13377656 Geometric and Group Theoretic Methods for Computer

8/2/2019 13377656 Geometric and Group Theoretic Methods for Computer

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Volume 11, (1992) number1 43-53

Geometric and Group-theoretic Methods for ComputerGraphic Studies of Islamic Symmetric Patterns

S. J. Abas and A. Salman

School ofMathematics, University College of North Wales, Bangor, Gwynedd, LL57 1UT,U.K.

AbstractOver the last several years the authors have used computer graphics to generate, study and analyze more than300Islamic geometrical repeat patterns. These patterns offer a rich source for exploitation by artists and arealso of interest to mathematicians, crystallographers, architects, archaeologists and others. They can serve aselegant test beds fo r research into hierarchical programming and texture mapping. The paper discusses theevolution of classical geometric methods for Islamic patterns and goes on to develop algorithms basedon grouptheory for efficient generation of all crystallographic repeat patterns using modern computer graphics.

1. Introduction

Visually appealing symmetric patterns and forms occur

everywhere and on every scale. They occur in elementaryparticles and in great swirling galaxies. They occur incrystals, flowers, bubbles, fish, birds, tigers, logos, flags,cathedrals and in myriads of other objects -natural aswell as those produced by humans.

Furthermore, the concepts of pattern and symmetryare not restricted to visually appealing forms. Pattern canrefer to any regularity that can be recognized by the brainand more generally, symmetry is defined mathematicallyin terms of invariance of properties of sets undertransformations1. With these generalizations the sets thatmight possess symmetric patterns are not confined tothose whose elements are geometrical entities. They may

be abstract in extreme. When the sets are restricted tothose whose elements are geometrical objects and thetransformations are restricted to isometries then werecover the familiar case which appeals a t the visual eye-catching level.

When generalized from its visual origins, symmetrybecomes a vast and unifying subject. It is seen to besignificant, not only in such areas as art, design, archi-tecture and archaeology, but also in many of the sciences.It becomes of importance in mathematics, theoreticalphysics, crystallography, chemistry, biology and manyothers. Indeed, symmetry turns out to be the very cornerstone of modern scientific thought2.

Studies of symmetric patterns, with which this paper isconcerned, can therefore have significances which go wellbeyond their immediate appeal. They provide a visual

lead into powerful abstract notions of pattern andsymmetry. The interested reader is referred to thepioneering treatments by Hermann Wey1and Shubnikov

and Koptsik3 and the more recent collection of papersgiven in reference4 as sources which discuss an enormous

range of applications of symmetry.

2. Islamic Geometrical Patterns

Unlike the arts of other cultures, Islamic art, underreligious injunction, set out deliberately to shun anthro-pomorphic forms. It was led into geometrical explorationof symmetry, an enterprise which resulted in an extra -

ordinarily large, complex and elegant collection ofperiodic patterns5, 6, 7, 8, 9.

Colour was another dimension which was system-atically and brilliantly explored by Islamic artists. Toquote Owen Jones5:

When we examine the system of colouringadopted by the Moors, we find that as withform, so with colour, they followed certain fixedprinciples, founded on observations of natureslaws.. . .

Most readers would be surprised to learn that it was torecord and display the colours of Islamic architecture


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44 S.J. Abas and A. Salman / Islamic Symmetric Patternsthat colour lithography was first employed in bookprinting in Britain. Owen Jonesstreatise on the Arab

palace of Alhambra in Granada, published in stagesduring 1842-46 was the first colour-printed book to beproduced in Britain10.

For these reasons, the geometric designs of Islamic arthave a universal and timeless significance which goesbeyond their original decorative and religious purposes.Modern computer graphics offers a powerful tool for thestudy of these designs and their further development.

In this paper, we shall first comment on the evolutionof geometric methods for Islamic patterns. This offersinsight into how simple geometric methods may beutilized with computer graphics to generate classicalIslamic patterns and also new patterns based on the same

methods. Next we shall develop algorithms which rely oninsights provided by group theory for their analysis andgeneration.

3.How did the sophisticated Islamic geometrical patternswhich are to be found on monuments dating from tenthcentury onwards evolve? Clearly, they did not evolvespontaneously.

The artists, artisans, architects and designers whocreated and perpetuated Islamic patterns and designswere secretive. They disclosed their methods only to a

chosen few. The long established tradition where themaster reveals his jealously guarded notebooks only to afew devotee apprentices is still very much the modeemployed in Islamic cultures of today11.

Although recent researches by Chorbachi12 have un-earthed a few documents in a few libraries and museums,no comprehensive treatise on the subject has come downfrom the past. Relatively recently, starting with thepublication of the pioneering work of Bourgoin6 in 1879,several authors7, 8, 9have published large collections ofIslamic patterns and offered their own analysis on themethods of constructions. The methods offered, however,are often unnecessarily elaborate and offer no expla -

nation as to how the patterns have evolved. The overallimpression that is created is that from the earliest of timesthe inventors of Islamic patterns where dedicated geo-meters inspired by theoretical compass/ruler based con-structions of the classical Greek geometry. No thought orcredit has been given to the practical experience oftilingwith real shapes.

If one asks the question as to how the Islamic patterns,or indeed patterns of any culture, originated, then itwould seem most logical to start with the practicalexperience of tiling and covering with simple naturallyoccurring shapes. The shapes would then be worked ongiving rise to triangles, rectangles, squares, hexagons and

Origins ofIslamic Geometrical Patterns

circles. The shapes would have been decorated withsimple colours and patterns. From this beginning, the

next stage, would be to experiment with multiple-shapedtiles, with shapes produced by overlapping tiles, and toinvent more pleasing decorations. As we shall showshortly, an enormous stock of patterns can be producedvery simply in this practical way without having to rely onelaborate geometrical constructions.

It is this practical experience rather than an ab initioinvolvement with compass/ruler based constructions,which seems to us to be the more satisfying explanation ofthe origins of Islamic patterns. It is this initial experiencewith tiling which would lead ultimately to sophisticatedemployment of geometrical ingenuity, such as the use ofcomplex hidden grids. We shall illustrate the full evol-

utionary range through a series of examples.

3.1. Patterns Based on Single-shaped Tiles

Many popular patterns to be found in Islamic culture,some of which pre-date Islam, can be made quite easily byusing just one simple tile. Figs. la and 1c show two suchexamples. Through varying the orientation of neigh-bouring pieces, or through removing pieces, relateddesigns emerge. For example, the design with star-shapedholes shown in Fig. 1b can be thought of as arising froma different placing of hexagonal tiles compared to thatused in Fig. l a or can be obtained by removing everyother tile from alternate rows in Fig. la . This immediately

suggests a tiling with two tiles, one hexagonal and onestar-shaped. Even, for such simple and obvious cases,some previous authors on Islamic designs have chosen toapproach them through grids and elaborate Greekinspired compass/ruler based constructions, see for

example Bowgoin6, p. 1, Wade9, (p. 28), El-Said and

Figure 1


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S.J. Abas and A. Salman / Islamic Symmetric Patterns 45

Parman8, (p. 11). Let us emphasize, that here we arereferring to the specialist writers on Islamic geometric

patterns. The excellent recent treatise on Tilings andPatterns by Grnbaum and Shephard26 does show atiling approach.

Many other more complicated patterns can be derived

relies only on an overlapping placement of S2. Themethod suggested, by El-Said and Parman is reproduced

in Fig. 4a.

3.3. Patterns from Geometrical Constructions on

Familiar Shapes

equally simply by using other shapes and using multipletile shapes, e.g. hexagonal with rectangular, and throughplacing simple linear decorations on tiles. For a numberof interesting examples of patterns produced by simplevariations in a single shape, the reader is referred toreference13.

3.2. Patterns from Overlapping Tiles

One would expect that the experience of storing andstacking simple tiles would naturally lead to someexperimentation with tiles that overlap. Many interestingtile shapes and patterns could be discovered in this way.Fig. 2 shows examples of these.

The top row shows how a square tile when rotated by45 degrees and placed on top of an identical tile gives rise

to an eight-pointed star shape. This is the most familiarshape in Islamic tiling. Let us refer to it as S1. When S1is placed as shown in the right diagram of the top row, weobtain the most frequently occurring star-cross pattern inIslamic decoration. Again, compare this explanation ofthe pattern with that of El-Said and Parman8, (p. 13).

By joining the vertices in S1, we obtain an octagonaltile S2. The bottom right diagram Fig. 2b, shows apattern which occurs frequently in Islamic carvings. It

The examples of patterns given in the last two sectionsrequire no knowledge of geometry whatsoever. Theycould be discovered purely through practical experiencewith tiling with simple shapes. It is only in the next stageofevolution that some geometrical construction is addedto familiar shapes. We shall illustrate this stage with twotypical examples.

Fig. 3a shows a shape obtained by an overlapping

placement of eight squares. Fig. 3b shows the patternproduced when we try to tile with the shape in Fig. 3a.This gives rise to octagonal holes, which can be filled invery obvious ways by extending the sides of the squareswhich form the vertices of the octagon. These extensionsare shown in Fig. 3c. With this addition we obtain thepattern in Fig. 3d. For comparison, Figs. 4b and 4c showthe construction methods for this pattern proposed by El-Said and Parman, and Bourgoin respectively. Bourgoinsconstruction is very difficult to comprehend and is quitetypical.

Figure 2 Figure 3


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46 S.J. Abas and A. Salman / Islamic Symmetric Patterns

Figure 4

As we remarked earlier, the eight-pointed star shape S1of Fig. 2 is a favoured characteristic shape in Islamic art.A class of patterns of Islamic art can be made by startingwith S1 and making symmetric geometrical extensionswhich give rise to a rectangle, a square or a hexagon. Theextended shape can then be used as a repeat pattern toform a tiling. The next example illustrates the procedure.

Fig. 5 shows a characteristic Islamic pattern. It can beobtained in the following way:

1. Start with the familiar eight-pointed star shape S1.

2 . Extend the sides to produce a bigger and sharper eight-pointed star.

3. Draw perpendiculars on the diagonals to obtain yetanother eight-pointed star made from two squares.

4.Draw a circle circumscribing the s tar.

5. Extend the sides of the star obtained in step 2 tointersect the sides of the squares produced in step 3 and

join the points of intersection to meet symmetrically onthe circle.

6. Draw a square circumscribing the circle and extend thelines obtained in step 5 to intersect the sides of the square.

Figure 5

7 . If we now discard the squares and the circle producedin steps 3 and 4 we obtain what we shall call the unit motif.

8. By filling and tiling with the unit motif we obtain thepattern shown at the bottom of Fig. 5.

3.4. Patterns on Concealed Grids

The most sophisticated patterns of Islamic art which datefrom later part of thirteenth century represent the laststage in evolution. They make use of concealed grids.These do require considerable geometrical ingenuity. Thegeneral procedure for producing such patterns is asfollows :

A grid of some sort is drawn. On the grid are placedpolygons and/or circles ofvarious sizes in some regularfashion. The circumferences of the figures are also dividedin some regular fashion and marked with points. Thepoints are joined together, again in some regular sym-metric manner. At this stage the pattern emerges and theconstruction lines are removed. Finally the lines in thepattern are often replaced with interlacing lines toenhance the effect.

The pattern in Fig. 6, which can be obtained asexplained below, is intended to illustrate the procedure

just described. The algorithm for the pattern given here isa simplified version of that published in14.


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Figure 6

1 . Draw a grid of heptagons as shown in Fig. 6a. Thisgives rise to a series of small squares s1.

2. Refer to Fig. 6a again. Centred on each of thesquaress1draw a circle c1 circumscribing s1and a squares2 of side d, where dis the smallest distance between twoof the nodes on the grid. From the nodes surrounding thesquares draw lines l1, l2 to the vertices of the heptagonwhich lie on the edge forming the square s1.

3. Refer to Fig. 6b. Use c1 to draw the octagon shownthen discard c1. Use l1, l2to cut offs2. Replace the twolines with the circle c2. From a point on the circumferenceofc1 draw a line l3to a vertex ofs1. The intersection ofl3with the line l4 joining two of the nodes defines theradius of the circle c3.

4. Fig. 6c shows how the pattern emerges by sym-metrically performing the above steps in the regionsurrounding one of the squares s1.

5. Fig. 6d shows the same when the operations areperformed on a larger region of the grid.

6 . Fig. 6e shows how an interlaced pattern is obtainedby replacing single lines with double ones in Fig. 6d. Suchinterlaced designs are a characteristic feature of Islamicgeometrical designs.

Our purpose in this section has been to give an insightinto the evolution of Islamic geometrical patterns. The

Figure 7

same steps, when coupled with modern computer gra-phics and CAD packages may be used much morepowerfully to explore and generate new patterns anddesigns. Fig. 7 shows an example produced using the

same technique as that employed by Islamic artists anddescribed in this section. More examples can be found inreference18.

4. Group Theoretic Approach

Symmetric patterns contain varying degrees of redundantinformation. Group theory shows that in two dimensions,symmetric periodic patterns can be analyzed into sev-enteen different types and provides the insight needed toidentify the minimum amount of information needed togenerate a particular symmetry type. It was in the earlypart of this century that Polya15and Speiser16 instigated


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48 S.J. Abas and A. Salman / Islamic Symmetric Patternsgroup theory based analysis of decorations and designs ofvarious cultures and Speiser drew particular attention to

Islamic art. In 1944 Edith Muller17

carried out amathematical symmetry analysis of designs in the Palaceof Alhambra, Spain. More recently, one of the authors(AS) has carried out a comprehensive study of this typeon Islamic geometric patterns18.

Using group theoretic insight, we shall now develop

algorithms which may be used with computer graphics togenerate two-dimensional symmetric periodic patterns ina highly efficient way. These algorithms, which were firstdiscussed at the Fourth International Conference onComputer Graphics held in 1990 at Dubrovnik, Yug-oslavia19, have been used by us to study more than 300Islamic patterns.

To help the reader understand the motivation behindthe symbolism to be introduced in section 4.1, we shallfirst explain the idea in simple non-mathematical terms.

Using group theoretic analysis, see for example20, wecategorise any two-dimensional periodic pattern into oneof17 types. These 17 types are known asperiodic groups,wallpaper groups orplane crystallographicgroups. Each ofthese types is now denoted by an internationally agreedsymbol.

The pattern in Fig. 8 is of the type denoted as p4mm.Once we recognize the group type we can pick out what is

the minimum essential information20

contained in apattern to be able to generate that pattern. We shall referto this portion as a template motif. A method to createthe pattern in Fig. 8 starting with the template motif isillustrated in Fig. 8a. A point to note is that the algorithmis not unique. An alternative one is illustrated in Fig.8b .

Basically, fixing the symmetry group determines a setof transformations. If we supply a template motif as datathen these transformations act to create the unit motif.The unit motif is then copied on a suitable net to producethe whole pattern.

4.1. Basic Mathematical Notions

Let us now outline the basic mathematical notionsrelevant to the subject under discussion. For a detaileddiscussion the reader may refer to21.

Two-dimensional Nets

Given two non-parallel vectors a two-dimen-sional net is the set of pointsThe vectors and v generate the net. A point ofN iscalled a node.

Figure 8

Transformations

A transformation T o n a set a is an action which changesthe initial state ofa to an image state o'. The interest liesin two kinds of sets. First, those whose elements aregeometrical entities, e.g. points, lines, polygons etc. Astate of such a set can be fixed by specifying the positionsand orientations and possibly other attributes, e.g.colours, styles, fill patterns etc. of the elements. The othertypes of sets that are of interest are those whose elementsthemselves are transformations.

We shall denote the action of the transformation bywriting If is another transformation then by

we shall mean

IsometryAn isometry A is a transformation which preservesdistances, i.e. if are any two points, then the distancebetween is equal to the distance between the images

and

Every isometry is one of four types : (i) a rotation abouta point called the centre of rotation (ii) a translation insome direction (iii) a reflection in a line (iv) a glidereflection which is a reflection in a line lcombined with atranslation parallel to 1. Isometries of type (i) and (ii) arecalled direct. In these cases if PQR form the vertices of atriangle named in a clockwise direction, then the same is


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S.J.Abas and A. Salman /Islamic Symmetric Patterns 49

true of the image triangle (AP)(AQ)(AR). In the cases (iii)and (iv) the corresponding vertices in the image triangleare traversed in an anti clockwise direction and these arecalled indirect isometries.

Symmetry

A symmetry is an isometry transformation which pro-duces an image state which is indistinguishable from theinitial state. If Xis a symmetry ofa then Apartfrom the trivial identity transformation which doesnothing, a symmetry transformation produces a per-mutation of the elements of

Symmetry Group

The symmetry groupofall the symmetries of The elements ofgroup, i.e. they satisfy the following :

of a set a is the set that consistsform a

(i) given any two elementsA , Einis in

their product AB

(ii) given any three elements A,B,C in A(BC)

(iii) there is a special element Iin called the identityelement such that for every elementA in E,.

(iv) given any element A in there exists an elementin called the inverse of A , such that

= (AB)C.

The orderof the symmetry group denoted bythe number of elements in

is

has symmetry, or is symmetric, if It isasymmetric if = 1,i.e.ifthe symmetry group containsonly the identity element 1.

has a greater degree of symmetry than if

4.2. Algorithms for Plane Crystallographic Groups

These groups have been discussed from mathematicalrather than algorithmic point of view. McGregor and

Watt22 used computer graphics to produce all the 17types, but they have not developed any formalism andtheir treatment is specialized. The algorithms given below

have the virtue that they can be generalized to treat highlycomplex structures involving many types of graphicalprimitives, to more than two-dimensions and to coloursymmetry.

Net types

Vectors v generate five different types of nets cat-u,egorized by their symmetry groups. These are: (i) aparallelogram net (ii) a rectangular net (iii) a

Figure 9

rhombic or a centred rectangular net (iv) a square netand (v) a hexagonal net which arises when the

angle between u, v is

For the purposes of writing down the algorithms weshall use the nota tion and the dimensions shown in Fig. 9.Note that the position vector of the point c in every caseis (u+v)/2, i.e. the centre of the cell, except in the case of

when it is (u+v)/3.

Action Set

By the action set we shall mean the following set ofisometries on Unless otherwise stated, bold subscripts

denote points.the identity.

a rotation by degrees anticlockwise around

the point p.

a translation (or a shift) by the vector r.

a reflection in the line passing through the

points p, q.

represents a half-turn around p i.e.

reflection in the line 1.

a glide reflection involving a translation by the

distance pq followed by a reflection in the linejoining the points p and q.

In writing down these symbols we shall also use twocoordinates to give a point when convenient. This shouldbe obvious. For example, will mean reflection inthe line joining (0,0) to (1,1) and will meanreflection in the line joining (0,0) to the point p.

Expressions

to formexpressions. These expressions are to be interpreted in thefollowing way.

We shall need to combine the elements of


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50 S.J. Abas and A. Salman / Islamic Symmetric Patterns

Let and We have already definedAp as the result of applying the transformation A top and

(AB)p to mean A(Bp). By (A +B)p we shall mean Ap

The expression depends on the group type of thepattern and is supplied in the table below. As we said

earlier, E is not unique and in general it is possible toBp. In expressions involving additions and products ofthe elements of the distributive and associative lawsapply. Thus A(B+ C)p =ABp+ ACp and we can write

A(B+C) =AB+AC. Similarly (A+B)(C+D) =AC+AD+BC+BD and A(B+C)D =ABDfACD.

At the start of this section we explained how a templatemotif when operated on by set of transformations whichdepend on the symmetry group of a pattern give rise tothe unit motif. In general, however, the template motifmay itself have symmetries and may be obtainable fromanother set of transformations which will be dependenton the pattern and not on the group type. To cater for

this we introduce the next two definitions.

write down several equivalent forms. The reader mayrefer to the symmetry diagrams in reference20 for eluci-dation.

Motif Element

A motif element is an ordered pair (E,m) where E is anexpression involving elements from the action set andm is a geometrical entity, e.g. a polyline.

Motif Set

A motifset isa set ofmotif elements

Motif

A motif is the set of geometric elementscontained in a motif set.

Template Motif

where is being used here to denote a union of sets.

Unit Motif

Given a template motif M, a unit motif is the setcreated from the action of an expression E on M, i.e. M

Periodic Pattern

A periodic pattern is created when a unitmotifM is copied on all the nodes of the net N, i.e.

or

Figure 10


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Figure 11

(a )

51S.J. Abas and A . Salman /Islamic Symmetric Patterns

(b )

(d)(c)


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52

5. Conclusion References

In this paper we have offered a logical framework for the H. Weyl, Symmetry, Princeton University Press,understanding of the origins and evolution of Islamic Princeton, New Jersey (1952). Paperback reprint,geometrical patterns. We have also given computer (1982).graphics algorithms for studies of Islamic geometric

J. Rosen, Symmetry at the Foundation of Science,patterns. The distribution of various symmetry groups in

Computers and Mathematics, vol 17, number 1-3,(1989).

Islamic art, found by us18 is plotted in Fig. 10.

The distribution in Fig. 10 is significant because of theA.V. Shubnikov, V.A. Koptsik, Symmetry in Science

recent interesting discovery that the symmetry responseand Art, Nauka, Moscow, (1972), Plenum Press,

of an individual seems to be strongly influenced by hisenvironment. Dorothy Washburn, an archaeologist at New York, (1974).

the University of Rochester, and Donald Crowe, a I. Hargettai, Symmetry I : Unifying Human Under-mathematician at the UniversityofWisconsin in the USA standing, Pergamon, Oxford, (1986) and Symmetryhave carried out extensive studies of symmetry in patterns 2, (1989).and designs produced in various cultures. Their recent

O. Jones,The Grammar of Ornament, Day and Son,bookSymmetries of Culture26, which has received much London (1856), recent reprint Studio Editions,publicity, announced their remarkable finding that the

choice of arrangements of motifs in cultures to produceLondon (1988).

symmetric designs is not at all random. They discovered J. Bourgoin,Arabic Geometrical Pattern and Design,that only certain symmetry types are preferred and Firmin-Didot, Paris (1879), Dover, new Yorkintuitively recognized as being right by each culture: (1973).

...the designs in any given culture are organized K. Critchlow, Islamic PatternsAn Analytical andby just a few symmetries rather than all classes Cosmological Approach, Thames and Hudson,of the plane pattern symmetries. London (1976).

I. El-Said, A. Parman, Geometrical Concepts in Islamic Art, World of Islam Festival Publ. Co.,London (1976).

D. Wade, Pattern in Islamic Art, Cassell & CollierMacmillan, London (1976).

10. C. Aslet, Art is Here: The Islamic Perspective,Leighton House, Country Life, vol 16, 1642-1643,(1983).

11. A. Paccard, Traditional Islamic Craft in MoroccanArchitecture, Editors Ateliers 74,74410 Saint-Jorioz,France, vol 1 and 2, (1980).

12. W.K. Chorbachi, In the Tower of Babel: BeyondSymmetry in Islamic Design, Computer Math.Applic, vol 17B, 751-789 (1989).

13 . B. Grnbaum, Z. Grnbaum, G.C. Shephard, Sym-metry in Moorish and Other Ornaments, Comp. &Math. With appls. vol 12B, 641-653 (1986), also tobe found in reference4 (1986).

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S.J. Abas and A. Salman / Islamic Symmetric Patterns

1 .

2.

3.

4.

5.

6.

7.

The discovery of Washburn and Crowe25 adds newsignificance to the study of symmetry type of each culture.Symmetry analysis of designs is now being adopted by

many archaeologists and art historians as a more ob-jective method of analyzing cultural style than methodsthat have been used in the past. Fig. 10 therefore sayssomething characteristic about Islamic culture.

Traditional as well as computer graphics based studiesof Islamic patterns can provide a rich source for artists,designers as well as scientists. They can also be used aseducational material to introduce geometry28, computergraphics and CAD in a very enjoyable way. A majorvirtue of symmetric periodic patterns is that they makeconcrete and visible the abstractions of group theory24

and can lead very pleasurably to the heart ofmodernabstract mathematics and to generalized notions of

symmetry which have wide application.

6. Acknowledgments

One of the authors (SJA) wishes to thank Mr T. Hewitt,Director of the Manchester University Computer Gra -phics Unit for allowing access to the facilities of the

8.

9.

Laboratory during the period when this paper waswritten. The other author (AS) wishes to thank Mr J.S.

318-319 (1936).

15. G. Polys, berdie Analogie der Kristallsymmetrie inSalman and Dr M.S. Salman for their finance and supportduring the work of this paper.

der Ebene, Zeitschrift fur Kristallographie, 60,

16. A. Speiser, Die Theorie der Gruppen von endlicherOrdnung,Second Edn. Springer, Berlin (1927); Third

278-282 (1924).


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22. J. McGregor, A. Watt, The Art of MicrocomputerGraphics, Addison-Wesley, Wokingham (1984).

University of Zurich) Baublatt, Ruschlikon, (1944).(b)El Estudio de Ornamentos como Applicacion de laTeoria de los Grupos de Orden Finito. Euclides 23. J. Niman, J. Norman,Mathematics and Islamic Ar t,(Madrid) 6 (1946) 42-52. Amer. Math. Monthly vol 85, 489-490 (1978).

18. A. Salman, Computer Graphics Studies of Islamic 24. E. Makovicky, M. Makovicky, Arabic GeometricalGeometrical Patterns and Designs. Ph.D. Thesis Patterns - a Treasury for Crystallographic Teaching,University of Wales, Cardiff(1991). Jahrbook fur Mineralogie Monatshefte, 2, 58-68

(1977).19. S.J. Abas, Computer Graphics Studies of Islamic

Geometrical Patterns, Proceeding of the Fourth 25. D. Washburn, D. Crowe, Symmetries of Culture,

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20. D. Schattschneider, The Plane Symmetry Groups: Freeman, New York (1987).

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13377656 Geometric and Group Theoretic Methods for Computer