cromwell star cross

8/9/2019 Cromwell Star Cross

1/21

A Modular Design System based onthe 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 pre-Islamic Star and Cross pattern. It can be used to generate a large family of traditionalIslamic patterns found in Central Asia and Iran. In other examples of Islamic modularsystems, 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 sometraditional 2-level geometric patterns as hierarchical structures of CAMS modules, cor-roborating 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 isformed from 8-pointed regular star polygons of type{8/2} arranged on a square grid. Theinterstitial spaces between the stars form crosses. Crosses with this pointed shape seemnot to have been used individually as emblems and are only found in the context of thispattern.

Use of the Star and Cross pattern is common among cultures that have developed ageometric 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 ingrey in the figure but filled with guilloche in the original) and the crosses are subdividedinto 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


2/21

(a) (b)

(c)

Figure 2: Variations on the Star and Cross design. Photograph reproduced fromwww.thejoyofshards.co.ukcourtesy of Rod Humby.

2


3/21

the large-scale framework in Cosmati work is provided by a compass-work design, so theuse 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 thatconforms to the silver ratio system of proportion. We show many examples of traditionalIslamic patterns that can be produced using this system; they are mostly from CentralAsia and Iran of the Seljuk, Timurid and Mughal periods and include field patterns, starpatterns, and 2-level patterns. This complements previous studies of Islamic geometricornament [3, 5, 6, 8, 13], which have found evidence to support the traditional use of amodular approach to pattern design, particularly in the construction of patterns based onthe 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 theproduction of 2-level designs via hierarchies of modules of different scales. We shall analyse2-level patterns in which the small-scale pattern either fills or outlines the compartmentsin the large-scale pattern; these two modes (filling or outlining) correspond to Type A andType B, respectively, in the classification of 2-level designs introduced by Bonner [3]. Weshall see that the principles for constructing 2-level patterns with the new system agreewith 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 oftenassembled to make repeating (periodic) units, but free-form arrangements that grow moreorganically are also possible. In general, the modular approach brings benefits through ease

of manufacture with the mass production of interchangeable parts. In geometric ornamentit also has benefits for composition: the eye finds unity in the economy of motifs and re-useof familiar forms as the same few shapes appear in a variety of contexts. It also means non-experts can produce acceptable results as modular design naturally creates a satisfactorydistribution 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 multiplesof 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 bothlengths, and the other tiles are equilateral with short edges. The tiles in the bottom row canbe 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 strictlylimited range of side lengths and angles produces a system that can generate an abundanceof different patterns, but the shared properties also give a visual consistency both within asingle pattern and across a collection.

These tiles can be assembled to create many traditional Islamic patterns from CentralAsia: Transoxiana (modern Uzbekistan), Khorasan (Afghanistan plus areas of neighbouringstates, including north-east Iran), and India. For convenience we shall refer to the tiles inFigure 3 as CAMS (Central Asian Modular System) tiles and to the patterns they generateas CAMS patterns.

Figure 4 shows some examples. The first two patterns are field patterns (do not containstars) and are pre-Islamic. Figure 4(a) can be found among the brickwork patterns on the

3


4/21


5/21

(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


6/21

(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 seeFigure 5. Those in the top row are dissections of the star into CAMS tiles. People whoanalyse patterns in terms of linear features rather than tile shapes may classify patternscontaining (e) as swastika designs. Figure 5(f) is from a design carved in relief with raisedflowers 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 ofthe 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 inspec-tion, we see that the Dutch bonnet tiles are mis-shaped: the edge lengths along the bottomshould 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; thelarge square is the only tile from Figure 3 that fits this space so Dutch bonnets are usedin opposing pairs, as in Figure 4(g). In Figure 6(b) this forced arrangement has beenbroken and plus-shaped regions have been introduced. These regions are of similar sizeto the CAMS tiles and share their geometric properties of edge length and angles, so aplus-shaped tile could easily be added to the set. It is in this way that modular systemsgrow and evolve: the modules are assembled in ways that leave vacant spaces, and the gaps

6


7/21

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 patternson surviving buildings; membership of the family has been determined by frequency of useand structural similarities of the tiles as much as by size, edge length, and angle. The mostcommon tiles are the star, house, large square, belt, bone and Dutch bonnet. The hexagontile appears in a few early patterns, but is rarely used later and perhaps should not beincluded 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 unlikelyto belong to any other. However, it is almost always used in groups of four arranged asFigure 5(e).

The Dutch bonnet is reminiscent of the motif on the trapezium tile in a modular systemfor producing patterns based on 10-pointed stars [6]; in that case it is a launch point for theconstruction of 2-point patterns. Figure 6(c) is a modification of Figure 4(g) that producesa similar effect it can be interpreted as overlapping house tiles. However, there seems tohave been no general means to produce 2-point CAMS patterns.

Documentary evidence for the historical use of other modular systems is provided bythe Topkap Scroll [8]. Panel 61 of the scroll, redrawn in Figure 7(a), is the only templatethat corresponds to a CAMS pattern. It is unusual in that the star centres lie on the edgesof the template, not in the corners [7].

Besides the geometrically pure form presented so far, CAMS patterns were also appliedin what would today be called a pixellated form the tile shapes are approximated ona square grid. An example from Panel 47 of the Topkap Scroll is shown in Figure 7(b).To see that the pixellated form is approximate note that the star is not equilateral (thediagonal edges are shorter than the other edges). However, the pattern is clearly closelyrelated to the modular system the true form is shown in Figure 4(j). The Topkap Scrollcontains four further examples of the approximated patterns: Panel 51d is the Star andCross pattern, Panel 43 is Figure 4(e), Panel 37b is the signature pattern of Figure 4(f) andPanel 1 is related to Figure 11(d) that we shall discuss in4. None of these panels providesany hint to its construction they are simply square grids with some of the squares filledin.

Glaze was rediscovered in Baghdad in the 9th century. Initially, it was expensive andwas only used for highlights. The pixellated patterns were an ideal application for the newtechnology: the black and white pixels were realised in blue glazed and unglazed bricks,respectively. CAMS patterns executed in this technique, known as bannai, can be seenon the muqarnas in the west iwan of the Friday Mosque in Isfahan see [19, Pl. 83] orphotograph IRA 0529 [22].

There are many more traditional examples of CAMS patterns. Is the fact that the pat-terns have CAMS tiles in common sufficient to conclude that the tiles were used historicallyas 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 edgeand compass: a set of lines and circles is drawn and a subset of the lines is chosen as thefigure. This was typical of Roman and Byzantine work; one circle-based form is even calledcompass-work. It is not necessary to invoke a modular system to reproduce many of the

7


8/21

(a) Panel 61 (b) Panel 47

Figure 7: Templates from the Topkap Scroll.

patterns in Figure 4. Books such as El-Said and Parman [17] use the Euclidean method toexplain how simple Islamic patterns can be constructed; it includes constructions for theStar 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 bedifferent 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 appearcluttered if the lack of consistency in size and shape is too obvious. The modular approachis a bottom-up method that proceeds by building up a pattern from a small set of simpleshapes. It enables a more experimental approach to composition so that a design can b egrown organically in an unplanned manner by continually attaching tiles to a patch. Thesheer variety of CAMS patterns, some of which are more free-form compositions, stronglysupports the modular system hypothesis of design. We shall find further evidence when weconsider the hierarchical properties of Islamic patterns.

3 Comparison with other modular systems

Although the CAMS tiles have not been described before, the suggestion that modularsystems underlie many Islamic geometric patterns is not new [3, 4, 5, 6, 8, 13, 18]. Forexample, Castera uses a modular system of 37 pieces [4, p. 1145] to produce the intricatepatterns of 8-pointed stars found in Spain and Morocco. Only the star, house and smalland large squares occur in both the CAMS and Castera sets of tiles. Modular design is alsonow well established as one of the traditional methods for the creation of geometric designsin the eastern Islamic world; it is supported by documentary evidence and has the powerto explain the structures of a variety of complex 2-level patterns.

8


9/21

pentagon decagon triangle trapezium bow-tie

Figure 8: A decorated modular system whose tiles have two edge lengths in the goldenratio.

In some widely used modular systems the modules provide an underlying structure forthe composition, but are not directly visible in the final design. We shall briefly describean 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 tilesin Figure 3, these tiles have two edge lengths but, in this case, the long and short edgesare 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 shortedges.

Notice that in this system the tiles are decorated with motifs, shown here in black. Allthe motifs meet the boundaries of the tiles in the midpoints of the edges. However, theincidence angles are different on the short sides and the long sides. The geometry is dictatedby the{10/4} regular star polygon used to decorate the decagon: it leads to an incidenceangle of 72 on the short edges and 36 on the long edges. The motif on the pentagon isthe

{5/2

}star.

As with the CAMS tiles, a design is created by assembling these modules to producean edge-to-edge tiling. Figure 9 shows the construction of a design from the ceiling of theChehel 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 producethe 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 thetiling across the joins between copies of the template. It is important to note that the tileboundaries do not appear in the finished product the black motifs form the foregroundregions of the pattern and the white areas in the corners of the tiles are fused to createbackground regions. In Figures 8 and 9 the motifs are filled in to aid the discussion, butthey 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 compositionthat 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 followingdiscussion.

The fact that the CAMS tiles do not carry motifs but appear themselves in the finishedpattern leads to some interesting structural differences in the families of CAMS patternsand PIC patterns.

The angles in the motif on a PIC module are of two kinds: those that meet an edgeof the module are determined by the incidence angle; the others are determined by

9


10/21

(a)

(b)

Figure 9: Construction of a ceiling design from the Chehel Sotoon Palace, Isfahan, usingthe modular system of Figure 8.

10


11/21

the angles in the corners of the module. In particular, all the angles in the motifscome from a small set. The angles in the corners of the background regions are thecomplements of the angles in the motifs. In most cases, these angles will be differentfrom the angles in the motifs, and the background regions will have different shapesfrom the foreground motifs. For example, in Figure 9 four of the motifs have an angleof 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 foregroundmotif and the other will be a background region. So it is unlikely that adjacent regionsin a PIC design will be congruent. However, there is no natural distinction intoforeground 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) arecongruent.

When two PIC modules share an edge, the midpoint of the edge will be a vertex inthe finished pattern, and the corners of two motifs will meet there. Hence, all thevertices in a PIC pattern will be 4-valent. However, Figures 4(a) and (l) show thatCAMS patterns can contain 3-valent vertices.

A pattern is said to have counter-change symmetry if its figure and ground can beinterchanged 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 reversethe colours. Figure 10(a) and (b) are star patterns; (b) is developed from the templatein Figure 7(a). Although these two patterns can be coloured in this way, I know ofno traditional Islamicstarpatterns that have been coloured to exhibit counter-changesymmetry. However, there are Islamic field patterns that display counter-changesymmetry. The CAMS example in Figure 10(c) is typical: all the tiles are congruentand it has the topology of a chessboard each tile is surrounded by four of the othercolour, and four tiles meet at each vertex.

By contrast, although a PIC design can be coloured in two colours so that adjacentshapes have different colours, this corresponds to the natural division into foreground(motifs) and background. If a PIC pattern had counter-change symmetry, the tilingformed 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 irreg-ular 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 begrouped together to form larger ones. We can see this process in the CAMS patternsof Figure 11. These examples all use the 4.8.8 Archimedean tiling as an organising principleand this underlying structure is highlighted in grey.

In Figure 11(a) CAMS tiles have been assembled to form the large squares and octagonsof the Archimedean tiling these large shapes are the higher-level modules. Although the

11


12/21

(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 beensubdivided as in Figure 5(h). Figure 11(b) is identical to part (a) except that the centralstar is subdivided as in Figure 5(d) notice that this destroys the 8-fold symmetry of thelarge octagon. Figure 11(c) is another field pattern constructed in a similar way, but withmore tiles per repeat unit. Figure 11(d) has a different method of construction: stars arecentred at the vertices of the Archimedean tiling so the large squares and octagons containsome tile fragments along their edges.

In these examples the large-scale structure is used purely to organise the compositionand is not highlighted in the finished artwork. Figure 12 shows designs from some jali screensat the tomb of Itimad-ad Daula in Agra, India see [14, pp. 8385] for photographs. In

these cases the grey lines in the figures correspond to thicker struts in the screen, producinga design with complementary patterns on two different scales. These 2-level patterns areType 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 bonetiles the classical field pattern of Figure 4(b). In Figure 12(a) the small-scale patternproduces a dissection of each large-scale bone into CAMS tiles. The central area of eachbone contains a small-scale star subdivided as shown in Figure 5(e); in the figure all thesechiral star centres have the same handedness, but they are not consistently oriented in theoriginal.

In Figure 12(a) and (b) each vertex in the large-scale pattern has a neighbourhood thatis 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 aneighbourhood with 8-fold symmetry. In (a) the edges of the large-scale bones coincide withedges 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 theprinciples they illustrate were widely used much earlier. In Islamic ornament when a patternis applied to fill a space, it is not simply cropped to fit in an arbitrary manner (as in theCosmati pavements, for example); rather, it is constructed to be compatible with geometricfeatures of the boundary. For example, Figure 13 shows a large CAMS cross filled withCAMS tiles and the modified octagonal configuration that we saw in Figure 12(b). Thisexample is taken from a painted strip of the Star and Cross pattern that runs around thesoffit of the arch in the portal of the Bala-Sar Madrasa, part of the Imam Reza Shrine

12


13/21

(a) IND 0821 (b)

(c) IND 1012 (d) IND 0423

Figure 11: CAMS patterns based on the 4.8.8 Archimedean tiling.

13


14/21


15/21

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 inthe south iwan of the Friday Mosque of Gawhar Shad in Mashhad, Iran (141618). Thecentre-line of the pathways defines the large-scale pattern it is the classical field patternof 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 inboth 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 correctgeometry 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 thecorner 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 toproduce a large white shape; in the centre of the path two houses are cut and fused toproduce a new black convex shape; on the inside of the path the two stars to be mitred areconcentric and have the same orientation and so are coincident the small-scale pattern isfully coherent along the boundaries of the square and house compartments. Except at thesemitred joints, the small-scale pattern is a proper edge-to-edge tiling of CAMS tiles. Exceptat the outside of the mitred joints, the corners of the compartments lie at the centres ofsmall-scale stars. Almost all the black tiles are house tiles, which gives a strong feeling ofconsistency to the design; the exceptions are the Dutch bonnets used in the frame and theconvex 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


16/21


17/21

(a)

(b)

Figure 15: Type B 2-level pattern from the Friday Mosque of Gawhar Shad, Mashhad

(141618).

17


18/21

Figure 16: Type B 2-level pattern from the Friday Mosque, Isfahan (1475/6).

18


19/21


20/21

[2] J. Bourgoin,Les Elements de lArt 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 Islamicgeometric ornament, Proc. ISAMA/Bridges: Mathematical Connections in Art, Musicand Science, (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 112.

[4] J.-M. Castera, Arabesques: Art Decoratif au Maroc, ACR Edition, 1996.

[5] P. R. Cromwell, The search for quasi-periodicity in Islamic 5-fold ornament, Math.Intelligencer31 no 1 (2009) 3656.

[6] P. R. Cromwell, Hybrid 1-point and 2-point constructions for some Islamic geometricdesigns,J. Math. and the Arts4 (2010) 2128.

[7] P. R. Cromwell, Islamic geometric designs from the Topkap Scroll I: unusual arrange-ments of stars, J. Math. and the Arts4 (2010) 7385.

[8] P. R. Cromwell, Islamic geometric designs from the Topkap Scroll II: a modular designsystem,J. Math. and the Arts4 (2010) 119136.

[9] P. R. Cromwell and E. Beltrami, The whirling kites of Isfahan: geometric variationson a theme, Math. Intelligencer 33 no 3 (2011) 8493.

[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. 177186.

[12] A. J. Lee, Islamic Star Patterns Notes, unpublished manuscript, 1975. Availablefrom http://www.tilingsearch.org/tony/index.htm(accessed Feb 2012)

[13] P. J. Lu and P. J. Steinhardt, Decagonal and quasi-crystalline tilings in medievalIslamic architecture, Science315 (23 Feb 2007) 11061110.

[14] G. Michell,The Majesty of Mughul Decoration: The Art and Architecture of IslamicIndia, Thames and Hudson, London, 2007.

[15] G. Necipoglu, The Topkap Scroll: Geometry and Ornament in Islamic Architecture,

Getty Center Publication, Santa Monica, 1995.

[16] P. PajaresAyuela, Cosmatesque Ornament: Flat Polychrome Geometric Patterns inArchitecture, W. W. Norton and Co., 2001.

[17] I. El-Said and A. Parman,Geometric Concepts in Islamic Art, World of Islam FestivalPublishing Company, London, 1976.

[18] R. Sarhangi, S. Jablan and R. Sazdanovic, Modularity in medieval Persian mosaics:textual, empirical, analytical, and theoretical considerations , Visual Mathematics 7no 1 (2005),

http://www.mi.sanu.ac.rs/vismath/sarhangi/(accessed Feb 2012)

20


21/21

[19] S. P. SeherrThoss and H. C. SeherrThoss,Design and Color in Islamic Architecture,Smithsonian Institution Press, Washington, 1968.

[20] B. Servatius and H. Servatius, Self-dual graphs,Discrete Math. 149 (1996) 223232.

[21] B. Servatius and H. Servatius, Symmetry, automorphisms, and self-duality of infiniteplanar graphs and tilings, Proc. International Scientific Conference on Mathematics,(Zilina, 1998), ed. V. Balint, 1998, pp. 83116.

[22] D. Wade,Pattern in Islamic Art: The Wade Photo-Archive,http://www.patterninislamicart.com/(accessed Feb 2012)

21

cromwell star cross

Documents