A background to two-dimensional design – geometry and pattern

In the West we have a familiar understanding of Islamic geometrical design. This first example typifies the kind of thing that comes to mind. However, it is not from Arabia but was made in France and is one of a pair of silver door panels, shown here on its side. Incidentally, the other panel of the pair, although having the same basic geometric construction, has different detailing. They are good examples of the combination of geometric and foliate design.

I have included these next two examples of Islamic geometry even though they are far less complicated and more crudely assembled than that above. I’ve shown them because they are, perhaps, more the type of example with which we are familiar in our daily lives. This standard and character is commonly found all over the Middle East in the decoration of everyday objects.

They represent the character of inlaid work that many decorative pieces, brought back from the Arab world, displayed. The lower example, in particular, is extremely poorly set out. Nevertheless, it has sufficient geometrical integrity for the pattern to be easily seen and readily comprehended. Incidentally, all of these first three examples are based on eight point geometry, a relatively easy framework to establish.

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Sacred geometry and geomancy

Before I write about Arabic geometry I think it would be useful to mention two other, related disciplines: sacred geometry and geomancy – both of which are related to geometry and both of which have strong associations in the Arabic and Islamic worlds.

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Sacred geometry

Since ancient times there has been a deep interest in forms that are considered to incorporate within their intrinsic relationships – both mathematical and geometric – a universal truth. Resonances were seen to be present from the smallest to largest elements of the natural world and, in this, a unity was perceived. It was believed that these geometries were derived from, or described, the basic laws of the universe.

It followed that, by studying or contemplating them, an understanding could be obtained of the origins of everything and, in this, a sacred truth. Conversely, it was believed that these geometries were based on creation itself and that patterns in every field – such as music, astronomy or cosmology and natural forms – were derived from them.

As an extension of this discipline it was believed that these geometries were sacred and, by incorporating them in, for instance, music, art and architecture, these works would have a harmony of proportions and a special sacred character.

Many of the geometries I describe later relate to this concept of sacred geometry, particularly those relating to the Golden Section and Fibonacci.

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Geomancy

Distinct from sacred geometries is geomancy, a tradition of divination, but which has a tradition in the Arabic and other worlds with a relationship with numbers, not geometry. The divination is composed of two elements: numbers and a body of knowledge governing interpretation. The only reason I mention it here is that some believe there is a relationship between geomancy and mathematics and, by extension, astrology and cosmology to which sacred geometries, as I’ve mentioned, relate.

This is not just common to the Arabic world but has been pursued in many parts of the world and, in fact, still is. The Arabic form was called ’ilm al raml or sand science, and related to the making of sixteen random lines on the ground and their interpretation.

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Decorative types

Studying the geometry behind traditional buildings in Qatar encouraged me to look at the geometry behind them, particularly that associated with the patterns of naqsh. While these traditional buildings and their decoration are relatively unsophisticated, the development of geometry is fascinating.

The Arab world is responsible for many disciplines we now take for granted. In the sciences and arts they gave to the world considerable scholarship and, in their development of design, introduced the visual strength of geometric structure in their use of pattern.

The decoration of nearly all buildings and artefacts is based upon a combination of:

• geometry,
• floral design, and
• calligraphy.

Calligraphy is not dealt with here as there are many resources on the Web as well, of course, elsewhere. I would recommend that anybody interested might visit the Islamicart and topleftpixel sites in the first instance. There are many other examples of calligraphic art which are worth looking at, such as this, dealing with novel developments of the art in an external setting. Here I will just look at the construction of natural geometries as these form the basis upon which artisans on site set out their designs and work.

This part of my writings has to do with Islamic geometry and design, but it is worth remembering that the geometries behind these designs pre-dated Islam. They appeared in many parts of the world, but it is likely that those originating in Mesopotamia were developed by many of the civilisations that followed in the region, spreading out from there with the advance of Islam.

Arabic geometry, at least the geometries I want to look at on this page, tend to have a significant degree of regularity in their use. At the foot of the page I deal with another form, though even that is based on regular elements. But it is interesting to see that, in Qatar, there are one or two examples of a new or developed geometric treatment which are worth noting. The Weill Cornell Medical College in Qatar has a number of interesting details in its architectural vocabulary. This walkway has a screen treatment which, while appearing to be irregular, will feel familiar to Arabs and those with experience of Arabic design culture. It seems a successful interpretation of traditional design both as a design motif and a signed route as well as a device to provide a small degree of protection from the sun.

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Geometric design

I’d like to begin with some notes on geometric design. This is a vast subject and better covered in many other studies. So here I would just like to introduce the concept of differences in pattern geometry. Let me start with one of the simpler geometries, six point geometry.

It is surprising how many variations can be made from a simple geometric shape. These first three illustrations show a pattern based on a study using six point geometry. The first diagram illustrates the basic construction, beginning on the left with the development from a circle of its basic, six point division; a simple exercise, easily made with only a pair of compasses. Moving towards the right, straight lines are added joining, at first, the intersections of the circles and, then, intersections of the straight lines with themselves. It is these selections which create the possibility for different patterns to evolve. In this first graphic I have shown an arbitrary pattern outlined in red as the result of this drawing and selection process. Incidentally, I have not shown all the creation lines for the small triangle which links the six-pointed stars, but on a larger scale it is easy to see how these are made.

The second graphic above shows how this geometric pattern develops when they are added together following the basic rules created by the selection process. The lowest of the three graphics illustrates how the decisions were taken to draw the straight lines on which the pattern is based. When I have the time I intend to develop this by illustrating different patterns created by varying the positioning of the straight lines.

As an illustration of how small changes can affect the overall pattern I have gone through the exercise again, this time I have organised a linear pattern more suited to tilework than to the more integrated pattern shown above.

The basic six-pointed star is arranged to touch at its horizontal and vertical points. As a six-point geometry produces an irregular appearance on the horizontal and vertical axes, then a different condition obtains when the stars touch horizontally and vertically. Note that I have made this a direct slide rather than move the stars across by half a unit. The second graphic illustrates the location of the straight lines on the pattern, the location of which can be seen to differ from those in the illustration above.

I have written more notes below relating to the types of pattern and how they are achieved. But here, as a demonstration of how different patterns can be produced by simple changes, I have taken the pattern above and moved each line of stars half way across and up to give a very different feeling, even though the basic star – and the horizontal line of stars – are exactly the same.

Here is just one more example. As a development of the preceding pattern, this pattern has been constructed with the same six-pointed star, but this time I have rotated copies around it through 60° instead of moving them 90° to the original star. Again it is possible to see how a very small difference can produce a dramatically different overall pattern.

There are an infinite number of ways in which Islamic geometries can be organised to form patterns, as these notes may demonstrate. These two illustrations show yet another pair of patterns, their construction being readily understood, the top one based on dodecagons and triangles, the lower on octagons and squares. This page is not a scientific approach intended to display a rational grouping or progression of patterns. There are many other sites which deal with the way in which these patterns come together, particularly with explanations of the mathematics underlying them. The purpose of this small excursion has been to demonstrate how small variations of a simple two-dimensional geometry produce very different patterns. I have, however, written more about the seventeen different forms of pattern below using a simple form to demonstrate the differences.

more to be written…

This geometric study, one of a number I made some time ago, shows the construction of the Egyptian door panel illustrated in the outline perspective further up the page. Based on ten point geometry it has the aesthetic advantage of being related to the Golden Section and is one of many ways that the geometry can be used to form different patterns on which are based constructions from a variety of materials.

Ten point geometry lends itself to a wide variety of design possibilities through relatively small variations in the underlying relationships. Many of the more interesting examples can be seen as panels on woodwork in Syria and Egypt, this being a study of an Egyptian panel. It is useful to see how little of the overall geometry is used for this running pattern.

And this study, based on a panel of ceramic tilework, was undertaken as an examination of its underlying geometry, particularly from the point of view of determining the relationships between the circles containing the ten points. Note how the underlying geometry is based on the central ten-point rose being turned through 18° compared with the study above.

The preceding black and white illustration shows the developed geometry for the study. Initially constructed as a drafting exercise with compass and straight edge, it looks considerably more clear in the drafted artwork than it does here reduced in size for the purpose of this essay. However, these three blue and white illustrations are based on it and should enable me to make a point.

The top illustration is of a ribbon pattern based on that extended geometry, and appears very much as a lace pattern. The middle drawing is a detail of that ribbon, giving a more clear idea of the way it works, and with a horizontal feel to it. The lowest of the three is the same pattern but rotated through 54° and has a very different feel to it, which is something a lay viewer might not expect. These studies, of which I have made many, illustrate a small number of the numerous possibilities there are for setting out patterns and, even without the addition of detailing and colour, the enormous opportunities for variation – with the possibility of three-dimensional or sculptural effects to create even more variations based on a simple ten-point geometry.

Here’s a brief exercise to demonstrate how basic two-dimensional patterns can be given a degree of form and depth. It’s exactly the same diagram as that above it. All I have done is give a hint of highlight top left and a heavier shadow, bottom right to give it a three-dimensional effect. The difference between this and its original drawing is dramatic and shows how easily these patterns can be developed.

And, here’s the real thing. These first two photographs illustrate the kind of detailing we commonly associate with Islamic geometry. They are taken from a sixteenth century Mamluk Egyptian minbar and are both based on ten point geometry. This underlying geometry can be constructed in many different ways and will produce variations that are implicitly understood as being related. The ways of altering the relationship between the elements of the geometry appears to be relatively simple, but every decision results in complex patterns that can appear quite different from each other. The actual detailing, here carried out in wood with painted elements, is capable of infinite variation though, having said that, local styles tended to work with a limited design palate and have a similar look about them. In reality designers tend to restrict their studies to a tried and tested series of designs which, nevertheless, are capable of an apparently infinite number of designs.

Just to illustrate how ubiquitous ten point geometry seems to be, here is an example of an Egyptian fourteenth century leather book binding. It is possible to see that it uses a different arrangement from the examples above, and how readily ten-point geometry lends itself to the creation of different patterns. This is a beautiful example of the book-binder’s art.

These next two photographs are of an old inlaid box made either in the Lebanon or Syria, probably between fifty and a hundred years ago. This form of inlaid work is very typical of a wide range of goods that are now produced for the tourist market but are based on traditional finishes on furniture, quran stands, boxes and the like. The techniques of manufacture have not changed in centuries though the materials may have. Now, for instance, instead of ivory, bone is used or even plastic for white elements such as on this box. The top photograph is based on six point geometry, the lower on eight point geometry though there is some twelve point geometry used in the side details. Again it can be seen how the geometries meld together and work at different scales as a unified design.

Compared with the eight and twelve point geometries of the two photographs above, here is a six point geometry based design which looks remarkably similar at first glance. Set out on a tambourine it is an easy geometry to work with, but is not favoured in some parts of the Arab world. What seems to me to be significant is that the pieces are more accurately constructed, but I don’t know if this is a result of the degree of craftsmanship or the ease of working with this particular geometry.

By way of contrasting materials, here is, firstly, part of a pavement inside one of the most beautiful of English abbeys, Fountains in the north of England. Construction began in 1132 AD with much of the construction being effected by lay brothers who, by carrying out the more practical work, relieved the Cistercian brothers of the more physical work on the development. The pavement is based on four point geometry and is constructed entirely of only three different tiles: a square, a lozenge and a triangle. I don’t know where the craftsmen came from who carried out this work, but the point of placing this illustration here is that geometries are universal.

Approximately two hundred years later the craftsmen working on the Alhambra in Granada, Spain, produced work which was a great deal more complex. This photo is a detail of one of the pavements at the Alhambra and you will be able to see that, while the geometrical basis of the Fountains Abbey pavement is four-point, this is eight-point, here shown turned through 22½° compared with the example above. The complexity is introduced on a relatively simple basis by the use of colour and the doubling of the structure lines while employing the technique of cutting the tiles in such a way as to imply the interweaving of the running lines.

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Cosmatesque design

These first three photographs may appear to have little in common with the subject of Arabic geometry, but I am including a note on this type of design here for three reasons. First, there are only two examples of this type of work in England, at locations to which there is relatively easy access, London and Canterbury; secondly, these are the only examples of this type of work north of Italy and, thirdly – and of particular relevance to the subject of this page – there is an unusual geometrical detail to be found in the design of the cosmatesque pavement of Westminster Abbey.

The first two photographs illustrate cosmatesque design work in the floors or pavements of two religious buildings. Both are located in Rome; that above is in the church of San Benedetto in Piscinula, and that below it to the side is part of the central guilloche in the church of Santa Maria in Trastavere. These two photographs, while both of Roman examples, illustrate the typical character of cosmatesque design in order that you may be able to see something of the difference between cosmatesque and Islamic design work. Although the first two of these photographs are of pavements, you should be aware that cosmatesque work is also found on vertical surfaces and columns. It may have no relevance to the geometry discussed on this page, but I have included this photograph of cosmatesque work applied to a barleysugar column, again in the Roman church of Santa Maria Trastavere. This should be seen in relationship to the photo above, both showing the typical running pattern which characterises cosmatesque design.

As mentioned above, there are only two locations in England where cosmatesque designs can be seen. These are before the high altar and around and on the tomb of Edward the Confessor in Westminster Abbey, London, and beside the tomb of St. Thomas à Becket in Canterbury Cathedral, Kent. There are no other designs like this in northern continental Europe, the style mainly being found in Italy where it originated with the work of the Cosmati family, members of which were involved in decorative work as architects, sculptors and mosaic craftsmen during the thirteenth century.

This image, like those above, illustrates a different type of geometric approach to the design of paving in Britain. Laid in 1268 under the supervision of the master craftsman, Petrus Oderisius, or Odoricus, this is a small element of the cosmatesque pavement in front of the High Altar in Westminster Abbey, London, and is characteristic of the work produced by the Italian Cosmati family, who developed their style in a move away from, though derived from, the predominating Byzantine work of the period. The distinctive character of this work can be seen in this detail and those above though, in this detail, the overall and linking patterns are not shown, just a single roundel.

You should note that the heavily worn state of the cosmatesque work in Westminster Abbey is due to a combination of the depredation of pilgrims, lack of proper maintenance and poor restoration compared with the Italian examples above it, there being the practice in Italy of keeping their pavements in good repair, but at the expense of destroying the original work. There have been three restorations of the pavement since it was laid in Westminster Abbey. Early in the 1660s the restitution of the monarchy saw the first restoration of the pavement; the second was in the early eighteenth century, and the late 1860s saw the third. Elsewhere I have touched on the fashions which have affected restoration work, not just in architectural areas but also in painting and archaeological work. Those with an interest in this subject should look elsewhere though it might be useful to make the general point that, in replacing work in a similar manner to the original, the history of a piece may be lost. It is around this conceptual difficulty that the issues relating to conservation and their resolution turn.

I should also mention that the cosmatesque pavement in Westminster Abbey has a significant element of mystery surrounding its conception, design and incorporation into the fabric of the building. A number of scholastic papers have been written on these areas and, for those with an interest in the political and symbolic background of the pavement, the papers by Foster and Sharp as well as the research papers edited by Grant and Mortimer might be pursued. There is also an illustrated book by Pajarez-Ayuela which mentions the work at Westminster and Canterbury. My notes here relate only to the geometry of a small part of the pavement, a single roundel on the edge of the pavement.

The chief design characteristic of cosmatesque work is that it has a vigorous style created through the medium of its containing shapes and the colours used. While constructed with the basic geometries, the curvilinear elements of design tend to be set on a plain ground which, in Italy were pale marbles, and in England, Purbeck marble – actually a hard limestone. These designs were established as simple geometric shapes which were surrounded and trimmed by bands and ribbons of mosaic created with semi-precious stones and glass. These pavements must have been extraordinary in their first viewings, particularly in England where the tradition for pavements was mainly stone flags or fired tiles and where the Westminster work was a considerable departure.

Cosmatesque work also has the characteristic, and one which further distinguishes it from Arabic geometrical patterns, in having considerable symbolic content. According to Foster, who based his opinion on both an analysis of the inscriptions on the pavement as well as a study of the symbolism of the overall geometry regulating the composition, the pavement represents a schematic description, or symbolic compendium of the whole of the universe.

This is very different conceptually from Arabic geometric work, containing within it messages or associations claimed to relate to Christianity, liturgy, cosmology, choreography and other aspects which may be complex if not obscure. Because of this, for instance, a full interpretation of the Westminster pavement is still awaited, though here are two descriptions believed to be contained, firstly, within the overall pavement design to the left and, secondly to the right, within the central quincunx, both according to Foster. Those interested in these aspects of geometry will again have to look elsewhere as these areas are complex and have little or nothing to do with the main subject of this site.

Arabic geometry, on which Islamic patterns are based may also have symbolism embodied within some works, but these relate in the main to numerology. In calligraphic geometrical work there is, of course, a specific meaning to the work contained within the calligraphy itself. But typical geometrical work, as well as that containing naturalistic elements does not have this. There are more notes written on this subject on the Islamic architecture pages.

This sketch illustrates the basic geometrical layout of the centre of the pavement before the high altar in Westminster Abbey. The first point to note is the use of the flowing, curvilinear geometry within which medallions of pattern are located and which geometry can be seen to show influence of the Roman work from which it has descended, albeit with the Greek and Byzantine influences which moved into Rome round about the eighth century. This grouping of a central circle with four smaller circles surrounding it is known as a quincunx, in particular, a poised quincunx – one of the diagonals being perpendicular to the principal axis of the ornamental composition which, in this case, coincides with the axis of the Abbey. In technical terms, the Westminster example is a decussate-quincunx-in-quincunx. The word ‘decussate’ means ‘ten’, this referring to the Latin form of ‘ten’ in the form of ‘X’.

The quincunx is usually formed of a central circle and four surrounding circles, though may have a square or rectangle in its centre. While simple geometry is used to establish the layout, you can see how different it appears from Islamic patterns in its loose form and, of course, the overlapping curved line. Islamic patterning, by contrast, tends to be far more intricate, particularly in designs constructed of mosaic.

This second sketch illustrates the geometric basis governing the whole of the pattern of the pavement, including its basic containing framework and the central pattern illustrated above. You can see it is based on √2 geometry. Commonly artisans constructed this geometry with the inner, poised, square having a side compared with the containing square in the ratio of 1 to √2. There is a geometric way of establishing this proportion – a right angle triangle with adjacent sides equal, will have a hypoteneuse of √2 to the adjacent sides. But artisans usually constructed it by measurement, utilising measures in the ratio of 12:17. This has been established at Westminster because the measurement of the respective sides of the squares is 3·57 metres and 5·05 metres, it being claimed that these are exactly 12 and 17 Roman feet, a Roman foot being just less than a British foot at that time, around 11·5 inches or 296 mm, which also shows that the craftsmen were Roman, or were working to Roman direction. However, the detailed restoration work being carried out on the pavement in 2008 may cause the claim for Roman authorship to be revised.

According to Pajarez-Ayuela, it should also be noted that, where ‘C’ is the diameter of the central roundel of the quincunx, and where ‘A’ is the width of the square circumscribed around the quincunx, in any cosmatesque pavement the ratio of ‘C’ to ‘A’ is always within the following limits: ⅓ ≤ C/A ≤ ½

Notice in the detail above right that the geometries which establish the patterns are discrete, they do not link with each other but sit adjacent to or within other geometric frameworks. This allows for elements of banding between the geometries which is a characteristic of cosmatesque work but which tends to differentiate it from many Islamic frameworks where continuous geometries are more likely to be the norm and where there is significant repetition.

As I mentioned earlier, the Westminster Abbey pavement has two features which I find of particular interest, the first being the wide variety of patterns used as infill within the overall framework. There is a considerable body of literature dealing with this, some of which is referenced above. Much of this literature investigates the historical and political setting which saw the introduction of the pavement to England as well as its symbolism. This really falls outside the intended nature of this page though it is worth noting that there are a number of other factors relating to cosmological and other causes which may govern both geometry and pattern in design. This part of the pavement is a significant contrast to the manner in which Arabic designs are put together. I can not recall seeing an Arabic design where different ground patterns are associated within a single geometric design.

The other feature I find very interesting – and the real reason the pavement occurs on this page – is the fact that one of the features of the pavement, a small roundel, has geometry based on eleven-point geometry. The hendecagon, or eleven-sided figure, has internal angles of 147·2727…°. Bearing in mind that it is not possible to construct one using compasses and a straight edge, it makes this geometry a strange choice to select for the basis of a decorative pattern. My understanding is that this particular feature was associated with the 1860s restorations, and it may be that it was altered and this geometry introduced at that point in time. This illustration of a hendecagon shows it divided, on the left, into twenty-two parts by lines running through its centre and, on the right, with all its chords drawn, illustrating its potential for complexity.

Illustrated to the near right, the roundel is comprised of six concentric circles of unequal widths containing triangles and a small number of lozenges. While all the internal divisions of this particular roundel have eleven sub-divisions, the outer circle is divided into thirty-nine parts, again a very unusual choice if simple geometry is required to construct it. To demonstrate the lack of geometrical congruity between the eleven and thirty-nine sub-divisions, to the right there is an illustration showing eleven sub-divisions with thirty-nine sub-divisions superimposed on them.

The only way in which the pattern is likely to have been constructed is by a process of trial and error. Looking at the whole of the work of the pavement it is noticeable that there are very different standards of expertise in the cutting and laying of the elements of the design which suggests that the different areas of the pavement were finished to significantly different standards – a rationale which seems extremely unlikely – or that the work was not carried out coevally. Those which are laid in this particular medallion are relatively coarsely cut and placed which suggests this work was carried out by a different artisan and, perhaps, that the stones making it up are those taken from a previous or different pattern. The inaccuracies of cutting may also have been responsible for some of the inaccuracy of laying out. I found it difficult to measure so have produced a sketch whose dimensions have been generalised. It is not a working drawing.

It should also be noted that the machinery used to cut and work hard materials will have developed over time, the more modern work utilising far better cutting equipment than would have been available to the original craftsmen. Despite this, it has been pointed out to me that some of the work which is believed to be original is of a very fine standard and, in this respect, stands in contrast to what is likely to be some of the later work. I am disappointed in being unable to illustrate the character of the roundel with a photograph of it as the Abbey has refused my request to do so, insisting on controlling exposure of the pavement to the public.

Here you can see a rough sketch of the pattern, not drawn to scale, but approximately accurate. I have shown all elements the same colour. In reality they are beige, rose, red, black and green, some idea of which can be seen a little higher up the page. Beside it is a diagram showing the circular divisions of each of the roundel with, lighter, the line of the smaller triangular elements of the design. I have kept the size of these diagrams small on purpose in order to mask the difficulties of aligning the different elements, but it is evident even here that there is a difficulty in establishing a satisfactory relationship between the outer ring of thirty-nine divisions and its neighbour of twenty-two, never mind the problems of setting out eleven, twenty-two and thirty-nine divisions. This outer ring bears an approximate relationship with the next inner ring of twenty-two – 22:39 – or √3. Considering the numerology and symbolism which others have argued to have gone into this pavement, it might be anticipated that there is a mathematical relationship between the rings. But 22:38 would be a more significant relationship and 22:35 or 22:36 closer to the Golden Section. The relationship of the outer ring to the inner rings of eleven sub-divisions would approximate pi if the outer ring had thirty-four or thirty-five sub-divisions – 11:34 or 11:35, and not thirty-nine.

So, we have a pattern, difficult to set out, which appears to be based on a slight but poor mathematical or geometrical relationship, and which suggests that the relationship is accidental, or that the numbers are significant and relate to something symbolic, the geometry of construction being incidental to the meaning. Nevertheless, I believe the roundel is fascinating whatever its original geometric intent. Perhaps it might be best to see it in a similar light to the eccentricities which exist in nature, but remember that this would never happen in an Arabic design.

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Representative design

Although many people in the West are aware of the fact, it is worth repeating that there is a prohibition against figurative art in Islam through an injunction in the hadith, though not in the holy Quran. This aims to avoid idolatry. For this reason it has been extremely unusual to see figurative art in public places though it is not uncommon in private and there is a long history of it in different parts of the Islamic world, both in secular and court settings. This tends to have been more common in Persia and India where the lack of perspective and shadows has kept the illustrations relatively free from direct representation. You will not see anthropomorphic art in mosques.

But, nowadays, there is a market for artists producing paintings of the past incorporating their client’s ancestors as well as illustrating stories. It is not common, but there are an increasing number of examples of figurative art being produced for public consumption. I assume this reflects demand and, particularly, the increasing influence of the Western world on general values. This example is of a good quality Persian carpet with the subject matter taken from the stories of Omar Khayyam. Carpets like this are woven for private consumption and can be seen in houses along with similar themes, often paired with pictures of landscapes.

Increasingly, however, there is a tendency to represent the past through art in public places – particularly where the past has been irretrievably lost – and this is seen in art associated with new offices and hotels as well as, and particularly on, roundabouts and in public spaces. There appear to be two reasons for it: firstly the marking of modern achievements of the State and, particularly the head of State and, secondly, the introduction of western forms in a demonstration of modernism. This art may be seen as a very important reminder of a shared history and a poignant reminder of a need to be seen to ‘progress’. However, it is with non-representational forms that Islamic art is most associated.

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Design variation

Over time designs were developed by different cultures around the world which were based on naturalistic and geometric forms. In the latter there appear, on first consideration, to be very few possibilities for variety as there are only a limited number of two-dimensional geometries on which forms or patterns can be based. But the reality is that the amount of variations on a single, geometric theme appear to be infinite, the variety being created through relatively small differences in the rules selected to form each design, as well as through a combination of different geometries. Even though this development of patterns based on two-dimensional geometry pre-dates Islam, this character of the decorative arts is now firmly associated with Islamic design.

I made the above animation to demonstrate something of the variety that can be found within two-dimensional geometry. It illustrates a number of design variations created by Bourgoin in his book on Arabic tessellations, published in 1879 with the title ‘Les Eléments de l’art arabe: let trait des entrelacs’, but now made more freely available in English, although without the original text. Bourgoin’s work has been used by many scholars in their investigations into the basis of interlace patterns, tessellations and the geometries used in Islamic designs, but many others, such as Issam El-Said, Critchlow, d’Avennes have also worked in, and developed this specialist area.

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Origin of design?

It is impossible to know the extent to which pre-Islamic geometries, and particularly those associated with ritual geometry lie behind the rich patterning with which Arabic buildings are now covered, but there may well be strong atavistic tendencies which recognise or at least are satisfied by immersion in the rhythms of the geometries. Within Islam there has developed an art which illustrates an infinite variety, and which permits and encourages contemplative reflection.

This represents the development of non-naturalistic Islamic art and is thought to be one of the most powerful forms of sacred art and not just an abstract art in the modern sense of the word. However we view Islamic patterns, there is a potent source of contemplation in observing its varied patterns. It is my own experience that, looking at the patterns within Islamic buildings, it is easy to see geometries forming and re-forming in an endless profusion. Yet the method by which these patterns were created is relatively simple.

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Unity

It is argued by Seyyid Hossein Nasr in his foreword to Keith Critchlow’s book on Islamic patterns that a doctrine of unity is central to Islam and that it manifests itself not in iconography but in geometry and rhythm, arabesques and calligraphy. More particularly he argues that a sacred – not just an abstract – art developed based on mathematics which goes to the very heart of Islam.

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The treatment of surfaces

Surface treatment of buildings in Islam appears to enliven the defining forms of the spaces. Interplay of form and decorative elements of the surface bring meaning and spiritual exercise for the observer, the more so with the voluntary or involuntary intellectual exercise of deconstructing the meaning behind the geometric framework. In this way the viewer is more than an observer; the viewer is a participant. At its best, this enlivening brings motion to static building, consolidating the harmonies and enriching the user of the building and, in the situation where the building is integrated with its setting, establishing a strong link with the geometries of nature.

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Symbolism in design

But more than this, there is evidence that there are symbolic messages contained within such geometric designs. Religious and numerical clues have been found by those carrying out refurbishment of Islamic buildings. These relate to the numbers relating to the number of names of God, and the derivation of the patterns contained within the overall scheme. The panel below, for instance, contains ninety-nine elements – the same number as there are for the names of God.

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Relationship between art and mathematics

In addition to issues relating to symbolism there is a very strong link between art and mathematics, particularly numbers. There are studies and considerable evidence of the intellectual interplay of mathematics and design such as occurs in the Alhambra in southern Spain. Much of this is based on Pythagorean mathematics and there are a number of theories explaining the manner in which mathematics is incorporated into buildings, both in their overall design as well as their decoration.

Put simply, Pythagoras believed that the intrinsic character of numbers reflected Nature. It followed that, if the character of Nature can be known, then the nature of numbers can be determined. Abstract concepts were held to be expressions of number; Justice, for instance, was thought to be four, and the Universe, ten.

more to be written…

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Geometry in plants

Wherever you look, plants, particularly flowers, tend to have only a small variety of geometries behind their two-dimensional geometry, although their variety seems to be infinite. These photographs of flowers illustrate some of the most common two-dimensional patterns found in nature, but not every flower is regular in this respect. Bear in mind that many plants have irregular geometric forms.

The first three flowers shown here have, respectively, two, four and eight petals and although four petals might be thought to be the simplest form of geometrical arrangement, it is the five and six petals which seem to be the more common in nature. The two, four and eight petalled flowers seem to be solid and stable in terms of their geometric appearance as we tend to equate them automatically with the character of the square or rectangle. This is particularly obvious in the four-petalled flower.

However, as I mentioned above, it is five- and six-pointed geometries which seem to be the more common in the natural world. I have to admit that I’m not sure which is the more common, five or six, but my impression is that it’s likely to be the six-pointed. Having said that it is the five- and ten-pointed geometries which, being based on the Golden Section, discussed below, might be argued to represent a more perfect proportion than plants based on other geometries.

Looking at the different forms five-pointed geometries take – illustrated with just two photos here – the plants seem to me to have a more exciting form than those based on four- or six-pointed geometries, though the reason for this is relatively simple and is discussed and illustrated below these photographs of flowers.

Compare the two photographs above illustrating five-point geometry with the four- and eight-pointed geometries above them, and the two based on six-pointed geometry below them. While the lowest photo shows balance and a calmness created by its regularity, the flowers based on five-pointed geometry do look to me more interesting in the irregularity of that geometry. Perhaps this is the reason I have more photographs of five-point geometry than I do of six-point geometry in my files – and why I have the feeling that there is more five-point geometry than six-point in nature. Anyhow, the point I am trying to make is that both five- and six-point geometries can be readily found, and that they create a slightly different feeling caused by their odd and even geometrical bases. But there are other geometrical bases for plant forms.

The more you look at the natural world, the more likely you are to see that everything is not as perfect as you might expect. But even when there is diversion from the norm, beauty may still be seen in nature’s variations or imperfections, particularly in its geometry. Here are two photos of primula flowers. The top one has the regular form with six-point geometry demonstrated in the disposition of its anthers and petals. The lower one, though, demonstrates the far more rare seven-point geometry as the basis for its petals. Even though the higher of these two photos demonstrates a more active balance in the shape of the petals – as discussed below – it can be seen that the lower, seven-point geometry is more interesting, as I argued in the previous paragraph. Both flowers are also enhanced by the contrasting colours of their petals and anthers – here the yellow, male element of the flower. And just to make the point again, here is a polyanthus with seven petals. It’s not an easy geometry to spot in nature but I have also noticed it a number of times with the division of the spokes on car wheels. Why this should be I have no idea. Perhaps somebody would tell me…

Six-point geometry can be discovered all over the natural world and is readily found almost wherever you choose to look. But it seems far less easy to spot three- and nine-point geometry which you might expect to be similarly apparent, though it turns out they are not. Unfortunately I am not a botanist and have no idea why this is so, though I suspect there may well be a sound reason for it. Obviously the majority of clover leaves fall into the three-point geometry form, and of which there are rare famous four-leafed forms. An example of the three-leafed form is shown in the upper of these two photographs, and below it there is a lewsia displaying its nine petals. Unfortunately, this particular specimen is not as regular as I would have liked but, nevertheless, it has a satisfying symmetry to it.

It follows that it is also very common in nature to find plants which are divided into ten divisions as is illustrated to the side here in this Passiflora caerulea. Note that although there are ten petals, the central part of the flower has five divisions and it has to be assumed that the two divisions – created by five- and ten-pointed geometries – may be found together in nature in the same plant.

Incidentally, there is an additional point to make here. Some plants combine a mixture of geometries at their base. Looking at this flower in more detail, not only is there five-point geometry, but an element of three divisional geometry where the brown stigmas sit over the green and yellow fruit pod. I believe too, that there are seventy-two purple filaments.

This complexity is reflected in man’s design work when five- and ten-pointed geometries may exist side by side and, as you will see from some of the examples further up the page, also can be worked with four- and six-pointed geometries to great effect. This is the basis of much of Islamic geometrical design. In fact Islamic designs may have a number of different bases for its geometry, their integration being a factor of the patterns selected, natural inter-relationship and the skill of the designer.

While a designer’s choice may be based on a number of different or accidental selections, nature is very often accidental itself as can be seen from some of the examples illustrated above. This group of four photographs are all of poppy heads, the first exhibiting ten divisions which is the norm for this species. Obviously this is similar to the five divisions but the ten divisions are very much a feature of traditional geometrical designs and, of course, are both related to the Golden Section or Mean as will be discussed later. It is interesting to note that although it takes a little time to count the number of divisions, this can be rapidly guessed at a glance and easily distinguished from twelve divisions with little or no practice. The second photograph is of one of nature’s many little eccentricities, a poppy head of eleven divisions, as is the third photograph, though it is of a poppy head seen hundreds of miles away from the first. This second poppy head of eleven divisions was found side by side on the same plant as the fourth photo which, as you can see, has twelve divisions.

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Shape and balance

One point to note, illustrated to the side here, is that flowers and plants that have an equal number of petals or elements are usually understood to have a more balanced form, whereas those with an odd number appear to be more active. If you compare the six and five sided figures you should see what I mean. Even sided figures look balanced, secure and strong while odd sided figures appear disbalanced, in motion, perhaps more exciting.

This is a normal psychological function of the manner in which we understand objects. If you look at these two shapes on the following two diagrams, the top diagram illustrates a shape with an odd number of sides, the lower and even number of sides.

The upper shape on the left seems more active while the lower one on the left feels solid and well balanced. But you can see in the right hand example of each shape that by balancing them on a point rather than a side, they both seem more active – or unstable, depending on how you view them. The point to bear in mind is that we automatically read two-dimensional shapes as having qualities or characteristics they don’t have in reality: an implied gravity. This character of shapes can be observed, consciously or unconsciously in the design of everything we see and affects the manner in which we understand them. It is something that designers bear in mind as an element of their design vocabulary. However, in repetitive geometric pattern we appear to lose that sense of weight and, in most patterned work we experience a weightlessness in the overall design. This is a quality which lends itself to contemplation and is very much in tune with an Islamic view of the world. More has been written on the Perception page. But I digress…

Pythagoras’ theories were developed in the Arabic world by, among others, a group known as the Brotherhood of Purity – Al Ikhwan as-Safa’ – in tenth century Basra. The Brotherhood placed emphasis on the numbers one, four and seven. While the intellectual advances in mathematics and numbers was effectively located on the east of the Mediterranean, the more utilitarian development of numbers was taking place in Spain. However, these skills and relationships travelled throughout the Islamic world and it has to be anticipated that they would have formed a basis for construction in most of the Islamic world.

I didn’t intend to write about Islamic geometry related to modern art and design. Nevertheless, I thought it useful to include at least one photograph to illustrate a modern artistic installation that is specifically based on Islamic geometry. The top of the two photographs is a straight elevation of half of the work, the lower was taken at an oblique angle. The piece is by Monir Farmanfarmaian and is very much related to her homeland, Iran. It is constructed of mirror mosaic of which there is a strong tradition there. This is only half the installation and each consists of three panels, each panel 1350mm wide by 1830mm high. It illustrates how, through geometry and the facetting of the mirror mosaics, a complex work can be constructed.

This third photo shows three separate details from the above work. It shows how the six-pointed geometry of this part of the work has resulted in three very different patterns. These designs are some of the most basic forms and can be found in Islamic work all over the Arab world, fabricated in a number of different materials. The advantage of mirror mosaic is that it reflects light as well as the colours of the space in which it is situated.

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Harmony

Thus the Islamic building can be seen to incorporate within it both the essential harmonies of nature together with various symbolic meanings, fixing them in its man-made structure. At their finest, these buildings are more powerful spiritually than are their Western counterparts. Where Western architecture takes its inspiration from traditional construction and theories of perfect proportions, Islamic architecture is created of a whole whose elements are defined through a series of relationships with nature and natural surroundings.

This surface treatment can be seen to have three characteristics, those:

• resolving themselves around the issues of symmetry, those
• which might be thought to relate to the metaphor of textiles, and those
• incorporating what Jay Bonner refers to as self-similar design.

This latter form – the repeating of the main pattern at a smaller scale in the background – is not found in traditional Qatari naqsh design but is typical of the more sophisticated work found in Persia, Turkey, Egypt, Morocco, and Andalusia.

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Limited design range in Qatar

No record has been passed down to us instructing in the theory of the design of Islamic patterns in Qatar. From talking with craftsmen from Iran and Qatar, it appears that they set out designs from a limited range of patterns they carry in their head and which are based on their experience rather than from any form of received instruction.

This means that the pool of design stays relatively stagnant, at least for a generation while that generation works. In Qatar the Ministry of Public Works employed a group of craftsmen who were responsible for most of the reconstructed works so it is reasonable to expect that the designs do not vary much. What is interesting is the way in which they have been able to break down the scale of the work they carry out while maintaining a reasonable degree of innovation within this very limited design area. Note the degree of symmetry about the vertical axis on the above example.

I had hoped to see some of the little eccentricities which enlivened naqsh in the older buildings, but that has not been the case. For instance I recall seeing in a corner of a majlis a small bird hidden away among the geometrical designs. It was a beautiful example of humour and lightness of design technique. Whether the idea for it came from the client or the craftsman, I don’t know, but nobody has been able to tell me which it was likely to have been.

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Symmetry

Symmetry can be defined in terms of grid and detailed design, and although there are only seventeen possible patterns, there are an infinite number of patterns that can be developed from them. All around us we see the manner in which these patterns are developed to adorn and decorate. It should be understood that a pattern is symmetric if there is at least one symmetry (rotation, translation, reflection or glide reflection) that leaves the pattern unchanged.

The geometries which are associated with patterning of finishes such as tiles are also often related to the three-dimensional forms of building in Islam just as it has been in some Western buildings. It might be, however, that there is a more important function for geometry in relating the building both in its proportions and spaces with the more spiritual functions of Islamic architecture – particularly, therefore, the architecture of the mosques and schools (madrassat), but not necessarily of the residential buildings that account for the major part of traditional Islamic urban developments.

As mentioned previously, the metaphor of textile in the decoration of Islamic architecture is one which appears to have some relevance. The form of the decoration, in this case, can be divided into two forms: free-flowing, and tailored. At its simplest free-flowing patterns can be seen to be draped over the underlying forms of a building, in the latter patterns are organised to be constrained by – or, alternatively, to define – architectural elements of a building.

It is likely that the sophistication suggested by the development of geometries in building complexes such as the Alhambra in Moorish Spain did not find its way to the Gulf. Perhaps more surprising is that there is seems to be no suggestion that the Turkish occupation had any design influence, though it may have come about in an unconscious manner. I know of no attempt to investigate it, and the majority of the old mosques in Qatar have been demolished to make way for newer, larger developments. The most likely design influences are, of course, from what is now Iran as the majority of builders appear to have originated there.

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Intellectual basis of design

It is generally recognised that the Alhambra was not the invention of its builders but the product of the intellectual workings of at least two of the Grand Viziers, Ibn Khaldun probably being the major contributor. Mathematics and geometry were normal considerations of intellectuals of the period. The integration of poetic writings with the geometric patterning and architecture suggest that the builders were strongly directed in their work.

On simpler buildings the builders would have been more easily able to integrate the two- and three-dimensional requirements of their buildings by themselves, and this is my experience observing them in the Gulf.

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First steps in Qatar

The first attempts at patterning in the Gulf are likely to have been associated with mud buildings, and an effort to relieve the large expanses of wall, particularly within buildings, perhaps the majlis being the first recipient of this treatment. In Qatar the tradition up to forty or so years ago was that patterns were drawn directly into the drying juss. This required the designer to work relatively quickly in order to complete the work easily.

In carrying out this work the craftsmen worked in the traditions they had been brought up with and, as they may not have moved far during their lives, there is some – though not much – regional variation in their work. The work in the Persian/Arabian Gulf can be seen to relate to that in the Oman and Dhofar though it is different from that of the Red Sea.

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Carved plasterwork

A development of this was the incorporation of carved naqsh into panels which were offered up to openings within buildings after completion. This permitted more formal, complex and accurate patterns to be produced. A refinement of this was to make them fretted to permit light and air to move through them, though they never ceased to be conceived and constructed as two-dimensional geometries. In some cases – particularly those carvings where panels were fretted – the two-dimensional geometry is cut straight through the panel, the face of which generally has no other decoration other than the holes to relieve it: but, on normal panel work, the cuts are made at an angle to give a similar, three-dimensional effect to that produced in Roman lettering carved into stone: a direct effect of the interaction between the material and the tools.

The decorative carving can be enlivened by simple, scratched markings which give more detail and interest to the face. In this sense it reinforces the decorative character of these panels which contrasts with the purely functional requirement of the fretted panels. Often these patterns are created when the face is marked with compasses, the metal point of the compass scribing a line which is not hidden by subsequent carving.

Gulf patterns are invariably relatively uncomplicated and it is only in the more sophisticated buildings elsewhere in the Islamic world that the more complex patterns were developed often with mixed geometries. Despite their simplicity there is still a sense of variety within the patterns still extant. One particular characteristic of patterns in the Gulf is the strong reliance on the circle and its derived six point geometries.

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Geometric basis of patterns

Execution of the work in the Gulf has, as a result of a number of factors, been relatively simple and has derived from the simple patterns based on four, five and six point geometries. I have not seen designs based on seven, nine or more complex geometries. Although geometric patterns are found in nature it is likely that they would not have been observed by designers in the Gulf as there is little to see; rather they would have developed from the simple tools needed to create the pattern geometries, though there is the likely relationship of designers working coevally on the other side of the Gulf.

The next part of these notes was researched and written in the 1990s. Since then I have discovered that a treatise – Kitab fi ma yahtaj ilayh al-sani min al-amal al handasiyya – was written by Abu al-Wafa (940-998) in Baghdad on the use of the straight edge and compass by artisans. It had been thought that it was an instruction manual for artisans but is now thought more likely to be a description of their work for intellectuals. It must be borne in mind that mathematics was more advanced in the Arab world than in the Christian West, and that it was treated not only as an intellectual exercise, but also as a functional system for organising a number of practical operations. The times of prayer, the division of inheritances and the direction of the qibla were such operations, but mathematics was also closely related to astronomy and astrology.

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Tools used in the design and layout of patterns

Essentially designs would be made with simple compasses and a straight edge, the same tools which can be observed in use by designers working in Iran and the Gulf to this day, though with the addition of a device to construct right angles and to develop parallel lines.

With these two basic tools it is easy to construct four and six point geometries and, even, five point geometries to form the basis for the designs commonly used for decoration. The extent to which the designs are then developed is apparently a matter for the master mason or individual craftsman, the main factor being the speed with which the work has to be executed, particularly when working in a medium such as juss which dries rapidly. Because of this, carved plaster work is relatively simple in Qatar and approximations rapidly executed in wet plaster. Where the work is made in plaster which is set, more care can be taken.

Within buildings, as well as on their faces, the designs of naqsh panels are always different. Sometimes there is reflection of designs facing each other within a majlis, for instance, but in the main an effort is made to ensure that no two designs within a single space are the same.

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Geometric basis

While some of the geometrical constructions are not found in Qatar, set out below are the basic methods by which 3, 4, 5, 6, 8 and 10 point patterns can be constructed using only a straight edge and compass, the standard site tools used to set out designs in naqsh. Approximations of 7 and 9 point patterns can be approximated, as can others, but there is rarely a need for them in Qatari naqsh work.

From these geometrical constructions, patterns with a greater complexity can be constructed. Patterns with 12, 14, 15, 16, 18 and 20 are readily established. I don’t know how to construct 13 and 19 point patterns, though there is a construction for 17. However, I digress…

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Three point geometry

Three point geometry can be constructed from six point geometry, it should be noted that it is not possible to subdivide an angle into three parts in Euclidean geometry. However, there is a construction which permits an angle being divided equally into three using only a straight edge and compass; that is by fitting the angle to a previously constructed construction.

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Four point geometry

Four point geometry is one of the easiest geometries to set out. It is often used in counterpoint with circular geometries. It is constructed by raising a perpendicular from the centre point of a horizontal line to the point where it cuts a circle described from that point, and joining the four points of intersection. Further sub-divisions into eight point geometry can be constructed by sub-dividing the sides of the square.

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√2 geometry

Related to this is the development of geometries based on the diagonal of a square – √2 where the side is 1. The geometries of the Alhambra depend, to some extent, on this geometry rather than on the Golden Section – 1:1.414 compared with 1:1.618. The second diagram illustrates the difference between the basic two proportions. Note that the √2 proportions are those of the International ‘A’ paper sizes – A4, the most commonly used, being 210mm x 297mm.

It has been said that the use of this geometry and the the complexity of the elements of design in this development, led to the intricacy and intellectual complexity of the Alhambra. This geometry is also used in the Gulf, but there isn’t the complexity of arrangement and interplay of proportions in these simpler buildings, the relationships being found only in two-dimensional patterns.

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Five point geometry

Five point geometries are slightly more difficult to construct, but can be relatively easily developed. They have an additional interest in that they are similar in proportion to the geometry derived from the Golden Section, having proportions between the minor and major chords of the five points circumscribed by a circle, of 1:1.618. From the five point geometry, ten point geometries are easily developed and form the basis for many of the more attractive patterns in Islamic decoration. They are not that common in the Gulf due, perhaps, to the difficulties associated with their construction.

There are many construction methods for basic five and ten point geometries; one of the simpler ones is illustrated here.

Begin with a line which is to form one of the faces of a five-sided figure. With a centre established at each end of the line, describe two circles whose radius is the length of the line. These two circles will intersect with each other twice. Draw a line between these two points. In this case it’s the vertical line.

With a centre based on the lower of the two points of intersection and with a radius established from that point to the ends of the original line, draw an arc which will intersect with the first two circles twice each.

Draw lines from the lower intersections of the new circle and the first two circles, extending them through the intersection of the third circle and the vertical line. These lines will intersect with the first two circles.

Draw lines from the ends of the first line to these two points of intersection and you will have created the next two sides of the pentagon. To obtain the final point necessary to complete the five sides, draw two arcs, their centres based on the previous points of intersection of lines and circles. Complete the five sides of the pentagon.

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Six point geometry

Six point geometry is by far the easiest geometry to construct, requiring only a simple form of compass to create the basis for accurate three-, six- and twelve-pointed forms. I have seen it drawn in Qatar and Iran using both a simple pair of wooden or metal dividers as well as with a string and two nails, one to fix the centre and one to describe the circumference. The string system can be used with nails, chalk or a stone to make a mark on a surface and the system tends to be used for larger circles. I should also add that, with a string and marker system, it is easily possible to draw simple ellipses, though this is very much a hit and miss operation and doesn’t really come within the area of these notes.

In order to construct a six-point design, draw a circle and, with the same radius, describe six circles centred on the points of intersection of each circle along the original circle.

Another way of constructing this geometry is by using seven mutually touching circles; a simple way of illustrating this is to have seven coins touching. This geometry can be simplified or developed into three-point or twelve-point geometry respectively. However, as it requires solids to draw the circles, it is unlikely to have been used traditionally. Having written that, I have seen circular wooden templates lying around in Iran where I watched craftsmen setting out simple geometric designs, but I’m not sure how – or even if – they were used in this manner. The method I have shown here requires the centre diameter of each circle being drawn in order to fix the points of each corner of the hexagon.

An important factor of six point geometry is its relationship with √3. In this illustration the rectangle with its short side coincident with two opposite sides of the hexagon has its long side in the proportion of √3:1 to the side of the hexagon.

The hexagon is one of the more important forms in Islamic geometry. It is simple to construct, it has the capability of producing, in repetition, an overall coverage of a surface, it contains the important relationship of 1:√3, and it bears a strong similarity to the circle, a symbol of creation in Islamic symbology. The hexagon, square and triangle are the basic shapes in this system, the square being associated with the earth and the triangle with human consciousness. In this manner it can be understood that there are a number of elements that would be apparent from an Islamic point of view.

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Seven point geometry

Seven point geometry is difficult to construct accurately, though there is a relatively simple way of making an approximation which is accurate within the normal working tolerances of traditional designers and craftsmen.

Perhaps, because of this basic problem and, particularly, the difficulty of relating other, simple geometries to it, I have never seen it used in Qatar.

On a horizontal line draw a circle and, with the centres on their intersection and the same radius, describe two arcs which cut the circle. Draw two vertical lines from the points of intersection. Construct a third, vertical line bisecting the circle. From its point of intersection with the circle draw a line which meets the junction of the horizontal line and circle. This line will be at 45° to the horizontal line.

With its centre on the intersection of the first arc and the circle, describe an arc from the point where the line joining the 45° line cuts one of the two vertical lines. The length of one of the sides of the heptagon will be from the point where this arc cuts the circle to the intersection of the central, vertical point of the circle. The additional points of the heptagon can be located by describing arcs with radius the length of this line.

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Nine point geometry

Although it may seem counter-intuitive, it is not possible to construct an accurate nine-point geometry. However, there is a method for making a good approximation.

First of all construct a six-point geometry within a circle as described above – by drawing a circle and, with the same radius centred on the circumference of that circle, draw six more circles. Join the intersections to produce a six-pointed star comprised of two interlocking equilateral triangles. Run a vertical line through the centre of the circle. From the point where that vertical line meets the horizontal line of one of the triangles forming the six-pointed star, draw a circle whose radius is half that of the original circle. With the same radius draw another circle with the centre on the junction of the vertical line and original circle. Where the those two similar circles meet, draw a horizontal line. This line will cut two of the sides of one of the equilateral triangles which form half of the six-pointed star.

From these two points of intersection, draw lines to the two points where the other equilateral triangle meets the horizontal side of the first equilateral triangle, and extend them to cut the original circle. The points where they cut that circle will be a very good approximation of a ninth of the circumference. With centre on one of the points of intersection with the circle and radius at the other point, draw a circle to cut the original circle and continue this around the circle to divide it into nine parts. Join these points of intersection to produce the nonagon.

The nonagon is an interesting development and occurs in many Islamic geometric patterns. This diagram, an extended nine-point rosette, is created with a single line joining the points on the circumference of the original circle, the line to be followed being found by the addition of lines joining the alternating points on the circumference.

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Ten point geometry

There are a number of ways of making a ten-point geometry, and I have amended the original drawing I made earlier to suggest what I believe is a simpler method.

With this method, first construct a pentagon as has been described above. With this as the basis, next find the centre of the pentagon by raising a perpendicular from the centre of each of the five sides of it. This is effected by drawing arcs, centred on the points of the pentagon, above and below each side of the pentagon. The junctions of these perpendiculars will give you not only the geometrical centre of the pentagon, but also an additional five points at their junction with the super-inscribed circle. Joining these points with the original points of the pentagon will give you the ten-pointed figure.

Just as there is with the nine-point geometric construction, there are a number of ways to develop the patterns used in Islamic geometrical work. Two are shown here. In the left hand diagram here, the straight red lines show how the internal points of the star are located and, again as with the nine-point star, the complete ten-point star is created with a single line. The right hand diagram illustrates another common construction, though here the star is made up of five similar patterns rather than a continuous line.

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The Golden mean or section

Over time a particular proportion of sides to a rectangle has been found to be attractive to the viewer. This proportion has been named the Golden Section, the Golden Rectangle or the Golden Mean.

There are a number of different ways to construct a rectangle with the proportions of the Golden Section. One is to locate side by side two equal squares, drawing their joint diagonal and dropping this down to meet the projected baseline giving an extended rectangle. Add to this rectangle a third square and divide in half the resultant rectangle. The larger of the two vertical rectangles left by cutting the second square with the vertical dividing line has the same proportions as does the rectangle formed by adding that rectangle to the first square.

These can be seen related to the Golden Section when the pentagram is combined with the construction of a Golden Section, creating a √5 rectangle which consists of reciprocal golden rectangles.

The proportions of the square to the rectangle are:

1:(√5+1)/2, or 1:1·618.

It is also interesting to see that there is a strong relationship between five point geometry, the Golden Mean or Section and Pythagorean triangles. Here I have shown a basic pentagon coincident with the lines of a Pythagorean triangle of adjacent sides 3 and 4, and hypotenuse, 5. Each of the sides of the pentagon are equal and relate to the extended side in the proportion of 8:5, or 1·618:1. Note that the proportions of 8:5 should not be confused with the measurements of the triangle, 3, 4 and 5.

There are so many aspects of this area of geometry to be discovered in Arabic geometry. This diagram, for instance, illustrates a construction where an eight-pointed star incorporates the proportions of the Golden Section within it, though admittedly not relating to the sides. Here A:B=B:A+B, the star being constructed within a grid of eighteen units width and height, the heavier, containing square delimiting a twenty-four unit square which has been the basis for significant investigation on this Russian site.

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Fibonacci series

I should just like to add a word about the Fibonacci series here. Although the sequence is named after him, it originated in the Indian sub-continent over two thousand years ago. Its importance here is that it relates to the Golden Section in that, if you take the proportion of one number to its neighbour, it approximates the Golden Section – the further along the sequence you go, the more accurate the approximation. This diagram illustrates the way the sequence is built up – 1 + 1 = 2 | 1 + 2 = 3 | 2 + 3 = 5 | 3 + 5 = 8 | 5 + 8 = 13 | 8 + 13 = 21 | 13 + 21 = 34 | 21 + 34 = 55 etc. At its simplest the relationship can be written as:

a : b is as b : a+b

While I’m at it, and I know this has little to do with Arabic geometry, I thought it might be useful to place here a reminder that these geometries were used in other parts of the world. To the side is a detail of the volute on a nineteenth century Ionic capital, developed from those which preceded it over two thousand years previously. They were developed with a strict geometry related to the understanding of natural forms at the time. Although similar to the geometry shown above, this particular example is not based on a Fibonacci series.

Later, these constructions were developed, again based on similar geometric principles. This sketch illustrates the basis on which a volute may be formed. What is significant is that although the volute looks as if it is a constant curve, it is not: it is a series of joined quadrants. Quadrants are formed of increasing size, each linked to its preceding quadrant with its centre further offset. I have drawn each quadrant a different colour to illustrate the sequence, though it can be readily imagined how variations can be established to produce the lines of the volute closer or further apart. Incidentally, the word ‘volute’ is derived from the Latin word for a scroll.

You can see in the above diagram one of the many forms commonly to be discovered in nature. Shells, fruit and flowers are often found with this type of geometry driving the arrangement of their forms, and which can be found exhibited in a number of ways. I should add that similar forms are used in design and architecture, and have been for thousands of years, the designers taking their geometrical vocabulary from that which they have seen in the nature around them. Designers have learned from this and have attempted to develop a codified series of proportions which, they feel, may improve our resonance or psychological comfort with buildings.

Based on this there have been a number of proportions suggested, perhaps the two best known being Leonardo da Vinci’s ‘Vitruvian Man’ and Le Corbusier’s modular systems.

But returning to the Fibonacci series, this may also be seen in the external surface of the pineapple above which has eight spirals in one direction and thirteen in the other; and the same is true for the pine cone in the photograph below it, underneath which is an immature pine cone on which you can also see the spirals. Incidentally, sunflower heads have thirty-four florets spiralling in one direction and fifty-five in the other direction, continuing the Fibonacci series. The lowest of these four photographs, of a cactus, while not based on a spiral configuration, shows it has a geometry based on thirty-four, again one of the numbers of the Fibonacci sequence.

There are many plants and other aspects of nature which exhibit this form of spiralling geometry. Here to the right is a dahlia, though I have to admit I’m not sure of the numbers involved due to the lack of accuracy of positioning in the petals of this lovely flower. I think it is thirteen and eight but, as I say, I’m not sure. However, you should be able to see the resemblance with the geometries of the pine cone and pineapple above. Geometry, whether it is as obvious as three-point, four-point and so on, or is related to Fibonacci proportions as these plants are, produces a seemingly endless variety of effects. Below the dahlia is a photograph I took of a cactus. Although you may not be able to see it easily, there are twenty-one spirals in each direction, again a number associated with the Fibonacci series.

Wherever you look there are patterns to explore. I have not meant this parts of my notes to be a series of photographs of spirals and designs associated with the Fibonacci series or other groupings, but I think it is useful to look at these for a minute to get an idea of how prevalent this is. Whether there is an identifiable pattern or whether it is a regular progression there is considerable difference in similar geometries though, of course, these all have a three-dimensional aspect and I am really only dealing with two-dimensional geometry here.

Just one more example before I stop. Here is a photograph of a South African polyphylla aloe which shows the almost perfect geometry of its spiral form with the leaves increasing in size as they move towards the outside of the plant. There will be a mathematical relationship governing this growth, though I am not able to say what it is. It is not formed in the same manner as lies behind the carpet below, but you should be able to see an interesting similarity.

I have added this photograph as it demonstrates what is to me an interesting point. This Persian carpet has been constructed with what appears to be a Fibonacci geometry driving the pattern. But there are thirty-two spirals, not the thirty-four you might anticipate from Fibonacci. So the design is based on four- or eight-point geometry from which the spirals are derived.

Here I have attempted to draw what I believe to be the geometrical construction behind the pattern. The inaccuracies are likely to be a combination of the angle at which I took the photograph, possible inaccuracies in the construction of the carpet and my inability to work out the exact points of geometrical derivation. However, I think it’s close enough to see the likely basis. Note that I have not shown the thirty-two divisional – or 11¼° construction – as they are just sub-divisions of 45° and would complicate the diagram.

The reason for briefly describing these geometries is to show that some of them are relatively complex, and that we tend to take them for granted. Despite this, it is instructive to watch craftsmen on both sides of the Gulf using these geometries with only a straight edge and string to construct complex patterns.

Many of these are undoubtedly traditional and easily learned but, from observation, changes and customisation still takes place, making each element of work unique while informed by and related to the country's heritage.

Having said that, it is undoubtedly true that the geometries used and patterns formed in Gulf design, particularly naqsh, are relatively simple. Naqsh is, after all, a relatively simple material, the setting out and carving traditionally carried out with drying juss mortar where speed is imperative. It is relatively easy to carve dry although, wet or dry, it is a relatively crude technique.

However, the designs produced and techniques used in this relatively simple craft differ along the Gulf. More sophisticated designs are found further south, the patterns there being more fluid and the elements of the design finer.

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Relationship of Arabic calligraphy to geometry

Although I don't want to deal with calligraphy here in any detail, it is relevant to note that Arabic calligraphy is firmly based on geometric proportions.

The most important form is that illustrated below and shows, on the right, the first letter of the Arabic alphabet, alif, which equates to the long 'a' in English. Note that it's proportions are 1:7. The style is known as al khatt al mansub and was designed by the great calligrapher Abu Ali ibn Muqlah. To the left of the alif is the letter 'ain which shows a cursive letter in the alphabet based on the same size dot.

Finally, there are two points which should be borne in mind with regard to calligraphy and its relationship with rigid geometry:

• there are a wide variety of manuscripts in Arabic, and the alif can differ in its proportions from 1:3 to 1:12, and
• traditionally, letters are formed with a pen which, while being held at an angle, is also varied by the scribe as he writes to create angles other than the basic 45°.

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Application of traditional patterns

Bearing this in mind, traditional patterns are only applied in four areas:

• naqsh panels,
• carved timbers, particularly the enf door posts,
• woven patterning in the rush ceilings, and
• painted patterns on boarded ceilings.

None of these patterns has developed to the extent seen in the repetitive patterns of Persian tilework. While the patterns found in the Gulf are non-figurative geometrical designs, they have not developed along with the mathematical complexity seen in Persia, north Africa or Andalusian Spain.

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Platonic solids

Towards the top of the page I mentioned briefly the related concepts of sacred geometry and geomancy. That geometry was essentially two-dimensional, but there are three-dimensional forms too, of course. Of these there are basically only five regular three-dimensional forms, the group being known collectively as the Platonic solids after the Greek philosopher and mathematician, Plato, who wrote extensively about them in his philosophical studies. My purpose in mentioning them here is only to introduce them as a development of two-dimensional geometry. Anybody wanting more information about them should look elsewhere.

Despite this naming, they are generally considered to have been discovered by Pythagoras or his group, though it is possible that Theaetetus, a contemporary of Plato, may have been responsible for the octahedron and icosahedron. While their discovery and naming is credited with Pythagoras and Plato, a point to bear in mind is that, like much else, there is considerable evidence that they were discovered centuries earlier in other parts of the world.

The five forms are known – in ascending order based on the number of sides – as a:

• tetrahedron,
• hexahedron,
• octahedron,
• dodecahedron, and
• icosahedron

These three-dimensional forms are considered to be the bases of all natural forms and, as such, are related to the very essence of the universe.

The five figures are composed as follows:

• the tetrahedron has four equilateral triangular faces and six edges,
• the hexahedron six square faces and twelve edges,
• the octahedron eight equilateral triangular faces and twelve edges,
• the dodecahedron twelve equilateral pentagons and thirty edges, and
• the icosahedron twenty equilateral triangles and thirty edges.

These forms might be considered while thinking of the two-dimensional geometries as well as the patterns derived from them. Two-dimensional geometries are often developed into either a three- or pseudo three-dimensional geometry by the use of shadow patterns or inter-weaving. My own experience, when working on or contemplating two-dimensional patterns, is that it is relatively easy to move into a third dimension view as the patterns form and re-form in front of you.

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Seventeen different patterns

Although there appear to be an infinite number of patterns to be seen around us, in reality there are only seventeen different ways in which patterns can be repeated two-dimensionally. Examples of nearly all of these patterns can be found in the Alhambra, Spain. It is stated by Grünbaum that there are four patterns missing from the Alhambra – pg, pgg, p2 and p3m1 – though the latter two are found coaevally in Toledo. He also states that the former two – pg and pgg – are not found in Islamic art at all, though this has been disputed. It is not my intent to argue this case here. Those with an interest should look elsewhere.

There are certainly less than this to be found in the Gulf, if for no other reason than there are not that many examples to analyse and, of these examples, few cover areas of the size required to see the repeats effectively. More than this, the patterns seen in Qatar tend to be iconic and are not designed specifically to cover large areas as might be found in, for instance, the glazed tilework on many buildings in Iran and further afield in the Indian sub-continent, Egypt and Morocco among others. It is possible that, if there had been a tradition in Qatar of tiling, there might have been a very different situation as craftsmen from Iran would most likely have made their influence apparent.

The seventeen different ways for patterns to be formed have been established mathematically and described notationally. They depend on taking an element and then repeating it by

• rotation,
• translation, or
• reflection.

There is a simplified diagrammatic explanation of these further up the page.

There are many sources of information for those interested in learning something of the mathematics governing the patterns. One useful reference is this which gives a simple view of the alternatives:

• the first group of four are made without rotation and are known as: p1, pm, pg and cm,

• the second group are constructed using rotations of 180°, without rotations of 60° or 90°: p2, pgg, pmm, cmm and pmg,

• the third group are constructed with rotations of 90°: p4, p4g and p4m,

• the fourth group use rotations of 120°: p3, p3ml and p3lm, and

• the fifth group are constructed with rotations of 60°: p6 and p6m.

The coding system is that of the International union of Crystallography, but alternative systems have been developed, such as those relating to topology, and other classifications have been put forward. I have not yet found a simple way to describe this to the layman other than this Open University programme.

With these basic arrangements there is an infinite number of ways in which patterns can be arranged together to give different effects. Shape, colour and texture are all used in Arabic design, as is the effect of three-dimensions in more sophisticated work.

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An eighteenth geometry

Before I finish with these basic notes on Islamic geometry, I should mention that there is one other set of patterns to add to the seventeen I noted above as being the total number of basic patterns. This pattern may not be found in Islamic designs, but is notable for the character of its non-repeating pattern. It was discovered in 1974 and subsequently patented by the mathematician Dr. Roger Penrose.

What is unusual about them is that, previously it was thought that only patterns based on two, three, four and six rotational symmetries could tile a plain surface, and that five- and ten-sided geometries could not.

Relying on two rhombi using angles based on π√2 – 36°, the basic angle of the Golden Section – they are assembled according to a set of rules he devised to ensure no repetition.

An interesting effect is created when running the eye over the basic geometry as the eye automatically finds familiar shapes which disappear as different shapes take over. This happens with the eye finding both two-dimensional shapes as well as three-dimensional shapes as the brain suggests three-dimensional shapes with which it is familiar.

The two rhombi are assembled into patterns using their two characteristics illustrated here. The rules require that:

• two adjacent vertices must be of the same colour, and
• two adjacent edges must have both arrows in the same alignment, or no arrows at all.

These basic two elements – in accordance with their assembling rules, can be grouped into eight permissible clusters. From these, non-repeating patterns can be constructed giving, in theory, an infinite and non-repeating design.

It is interesting to speculate on how this patterning would have been used by Arab designers had the basic geometry been discovered a thousand or so years ago. My feeling is that the asymmetry would be admirably suited to the premise of man’s inability to know everything, and the infinity demonstrable in two dimensional design. It would have been an ideal way in which to cover plain areas of walls in a non-repeating pattern.

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Decagonal and quasi-crystalline tiling

Having written the above note some time ago there has been, in February 2007, a significant paper written on the possibility that medieval Islamic artisans produced their geometries with the aid of girih tiles. These tiles are equilateral polygons decorated with straplines which, it is argued, permitted the creation of complex periodic patterns. The tiles were of five shapes: hexagon, bow tie, rhombus, pentagon and decagon.

Here are the five tiles:

Termed ‘quasi-crystalline’ patterns, they are patterns which fill all of a space, but without the translational symmetry characteristic of true crystals. Essentially this means that attempting to match an exact copy of the pattern over itself will never produce a precise match. In this they are similar to the tiles to which Dr. Penrose put his name in the 1970s. The thesis is that Islamic designers developed these geometries between the thirteenth and sixteen