments discussed here carry patterns based on the spiral — an ancient and universal element of geometric ornament. Although spirals are easy to recognise, they have a diversity of forms that makes it surprisingly difficult to identify their essential features. A few naturally occurring spirals admit simple mathematical descriptions with parametric formulae but, for the general case, it may be easier to define spirals in terms of their visual impact rather than any geometric properties. The Fraser spiral is an optical illusion in which concentric circles of twisted cords produce a spiralling effect — spirals are as much a product of our perception as of their geometry. Figure 31(a) shows a 3-armed spiral constructed by shading cells in a triangular grid — the light and dark lines are of equal width. In Islamic art motifs are rarely used in isolation. Figure 31(b) and (c) illustrate the effect of combining spirals: two mirror-image spirals lead to arrow shapes, and two of the same handedness produce zig-zags or Z shapes. The basic spiral form admits many variations: the number of arms, whether they are smooth curves or angular, how much they wind around the centre before reaching the boundary of the motif and connecting to something else. When spirals are combined, we have the additional options of mixing mirror-image forms and even different types. Seljuk brickwork includes some simple patterns that can be produced by arranging spirals. Figure 31(d) is from the spandrels of the small mihrab in the Golpayegan Mosque. The grey trapezia show the bricks used in the original; they are all congruent and are set out from the wall, making a relief pattern. The red triangles break up the pattern into its structural units and show mirror-image pairs of stubby 3-armed spirals. A variant of this pattern is used on some of the corner columns of the west tower at Kharraqan. Figure 31(e) is another relief brickwork pattern from Kharraqan. It is an assembly of 6-armed spirals, as shown by the overlaid hexagonal grid. Figure 31(f) is more interesting and shows the structure underlying a pattern from the Round Tower at Maragha — Figure 4(b). It is composed of 3-armed and 4-armed spirals, arranged according to the 3.3.4.3.4 Archimedean tiling (shown overlaid in red). Each 4- armed spiral lies on a grid of equidistant lines parallel to the edges of its boundary square and the spiral lines pass through the centre of the square. To preserve the equal line-width of the dark and light lines in the pattern, a similar grid is constructed in the triangles by drawing equidistant lines parallel to the boundary. In this case, the lines do not meet naturally in the middle — the 3-armed spirals are constructed by swirling in from the boundary until they meet, rather than swirling out from the centre. Figure 31(g) is based on a pattern in the lower part of the mihrab in the Barsian Mosque. It looks rather like the starting point for one of M. C. Escher’s tessellations of crawling animals. It has 5-armed and 8-armed spirals — the overlay in red shows how they fit together. The spiral nature of this pattern is not immediately apparent because our visual system is drawn to the (finite) shapes rather than the regular structure in the (infinite) edge network. Figure 6(c) shows a common star pattern made up of 5-pointed and 8-pointed stars; the arrangement of irregular pentagons and regular octagons forming the red tiling reveals the underlying structure common to both patterns. The original panel in the Barsian mihrab is a narrow vertical strip and the 8-fold centres lie on the edges of the panel — it does not contain a complete 8-armed spiral. The process we have just described for creating patterns is line-based, not shape-based: we have taken linear motifs and connected them together. The resulting edge network

43 deﬁnes a collection of regions, but the shapes of the regions are an emergent feature of the pattern. Some of the simpler patterns, such as Figure 31(e), can be made in other ways — just colouring cells in a square or triangular grid, for example. However, in the last two examples, the regions have very irregular shapes, and the spiral structure around the vertices in the edge network would not have arisen by chance. So it seems likely that these examples were produced by assembling spirals; with this perspective the simpler ones also belong to the same family. Spiral-based patterns are common in Seljuk ornament in Iran, Turkey and Syria, but seem rare elsewhere. Examples are found on most of the buildings discussed here. The simple cases include 3-armed spirals on a triangular grid in Figure 2(a) and swastikas on a square grid in Figure 3(a). Figure 4(b) is built from 3-armed and 4-armed spirals, and spiral motifs are combined with other elements in Figure 3(b), Figure 7(a) and (b). The spiral patterns are not the primary ornament — they are usually the outermost part of a panel, either sitting as spandrels over an arch or in a border forming a frame. They may have an apotropaic function (like the knotwork around portals in the medieval cathedrals of Europe): talismans that give protection from the evil eye whose gaze gets lost in the labyrinth created by the maze of spirals.

5 Observations

5.1 Pattern re-use The patterns in Figures 2–9 are almost all different. There is one example of a pattern that appears with different orientations: Figure 9(a) is the same as Figure 6(c) rotated by 45◦. The pattern in Figure 3(d) from one of the main panels in the Kuseyir mausoleum is re-used in the tympanum above the doorway, and also with slightly different geometry in the tympanum above the doorway of the mausoleum of Mu’mina Khatun. These buildings were designed by Ajemi ibn Abubekr Nakhchivani, who presumably liked this pattern as it has such a prominent position in both monuments. Although there is little copying or re-use of patterns from one building to another, the strategy for re-use within the ornamental scheme on individual buildings varies. On circular buildings, the main panels of ornament are placed over the doorway, and either a wide border (Round Tower) or a sequence of nested borders (Gunbad-i Seh) frame the entrance on the fa¸cade of the pishtaq. On polygonal buildings, each face typically carries a large panel. On the Kuseyir and Mu’mina Khatun mausoleums, these primary panels are different on different faces. However, the simple patterns used as frames or fillers are re-used — for example, all the spandrels on the Mu’mina Khatun mausoleum use the same pattern. The Gulustan mausoleum has only three different patterns, which are replicated around the twelve faces of the tower. As we noted earlier, the Gunbad-i Qabud uses very few patterns in essentially the same arrangement on all its faces. Even when patterns on a building are not identical, they quite often have features in common. Sometimes it looks as if the designer is exploring the possibilities of a particular method from different starting points. The resulting variations on a theme give harmony to the overall design scheme as there is a family resemblance between different patterns. We have seen several examples of this: a single process applied to two Archimedean tilings generates two patterns at Erbil (Figure 10), two patterns on the Gunbad-i Seh can be derived

44 (a) (b) (c)

(d) (e)

(f) (g)

Figure 31. Constructing patterns from spiral motifs.

45 from each other (Figure 11), three patterns on the Kuseyir mausoleum can be generated by doubling the edges of a tessellation (Figure 3), and many of the patterns on the mausoleum of Mu’mina Khatun contain unusual collections of stars (Figure 6). Many of the patterns on the latter monument are also linked by having octagonal regions outlined in blue; this shared feature has not arisen by chance and the patterns were selected or engineered to produce it.

5.2 Archimedean tilings In many places in the discussion we have found evidence for the application of Archimedean tilings in the patterns, either as a visible feature in the completed design, or indirectly as a structural element used to organise other motifs. There are eight Archimedean tilings and we have seen six of them in Seljuk patterns:

• 3.3.3.3.6: this tiling was not found,7 possibly because it does not have mirror symmetry. • 3.3.3.4.4: this tiling was not found. It consists of alternating rows of a single kind of shape — a line of squares, then a line of triangles, and so on. • 3.3.4.3.4: provides the tessellation for the parallel offset pattern in Figure 13(f), and the structure in Figures 16(a) and 31(f). • 3.4.6.4: provides the visible substrate for the complementary pattern in Figure 30(b). • 3.6.3.6: provides the tessellation for the parallel offset pattern in Figure 3(e), and the substrate for the complementary pattern in Figure 30(b). • 3.12.12: provides the starting point in Figure 10(a), and the structure in Figure 21(b). • 4.6.12: provides the starting point in Figure 10(b). • 4.8.8: provides the tessellation for the parallel offset pattern in Figure 13(d).

Although we now think of tiling as a mathematical topic, there is no evidence that the Greek geometers took any interest in it. These eight tilings are called after Archimedes by analogy with the Archimedean solids,8 not because he studied them. The first written account and enumeration of them is in Kepler’s Harmonices Mundi of 1619. However, architects and craftsmen in the Classical period were aware of the ornamental uses of regular polygons. Roman mosaics use the 3.4.6.4 and 4.8.8 tilings, and also a modified form of the 3.3.4.3.4 tiling in which the pairs of adjacent triangles are replaced by rhombi. Besides exploiting the Archimedean tilings, Seljuk designers may have also explored uses of poorly fitting arrangements of regular polygons, as shown in Figure 20.

5.3 Broken symmetry Figure 14 shows patterns formed by arranging star motifs on grids of triangles, squares or rhombi, using regular stars with six, eight, ten or twelve points. These simple arrangements do not occur in the source materials. Instead, we find many examples of patterns with stars that are more awkward to use: 7A cheiral pattern that corresponds to this tiling can be found in the ibn Tulun Mosque in Cairo [25, EGY 0511]. Pairs of triangles are fused to form rhombi. 8The discovery of thirteen polyhedra with regular faces of different kinds is attributed to Archimedes in book five of the Mathematical Collection written by Pappus in the fourth century a.d.

46 • seven-pointed stars in Figures 6(b) and (g), 16(a), and 8(d) and (e) • eleven-pointed stars in Figures 6(i), 7(g) and 22(a) • thirteen-pointed stars in Figures 6(i) and 22(b) • fourteen-pointed stars in Figure 7(g).

Fully symmetric versions of these stars with equally spaced spikes are not constructable with Euclidean tools — that is, we cannot make one with a lines-and-circles construction. This limitation, which is a consequence of the Euclidean axiom system modelled on a rule and compasses, is only of concern to pure mathematicians, and does not seem to have hindered craftsmen in producing geometric patterns. More of a problem for designers is the fact that the angles in these stars are not compatible with the standard grids used for organising motifs. Moreover, the stars are not compatible with each other, and yet we find patterns containing varied combinations of stars where the lines flow naturally and nothing seems out of place. We have seen several families of star motifs: regular stars and parallel-sided stars are distinct but overlapping subsets of wheel-based stars, and cut-and-paste methods can produce yet more stars. These different kinds and techniques satisfy different needs of the designer. Stars with equally spaced spikes are easy to create as free-standing motifs, which can then be arranged in a composition. On the other hand, irregular stars are created in situ in response to a particular design problem so that their alignment with neighbouring motifs can be controlled — they are constructed as part of the process of laying out the final form of a design. It is the use of irregular stars that enables designers to make patterns that would not be geometrically consistent if all the component parts were ideal forms. If you try to combine incompatible motifs, either they do not line up correctly and leave gaps, or they must be deformed to fit the space available. Our visual system is far more alert to small errors in alignment than to broken rotational symmetry, so it is better from a design viewpoint to allow slightly irregular spacing of the spikes in star motifs to ensure that spikes of neighbouring stars line up with each other. This approach seems to have been used in many surviving Seljuk patterns, and I have adopted it in the reconstructions in the figures. In constellation patterns the positions and sizes of the stars are controlled by a framework of (quasi)tangent circles (see Figure 23). The small circles, which generate the 5-pointed stars, are inscribed in triangles formed from the radii defining the spikes of the primary stars. Once this framework is set up, essentially everything else is determined by a single parameter which adjusts the spikiness of the pattern: we just need to set the sizes of the inscribed circles of the 5-pointed stars. This determines the incircles of the primary stars, and the pattern lines are determined by the incircles and the spokes. The examples from the Barsian Mosque in Figure 22(a) and (b) show some initial ex- periments of this kind, but the arrangements of stars only cover the available space and it is not obvious how to extend them. Figures 6(i) and 7(g) show that, by the end of the twelfth century, designers had worked out how to produce periodic constellation patterns with unusual combinations of primary stars. Geometric analysis of these star patterns reveals that what is apparently shown is impossible. However, the errors are small and well concealed — the unavoidable distortion is distributed around the pattern and not concentrated in small areas where accumulated errors would be larger and more obvious. The deception is subtle and unobtrusive.

47 5.4 Designing with colour The impact of coloured glaze on architecture is reflected in the colloquial names of some buildings. Many towns have a Blue Mosque, for example. Of the five medieval tomb towers in Maragha, the Gunbad-i Qabud is the blue tomb (qabud is Persian for blue). Similarly, in Turkish sır¸calı refers to the glaze, and gök to the blue colour. From the few examples of coloured patterns in our data set, we can see that colour started to become an integral part of the design process, and not merely something added as an afterthought. Its function evolved as tile production increased.

• At ﬁrst, colour was applied sparingly as small point-like highlights, either as ceramic inserts, or as bricks with one glazed side (banna’i).

• The next step was to colour linear features using thin straight strips of glazed ceramic, cut to length and mitred at the joins. For example, on the mausoleum of Mu’mina Khatun, such strips are used to frame a panel, to draw an outline around the boundary of a compartment, or to highlight a ring, polygon or star component added as an embellishment. Some of the patterns on the Gunbad-i Qabud are composed of two complementary parts, each a complete pattern in its own right; colour is used to separate and identify the two parts. The ﬁrst examples of such patterns may have been discovered by selectively colouring components in known patterns (see Figure 27). In other cases, it seems that complementary parts were constructed deliberately and systematically by threading extra components through existing patterns.

• When used in larger quantities, glazed strips and inserts can form a coloured background for a pattern realised in stucco or terracotta. On the Mudhafaria Minaret at Erbil the effect is reminiscent of cloisonnéenamelwork with the blue inserts filling the spaces between the unglazed walls of the compartments. In Figure 2(a) from the Gunbad-i Surkh the glazed and unglazed strips have the same width; the blue inlay is sunk leaving the terracotta in relief.

• Glazed tiles can also be positioned to form a continuous coloured surface. For example, the dome on the mausoleum of Oljeitu at Solyaniyeh was covered with turquoise tiles and was visible across the plain as one approached the city.

The introduction of opaque turquoise glaze to augment geometric patterns seems to have occurred in northern Iran and Azerbaijan. A ceramics industry developed in this area [3, p. 220] and Kashan became a major centre for the production of glazed and decorated tiles. Konya, the capital of the Seljuks in Anatolia, was home to many refugees ﬂeeing the Mongol invasion of Iran and neighbouring lands. It also became a cultural centre known for its coloured tilework. All the examples in our data set are based on the addition of a single colour. Tech- nological advances in the thirteenth century led to a broader palette as black, white, dark blue and light blue glazes supplemented the turquoise. This development made it possible to have continuously tiled, patterned surfaces. Shapes were cut from monochromatic tiles and assembled as a mosaic. In Konya, early examples include patterns of stars and polygons made with black lines on a turquoise ground. Sometimes areas of plaster are left

48 showing through to provide another colour (white or beige) and a different texture. Besides these line-based designs, tessellations (that is shape-based designs) were also produced. The tessellations typically have small repeat units containing a few simple shapes, each shape is cut from a coloured tile, and adjacent tiles have contrasting colours so the shapes are immediately visible. By the fourteenth century the Ilkhanate rulers in Iran had established their capital in Tabriz, and cultural activities had resumed. This included large architectural projects decorated with coloured patterns. Intricate examples of threading complementary linear patterns through each other can be found in the mausoleum of Oljeitu (1302), the shrine complex of Sheikh Abd al-Samad in Natanz (1308), and the Friday Mosque in Varamin (1322). Tessellations are used on the muqarnas ceiling of the iwan in the Abd al-Samad shrine, and the large panel above the entrance to the Gunbad-i Ghaffariya at Maragha (1328). Colour played an increasingly prominent role in the decoration of religious buildings and whole fa¸cades were covered with glazed tiles, as can be seen further south in the Friday Mosques at Kerman and Yazd. The cut-tile technique is time consuming and expensive. In the Timurid period a different method of application was developed: a design was spread over an array of square tiles, and each tile was painted with its piece of the picture. The painted tiles were glazed and fired, then assembled to form the panel. A few more colours had been added to the palette (yellow, green, and shades of brown from sandy to dark orange), and this technique was known as haft rangi (seven colours). It provided a cost- effective way to create large panels of ornament, and floral designs representing the gardens of paradise became popular. Cut-tile mosaic continued to be used for geometric panels.

6 Concluding remarks

In this paper we have explored Islamic geometric ornament, focussing in detail on Seljuk patterns from a small group of buildings in the area around Azerbaijan. The analysis has included most of the patterns in our source materials, but the patterns in Figures 3(c) and (d), 4(a), and 7(e) and (f) have not been discussed. Part of our approach has been to consider the question ‘how could someone have come up with this?’ and we have described several possible sources of inspiration, and a range of methods for experimenting with ideas. Patterns can be derived from existing geometric structures, either by inserting geometric motifs such as stars directly into tessellations and making modifications to accommodate them, by applying an algorithm to tessellations that automatically generates stars centred at the vertices, or by threading additional components through a given pattern. Tessellations and lattices also provide ready-made frameworks for coordinating the layout of preformed motifs such as spirals, stars or rosettes. Placing circles at the vertices of an Archimedean tiling and other equilateral tessellation produces a packing of equal circles, which gives a foundation for compositions using a single kind of star. Generating ideas and implementing them can be separate processes. We have seen how a set of polygonal tiles decorated with star motifs can be used as an aid to explore arrangements of different stars in an empirical manner — the composition grows like a crystal. A technique like the wheel construction can be used to turn the resulting rough assemblies into effective, practicable designs. With this approach, motifs can be arranged to custom-fit a particular space with an unusual shape (like an arch or spandrel), in which

49 case the ornament will cover the space available, but may not extend in an obvious way because the panel does not contain a repeat unit. Judicious cropping can be applied to a large patch of ornament to produce a template that can be repeated to cover the plane in a satisfactory way. It is diﬃcult to assess whether any of these ideas are Seljuk innovations because few artefacts from pre-Seljuk Iran survive for comparison. However, we note that:

• Spiral motifs were popular in Seljuk ornament but are rare in the rest of the Islamic world. • Complex star patterns, the uniquely Islamic form of geometric ornament, seem to have ﬁrst appeared in the Seljuk period. These include constellation patterns and rosette patterns. • The re-introduction of techniques for making coloured glaze brought a new material that spread across the north of Iran in the Seljuk period, and led to new forms of design.

We have seen that stars come in many forms. Some, like the regular stars, have a well- deﬁned shape that exists independently of any pattern; they are preformed motifs that can be arranged to form a composition. Some are by-products of processes such as doubling edges in a tessellation, and the designer has little control over their properties. Others are constructed where they are required, using techniques like the wheel construction to ﬁnesse the alignment of neighbouring stars. The dissection (cut-and-paste) technique can also produce irregular stars when the need arises. The mausoleums of Yusif ibn Kuseyir and Mu’mina Khatun are separated by 25 years. During that time, Ajemi ibn Abubekr Nakhchivani, who designed both buildings, mastered the new techniques for designing with stars and created some exemplary patterns. Other writers have traced the development and spread of coloured glaze during the twelfth century [26], and studied it as part of the history of ceramics or technology. We have looked at the impact of colour on design, the functions it performs, and how patterns changed in response to its introduction. We have seen that, initially, it was used to add highlights to plain patterns. However, the Gunbad-i Qabud presents a new mode in which colour is an intrinsic part of the design and is fundamental to the viewer’s interpretation of the pattern; in many of its upper designs two independent yet complementary patterns are presented together, overlapping or interweaving, and separated through colour. Unfor- tunately, some inscriptions on the Gunbad-i Qabud are damaged and we do not know the name of the architect.

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