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Islamic Geometry In Architecture

www.elsevier.co

Available online at www.sciencedirect.com

RESEARCH ARTICLE

of design proce

Loai M. Dabbourn

Department of Architectural Engi

Received 24 June 2012; received in

KEYWORDSGeometry;Proportions;

time they mediate between the two poles of order and diversity as well as among different

relationship between cosmology and geometry is manifested in Islamic patterns where

universal language, constituting one of the most important multicultural symbols in design.V. All

relations of magnitudes such as points, lines, surfaces, orPeer review under responsibility of Southeast University.

Frontiers of Architectural Research (2012) 1, 3803912095-2635 & 2012. Higher Education Press Limited Company. Production and hosting by Elsevier B.V. All rights reserved.solids in space and the way the parts of a particular objectt together (Concise Oxford English Dictionary, 1999).Geometry is well known based on two Greek words geo,meaning the earth, and metry, meaning to measure.

http://dx.doi.org/10.1016/j.foar.2012.08.0051. Introduction

The word geometry refers to the science of properties and

nTel.: +962 796978940.

E-mail addresses: [email protected],[email protected]& 2012. Higher Education Press Limited Company. Production and hosting by Elsevier B.rights reserved.geometry acts as the vocabulary underpinning the pattern language. These ndings supportthe argument that geometry was independently discovered and applied by Islamic culture as aesthetic levels. This paper analyzes geometric principles and design features of traditionalIslamic patterns, involving geometric proportion analysis. It also describes the syntheses ofIslamic patterns from the perspectives of Islamic cosmology, philosophy and metaphysics ofgeometry, which can be seen as following a pre-established principle of geometric andproportional design with high accuracy and great precision. Furthermore, the underlyingIslamic patternsss for Islamic geometric patterns

neering and Building Systems, Alzaytoonah University of Jordan, Amman 11733, Jordan

revised form 23 August 2012; accepted 24 August 2012

AbstractThis paper discusses geometric proportions which serve as a design tool, especially for Islamicpattern design. The key role of geometry in Islamic patterns is discussed with relation toproportions which act as the underlying structure of the design process. Evaluations based onspatial structure and esthetic criteria are integrated into the conceptual phase of the designprocess. Proportions serve two objectives; rstly, in pattern design, proportions are stronglylinked to geometry, which can be viewed as a self-guiding method of esthetically proven design.Secondly, geometric proportions regulate primarily the order of patterns; while at the sameGeometric proportions: The underlying structurem/locate/foar

So geometry literally means the measurable earth orearthly measurements. Geometry, according to Plato,can also refer to the science of stereo-metry or spatialmeasurements. The very laws of how things are allowed tobe by the Divine will are the laws of Harmony of Being.Tradition holds that, as engraved over the entrance toPlatos Academy, Only he who is familiar with geometryshall be admitted here (Critchlow, 1976).

Geometry has existed in many buildings and design formsacross centuries. The rst conrmed record of geometricknowledge and its relationship to astronomy, human and

Geometric proportions 381music can be traced back to the ancient Greeks, inparticular, Pythagoras and Plato. However, prior to theGreeks, many old civilizations have left clear geometricfootprints by applying sacred geometry in their construc-tions. The prehistoric carved stones found in Scotlandrepresenting the Platonic Solids.1 can be dated back toover a thousand years earlier than the Greeks.2 The Greeksmay well have been the rst to have offered geometry tothe public, but they were, by no means, the rst to realizeit (Critchlow, 1987; Hecht, 1988; Critchlow, 1987).

Geometric ratios and proportions existed and wereemployed in the design of ancient sacred and ceremonialsites in all traditional civilizations. They were invariably builtwith dimensions that incorporate mathematical numbers,constants and ratios such as the golden/sacred mean,and the use of geometries based on proportional roots,proportional rectangles and Pythagorean triangle (Pennick,1994). For example, Christian cathedrals incorporated sym-bolic geometry, while the design of Hindu and Buddha templeswas based on sacred geometry. In Islamic art, geometricelements have been employed since its origin and were usedto create unique geometric formations, serving as the under-lying structure of Islamic design process.

Geometry or proportional geometry is a sacred art formdue to its fundamental association with the Creationsprincipal laws. The visual expression of the order of theselaws is best represented through the discipline of geometry.Geometry is both quantitative and qualitative in nature. Itsquantitative dimension regulates the order and constructionof design forms. Its qualitative nature sets the proportionsof design forms and represents an expression of the order ofthe universe as a visual representation of the truth. Eachgure or geometric shape, when seen from the perspectiveof its symbolic meaning, represents an echo of unity and areection of the values and principles within the largerframe beyond that unity (universal unity) (Ardalan andBakhtiar, 2000). Seyyid Hossein Nasr argued that geometryand rhythm manifest a doctrine of unity which is central toIslam, upon which Islamic art developed based on

1The Platonic solids are the names given to the ve three-dimensional solidstetrahedron , hexahedron, octahedron, dode-cahedron, and icosahedrons(perfect shapes) formed when divid-ing a sphere based on the exact shape and angle. For the Greeks,these ve solids were the core patterns of physical creation. Theywere seen as the archetypal patterns behind the four elements(earth, air, re, and water), while the fth was held to be thepattern behind the life force itself, the ether.

2Small carved stones found in Scotland conform to the vePlatonic solids, which suggest an association with this mathematicalprinciple.mathematical ratios and proportions which represent thevery heart of Islam (Critchlow, 1976).

Geometry is the blueprint of the Creation and thegenerator of all forms. It is a science that deals withnumbers in space on four basic levels: the rst one isarithmetic (pure numbers), i.e., any measurement orproportion is a geometrical measurement. The second levelis numbers in space which represent proportional geometry.They reect the meanings and Ideas. The third levelis numbers in time, which is the foundation of music. Thefourth level is numbers in space and time which representthe cosmology of the universe. Geometry explores andexplains the patterns that unify and reveal the structureof the Creation within all natural patterns of growthor movement and their conformity with geometric shapes.All life forms emerge out of timeless geometric codes.Viewing and contemplating these codes allow us to under-stand the wisdom of the inner workings of the universe inorder to fully comprehend and appreciate its beauty basedon the concept of proportions. Ikhwan Al-Safa wrote, Oneof our aimsy consists of demonstrating clearly that thewhole world is composed in conformity with arithmetical,geometrical, and musical relations. There, we haveexplained in detail the reality of universal harmony(Nasr, 1978).

Geometric proportions in architectural patterns repre-sent a design language, as words do in a spoken language.They determine the frameworks within which elements maybe arranged into a pattern, a relation between one elementand another, and a proportional relation within one ele-ment. They address and reect the natural laws that governthe basic harmonies of nature, being describable by meansof mathematics and geometry.

Islamic art favors geometry because of its intelligibleaspect. Geometry possesses abstraction and the capacity toexpress and reveal objectively immutable and spiritualtruths. Geometry involves proportions and prime roots thatare considered the most beautiful proportions (the propor-tions of beauty). As such, beauty, for the Muslim artists, isobjective and self-expression of truths, which is the essen-tial nature of beauty as Plato stated, Beauty is thesplendor of the truth (Burckhardt, 1987).

The language of Islamic design springs primarily from theabstract. Therefore, geometry in general and geometricalproportion in particular carry great importance in the designprocess of Islamic art, presented in its primary elements,geometry, biomorphic forms, and calligraphy, which are allbased on and ruled by the underlying geometric laws(proportions). It follows that the Islamic design principlesare reected in geometry. In other words, geometry plays acentral role in Islamic design.

This paper examines Islamic pattern design, which can beviewed as a key method of expressing Islamic cosmologicalesthetics. In addition to representing cosmological andphilosophical structures at the level of form, geometricpatterns can also be seen as efcient and powerful repre-sentations of some of the central concepts that characterizeIslamic debates on the divine nature. Islamic pattern wasdeveloped as visual tools for contemplating the underlyingmathematical nature of the universe, leading to the natureof beauty, by virtue of Gods creation of the universe (thenature of the divine itself).

2. Geometric proportional systems

In architecture dating back to prehistory, particularly in

comprehend the order and harmony inherent in the Crea-tion. It is the visual representation of the mathematicalpatterns found everywhere in man, nature, and cosmos.These patterns, with their esthetic and philosophicalvalues, are found within all aspects of Islamic designprocess.

The understanding of geometry as an underlying part ofour existence is nothing new. In fact, the golden mean and

L.M. Dabbour382Islamic arts and architecture, the most important geometricproportional systems are: the proportions of the golden meanand the primary three proportional roots O2, O3 and O5,on which all Islamic arts and architectural forms, especiallytheir geometric pattern design, are based.

The golden mean proportion is a proportional systemwhereby two elements are related to each other by a setproportion. Two segments of a line not equal to each otherare related in a proportion: a/b=(a+b)/a (Figure 1). It isthis unique point that divides the single line into segmentswith qualitative proportions. It is a reection of multiplicitywithin unity in terms of geometry. If the line is divided intoequal halves, the two segments would be a monotonousrepetition of one and the same thing, neither multiplicitynor unity within the geometry.

The proportional rectangles or the proportional roots arebased on the geometry of the polygons.

For example, the square includes O2,3 the hexagonincludes O3, and the pentagon includes the golden meanwhich numerically equals to [(O5+1)/2]=1.61803 (Figure 2;Hambidge, 1920; El-Said, 2001)

Figure 3 shows the successive constructions of the rootproportions in proportional rectangles based on the diagonalof the square.

The Muslim artists created these geometric proportionsfrom the circle of Unity. As one of the most common shapes innature, it was reected symbolically in the signs of theCreation, just like the sun being one of the signs of divinity(the universal symbol) (Guenon, 1995; Figure 4). The circle isan obvious example of a basic geometry, constituting all theproportional geometries inherent in traditional architecture.

The circle of Unity is the most signicant form, for itcontains a circumference revolving around a xed center. It istherefore extremely important to understand that all othergeometric shapes can be determined from a circle, fromwhich come the full polygons, including series of roots andproportions. Figure 5 shows a symbolic relationship betweenthe absolute and the relative, represented by two circlesoverlapped. This construction is known as the Vesica Pisces.The realm enclosed within this overlap is, according toFrithjof Schuon, the Relatively Absolute (Schuon, 2003).The absolute represents the metaphysical and the relativerepresents illusion. The geometric proportions within theRelatively Absolute sphere indicate that it is connectedto truth. The Vesica Pisces is one of the key starting blocksfrom which sacred geometry was applied in the geometricmatrix used for the design in all religious buildings.

Within the Vesica Pisces realms, the primary proportionalroots, i.e., O2, O3, and O5, as well as the golden mean, areillustrated, all of which are found within the realm of theRelatively Absolute. When the drawing is further inter-sected by straight lines, an equilateral triangle, a hexagon,a pentagon, a square and so on, have emerged.

The sacred mean or Phi (), the divine proportion, andalso golden section or golden ratio, is one of the key-stones

3O2 proportion is that of the internationally used A papersizesA4 , the most commonly used, being 210 mm 297 mm.of sacred geometry. The mathematics of the golden ratioproportion is generated by the formula (O5+1/2) (or1:1.618).

The mathematics of the golden ratio is related to LeonardoFibonacci, the founder of the Fibonacci sequence, a nume-rical series found frequently in the natural world (Lawlor,1989). It simply follows the rule that the next number is thesum of the previous two numbers, as follows: 1, 1, 2, 3, 5, 8,13, 21, 34, 55, 89, 144 etc. If we divide any number in theFibonacci sequence by the one before it, for example 144/89,or 89/55, the answer is always close to 1.61803, which is thevalue of the golden ratio. The Fibonacci sequence, or thedescription of the golden mean features can be seen every-where in the forms of nature, music, and the arts. One of themathematical products of the sacred mean is the spiral,commonly found in nature (Rawles, 1997; Schneider, 1994;Figure 6).

Proportional geometry regulates primarily the order ofdesign process. It is needed to create the micro cosmos bywhich we acknowledge beauty in design forms. The beautyof proportions is based on the simple geometry of regularpolygons with a set of ratios of side lengths to diagonals(such as O2, O3, O4, O5 andy), which can be translatedinto the design process.

These geometric proportions are used to quantify esth-etic qualities. This process is, to a great extent, based onobjective facts. Judgments on beauty versus ugliness can bebased on systematic arguments characterized by unifyingprinciples or patterns. They can be achieved by means ofalgorithms of visual math as an objective ground which canbe founded on geometry and proportions, which are part oftheir elementary pillars. As Aristotle stated, beauty is tomaintain the just measure (Lawlor, 1989). Beauty inproportions has a key function in constituting them becauseit forms the underlying structure of order versus diversity,establishing systematic design principles.

The objective criteria of beauty revealed by means ofgeometric analysis aim to establish sets of design tools,which would increase, rather than restrain, the freedom ofdesign, and to nd a system path to a design language thatincludes beauty rather than systematically excludes it.

3. Geometry in man, nature, and cosmos

Geometry is an unseen presence in Islamic art and archi-tecture. It is the visual means by which human mind can

Figure 1 The golden mean proportion: a/b=(a+b)/A=1.61803.

) th

rtio

Geometric proportions 383Figure 2 The proportional roots: (a) the O2 proportion, (b

Square 2 3

2 3 4Figure 3 The root propoother forms of geometric proportions can be seen imbeddedin man, nature, and the universe, either represented asmicro or macro cosmos.

Since ancient times, there has been a deep interest informs that are considered to incorporate within theirintrinsic relationships mathematical and geometrical pro-portions (universal truth). It was believed that thesegeometries were derived from the basic laws of theuniverse. Kepler stated the belief that geometry is under-pinning the cosmos, which was based on Platos ideas thatGod created the universe according to a geometric plan-God geometrizes continually (Lawlor, 1989). The structureof the universe is determined by and revealed as certainmathematical and geometric constants which represent aconrmation that proportions are the underlying fabric ofnature. This can be seen in man, nature, and cosmos.Beyond this discipline, it was believed that these geome-tries were based on the Creation itself (sacred) and, byincorporating them in music, art and architecture, the fruitsfrom these elds would have a harmony of proportions and aspecial sacred characterharmony of being. By studyingor contemplating these geometric proportions, an under-standing could be obtained towards the sacred truth.

Figure 4 Unity in multiplicity and multiplicity in unity primarycircle symbolizing wholeness, completion, unity and innity.3.1. Man

Geometric proportions are one of the denitive geometricqualities of life itself. According to the Holy Quran, man haswithin himself all what is reected in the universethe bestproportions. Man is the core of Gods creatures; he possessesthe most harmonious proportions, reecting of the Divineharmony of being (Lings, 1976; Guenon, 1994). We haveindeed created man in best of formsproportions (The HolyQuran, Chapter 95, verse 4). Leonardo da Vinci illustrated themathematical proportions of human body, showing thathuman being exhibits clearly golden mean proportions in his

4

5ns based on the square.e O3 proportion, and (c) the golden mean (Phi) proportion.body based on ratios of 1.618. The Vitruvian Man drawn byLeonardo Da Vinci is based on Vitruvius, who had written thathuman proportions should have a relationship in architecture.Vitruvius believed that if human proportions could be incor-porated into buildings, they would become perfect in theirgeometry. According to Vitruvius, the distance from ngertipto ngertip should be the same as that from head to toe. Thesacred mean rules can be seen in the ratios of body partsthroughout the human body. Man reects these principles in alanguage of sacred geometry (Guenon, 1995; Figure 7).

According to Robert Lawlor, The human body contains in itsproportions all the important geometric geodesic measures andfunctionsy the proportions of ideal man are at the center of acircle of invariant cosmic relationships (Lawlor, 1982).

3.2. Nature

The study of geometric proportions has its roots in thestudy of nature and matter (Skinner, 2009). Many formsobserved in nature can be related to geometry. Forexample, the snowake crystals and honey cells constructhexagonal geometries. Radium crystal and sound vibra-tions that carry sound information are based on O2proportions, showing the octagonal geometry. Flowers in

L.M. Dabbour384nature tend to have a variety of geometries, althoughtheir variety seems to be innite. Four, ve, and sixpointed geometries seem to be more common in nature.There are many plants and other aspects of nature whichexhibit the form of spiraling geometry, which is related toFibonacci proportions based on the golden mean. Thus thegolden mean is the ngerprint of the Creation (Figure 8).

Figure 5 Roots and the golde

Figure 6 Spirals, the mathematical products of

Figure 7 Vitruvian man, with proportions ba3.3. Cosmos

The mathematical harmony of the universe can be seenfrom the proportions of the planets in our solar system. Forexample, the ratio of the sacred mean can be seen in therotations of the Venus and the Earth around the Sun in thatfor each ve years that the Earth rotates around the Sun,

n mean within the Vesica.

Fibonacci sequence and the sacred mean.

sed on the sacred mean (Critchlow, 1976).

Geometric proportions 385the Venus rotates around it eight times. The connectionbetween 5 and 8, both of which are Fibonacci numbers, isthe golden mean proportion (8/5=1.6). The result of thismotion is that the Venus draws a pentagon around thesun every eight years (Skinner, 2009; Figure 9A).

Figure 9B shows that a circle is drawn, which representsthe Venus mean orbit. A pentagon is constructed inside itand a small circle placed through the arm-crossing points.The radius of this small circle divides the radius of the largeone into golden sections and can be used to space the

Figure 8 Geometric proportioVenus orbit from the Earths orbit. It can be seen from theagreement between eightfold and vefold geometries thateight touching circles are drawn from the Venus meanorbit. In turn, the circumference circle is enclosing theseeight circles, dening the Earths mean orbit (Martineau,2001; Figure 9C).

The conclusion from these diagrams shows that, mathema-tically speaking, the ratio of the mean orbits of the Venuss tothe Earth is O2 proportion. The geometric representation ofthese orbits creates the golden mean proportion.

ns in nature (Singer, 2008).

The same proportions can be observed between the Earthand the Mercury. Two circles dened by the arm-crossingpoints of the pentagon are both the Earths and theMercurys mean orbit. Their sizes are their relative physicalbodies (Martineau, 2001; Figure 10).

Sacred geometry describes the assertions of a mathematicalorder of the intrinsic nature of the universe. Scientists see thesame geometric and mathematical patterns asarising directly from natural principles. Traditional art hasrules that apply to cosmic laws and universal principles.(Schuon, 1984)

These and other similarities are seen in sacred geometryto serve as the further proof of the cosmic signicance ofgeometric forms. Islamic architecture is created based onthe essential harmonies of nature together with varioussymbolic meanings and theories of perfect proportions. Thedesigners based their geometrical vocabulary on what theyhave seen in the nature around them, in an attempt todevelop a codied series of proportions which may improveour psychological comfort with buildings (Skinner, 2009).

It has been proposed that the understanding of sacredgeometry was extended into time and space. We may neverknow for sure whether such geometry was identied rstfrom the observation of natural formations, or alternatively,it came as a result of an intellectual quest. However,whichever it would be, it is clear that these mathematicalcodes derived from nature began to be applied in the designof many important man-made structures.

Geometry constitutes a graphic description of the uni-verse. Geometric analysis is an attempt to understand the

causes and the reality of the nature, the principles govern-ing existence, the material universe, and the perception ofphysical phenomena, as well as to analyze human behaviors.

God speaks through these forms. Virgin nature was therst book, which made the Truth visible. From the bosomof nature, man seeks to transcend nature and nature herselfcan be an aid in this process, provided man can learn tocontemplate her, not as an independent domain of realitybut as a mirror reecting a higher reality, a vast panoramaof symbols which speaks to man and has meaning for him.(Nasr, 1997)

liv

4.

ea

L.M. Dabbour386Figure 9 Proportions of the Venus and the Earths meanorbits are based on the sacred mean (Martineau, 2001).

Figure 10 Proportions of the Mercurys and the Earths mGeometric proportions are used by Muslim designers as themost valuable tool of design process to produce orderedpatterns that govern esthetic beauty in designed spaces,surfaces and objects. They present us the Principles ofNature in their own language. By applying the golden mean(Phi) and the root proportions, geometric patterns areproduced to be applied to the elds of building crafts, suchas tiles, mosaic, plaster and wood, as well as to the eld ofbook art.

By diminishing the polygon contained in the circle ofUnity toward the center until the stars, which control theshape of the pattern, appear, the Muslim artists preservethe esthetics of these proportions and grant them exquisitebeauty at a higher level than direct, material expression.

The system of geometric design starts from the circle (thebasic unit), from which the pattern starts to unfold,creating harmonious divisions of the circle in four stages:

1) The planning stage: starting by determining the propor-tional systems based on the unit pattern structure withinthe circle of Unity. The decision is based on the symbolicmeanings underlying the geometric pattern and itsrelation to micromacro cosmos.

2) The division phase: construction of the basic geometricpattern.

3) Pattern order and structure: initiating the crossing linesto create the artistic shape of the pattern on naturaljunctions formed by those lines. This gives rise to a series

n orbits are based on the sacred mean (Martineau, 2001).stunliving forms.

Geometric proportions as a tool of design:dy modelwhnoportional roots govern the proportion of our world,ich can be found in the proportions of all living andareproGeometry is the very basis of our reality. Therefore, wee in a coherent world governed by underlying laws. Theyalways manifested in our world. The golden mean and

of points that can be used in the development of patterns.This type of framework forms the driving geometry forthe relatively simple Islamic patterns with which we arefamiliar.

4) Desired pattern revealing: establishing the geometricvariations of the pattern and dening its boarder lines.It is derived from all the vital proportion systems basedon a single unit. The process can be repeated inde-nitely, presenting the same center everywhere andnowhere. It is ratio, rather than measurement, thatdetermines the relative lengths of crucial dimensions.

The innitely repeating geometrical patterns and therules of geometric construction are a reection of theunchanging laws of God. They are based on estheticproportions as seen in the rhythm of the Creation and canbe presented according to the design process in the follow-ing study models.

4.1. Fourfold to eightfold pattern

space, is a symbol of the divine throne and its applicationwithin architecture is reected in a Moroccan ceramic (Zilij)panel and in the harmonic diminishing of the eight-fold starin the Alhambra Palace, The Hall of The Abencerrajes, andthe Hall of the Two Sisters (Figure 13).

4.2. Fivefold to tenfold pattern

This part of geometric study shows a construction of thegeometric panel based on proportional geometry. Itsesthetic advantage is related to the golden section, showingone of the many ways to apply the geometry to formdifferent patterns using a variety of materials.

Figures 14 and 15 show the construction systems of a ve-to-tenfold pattern which appears on the body of Salah Al-Dinwooden Minbar (Pulpit) at Al-Aqsa mosque (Jerusalem), on awooden door from Mumluk period (Cairo) and on a frontispieceat a Mamluk Quran. Ten pointed geometry lends itself to a widevariety of design possibilities through relatively small variationsin the underlying relationships and to the creation of different

The complexity is introduced on a relatively simple basis by

in thn noatterns

age

Geometric proportions 387Figure 11 Construction stThe two simplest geometric constructions which form thebasis for many of the patterns to be found in Islamicdecorations are the constructions of four and eight pointedgeometries respectively. These relatively simple geometri-cal constructions are easy to develop by connecting variousintersections, forming the basis for a variety of patternmaking.

Figure 11 shows how this kind of geometric patterns developswhen they are added together following the basic rules createdby the selection process. By illustrating how small changes canaffect the overall pattern according to the types of pattern andhow they are achieved by simple changes, it is possible to seehow a very small differentiation can produce a dramaticallydifferent overall pattern.

Figure 12 illustrates the construction of eight pointedgeometry based on subdividing the square and O2 rectan-gles. The octagon, associated with the eight directions of

Changing intersectiodifferent pconstructio

Stage 1The planning stage

Stage 2The division phases of four pointed patterns.applying colors and doubling of the structure lines while

e selected des creates n

Stage 3Order and structure

Stage 4Pattern revealingpatterns at different scales as a unied design.

4.3. Sixfold to twelvefold pattern

Twelve pointed geometry can be constructed from sixpointed geometry. The hexagon is one of the most impor-tant forms in Islamic geometry. It is simple to construct, ithas the capability of producing, by means of repetition, anoverall coverage of a surface, it contains the importantrelationship of O3 proportions, and it bears a strongsimilarity to the circle, a symbol of the Creation in Islam.

Figure 16 is a study based on variations of the panelsof ceramic tile. Their underlying geometry is based on therelationships between the circles containing the 612 pointedgeometry, which is derived from the central six-pointed roseand a combination of a square, a lozenge and a triangle.

se

L.M. Dabbour388Stage 1The planning stage

Stage 2 The division phaemploying the technique of cutting the tiles in such a way as toimply the interweaving of the running lines.

The pattern is both logical and rhythmical, i.e., bothmathematical and melodious, which is the most signicantcharacteristics of the spirit of Islam, showing its balancingu-nity in multiplicity or multiplicity in unity within a series ofregular geometrical gures contained within a circle, or a seriesof regular polyhedral contained within a sphere (Burkhardt,1986; Burckhardt, 1987).

Figure 12 Construction stage of eight poin

Figure 13 Applications of the octagon based on eight pointed paeight will, that Day, bear the throne of thy Lord above them (TheStage 4Pattern revealing

Stage 3 Order and structureFigure 17 shows how a relatively simple pattern based on six-pointed geometry might be developed into so many potentialchoices for the designer, which accounts for the great numberof design variations that are possible.

It is surprising that how many variations can be madefrom a simple geometric shape. Figure 18 illustratesa pattern based on a study using six-fold geometry. Therst diagram illustrates the basic construction, beginningwith the development from a circle. The second one shows

ted patterns based on O2 proportions.

tterns in architecture And the angels will be on its sides, andHoly Quran, Chapter 69, verse 17).

Geometric proportions 389Stage 1The planning stage

Stage 2The division phasethe six pointed divisions. It is these selections shown in thethird diagram that create the possibility for differentpatterns to evolve. The fourth is the result of this drawingand selecting process.

Figure 14 Construction of a ve-to-tenfold pa

Figure 15 Applications of a ve-to-tenfold pat

Figure 16 A study of 6Stage 3Order and structure

Stage 4Pattern revealingThe Ikhwan Al-Safa (the Brothers of Purity) stressed theimportant symbolism of number six, regarding number 6 asa perfect number. The number 6 consists of 1 2 3; at thesame time also 1+2+3. Any number with this quality is

ttern based on the golden mean proportion.

tern based on the golden mean proportion.

12 pointed patterns.

L.M. Dabbour390considered a perfect number. Also, it was on the sixth dayof the Creation that God created man. There are six directionsof space as well. The six-sided hexagon is therefore a symbol ofthe universal man (Ardalan and Bakhtiar, 2000).

Figure 18 shows the signicant study on tile patterns andtheir underlying geometry, particularly the visual strengthof the geometric structure in the application of thesepatterns. The construction of such natural geometries formsthe basis upon which artisans on building sites set out theirdesigns and work.

Figure 17 Potential variations of 612 pointed patterns based on

Figure 18 Analysis stages: example of s

Figure 19 Applications ofFigure 19 shows the applications of 612 pointed patternsbased on O3 proportions in a wooden panel and a paperbased geometric stars (medallion).

These studies illustrate only a small number of numerouspossibilities out of the variations of proportional patternsbased on simple polygon geometries. The ways of alteringthe relationship between the elements of these geometriesappear to be relatively simple. However, every decisionresults in diversied complex patterns that are quite diff-erent from each other.

the same structure as in Figure 16 based on O3 proportions.

tep by step unfolding of the pattern.

612 pointed patterns

Islamic patterns, by their very nature, do not have anydirection underlain in their geometry. Rather, they tend tobe formed in elds with no apparent ending. The framing ofpatterns tends to be controlled by proportion systems. Inother words, it is an objectively intellectual qualitativegeometry, reecting Truth in space. Ibn Khaldun states:Geometry enlightens the intellect and sets ones mindright. All its proofs are very clear and orderly. It is hardly

Burckhardt, Titus, 1987. Mirror of the Intellect. State University ofNew York Press, New York, USA, p. 219,227.

Burkhardt, Titus, 1986. Sacred Art in East and West: Its Principlesand Methods. Perennial Books, Indiana, USA, p. 101.

Concise Oxford English Dictionary, 1999. Oxford University PressInc., USA.

Critchlow, Keith, 1976. Islamic Patterns: An Analytical and Cosmo-logical Approach. Schocken Books, Thames and Hudson, London,UK, p. 4,9.

Critchlow, Keith, 1987. Order in Space: A Design Source Book.Thames and Hudson, USA, p. 9.

El-Said, Isasam, 2001. Islamic Art and Architecture: The Systems of

Guenon, Rene, 1995. The Reign of Quantity and the Sign of theTimes. Sophia Perennis, Ghent, p. 8,170.

Guenon, Rene, 1994. The Great Triad. South Asia Books, New York,

Geometric proportions 3915. Conclusion

We can conclude that proportion is a characteristic of (tradi-tional) Islamic patterns, serving as a tool of self-guiding processfor esthetically proven design. Muslim builders knew of theautonomy or inner laws in constructing spaces and patterns.The relations among various parts can be seen everywhere inancient buildings. In other words, buildings were designed withparticular care, exhibiting mastery in pattern design of theirspaces and surfaces.

Geometric proportions regulate the order of patterns basedon mathematical ratios, which is inuential in understandingthe universe, man and nature; these proportions are, by theirnature, the essential ingredients in sacred geometry, asPythagoreans stated that these harmonic ratios render musicin a balanced pattern.

Geometric proportion is strongly linked to Islamic cosmology,philosophy and metaphysical dimensions. It brings meanings andspiritual exercises for the viewer, deconstructing the meaningsunderpinning the geometric framework. As such, the viewer ismore than an observer; he or she is a participant in theestablishment of a strong link with geometry symbolizing man,nature and cosmos.

The geometry as well as its rhythm shown in Islamic patternsillustrates an innite variety, permitting and encouragingcontemplative reections. It is one of the most powerful formsof sacred art It is a source of contemplation that allows ourminds to wander and contemplate the innite. These patternswere developed with strict geometric rules, relating to theunderstanding of natural forms based on geometric ratios in thecosmos, set as the prime mover behind design.

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basis for the periodicity of the elements. 21st Century Science,18.

Ibn, Khaldun, 1958. The Muqaddimah Franz Rosenthal. PantheonBooks, New York, USA 378.

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The Holy Quran. Chapter 95, verse 4 (95:4).error. (Ibn Khaldun, 1958) Geometric Design. Garnet Publishing Ltd., UK, p. 15.possible for errors to enter into geometric reasoning,because it is well arranged and orderly. Thus the mind thatconstantly applies itself to geometry is unlikely to fall into

Geometric proportions: The underlying structure of design process for Islamic geometric patternsIntroductionGeometric proportional systemsGeometry in man, nature, and cosmosManNatureCosmos

Geometric proportions as a tool of design: study modelFourfold to eightfold patternFivefold to tenfold patternSixfold to twelvefold pattern

ConclusionReferences