Islamic Geometric Designs from the Topkapı Scroll I: Unusual Arrangements of Stars

8/17/2019 Islamic Geometric Designs from the Topkapı Scroll I: Unusual Arrangements of Stars

1/13

Islamic Geometric Designs from the Topkapı Scroll

I: Unusual Arrangements of Stars

Peter R. Cromwell

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

The Topkapı Scroll is an important documentary source for the study of Islamic geo-metric ornament. Here we give a mathematical analysis of some its exemplary designsillustrating a variety of methods of construction. We also consider how the techniquesrestrict the range of symmetry types produced. The lack of 3-fold symmetry is related tothe exclusive use of rectangular templates. Original designs are included to demonstratethe application of traditional methods with a triangular template.

1 Introduction

Islamic geometric art is a distinctive idiom characterised by networks of interlocking stars andpolygons, high levels of symmetry on both local and global scales, and various forms of repetition.The designs are usually laid out on some form of grid but, in the finished product, much of theunderlying geometry is concealed from the viewer who can only marvel at the complex interplayof complementary shapes. Although the fundamental methods of construction are not compli-cated in themselves, they are capable of elaboration and variation as a form of geometric game,and this flexibility provides an extremely rich source of designs limited only by the imagination

and ingenuity of the designer. By the thirteenth century artists were producing designs of greatcomplexity, including elegant combinations of seemingly incompatible stars, and designs containingcomplementary patterns of different scales.

Although we have a few manuals that document workshop geometry of the time, there areno contemporary sources that explain the design principles of Islamic art. The master craftsmenguarded their methods as trade secrets but they did make records in the form of scrolls containingtemplates and other aides-mémoire for pattern production. Fortunately, an outstanding exampledating from around 1500 has survived in the library of the Topkapı Palace, Istanbul.

The Topkapı Scroll (manuscript H.1956) is an important documentary source for the study of Islamic geometric ornament. It contains a series of geometric figures drawn on individual pages,glued end to end to form a continuous sheet about 33cm high and almost 30m long. It is not a ‘how

to’ manual as there is no text, but it is more than a pattern book as it shows construction lines.Necipoğlu produced a detailed study [13] that examines the artistic, historical and philosophicalcontext of the scroll, but she says little about the mathematical properties of individual patternsor their constructions. Her book contains a half-size colour reproduction of the scroll, and alsoincludes annotations to show the construction lines and marks scored into the paper with a stylusthat are not visible in the photographs. References in this article to numbered panels of the TopkapıScroll use the numbering in [13].

The figures in the scroll are a mixture of calligraphy, designs for use in architecture (muqarnas,brick patterns, domes), and templates for 2-dimensional geometric patterns that have more generalapplication. We shall concentrate on the latter class. When the templates are repeated to producefull patterns they reveal advanced designs that display virtuosity and mastery of the style, and

1


2/13

include some patterns that I have not seen elsewhere. In this article I provide mathematical analysisof some of the more remarkable designs, focussing on star patterns with unusual properties.

2 Designs generated by reflections

A high proportion of Islamic patterns have mirror symmetry. Drawing all the mirror lines ona pattern divides it up into regions, each of which can be used as a template to regenerate thewhole pattern. So long as the mirror lines are not all parallel, the regions will be finite. Theseven possibilities are shown in Figure 1. The left-handed and right-handed tiles are differentiatedby shading. The tiles are marked to destroy some of their inherent symmetry; in (d), the tilesthemselves are asymmetrical so no marks are required.

Each tiling is labelled with its symmetry type. The first four designs are produced by replicatingan asymmetric motif and are called primitive . If the decoration on the tiles is not asymmetric, the

symmetry type of the design may change. In Figure 1(e–g) the tiles are decorated with motifs thathave rotational symmetry. The rectangular tiles with primitive type pmm give rise to two otherarrangements: a motif with 2-fold rotational symmetry produces a design with cmm symmetrytype; if the rectangular tile is made square, the motif may have 4-fold symmetry and this leads totype p4g. If the equilateral triangle tiles in primitive type p3m1 carry a motif with 3-fold symmetry,the design will have p31m symmetry type.

The frequency distribution of symmetry types in Islamic art is extremely uneven. The simplemethod of arranging suitable star motifs at the vertices of the standard triangular, square or rhombiclattices gives symmetry types p6m, p4m and cmm — the most common types by a large margin.Conversely, the 3-fold groups p3, p3m1 and p31m are very rare. Examples are shown in Figure 2 butthe first two are rather contrived. In (a) the symmetry type is p3 only because decorative markings

separate the tiles into two classes — the unmarked tiling (known as the Andalusian pattern) hastype p6. The pattern in (b) is from Jones [9, Pl. 44 #15] where it is described as being ‘from aPersian manuscript’ and (c) is from Bourgoin’s collection [1, Pl. 7]. This scarcity is partly due to atheoretical limitation: many Islamic tilings are isohedral (tile transitive under the symmetry group)and there are no isohedral tilings with unmarked tiles that have symmetry group p3m1 [6].

About 40% of the panels in the Topkapı Scroll are templates for 2-dimensional geometric pat-terns, mostly star patterns. The templates are rectangular and are replicated by reflection in thesides of the rectangle. When a design is being created from a stencil, the mirror image tile isobtained simply by turning the stencil over. Some of the designs have symmetry type p4m or p6mand, in these cases, the template contains more than one copy of the fundamental region (panels30 and 63, discussed later, provide examples). This preference for generating designs by reflecting

rectangular templates is another factor that precludes the production of 3-fold symmetry.

3 Designs with symmetry p4g and p31m

After p6m, p4m and cmm, one of the most common symmetry types found in Islamic ornamentis p4g. As shown in Figure 1(f), its template is a square with a motif that has 4-fold rotationalsymmetry. There are two common configurations for generating star patterns with p4g symmetry:

(i ) place stars on the edges of the square so that a mirror line of each star lies on an edge

(ii ) place stars with 4n+2 points at the corners of the square — the stars are aligned spike-to-dentalong the edges of the square.

2


3/13

(a) pmm (b) p3m1

(c) p4m (d) p6m

(e) cmm (f) p4g (g) p31m

Figure 1: The tilings generated by reflections in the sides of a tile.

3


4/13

(a) p3 (b) p3m1 (c) p31m

Figure 2: Some Islamic patterns with 3-fold symmetry.

(a) (b)

(c) (d)

Figure 3: Designs with symmetry group p4g from panels 61 and 68a of the Topkapı Scroll.

4


5/13

(a) (b)

(c) (d)

Figure 4: Original designs with symmetry group p31m.

As an example of each type, Figure 3(a) and (b) show the templates in panels 61 and 68a of the scroll, respectively, and parts (c) and (d) show the designs they generate. The design of panel 61 has an elegant simplicity: all the angles are 135◦, all lines cross at 90◦, and there are onlythree shapes of region: a regular {8/2} star, a pentagon with two right angles, and a ‘dog-bone’.Panel 81a is another type (i ) example; it uses irregular 7-pointed stars. Other panels with type (ii )configurations are 59 (having 6-pointed stars) and 66 and 72c (both having 10-pointed stars).

Of the p4g designs in Bourgoin, plate 170 is a type (i ) design that has regular 7-pointed stars(its construction is explained by Hankin in [8]), and plates 27–30 are all type ( ii ) with 6-pointedstars. The template in panel 59 of the scroll generates the design in plate 27 of Bourgoin.

Figure 4 shows that it is possible to perform equivalent constructions using an equilateraltriangle instead of a square as the foundation. In (a) 12-pointed stars are placed along the edgesof the template and in (b) 9-pointed stars are placed at the corners. In the second case we can useany star with 6n + 3 points. Parts (c) and (d) show the designs they generate — the template isrotated by 90◦in each case. I do no know of any traditional patterns of this kind.

5


6/13

Figure 5: Design with symmetry group p4g from panel 35 of the Topkapı Scroll.

(a) (b)

Figure 6: Two designs with the same 12-pointed rose motif arranged on the vertices of a square

lattice and a triangular lattice.6


7/13

(a) (b) (c)

Figure 7: Structure of panel 35 of the Topkapı Scroll.

Panel 35 of the scroll shows a more complex design (see Figure 5) with p4g symmetry containing8-pointed and 12-pointed stars. It is not a template like the other panels but shows a large pieceof the design with a line drawn over it to illustrate how to fit a pattern into an arch. Figure 6shows two simpler but related designs. They contain the same 12-pointed rose motifs arrangedon the vertices of a square lattice (symmetry type p4m) and a triangular lattice ( p6m) and arequite similar to Bourgoin’s plates 118 and 78, respectively. The shaded rectangle in part (b) is thetemplate in panel 63 of the scroll. The shaded square in part (a) can be used as the template forthe design in panel 35 after some minor modifications in the corners, showing that Figure 5 can beclassified as type (i ).

However, the underlying structure of this design has a simpler interpretation. Observe that the

centres of adjacent 12-pointed stars in Figure 6(a) determine a square that contains an 8-pointedstar, and that the centres of adjacent stars in Figure 6(b) determine an equilateral triangle with a3-pronged region in the middle. These regions are shown in Figure 7(a) and (b); panel 35 can beconstructed by arranging them according to Figure 7(c). From this viewpoint, panel 35 is basedon the Archimedean tiling with vertex type 3.3.4.3.4 with the 12-pointed stars are centred at thevertices.

A similar procedure underlies the design in panel 38 — the most colourful and detailed panelin the scroll. It is an example of a kind of 2-level pattern in which a network of pathways forminga large scale pattern is paved with a small scale pattern. Panel 49 contains a similar example. Thenetwork of pathways is shown in white in Figure 8 — the illustration in the scroll corresponds to thebottom-right quarter. This kind of work is usually executed in low relief where the regions shown

in grey are raised up and capped with a decorative tile, and the smaller scale pattern runs along thechannels in between. The rectangular format with a frame is typical. In this case the small scalepattern is supplied by applying the motifs on the square and the triangle in Figure 7(a) and (b) tothe white regions in Figure 8. In the scroll panel 38 is coloured in a consistent manner: the regularstars have pale blue centres surrounded by black kites, the 12-pointed roses have white petals, the8-pointed roses have yellow petals, the irregular 5-pointed stars are red, and other regions are palepink. Most of the regions also carry a black motif that looks a bit like a snowflake or a bird’sfootprint.

Panels containing examples of a different form of 2-level pattern are discussed in part ii [?] and[4].

7


8/13

Figure 8: Structure of panel 38 of the Topkapı Scroll.

4 Designs with unusual star combinations

In the simpler kinds of star pattern, star and rose motifs are distributed like the vertices in alattice and connected together. Some combinations of stars fit quite naturally with this approach:8-pointed and 12-pointed stars arranged on a square lattice like the black and white squares ona chessboard, for example, or 9-pointed and 12-pointed stars on a triangular lattice. Other com-binations of stars have symmetries that are seemingly incompatible with each other and with thestandard grids and lattices, yet artists still found ways to incorporate them into elegant designs.

Figure 9(c) shows a design generated from panel 30 of the Topkapı Scroll. The Archimedeantiling with vertex type 4.8.8 provides the geometry underpinning the arrangement of the stars: thereare 13-pointed stars centred at the vertices and 16-pointed stars in the middles of the octagons.The 16-pointed stars are regular but the 13-pointed stars are not — the angles between adjacentspikes are not constant. The angle in the corner of a regular octagon is 135◦; in the figure thisangle is spanned by five spikes of a 13-pointed star. If the star were regular, this span would be5/13 ·360◦ ≈ 138.5◦. To conceal this discrepancy the spikes lying in the octagons are closer than theyshould be. Correspondingly, the spikes lying in the squares are further apart than than they shouldbe: 30◦ (the angle for a 12-pointed star) instead of about 27.7◦. This deviation from regularityis only noticeable after some thought; the pattern has so many other symmetries that this lack of local symmetry does not draw the eye’s attention.

Figure 9(b) shows how the deception is produced. The proportions of the two stars can be fixedby drawing a regular 16-gon and a regular 13-gon with a common edge. From this we can deducethat the line DE divides the line AF in the ratio of about tan(180/16) : tan(180/13) ≈ 5 : 4. Inthe underlying Archimedean tiling, BF becomes the lines between two octagons, and FG becomesthe lines between octagons and squares. Triangle AFG is the reflection of triangle AFB in the line

8


9/13

A B

C

D

EF

G

(a) (b)

(c)

Figure 9: A design with 13-pointed and 16-pointed stars from panel 30 of the Topkapı Scroll.

9


10/13

AF. The rays sent out from A to the line DE are equally spaced at angles of 11 1/4◦, making the16-pointed star regular. They meet rays sent out from F which, together with rays FB, FC andFG, determine the geometry of the 13-pointed star. In the scroll, panel 30 is a square templateformed by reflecting Figure 9(b) in line AC. Necipoğlu’s annotations show uninked lines scored inthe scroll radiating out from the star centres like this.

Figure 10(d) shows a design generated from panel 42 of the Topkapı Scroll. Its constructioncan be explained as a classic application of the ‘polygons in contact’ (PIC) method to combine9-pointed and 11-pointed stars. In Figure 10(a) lines of 11-gons that zig-zag down the page arearranged in mirror-image pairs across the page. The 11-gons are regular and the hexagons thatconnect them are equilateral. To generate the motifs, a pair of lines is placed symmetrically atthe mid-point of each solid edge; these lines are extended until they meet other similarly producedlines. If the angle between the line pairs is chosen carefully, the tips of the arrowhead motifs touchthe broken lines and meet in the centres of the hexagons. This produces Figure 10(b).

To complete the design we must fill the central region bounded by the broken lines in Fig-ure 10(a). Each of the four lines meeting at P passes through the centre and a vertex of one of the 11-gons. The angle between two of these lines is about 81.8◦. In Figure 10(c) a regular 9-gonhas been inserted into the space with its centre at P. Its vertices do not lie on the four radii sothe star motifs in the final design will not be perfectly aligned, but the error is less than 1◦. Thetiling for the PIC construction is completed by connecting the vertices of the 9-gons and 11-gonsas shown. Figure 10(d) shows the finished design. The star motifs arising from the 9-gons and11-gons are regular but their rose-like extensions that come from the connecting cell structure of irregular pentagons and hexagons are distorted.

The PIC technique was proposed by Hankin [7], developed and refined by Bonner [2, 3], andhas been computerised by Kaplan [10, 11] and made available as an on-line tool [12]. When theregular polygons in the tiling that lead to star motifs in the final design are properly aligned, aproperty guaranteed when they have edges in common for example, the strict PIC method appliedby computer works well. However, when they are misaligned or are supplemented by a large matrixof smaller irregular polygons, small adjustments to the method, such as moving the crossings awayfrom the mid-points of edges or varying the incidence angle, can improve the visual appearanceof the design. In the scroll, the region of the design shown in panel 42 corresponds to the shadedrectangle in Figure 10(c). The annotations reveal bunches of radiating lines connecting the starcentres similar to those in Figure 9(b). This suggests that medieval artists also made adjustmentsand valued alignment more than regularity.

As a final example Figure 11 shows an original design having the elusive p3m1 symmetry type.The three kinds of 3-fold rotation centres are associated with 6-pointed, 12-pointed and 18-pointedstars, and 9-pointed stars straddle the mirror lines — all stars are regular and perfectly alignedwith their neighbours. Although its construction follows many of the principles of Islamic geometricart, should it or the designs in Figure 4 qualify as Islamic art?

Design structure is an important feature of a culture’s stylistic tradition. Some anthropologistsuse symmetry type as an objectively determined attribute associated with tribal identity. Thecultural variation observed in the frequency distribution of symmetry types may be the result of a conscious choice to avoid designs used by neighbouring tribes and thus maintain tribal identity,or an unconscious consequence of some aspect of the method of construction that favours onesymmetry type over another. In the case of Islamic geometric patterns, the underlying principles of the style include linearity of motifs (stars and polygons), division into equal parts (regular polygons,repetition), symmetry, and complexity. These properties alone are sufficient to give Islamic art a

10


11/13

P

(a) (b)

(c) (d)

Figure 10: A design with 9-pointed and 11-pointed stars from panel 42 of the Topkapı Scroll.

11


12/13

© 2008

P. R. Cromwell

Figure 11: An original design with symmetry type p3m1 containing regular 6-, 9-, 12- and 18-pointed stars.

12


13/13

distinctive quality that sets it apart from that of other cultures. The lack of patterns with 3-fold symmetry may be due to something as simple as a preference for rectangular templates. If using triangular templates can be accepted as a stylistically appropriate extension of the traditionaltechniques, it opens the way for the creation of many new patterns.

References

[1] 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.

[2] J. Bonner, ‘Three traditions of self-similarity in fourteenth and fifteenth century Islamic geo-metric ornament’, Proc. ISAMA/Bridges: Mathematical Connections in Art, Music and Sci-ence , (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1–12.

[3] J. Bonner, Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Derivation, unpublished manuscript.

[4] P. R. Cromwell, ‘The search for quasi-periodicity in Islamic 5-fold ornament’, Math. Intelli-gencer 31 no 1 (2009) 36–56.

[5] P. R. Cromwell, ‘Islamic geometric designs from the Topkapı Scroll II: a modular design sys-tem’, J. Math. and the Arts 4 (2010) 119–136.

[6] B. Grünbaum and G. C. Shephard, ‘The 81 types of isohedral tilings of the plane’, Math. Proc.Cambridge Philos. Soc. 82 (1977) 177–196.

[7] E. H. Hankin, The Drawing of Geometric Patterns in Saracenic Art, Memoirs of the Archae-ological Society of India 15, Government of India, Calcutta, 1925.

[8] E. H. Hankin, ‘Some difficult Saracenic designs II’, Math. Gazette 18 (1934) 165–168.

[9] O. Jones, The Grammar of Ornament, Messrs Day and Son, London, 1856. Reprinted byStudio Editions, London, 1986.

[10] C. S. Kaplan, Computer Graphics and Geometric Ornamental Design, Ph.D. thesis, Univ.Washington, 2002.

[11] C. S. Kaplan, ‘Islamic star patterns from polygons in contact’, Graphics Interface 2005 , ACMInternational Conference Proceeding Series 112, 2005, pp. 177–186.

[12] C. S. Kaplan, taprats , computer-generated Islamic star patterns,http://www.cgl.uwaterloo.ca/~csk/washington/taprats/

[13] G. Necipoğlu, The Topkapı Scroll: Geometry and Ornament in Islamic Architecture , GettyCenter Publication, Santa Monica, 1995.

13

Documents

Islamic Geometric Designs from the Topkapı Scroll I: Unusual Arrangements of Stars