A Modular Design System based on the Star and Cross Pattern Peter R. Cromwell Pure Mathematics Division, Mathematical Sciences Building, University of Liverpool, Peach Street, Liverpool L69 7ZL, England. We introduce a modular design system, which we call CAMS, that is based on the preIslamic Star and Cross pattern. It can be used to generate a large family of traditional Islamic patterns found in Central Asia and Iran. In other examples of Islamic modular systems, the modules form a substrate that is used in construction and then deleted; in this case the modules themselves form the finished pattern. We also analyse some traditional 2-level geometric patterns as hierarchical structures of CAMS modules, corroborating the principles of 2-level pattern construction found in other Islamic modular systems. 1 Introduction Figure 1 shows a simple geometric design known as the Star and Cross pattern. It is formed from 8-pointed regular star polygons of type {8/2} arranged on a square grid. The interstitial spaces between the stars form crosses. Crosses with this pointed shape seem not to have been used individually as emblems and are only found in the context of this pattern. Use of the Star and Cross pattern is common among cultures that have developed a geometric style of ornament. For example, in the Roman mosaic from Cherchel, Algeria [1, Pl. 177d] shown in Figure 2(a), the stars are decorated with interlaced squares (shown in grey in the figure but filled with guilloche in the original) and the crosses are subdivided into squares and rhombi. Figure 2(b) shows another non-Islamic treatment of the Star and Cross pattern — it is from a sketch of a 13th-century Cosmati pavement in Rome, Italy [16, Fig. 4-105]. Figure 2(c) shows part of the Cosmatesque floor in Monreale Cathedral, Sicily. In general, Figure 1: The Star and Cross pattern. 1 (a) (b) (c) Figure 2: Variations on the Star and Cross design. www.thejoyofshards.co.uk courtesy of Rod Humby. 2 Photograph reproduced from the large-scale framework in Cosmati work is provided by a compass-work design, so the use of the angular Star and Cross pattern here is evidence of Islamic influence. In this paper we introduce a modular system based on the Star and Cross pattern that conforms to the silver ratio system of proportion. We show many examples of traditional Islamic patterns that can be produced using this system; they are mostly from Central Asia and Iran of the Seljuk, Timurid and Mughal periods and include field patterns, star patterns, and 2-level patterns. This complements previous studies of Islamic geometric ornament [3, 5, 6, 8, 13], which have found evidence to support the traditional use of a modular approach to pattern design, particularly in the construction of patterns based on the geometry of 10-pointed stars and the golden ratio. One of the benefits of the modular method is that it provides a natural route to the production of 2-level designs via hierarchies of modules of different scales. We shall analyse 2-level patterns in which the small-scale pattern either fills or outlines the compartments in the large-scale pattern; these two modes (filling or outlining) correspond to Type A and Type B, respectively, in the classification of 2-level designs introduced by Bonner [3]. We shall see that the principles for constructing 2-level patterns with the new system agree with those found previously in the 10-point setting. 2 CAMS: an early modular design system The construction of a large variety of different structures from a small set of basic elements (modules) is known as modular design. Bricks are the simplest example. Modules are often assembled to make repeating (periodic) units, but free-form arrangements that grow more organically are also possible. In general, the modular approach brings benefits through ease of manufacture with the mass production of interchangeable parts. In geometric ornament it also has benefits for composition: the eye finds unity in the economy of motifs and re-use of familiar forms as the same few shapes appear in a variety of contexts. It also means nonexperts can produce acceptable results as modular design naturally creates a satisfactory distribution of related elements. Figure 3 shows a set of tiles that form a modular system for creating geometric patterns. I have named the individual tiles for ease of reference. The internal angles are all multiples √ of 45◦ and the tiles have two edge lengths: the long and short edges are in the ratio 2 : 1. In the top row, the large square is equilateral with long edges, the house has edges of both lengths, and the other tiles are equilateral with short edges. The tiles in the bottom row can be seen as composites of the square, house and hexagon tiles, as shown by the dotted lines. (We shall see later that the star is also a composite figure, in many ways.) The strictly limited range of side lengths and angles produces a system that can generate an abundance of different patterns, but the shared properties also give a visual consistency both within a single pattern and across a collection. These tiles can be assembled to create many traditional Islamic patterns from Central Asia: Transoxiana (modern Uzbekistan), Khorasan (Afghanistan plus areas of neighbouring states, including north-east Iran), and India. For convenience we shall refer to the tiles in Figure 3 as CAMS (Central Asian Modular System) tiles and to the patterns they generate as CAMS patterns. Figure 4 shows some examples. The first two patterns are field patterns (do not contain stars) and are pre-Islamic. Figure 4(a) can be found among the brickwork patterns on the 3 small square flask hexagon chevron bone large square Dutch bonnet belt house dagger star cross Figure 3: Tiles of the Central Asian modular design system (CAMS). tomb tower at Damavand, Iran (mid 11th century) [19, Pl. 8], along with the Star and Cross pattern. Figure 4(c) is another field pattern; by looking at the lines rather than the tiles, it can be interpreted as overlapping regular octagons and seen as a polygonal version of a common pre-Islamic arrangement of overlapping circles. Figure 4(d) and (e) are obtained from the Star and Cross pattern by subdividing the crosses. Figure 4(e) is very common. It can be found in the remains of the 9th-century al-Tariq Mosque (Masjid-i Nuh Gunbad) at Balkh, Afghanistan. The crosses are oriented diagonally and the tiles are outlined in alternating interlace; unusually for Islamic patterns, the ribbons are not truncated at the edge of the pattern, and the design is adapted so that what would be loose ends are connected and turn back to re-enter the composition. The al-Tariq Mosque also contains the field pattern of Figure 4(a). The pattern in Figure 4(f) is common and widespread. It appears in early brickwork patterns of the Seljuk period — see, for example, the minarets at Damghan (mid 11th century) and Saveh (1110), both in Iran, and the 12th-century Minaret of Jam in Afghanistan. In one of the ceilings at the Mausoleum of Oljeitu at Sultaniya, Iran (1304–13), the design is executed in carved stucco with the tile outlines painted red — see photograph IRA 2631 in Wade’s collection [22]. It is the only pattern from this modular system to appear in Bourgoin’s collection [2, Pl. 67] from his travels in Egypt and Syria. Lee observes [12, p. 166] that Figure 4(f) can be obtained from Figure 4(c) by placing the 8-pointed stars of the Star and Cross pattern at the centres of some of the large overlapping octagons in (c) and adapting things to fit. In this sense, (f) can be seen as a blend of two simple early patterns. Its popularity may be an indicator that it was the first of a new system of patterns, a sort of signature piece, distinct from what came before but still clearly derived from it. Figure 4(g) is another early CAMS pattern. A brickwork version fills the pishtaq framing the entrance to the 12th-century Middle Mausoleum at Uzgen, Kyrgyzstan. The other examples are from a variety of later sources. Figure 4(l) is from a sandstone balustrade at the Amber Fort, India (1660) — see photograph IND 0914 [22]. Although it is not immediately apparent, this field pattern is derived from the Star and Cross pattern by subdivision: the pattern is rotated by 45◦ , the crosses are divided into a central square and four house tiles as in Figure 4(e), and the stars are divided into four flasks as shown in Figure 5(e); the handedness of the star filling switches in alternate rows. 4 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 4: Traditional patterns that can be built from the CAMS tiles. 5 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 5: Subdivisions of the star tile found in traditional patterns. (a) (b) (c) Figure 6: Traditional patterns closely related to CAMS. Many other versions of the {8/2} star motif can be found in traditional patterns — see Figure 5. Those in the top row are dissections of the star into CAMS tiles. People who analyse patterns in terms of linear features rather than tile shapes may classify patterns containing (e) as swastika designs. Figure 5(f) is from a design carved in relief with raised flowers in the centre. The remaining designs are dissections that introduce new shapes: (g) contains the {8/3} star; (h) is another subdivision that preserves the 8-fold symmetry of the star; (i) is derived from (h) by deleting lines. Figure 6 shows some interesting variants of CAMS patterns that require modified or additional tiles. At first glance (a) appears to be a genuine CAMS pattern but, on inspection, we see that the Dutch bonnet tiles are mis-shaped: the edge lengths along the bottom should be short-long-long-short, but here the sequence is long-short-short-long. The Dutch bonnet tile has two adjacent long edges meeting at a re-entrant corner; the large square is the only tile from Figure 3 that fits this space so Dutch bonnets are used in opposing pairs, as in Figure 4(g). In Figure 6(b) this forced arrangement has been broken and plus-shaped regions have been introduced. These regions are of similar size to the CAMS tiles and share their geometric properties of edge length and angles, so a plus-shaped tile could easily be added to the set. It is in this way that modular systems grow and evolve: the modules are assembled in ways that leave vacant spaces, and the gaps 6 become new modules. This ease of extensibility means that we cannot be definitive about which tiles are CAMS tiles. There is no medieval instruction manual containing a diagram like Figure 3. The CAMS tiles have been abstracted empirically after surveying many traditional patterns on surviving buildings; membership of the family has been determined by frequency of use and structural similarities of the tiles as much as by size, edge length, and angle. The most common tiles are the star, house, large square, belt, bone and Dutch bonnet. The hexagon tile appears in a few early patterns, but is rarely used later and perhaps should not be included in the CAMS set. The flask tile is formed by fusing the hexagon with a square; it has a more distinctive shape that has features characteristic of this set and is unlikely to belong to any other. However, it is almost always used in groups of four arranged as Figure 5(e). The Dutch bonnet is reminiscent of the motif on the trapezium tile in a modular system for producing patterns based on 10-pointed stars [6]; in that case it is a launch point for the construction of 2-point patterns. Figure 6(c) is a modification of Figure 4(g) that produces a similar effect — it can be interpreted as overlapping house tiles. However, there seems to have been no general means to produce 2-point CAMS patterns. Documentary evidence for the historical use of other modular systems is provided by the Topkapı Scroll [8]. Panel 61 of the scroll, redrawn in Figure 7(a), is the only template that corresponds to a CAMS pattern. It is unusual in that the star centres lie on the edges of the template, not in the corners [7]. Besides the geometrically pure form presented so far, CAMS patterns were also applied in what would today be called a pixellated form — the tile shapes are approximated on a square grid. An example from Panel 47 of the Topkapı Scroll is shown in Figure 7(b). To see that the pixellated form is approximate note that the star is not equilateral (the diagonal edges are shorter than the other edges). However, the pattern is clearly closely related to the modular system — the true form is shown in Figure 4(j). The Topkapı Scroll contains four further examples of the approximated patterns: Panel 51d is the Star and Cross pattern, Panel 43 is Figure 4(e), Panel 37b is the signature pattern of Figure 4(f) and Panel 1 is related to Figure 11(d) that we shall discuss in §4. None of these panels provides any hint to its construction — they are simply square grids with some of the squares filled in. Glaze was rediscovered in Baghdad in the 9th century. Initially, it was expensive and was only used for highlights. The pixellated patterns were an ideal application for the new technology: the black and white pixels were realised in blue glazed and unglazed bricks, respectively. CAMS patterns executed in this technique, known as banna’i, can be seen on the muqarnas in the west iwan of the Friday Mosque in Isfahan — see [19, Pl. 83] or photograph IRA 0529 [22]. There are many more traditional examples of CAMS patterns. Is the fact that the patterns have CAMS tiles in common sufficient to conclude that the tiles were used historically as a modular system? Islamic ornament drew on the classical and Sassanian traditions for its early work. Much of the early ornament can be constructed in the Euclidean sense with straight edge and compass: a set of lines and circles is drawn and a subset of the lines is chosen as the figure. This was typical of Roman and Byzantine work; one circle-based form is even called compass-work. It is not necessary to invoke a modular system to reproduce many of the 7 (a) Panel 61 (b) Panel 47 Figure 7: Templates from the Topkapı Scroll. patterns in Figure 4. Books such as El-Said and Parman [17] use the Euclidean method to explain how simple Islamic patterns can be constructed; it includes constructions for the Star and Cross pattern and Figures 4(c) and (f) as its Figures 15, 19 and 31, respectively. While many patterns can be reproduced in this way, reproduction and discovery may be different processes. Euclidean construction is a top-down method that focusses on lines. It produces a dissection of a space into pieces, often of many different shapes, and the result can appear cluttered if the lack of consistency in size and shape is too obvious. The modular approach is a bottom-up method that proceeds by building up a pattern from a small set of simple shapes. It enables a more experimental approach to composition so that a design can be grown organically in an unplanned manner by continually attaching tiles to a patch. The sheer variety of CAMS patterns, some of which are more free-form compositions, strongly supports the modular system hypothesis of design. We shall find further evidence when we consider the hierarchical properties of Islamic patterns. 3 Comparison with other modular systems Although the CAMS tiles have not been described before, the suggestion that modular systems underlie many Islamic geometric patterns is not new [3, 4, 5, 6, 8, 13, 18]. For example, Castéra uses a modular system of 37 pieces [4, p. 114–5] to produce the intricate patterns of 8-pointed stars found in Spain and Morocco. Only the star, house and small and large squares occur in both the CAMS and Castéra sets of tiles. Modular design is also now well established as one of the traditional methods for the creation of geometric designs in the eastern Islamic world; it is supported by documentary evidence and has the power to explain the structures of a variety of complex 2-level patterns. 8 pentagon decagon triangle trapezium bow-tie Figure 8: A decorated modular system whose tiles have two edge lengths in the golden ratio. In some widely used modular systems the modules provide an underlying structure for the composition, but are not directly visible in the final design. We shall briefly describe an example of such a system to provide a basis for the comparison with CAMS. Figure 8 shows a set of tiles whose interior angles are all multiples of 36◦ . Like the tiles in Figure 3, these tiles have two edge lengths but, in this case, the long and short edges are in the golden ratio. The pentagon and decagon are regular polygons with short edges, the bow-tie is equilateral with long edges, and the two other tiles have both long and short edges. Notice that in this system the tiles are decorated with motifs, shown here in black. All the motifs meet the boundaries of the tiles in the midpoints of the edges. However, the incidence angles are different on the short sides and the long sides. The geometry is dictated by the {10/4} regular star polygon used to decorate the decagon: it leads to an incidence angle of 72◦ on the short edges and 36◦ on the long edges. The motif on the pentagon is the {5/2} star. As with the CAMS tiles, a design is created by assembling these modules to produce an edge-to-edge tiling. Figure 9 shows the construction of a design from the ceiling of the Chehel Sotoon Palace in Isfahan: the template shown in (a) is composed of these tiles; the template is repeated four times by reflection in the bottom and left sides to produce the standard quartering shown in (b). Where a tile meets the boundary of the template, it intersects along an edge or a mirror line of the tile and this ensures continuity of the tiling across the joins between copies of the template. It is important to note that the tile boundaries do not appear in the finished product — the black motifs form the ‘foreground’ regions of the pattern and the white areas in the corners of the tiles are fused to create ‘background’ regions. In Figures 8 and 9 the motifs are filled in to aid the discussion, but they may be outlined in simple lines or interlacing. The somewhat irregular arrangement in the template contains more than 60 tiles and is a good example of the free-form composition that modular systems can produce. As these decorated modules can be viewed as a special case of the ‘polygons in contact’ (PIC) method of construction [11], we shall refer to them as PIC modules in the following discussion. The fact that the CAMS tiles do not carry motifs but appear themselves in the finished pattern leads to some interesting structural differences in the families of CAMS patterns and PIC patterns. • The angles in the motif on a PIC module are of two kinds: those that meet an edge of the module are determined by the incidence angle; the others are determined by 9 (a) (b) Figure 9: Construction of a ceiling design from the Chehel Sotoon Palace, Isfahan, using the modular system of Figure 8. 10 the angles in the corners of the module. In particular, all the angles in the motifs come from a small set. The angles in the corners of the background regions are the complements of the angles in the motifs. In most cases, these angles will be different from the angles in the motifs, and the background regions will have different shapes from the foreground motifs. For example, in Figure 9 four of the motifs have an angle of 36◦ , but this angle does not appear in any background region. When two regions of a PIC pattern share an edge, one of them will be a foreground motif and the other will be a background region. So it is unlikely that adjacent regions in a PIC design will be congruent. However, there is no natural distinction into foreground and background in CAMS patterns and adjacent tiles can be congruent. Figure 4(i) contains pairs of adjacent house tiles, and all the tiles in Figure 4(b) are congruent. • When two PIC modules share an edge, the midpoint of the edge will be a vertex in the finished pattern, and the corners of two motifs will meet there. Hence, all the vertices in a PIC pattern will be 4-valent. However, Figures 4(a) and (l) show that CAMS patterns can contain 3-valent vertices. • A pattern is said to have counter-change symmetry if its figure and ground can be interchanged by a geometric symmetry operation. This is also known as antisymmetry. Figure 10 shows three CAMS patterns coloured to exhibit counter-change symmetry — the black and white regions are identical, and there exist symmetries that reverse the colours. Figure 10(a) and (b) are star patterns; (b) is developed from the template in Figure 7(a). Although these two patterns can be coloured in this way, I know of no traditional Islamic star patterns that have been coloured to exhibit counter-change symmetry. However, there are Islamic field patterns that display counter-change symmetry. The CAMS example in Figure 10(c) is typical: all the tiles are congruent and it has the topology of a chessboard — each tile is surrounded by four of the other colour, and four tiles meet at each vertex. By contrast, although a PIC design can be coloured in two colours so that adjacent shapes have different colours, this corresponds to the natural division into foreground (motifs) and background. If a PIC pattern had counter-change symmetry, the tiling formed by the module edges would be self-dual. Self-dual tilings do exist [20, 21], but they are not suitable for constructing Islamic star patterns (the tiles are irregular polygons and the dual tilings are not situated so that dual edges cross at their midpoints). So PIC patterns cannot be coloured to show counter-change symmetry. 4 Type A 2-level patterns Modular systems naturally lead to structural hierarchies as the basic modules can be grouped together to form larger ones. We can see this process in the CAMS patterns of Figure 11. These examples all use the 4.8.8 Archimedean tiling as an organising principle and this underlying structure is highlighted in grey. In Figure 11(a) CAMS tiles have been assembled to form the large squares and octagons of the Archimedean tiling — these large shapes are the higher-level modules. Although the 11 (a) (b) (c) Figure 10: Traditional CAMS patterns with counter-change symmetry. pattern is a field pattern, the centre of each large octagon contains a star that has been subdivided as in Figure 5(h). Figure 11(b) is identical to part (a) except that the central star is subdivided as in Figure 5(d) — notice that this destroys the 8-fold symmetry of the large octagon. Figure 11(c) is another field pattern constructed in a similar way, but with more tiles per repeat unit. Figure 11(d) has a different method of construction: stars are centred at the vertices of the Archimedean tiling so the large squares and octagons contain some tile fragments along their edges. In these examples the large-scale structure is used purely to organise the composition and is not highlighted in the finished artwork. Figure 12 shows designs from some jali screens at the tomb of Itimad-ad Daula in Agra, India — see [14, pp. 83–85] for photographs. In these cases the grey lines in the figures correspond to thicker struts in the screen, producing a design with complementary patterns on two different scales. These 2-level patterns are Type A: the small-scale pattern fills the regions of the large-scale pattern. The large-scale pattern in both examples of Figure 12 is a monohedral tiling of bone tiles — the classical field pattern of Figure 4(b). In Figure 12(a) the small-scale pattern produces a dissection of each large-scale bone into CAMS tiles. The central area of each bone contains a small-scale star subdivided as shown in Figure 5(e); in the figure all these chiral star centres have the same handedness, but they are not consistently oriented in the original. In Figure 12(a) and (b) each vertex in the large-scale pattern has a neighbourhood that is a regular octagon formed from small-scale tiles. In (a) this neighbourhood is formed from eight house tiles and four squares; in (b) the tiles have been modified to produce a neighbourhood with 8-fold symmetry. In (a) the edges of the large-scale bones coincide with edges of the small-scale tiles; in (b) they coincide with mirror-lines of the small-scale tiles. The examples in Figures 11 and 12 are from Mughal India, and hence quite late, but the principles they illustrate were widely used much earlier. In Islamic ornament when a pattern is applied to fill a space, it is not simply cropped to fit in an arbitrary manner (as in the Cosmati pavements, for example); rather, it is constructed to be compatible with geometric features of the boundary. For example, Figure 13 shows a large CAMS cross filled with CAMS tiles and the modified octagonal configuration that we saw in Figure 12(b). This example is taken from a painted strip of the Star and Cross pattern that runs around the soffit of the arch in the portal of the Bala-Sar Madrasa, part of the Imam Reza Shrine 12 (a) IND 0821 (b) (c) IND 1012 (d) IND 0423 Figure 11: CAMS patterns based on the 4.8.8 Archimedean tiling. 13 (a) IND 0335 (b) Figure 12: Type A 2-level patterns from carved jali screens at the tomb of Itimad-ad Daula, Agra (1622–28). complex at Mashhad, Iran. The cross is marked out by placing a star centred at each corner and a bone length-ways along each edge; the interior is completed in a way that preserves the symmetry of the cross. The accompanying large-scale stars are filled with arabesques. As a final example of a Type A 2-level pattern Figure 14 shows a construction typical of the decoration applied with cut ceramic tiles. The large-scale pattern, in grey and white, is the divided Star and Cross pattern of Figure 4(e). The small-scale pattern is also composed of CAMS tiles and includes chevron tiles. Notice that the filling of the large-scale star does not preserve its 8-fold symmetry. This design is adapted from a spandrel in the Seyyed Mosque in Isfahan (1840) — see photograph IRA 1617 [22] for the original. Although this building dates from the relatively recent Qajar period, the design follows the traditional practice of placing a small-scale star centred on every vertex of the large-scale pattern. 5 Type B 2-level patterns Type B 2-level patterns are a continuation of the classical method of framing or outlining the compartments containing the primary motifs. For example, in a Roman mosaic from SaintRomain-en-Gal, Vienna [10, Fig. 78], a geometric framework of regular hexagons divides the composition into compartments displaying figurative designs of flowers; the pathways that outline the compartments are covered with a simple arrangement of 6-pointed stars divided into rhombi, and hexagons. The catalogue of designs from Roman mosaics [1] has other examples in which Archimedean tilings are used to provide the compartmental framework: 3.4.6.4 (Fig. 205), 3.6.3.6 (Fig. 209) and 4.8.8 (Fig. 164). In these classical applications of the outlining method, the geometry always plays a strictly supporting role — it is just used to organise the space. In the Type B examples described here, the large-scale patterns are standard Islamic geometric patterns and have the higher level of complexity one would 14 Figure 13: Filling of a CAMS cross from the Bala-Sar Madrasa, Mashhad. expect. Figure 15(a) shows a Type B 2-level design in which both the large- and small-scale patterns are based on common CAMS patterns. It runs around the soffit of an arch in the south iwan of the Friday Mosque of Gawhar Shad in Mashhad, Iran (1416–18). The centre-line of the pathways defines the large-scale pattern — it is the classical field pattern of Figure 4(c) and is easily recognised. The small-scale pattern is a mix of Figures 4(c) and (e). The same few tiles are used in each case: the large square, house and belt are in both the large- and small-scale patterns, and the star is also used in the small-scale pattern. The grey compartments shaped like the large √ square and house tiles have the correct geometry for the tiles: the base of the house is 2 times the length of its other sides. The path is covered by wrapping the small-scale pattern around these compartments. At the corner of a (large) house where the roof meets a wall the small-scale pattern is mitred, as shown in Figure 15(b). On the outside of the path two stars overlap and are fused to produce a large white shape; in the centre of the path two houses are cut and fused to produce a new black convex shape; on the inside of the path the two stars to be mitred are concentric and have the same orientation and so are coincident — the small-scale pattern is fully coherent along the boundaries of the square and house compartments. Except at these mitred joints, the small-scale pattern is a proper edge-to-edge tiling of CAMS tiles. Except at the outside of the mitred joints, the corners of the compartments lie at the centres of small-scale stars. Almost all the black tiles are house tiles, which gives a strong feeling of consistency to the design; the exceptions are the Dutch bonnets used in the frame and the convex shape produced by the mitring. Figure 16 shows the design from a relief panel in the south iwan of the Friday Mosque, Isfahan (1475/6). The small-scale pattern is assembled from CAMS tiles. In fact, the con- 15 Figure 14: Type A 2-level pattern based on a design from the Seyyed Mosque, Isfahan (1840). 16 (a) (b) Figure 15: Type B 2-level pattern from the Friday Mosque of Gawhar Shad, Mashhad (1416–18). 17 Figure 16: Type B 2-level pattern from the Friday Mosque, Isfahan (1475/6). 18 struction is based on the same templates as Figure 11(d): the large square and a triangular sector of the large octagon in the latter figure can be arranged as shown in the bottom-left of Figure 16 to produce the central part of the panel. All the corners of the grey compartments coincide with star centres of the small-scale pattern. The thick black lines in the bottom-right of Figure 16 outline the equilateral polygons that form the skeleton of the large-scale pattern. In Figure 15 the skeleton runs along the centre-line of the white pathways, but here the skeleton is noticeably off-centre. This gives the impression that the paths are not properly aligned and that the large-scale pattern is not well-defined. 6 Conclusions This paper has introduced a modular system derived from the Star and Cross pattern. The family of patterns that can be produced by assembling the modules includes many traditional Islamic patterns from Central Asia and Iran. Some of the simpler patterns, including the Star and Cross pattern itself, are pre-Islamic. The earliest Islamic examples of CAMS patterns are found among the Seljuk brickwork designs on 11th- and 12th-century minarets. The restriction to 45◦ and 90◦ angles in the modules may indicate that CAMS was developed in this medium as these are the angles most compatible with working in brick. As technical innovations provided new media, CAMS patterns were adapted and developed. The rediscovery of glaze led initially to approximate or pixellated versions of the patterns being expressed in banna’i, and later to complex 2-level patterns in cut tile work as more colours became available. We have also seen that grouping modules together to form larger ones introduces hierarchical structure into a composition, either as a hidden organising principle or to generate a complementary pattern on a different scale. Traditional examples include carved screens from Mughal India and the more complex patterns possible with glazed tiles found in Iran. The Iranian formula for constructing 2-level patterns that has been observed in other modular systems [5, 8, 9] also applies to the CAMS examples: the large-scale pattern is easily recognisable, usually simple and very familiar; in Type A patterns the corners and intersections of the large-scale pattern are key points and in Type B patterns the corners of the pathways are key points; small-scale stars are centred at the key points and the remaining edge segments of the large-scale pattern are covered by edges or mirror-lines of small-scale modules; the regions are infilled in a consistent manner (congruent regions generally have the same filling) but the filling need not preserve the symmetry of a region. The CAMS modules are undecorated and hence easier to produce than other modular systems. Classical patterns provide some of the basic module shapes, and a range of CAMS patterns is found among early brickwork ornament. These properties lead me to propose that CAMS is one of the earliest Islamic modular systems for producing star patterns. References [1] C. Balmelle and R. Prudhomme, Le Décor Géometrique de la Mosaı̈que Romaine: vol 1 Répertoire Graphique et Descriptif des Compositions Linéaires et Isotropes, Picard, 1985. 19 [2] J. Bourgoin, Les Eléments de l’Art Arabe: Le Trait des Entrelacs, Firmin-Didot, Paris, 1879. Plates reprinted in Arabic Geometric Pattern and Design, Dover Publications, New York, 1973. [3] J. Bonner, ‘Three traditions of self-similarity in fourteenth and fifteenth century Islamic geometric ornament’, Proc. ISAMA/Bridges: Mathematical Connections in Art, Music and Science, (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1–12. [4] J.-M. Castéra, Arabesques: Art Décoratif au Maroc, ACR Edition, 1996. [5] P. R. Cromwell, ‘The search for quasi-periodicity in Islamic 5-fold ornament’, Math. Intelligencer 31 no 1 (2009) 36–56. [6] P. R. Cromwell, ‘Hybrid 1-point and 2-point constructions for some Islamic geometric designs’, J. Math. and the Arts 4 (2010) 21–28. [7] P. R. Cromwell, ‘Islamic geometric designs from the Topkapı Scroll I: unusual arrangements of stars’, J. Math. and the Arts 4 (2010) 73–85. [8] P. R. Cromwell, ‘Islamic geometric designs from the Topkapı Scroll II: a modular design system’, J. Math. and the Arts 4 (2010) 119–136. [9] P. R. Cromwell and E. Beltrami, ‘The whirling kites of Isfahan: geometric variations on a theme’, Math. Intelligencer 33 no 3 (2011) 84–93. [10] K. M. D. Dunbabin, Mosaics of the Greek and Roman World, Cambridge Univ. Press, 1999. [11] C. S. Kaplan, ‘Islamic star patterns from polygons in contact’, Graphics Interface 2005, ACM International Conference Proceeding Series 112, 2005, pp. 177–186. [12] A. J. Lee, Islamic Star Patterns — Notes, unpublished manuscript, 1975. Available from http://www.tilingsearch.org/tony/index.htm (accessed Feb 2012) [13] P. J. Lu and P. J. Steinhardt, ‘Decagonal and quasi-crystalline tilings in medieval Islamic architecture’, Science 315 (23 Feb 2007) 1106–1110. [14] G. Michell, The Majesty of Mughul Decoration: The Art and Architecture of Islamic India, Thames and Hudson, London, 2007. [15] G. Necipoğlu, The Topkapı Scroll: Geometry and Ornament in Islamic Architecture, Getty Center Publication, Santa Monica, 1995. [16] P. Pajares–Ayuela, Cosmatesque Ornament: Flat Polychrome Geometric Patterns in Architecture, W. W. Norton and Co., 2001. [17] I. El-Said and A. Parman, Geometric Concepts in Islamic Art, World of Islam Festival Publishing Company, London, 1976. [18] R. Sarhangi, S. Jablan and R. Sazdanovic, ‘Modularity in medieval Persian mosaics: textual, empirical, analytical, and theoretical considerations ’, Visual Mathematics 7 no 1 (2005), http://www.mi.sanu.ac.rs/vismath/sarhangi/ (accessed Feb 2012) 20 [19] S. P. Seherr–Thoss and H. C. Seherr–Thoss, Design and Color in Islamic Architecture, Smithsonian Institution Press, Washington, 1968. [20] B. Servatius and H. Servatius, ‘Self-dual graphs’, Discrete Math. 149 (1996) 223–232. [21] B. Servatius and H. Servatius, ‘Symmetry, automorphisms, and self-duality of infinite planar graphs and tilings’, Proc. International Scientific Conference on Mathematics, (Zilina, 1998), ed. V. Balint, 1998, pp. 83–116. [22] D. Wade, Pattern in Islamic Art: The Wade Photo-Archive, http://www.patterninislamicart.com/ (accessed Feb 2012) 21

© Copyright 2018