A background to two-dimensional design – geometry and pattern

A brief introduction to the history of Arabic geometry is set out further down the page. But I believe it would be useful to begin with a note on the understanding we in the West have of Arabic or, perhaps more accurately, Islamic geometrical design as it is a wide field of study and there are certain misconceptions. At its simplest, Arab scholarship enlarged our knowledge of underlying geometry, and Muslim craftsmen, not all of them Arabs, relied on this work in producing their characteristic fields of tiles, mosaic, plaster and wooden patterns that were applied to their buildings.

To take the argument a little further, the distinction might suggest that the work is considered Islamic when employed in the service of religious buildings. However, with Islam governing all aspects of Muslim life, the argument would suggest that geometric designs, with their inherent character supporting introversion and reflection, should all be considered Islamic.

The first example, above, typifies the kind of design that comes to mind when thinking of Arabic geometric designs. However, the example is not from Arabia but was made in France and is one of a pair of silver door panels, shown above on its side. Incidentally, the other panel of the pair, although having the same basic geometric construction, has different detailing as is illustrated here.

In a very similar manner, this photograph also illustrates the centre of a door panel. Based this time on twelve-point geometry rather than the eight-point shown above, there is an evident family feel to the design of the two panels, one that is familiar to many people as characterising their recognition of Arabic or Islamic design. Again the pattern is formed by the use of geometrically arranged ribs set on the same plane, containing foliate decoration, but here with contrasting materials used to emphasise the distinction between the two characters of Islamic design – lineal geometry and foliate decoration.

I have included these next two examples of Arabic 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 example to the side, 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.

This photograph, too, shows an example of eight-point geometry though it may not appear at first sight, and to Western eyes, particularly Islamic in character. Compare it with the examples above. It is in fact an example of tatting, a practice very similar to crochet work, and was designed in conformance with traditional English styling for this particular character of knotting. What is notable here is that while the two-dimensional design pattern depends to some extent upon Western traditions, there is still a hint of natural Islamic geometry to be seen in the design with the natural, cursive forms following, and contrasting with, a more rigid geometric setting out. But, on more detailed examination, there can be seen to be an active interplay between geometrical layout and constructed detail, a characteristic of much Arabic and Islamic geometrical work, and the reason for its being used here as an illustration.

This next group of illustrations are based on a trio of panels displayed together in the Qatar Museum of Islamic Art. Each of them is a beautiful example of Islamic or Arabic geometric artistry, and I believe they are well worth study to determine the geometry on which they were based. Further down the page there are illustrations of how simple geometries are constructed – for those who are not familiar with geometrical systems – but here I have shown them in their complexity as patterns I find attractive.

Here is the first panel showing its design and framing, the end frames featuring a small running design. The most probable reason for this extension to the length of the panel is the need to make up the difference between the proportion dictated by the geometry and the requirements of the opening into which the panel would be used.

The first panel is based on a run of circles containing figures set out on the basis of twelve-point geometry, the circles mutually touching on their circumferences and located along a common straight line. In fact there is a central circle flanked by two semi-circles providing the basis for the pattern but, in common with many such patterns, the geometry has to be extended beyond the panel in order to establish all the necessary lines.

In order to construct the basic geometry of the panel, diagonals are drawn at 30° intervals through the centre of each circle, cutting the circumference and being extended. A line is drawn between the intersections of each diagonal with the circumference and extended in both directions.

From the points of intersection of each of these extended lines outside the circle, new lines are drawn across the circle to meet each fifth point of intersection. This produces pairs of parallel lines each side of the circles’s centre. Where these lines cross the lines drawn around the circumference, they will produce the setting out for the red lines which comprise three regular pointed crosses for each circle, producing the familiar twelve-pointed dodecagon.

The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.

This second panel has, like that above it, decorative panels at each end, producing a balanced frame to the internal, main pattern. The decorative panels are not running designs as are those on the first panel, but are carefully structured to reflect the proportions of the main design of the panel.

This design bears a resemblance to the first panel in that the two central figures are again based on twelve-point geometry. But this design is created by two small differences in the layout geometry: firstly the circles do not touch and, secondly, their mutual arrangement is designed to be at 45° to the basis of the pattern of the first panel. It is in such relatively small alterations to basic geometry that the apparently infinite variations of Islamic designs are created.

The basic geometry relating to the divisioning of the circle is the same as for the first panel, as well as for the setting out of the lines extending from the intersection of the circumference of the circle with the diagonals. But there is a major change establishing the distance between the circles. This is effected by the lines coinciding with the similar lines of the adjacent circle. The whole arrangement is then turned through 45° to produce the framework for the design of the panel.

This third sketch, based on the layout of the panel, illustrates how the overall pattern produces a very different design from the similar study of the first panel shown above, here each group of four circles having a fifth circle between, but not touching them.

The third of these three panels is very different from the two above. Whereas the central circle is based on twelve-point geometry, its two flanking circles are based on ten-point geometry, and it appears that it is from those that the basic setting out relationships are derived. The distortion of the photograph has been corrected but this has made it difficult to fit the geometry to the image. However, it does allow you to see some of the eccentricities, such as the irregular five-pointed stars between the central and flanking circles created by the slight differences inherent in the two geometries.

Mixing geometries of different bases is not uncommon in Islamic design. The process adds to the possibilities for invention and so enlivens the designs in which it is deployed. In so doing, the underlying construction can become more complex, particularly in establishing the linking shapes. Because of this I have not displayed the continuation of pattern over a wider field, but hope to do so later.

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Carpets and rugs

In addition to finding Islamic geometries on metalwork, tilework, plasterwork and woodwork, we also associate it with carpets and rugs, particularly with Persian carpets that have a long tradition of use in the area of the Gulf and beyond. One of a pair, the carpet illustrated here is one of the most famous Persian carpets, the Aradbil, named after the area from which it originates. It is approximately 1040 x 530 centimetres and was woven at a density of 49 knots per sq.cm. around 1540 AD. Its central feature is based on eight or sixteen point geometry set in a cursive and floral field with decorative surrounding frame. There is an inscription at the top of the carpet, on the right in this photograph.

Looking at the Ardabil carpet, even in this extremely small photograph, you can see how well proportioned it is. Here I have illustrated the geometry upon which the design of the carpet has been based. I will show a fuller geometry, but the key point to note in this sketch is the use of a square in the centre flanked by two rectangles, each having the proportion 1:√2. You can also see how the outer border of the pattern has been established but, again, this should be more clear in the illustration below.

In this study you can see a fuller geometric basis, one that shows the wider setting outside the carpet. This illustrates a point made elsewhere: that one of the chief characteristics of Islamic designs is their continuity outside their encompassing frame. While there are no lineal devices within the carpet to suggest this, there is an implicit continuity outside the carpet that can be sought, and responds to contemplation and reflection.

I have not drawn the stages of construction as they are easily worked out by examining this sketch. Essentially they derive from the central small square upon which a circle is drawn of radius equal to half the diagonal of the square. Circles of the same radius are drawn on the extended lines forming the central square. Connections between the eight encircling circles automatically produce the 1:√2 rectangles that flank the central square. It is also evident from the sketch how the outer edge of the frame is defined, the frame being the difference of the inscribed and exscribed circles relating to the small square.

While the upper sketch illustrates the manner in which the basic shape and the setting out of the main elements of the carpet were effected, I have not yet been able to work out the basis of the setting out of the smaller elements of the pattern, nor of the width of the bands defining the borders, but believe there must have been a rationale to establishing all of them, and that this might be significant. What I have noticed is that the inner motif of the central circular feature has a radius equal to the width of the blue lines defining the border of the carpet. The four quadrant patterns found in the corners of the carpet have the same outer radius as that of the central circular feature, though the inner motif has a larger radius.

In this sketch, I have expanded the geometry of the field in order to create something of the feeling for an infinite pattern within which the carpet pattern sits. In this latter pattern, the frame is derived from the geometry of the overall continuing geometrical field, and might be considered to be placed anywhere within this continuum creating, in effect, a sample from this continuum.

The key feature of carpets is that they are knotted, a time-consuming and skilled exercise, quality depending upon the number of knots to the square inch. While Persian carpets are objects of desire as well as functional and transportable items of a household, traditionally it was the woven kilim that was the less expensive and more versatile rug to be found in the Gulf. Usually brought over from Iran the most commonly found in Qatar seems to have been those woven by the Qashqai from the region around Shiraz in present day Iran.

Due to the relatively coarse manner in which they were woven, these rugs tended to have simple geometries, a hint of which can be seen in these photographs relating to traditional weaving in the peninsula though, as you can see, they are carried out on a narrow, portable loom producing strips that are later sewn together to produce runners. However it does illustrate the mechanical geometrical form the weaving creates, and also something of the limitations this type of weaving presents. In these two photographs of the corner of a small kilim and a larger kilim, you can see clearly how the woven form creates patterns based solely on right angled geometry, with implied diagonal lines being created by stepping the weave along the warp. The warp threads are those running the length of the carpet, over and under which the weft threads are woven.

Here, the photograph is of the edge of a Qashqai kilim, seen on its side. Although the colouring is not that typical of this type of kilim, the simple running edge detail can clearly be seen to have its patterning derived from 45° geometry, itself derived from the manner of weaving the piece. The patterning in the centre of the kilim also makes use of 45° geometry.

While this type of weaving produced a characteristic pattern responding to the governing geometry of woven rugs, a secondary development saw the use of stepping as a way of varying the 45° geometry. This third photograph illustrates clearly how the angled geometries steeper than 45° are woven. It also shows the characteristic slots created by the return of the weft threads.

In this detail of the centre of a Qashqai rug, the characteristic hexagonal form has been used as a feature. This hexagon is obviously not a true geometry with internal angles of 120°, but is an approximation created by the weaver. Nevertheless, the organisation of the pattern within the hexagon continues the feeling of six-point geometry with its incorporation of triangular motifs.

Stepping the weaving is the only method available to create the lines of geometry on a kilim other than those on the horizontal and vertical. Because of this the angles are either right angles or are formed by using different ratios of horizontal weft to vertical warp threads. Note that taking the weft around the warp leads to a series of slits being formed in the kilim – one of its features. This diagram illustrates the angles formed by using ratios of 1:1, 2:1 or 3:1 in the stepping of the pattern. Note that the angle of 72° is related to five-point geometry, and that 63° is an approximation related to six-point geometry.

The stepped element of pattern is a standard feature of woven rugs and can be found all over the world in artefacts, including carpets, using a woven method of manufacture. In Iran and areas adjacent to it in the north and east, there was also a hexagonal pattern developed, similar in some respects to the gul pattern of Turkmenistan tufted carpets, though the latter is commonly octagonal. The kilim shown here is particularly eccentric in its weaving but is here because it displays variations created, most probably, by a novice. The kilim has none of the strong colouring that characterises a Qashqai.

Finally, with regard to rugs, I am including a detail of an embroidered Iraqi kilim to illustrate two points. Firstly, not all carpets and rugs are geometrically based; this kilim is completely covered with non-geometrical images. The second point is, of course, that there are still figurative works to be found in the Islamic world. At the top of the photograph a line of simplified figures can be seen and, below them and on its side, is what appears to be a dancing figure. Elsewhere on the same kilim are stylised birds and camels.

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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. The aim is to avoid idolatry. For this reason you will not see anthropomorphic art in mosques and 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.

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 featuring water, trees and mountains.

This might be thought to cater for the upper end of the market, but there have always been works produced for less expensive enjoyment as in this example above.

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 work associated with new offices and hotels as well as, and particularly on, roundabouts and in public spaces. Although differing in concept, it is also notable that there are a significant number of portraits of the Ruler and other members of the State or Government in government buildings. 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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Sacred geometry and geomancy

Before I continue the notes on 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.

Further down the page I have set out ways of drawing each of the basic geometries but, first, let me deal with the two simplest geometric constructions which form the basis for many of the patterns to be found in Islamic decoration. They are respectively the constructions for six and four point geometries. You may find the methods slightly different from the constructions shown further down, but they all produce the correct results. These next two diagrams read left to right.

The construction of six point geometry is by far the simplest construction. Draw a straight line and, with the centre of your compasses on the line, draw a circle. With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius. With the compasses centred at one of the two points where this circle cuts the first circle, draw the next circle, continuing this process until there are six circles mutually intersecting and centred on the original circle. You will now have a basic construction for patterns derived from six-point geometry.

The construction of four point geometry is only marginally more complicated. Again, draw a straight line and, with the centre of your compasses on the line, draw a circle. With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius. Repeat this with the compasses centred at the other point where the first circle cuts the line. You will now have three circles, all centred on the line. With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line. Where these arcs intersect, draw a line. This will intersect with the first line at right angles. With the compasses centred on the two points where this vertical line intersects the first circle, draw two circles with the same radius as the previous three circles. You will now have a basic construction for patterns derived from four-point geometry.

These relatively simple geometrical constructions are easy to develop for pattern making. With a pair of compasses and a straight edge it is easy to create the circular forms and then, by joining various intersections, to produce the basis for a variety of pattern making. The points selected may be the obvious intersections or, as illustrated lower in the page, more unusually related points. These diagrams show the simplest connections.

In this first exercise I have taken the basic seven-circle rose from the six-point geometry construction illustrated above and shown how the basic pattern lines evolve. The first illustration shows, on the left, the basic circle rose with, to the right, the addition of lines joining the intersections of the surrounding circles with the basic circle to create a regular hexagon.

These two illustrations show how this simple exercise can be extended. On the left every second point of intersection of the surrounding circles with the basic circle has been connected, creating two interlocking, regular isosceles triangles, creating a regular six-pointed star. To the right it has been amalgamated with the hexagon to produce a basic pattern former that is found in many examples of patterning.

In this pair of illustrations the basic six-point geometry has been developed to form a basis for twelve-point patterns. Firstly, every second pair of the points of intersection of the surrounding circles with the basic circle, are joined as in the isosceles triangles example above, and then the points of the triangles’ mutual intersection are joined and extended to cut the basic circle.

In the example above right, every fifth point on the central circle has been connected with what is, in effect, a continuous line to create a twelve-pointed star. In this pair of illustrations, that on the left has had added to it lines joining every fourth point to create a more complex pattern and, to the right, every point has been linked to create the most complex pattern.

Here is a detail of the above right illustration which gives more of an indication of the complexity of the underlying geometrical web formed by the development of the circular patterns. Looking at it you will see that there are natural junctions formed by the lines. This gives a series of points that can be used in the development of patterns. This type of framework forms the driving geometry for the relatively simple Islamic patterns with which we are familiar.

While there can be seen to be considerable room for variation in the dividing of a single circle, the more common basis for a pattern is a simple grid, here the development of the circle suited to six-point geometrical patterning. Even at this scale the eye finds it difficult to rest, but is continually moving around the circles. The addition of straight lines just develops and guides this movement.

Here, for comparison, is a basic grid based on the four point geometry illustrated above. Note that the circles in these two diagrams are the same size, but the patterns have markedly different visual densities. I have not gone through the exercise to demonstrate how the four-point geometry is developed as I have with the six-point above. But it will be obvious how squares, octagons and eight and sixteen point stars can be readily developed.

A slightly different development of this illustration is shown in the next diagram and is repeated here to show how a relatively simple pattern based on six-pointed geometry might be developed. The basic grid shown in the six-point study just above, has been turned 30° counter-clockwise and a number of straight construction lines selected on which the pattern has been produced. There are so many potential choices for the designer: this accounts for the great number of design variations that are possible.

As I wrote, 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 four reasons. First, the expression of religious art through geometry is intrinsically interesting; secondly, there are only two examples of this type of work in England, at locations to which there is relatively easy access, London and Canterbury; thirdly, these are the only examples of this type of work north of Italy and, fourthly – 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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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 and Islamic 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 depending, as it does, on clear geometric development.

Although it may seem counter-intuitive, there seems to me to be a connection in this with Japanese Zen Bhuddist philosophy. Contemplation of a restricted vocabulary of objects – traditionally, raked sand and rocks, planting and containing walls – can produce in the viewer the effect where time and space collapse or dissolve in a similar manner to that experienced in the contemplation of Arabic tilework. This suggests that there is a need in all of us to find a mechanism that will permit us to escape the immediate pressures of life, and allow our minds to wander. In Zen this becomes a form of meditation that is designed to encourage and lead to enlightenment. The major difference between Zen and Islamic contemplation would be the focus Zen has on the natural world with its three-dimensional forms and their association with nature. In Islam, tilework is not dissimilar in its effect, although may not have specifically developed for this purpose. Perhaps tilework has evolved as a subconscious exercise which, while bound by the rules governing Islamic illustration and decoration, produces a very simple palate of pattern and colour, allowing the mind to contemplate the infinite.

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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.

Yet the patterns on buildings are generally two-dimensional, there being no real perspective design work in Islamic art, particularly in pattern design. There may, however, be patterns placed on patterns which can give an illusion of depth and, in timber and naqsh work there may be actual three dimensional construction, though this is really a projection of two-dimensional pattern. Sometimes the illusion of depth can be found by the implied weaving of pattern lines above and below each other.

Moreover, it is a common feature that containing frames tend to appear arbitrary, implying the continuity of pattern beyond the frame that the brain sees, comprehends and mentally extends as part of its normal working. This is in sharp contradistinction to Western art, where the frame generally comes first and the work is formed within it, often having a direct relationship with the frame.

Reading patterns

Although I have written notes about it elsewhere, it would be useful to place a brief note here on the reading of patterns. The notes on the page dealing with perception essentially argue that Arabs will tend to read what they see in front of them from right to left, compared with Westerners who will read from left to right. Both these patterns of reading apply not just to writing, but to the whole of the visual field in front of the viewer.

Arabic or Islamic patterns, by their very nature, do not have a direction implicit in their geometry but tend to be formed in fields with no apparent ending. As I noted above, the framing of patterns, whether on a document, piece of furniture or on a wall, tends to be arbitrary, as is illustrated in this drawing to the right where the pattern stops at the top and bottom.

In this sense, even if the viewer’s inspection enters the piece from the right or left, the piece will be read in an irregular manner, the eye following lines or elements traced out in the pattern, and reading both explicit and implicit patterning within, and even outside, the pattern as the subconscious search for unity takes place. The upper photograph here illustrates the way a field pattern may be seen.

In the lower photograph, where the geometric pattern is focussed and not in the form of a field, then the eye is likely to move first to the centre of the design as this is the point from which the geometry is driven. In most cases this will mean the centre of a circle or, as here, the point derived from a circle, as most Islamic geometry is derived from the circle or from a square, the latter implicitly also associated with the circle, and with the grids formed usually being based on equilateral triangles or squares.

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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. Not every flower is regular in this respect, but I have generally sought out plants which display a relatively simple two-dimensional geometrical form. Bear in mind that many plants have irregular geometric forms and that some combine different geometries as may be seen below where three, five and ten can be seen to form the basis of the flower’s pattern. Three-dimensional forms make the form of plants even more complex, of course.

Two, four and eight point

These first four photographs, shown to the side, illustrate flowers that have, respectively, two, four, four and eight petals. Despite this, and although four petals might be thought to be the simplest form of geometrical arrangement, it is flowers with five and six petals – and, of course, plants and other natural forms – that seem to be the more common and appear to predominate in the world around us.

When you look at the two, four and eight petalled flowers their shapes appear to be far more balanced and stable in terms of their geometric appearance because we tend to equate them automatically with the character of the square or rectangle, symbols of stability in our experience of two and, particularly, of three-dimensional forms based on them. In fact in English we use the term ‘foursquare’ to mean solid, honest and forthright. This quality is especially obvious in the two photographs of the four-petalled flowers, though notice that the white flower, by virtue of its being photographed at a slight angle, looks more active or lively than the horizontally located green leafed plant below it. This effect is something I will discuss briefly a little further down the page.

The eight-petalled white flower has also a stability apparent in it though, by virtue of the photograph being taken with single petals top and bottom – as opposed to a pair of petals top and bottom, again an effect discussed further down the page – appears to be both stable and active. This is not to say that other geometries appear less stable, but it is a characteristic of evenly numbered geometries that they appear more relaxed and balanced due in the main to their symmetry.

Five point

However, as I mentioned above, it is five- and six-pointed geometries that seem to be the most commonly seen and experienced in the natural world. I have to admit that I’m not sure whether five or six divisions are the more common, 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 proportions of the Golden Section, discussed below, might be argued to represent a more perfect proportion than plants based on other geometries and, therefore, more beautiful in that arrangement.

Looking at the different forms five-pointed geometries take – illustrated with these four photos to the side – the plants do seem to me to have a more interesting or exciting form than those based on four- or six-pointed geometries, though the reason for this might be relatively simple and is discussed and illustrated below these photographs of flowers, having to do with the regularity and balance of even and odd numbers.

Odd numbered divisions of petals – or any other natural or mechanical form for that matter – have an intrinsic disbalance that creates a more interesting or visually stimulating appearance for the viewer. The lower two photographs of blue and white flowers illustrate the phenomenon well, particularly if you compare them with the six-point geometries of the patterns in the next group of photographs illustrating six-point geometries. They are both very small plants with the flowers occurring in clusters, but the flowers both show how the sub-divisioning works.

Although I may seem to be making a case for one form of geometry being more attractive than another, that is not the case. With reference to Arabic geometry don’t forget that typical patterns can mix more than one type in their overall form.

Six point

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.

It may seem unusual to see snow crystals on a page dealing with Islamic geometry, but I wanted to emphasise the point that these geometries govern designs to be found all over the natural world. Snow crystals take many different forms, but the majority of them are based on six-point geometry though, as the crystal on the right demonstrates, they may also have a basis in twelve-point geometry.

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. This to the right has the regular form with six-point geometry demonstrated in the disposition of its anthers and petals.

Seven point

This primula, 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…

Three and nine point

It appears that six-point geometry can be discovered all over the natural world, and is readily found almost wherever you choose to look for it. But it seems far less easy to spot three- and nine-point geometry which you might expect to be similarly readily apparent, though it appears to be that they are not. Unfortunately I know little about botany and have no idea why this apparent lack of three- and nine-point geometry might be 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 are two plants that exhibit nine-point geometry. Firstly, there is a lewsia flower displaying its nine petals. Unfortunately, this particular specimen is not as regular as I would have liked but, nevertheless, it has a satisfying balance to its symmetry. Below it is another plant, which I believe is a celandine, that also shows nine-point geometry as its basis, and in this case, with a more geometrically accurate display.

Ten point

There are a significant number of plants that exhibit five-point divisions in their geometries, a couple of which are illustrated further up the page. It follows that it is also very common in nature to find plants which are divided into ten divisions. Here a photograph illustrates the ten segments of a peeled and opened tangerine, though I should add that it is relatively common to find variations to this rule.

Below it are two photographs of the petals of a 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. This may seem obvious and will have much to do with the sizing of elements of the 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 would mean that there are three-, five- and ten-point geometries found together – assuming that seventy-two is related to three-point geometry. How the seventy-two is made up, I have no idea; it could, of course, be based on three-, four-, six-, eight-, nine-, twelve-…

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.

Eleven point

While a designer is free to make selections based on a variety of rules of his or her own determination – which may be either deliberate or even accidental, but which imply a degree of control or selection and conscious decision-making – nature often produces accidents in its geometries due to a variety of reasons over which there is little or no control. A few examples of this can be seen illustrated both above, and below.

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 and accurately guessed at a glance and easily distinguished from twelve divisions with little or no practice.

The second and third photographs are examples of one of nature’s many little eccentricities, poppy heads of eleven divisions found hundreds of miles away from each other. To illustrate how these eccentricities seem to be found everywhere, the lower two photographs are of poppy heads found side by side on the same plant. The upper of the two can be seen to have eleven divisions, whereas the lower is divided into twelve.

Fourteen point

This photograph of a large daisy appears to show fourteen petals. Looking at it carefully I decided it hadn’t originally had fifteen, but that fourteen was its normal state. However, it is obvious from some of the photos seen here that there can be a number of different divisions of the same type of plant, and that few plants with many divisions consistently show the same number. Variations happen in nature and this, together with natural eccentricity of shape and placement bring increased interest to the plant, as in this example.

Other divisioning

As the number of divisions of a plant increase there seems to be an increasing incidence of eccentricity. That, coupled with natural departures from the perfect form or placement, makes it increasingly difficult to count the number of divisions. These two photographs, of a dandelion and a large daisy, illustrate how there appears to be a strong geometry driving the form, yet how the eccentricities make it virtually impossible to suggest a definite figure for the number of divisions of either of the flowers.

What I find interesting in these cases is the resemblance there is to the conscious or subconscious study of Islamic patterning; not in the accuracy of their geometry, but in the irregularity and lack of conformity, a similar characteristic to the manner in which Islamic patterns encourage the eye and mind to wander in exploration of the geometry driving their forms. Although flowers have a discrete number of petals and Islamic patterns have implied continuity outside their actual fields, there still seems to be a similarity.

But not all elements in nature, in this particular case, flowers, have geometries radiating from a single point in straight lines. This is discussed below, but I will add this photograph of a flower where its leaves can be seen to be spiralling out. Obviously this has a lot to do with the mechanical requirements the flower needs to move its petals from a closed to an open position, but in doing so it provides a more complex appearance for us to appreciate even if, as in this case, the geometry is not strictly regular in the terms of the flowers I have illustrated above.

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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

The construction of patterns has been the subject of research and investigation for centuries and, while I have witnessed the use of straight edge and string in the setting out of Islamic patterns, it is evident that much is carried in the heads of craftsmen that is not immediately evident in watching them work. Nowadays we have greater resources available to investigate pattern conceptualisation and construction; we also have the capability to examine non-Euclidian geometry and create designs which would not have been possible six hundred years ago, as has been effected in this study which has introduced the possibility of extending the beauty of traditional geometries into a third dimension. It is interesting to speculate on how an understanding of this mathematical approach might have altered the artefacts produced by Islamic craftsman in the past, particularly bearing in mind the use of pattern to cover large planar surfaces, and the manner in which this is said to have aided contemplation.

more to be written…

On a more prosaic level, 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 never seen designs based on seven, nine or more complex geometries in Qatar. 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, and it is true that many craftsmen worked on both sides of the Gulf, some of them moving up to Qatar from Dubai.

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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The beginnings of Arabic geometry

The first Arabic writings on geometry can be dated from the ninth century with the publication of Muhammd ibn Musa al Khwaarizmi’s treatise on algebra in which there is a considerable section dealing with geometry in his chapter on measurement – bab al misaaha. Following this publication there was a extensive body of work produced in Arabic, much of it dependent upon Greek and other earlier writings, but increasingly incorporating original thinking. They might be loosely divided into three general categories:

• translations of Greek and other languages as well as original work derived from them. The main source was Euclid’s ‘Elements’, but works by Archimedes, Apollonius, Theodosius and Menelaus were also referred to,
• geometric contributions to other sciences, particularly astronomy, and
• manuals for those working in the field of surveying and building construction.

Considerable advances were made by Arabs in sciences generally, and in medicine, astronomy and mathematics more specifically. Apart from medical instruments, perhaps one of the most familiar objects to us are the attractive designs of the alustrlaab or astrolabe, used both to make astronomical observations, surveying, and marking prayer times as well as in way-finding. It was also used for astrological purposes both in the Islamic world and, later, in Europe. A form of astrolabe was developed for use in navigating at sea to take account of the movement of the sea.

Much of Khwaarizmi’s work – who died around 850 – was based on Greek writings, but it is interesting to note that other sources were the Chinese astronomer, Chang Heng, (78 to 139), and the Indian astronomer, Brahmagupta (born 598), both of whom had calculated the value of pi, their methods and results being slightly different from Khwaarizmi.

Later, Abu al-Wafaa noted a number of geometric rules that would have assisted artisans in their work and, following him, al-Karaji wrote similarly on geometric constructions in his al-kaafi fi ’ilm al-hisaab – The Sufficient in the Science of Arithmetic. The importance of these works, and others that came later, is that an interface between algebra and geometry was established. He is considered to have been the first writer to free algebra from the constraints of geometry, itself an outcome of preceding Greek mathematical work.

In the middle of the ninth century, kitaab ma’rifa misaahat al-ashkaal al-basita wa al-kuriyya – The Book of the Knowledge of Measuring Plane and Spherical Figures – was published, the work of three brothers, Muhammad, Ahmad and al-Hassan Banu Musa. But it wasn’t until centuries later that Jamshid al-Kaashi, who died in the early fifteenth century, surpassed the accuracy of calculating pi by a method which, while based on Archimedes’ work, used as a basis significantly more polygons inscribed within a circle than Archimedes – 3x2²⁸ – 805,306,368, compared with the latter’s 3x2⁵ – 96. The work in which this was published was titled al-risaala al-muhitiyya – Treatise on the Circumference. The result turned out to be correct to sixteen decimal places, the same accuracy being attained, one hundred and fifty years later, by the Dutch scientist, van Roomen using, as a basis, inscribed and circumscribed polygons of 2³⁰. It is notable that Arab mathematicians believed pi to be irrational, but it wasn’t until 1766 that Lambert succeeded in proving this to be true. It is held that the interest of Arabic mathematicians in these fields of algebra and geometry was responsible for restoring these areas of science to the heights they had enjoyed in Babylon and, later, Greece and its territories.

While the work described above dealt in considerable accuracy with geometry, both two- and three-dimensional, and its relationship, particularly, with algebra, more practical methods were being developed in order to carry out day-to-day work associated with surveying and the construction of building works.

Today we have an interest in the use of compasses and straight edges to establish shapes, but evidence is that from an early time, a knotted string was the medium for establishing at least a right angle. For instance, divided into twelve equal parts, a knotted string was capable of immediately producing the right angle of a triangle of three, four and five sides, a figure that was held to have magical properties. Note in the diagram that the circle inscribed within the triangle has the same diameter as the circle, with centre ‘C’. It is considered probable that other strings were used to fulfill a range of simple geometric functions associated with the need to survey and set out constructions. Moreover, by this time it is probable that simple compasses and, perhaps, marked rulers, were used in relatively simple works.

The Greeks ascribed the invention of compasses to Thales, who died over two and a half thousand years ago, and much of the work of geometry carried out by the Greeks was based on methods involving compasses as well as rulers with special markings. Archimedes, for instance, used a ruler with two marked points to trisect angles. As I wrote earlier, much of the work of Arab mathematicians is important in that they developed their thinking on the basis of previous Greek and earlier mathematicians and, in the process of transcribing and drawing this opus together – as well as adding to that knowledge – spread their work through the medium of the Arabic language, the lingua franca of Islam.

Much of earlier Greek work has been lost to us and we must be grateful to Arab mathematicians who carried out work based on Greek traditions. For instance, Thaabit ibn Qurra ibn Marwan al-Sabi al-Harrani, a student of the Banu Musa brothers and writing in the ninth century, re-established work of Socrates and Archimedes, from the latter setting out a proof for a heptagon, the original having been lost. Abu Nasr al-Faaraabi and the previously mentioned Abu al-Wafaa both produced books relating to geometrical constructions, with the book by al-Wafaa containing most of the work of al-Faaraabi. This established a number of practical operations such as the setting out of simple constructions with ruler and compasses, in particular, the creation of four, five, six, eight and ten sided figures. But he also showed how an approximation of a seven-sided figure might be constructed and, by using a method for trisecting an angle, a nine-sided figure. In addition he illlustrated how a number of other, more complex, three-dimensional constructions might be effected.

Arab mathematicians continued to move forward the understanding of earlier mathematicians through examination, development and invention. Basic concepts were readdressed and redefined in order to produce a more sound foundation for geometry as well as other areas of mathematics. Issues relating to parallel lines, for instance, were worked on for hundreds of years and formed the basis for later European work. Geometric transformations, projections – an area relating to determining the direction of the qibla – a particular interest of Abu Rayhan Muhammad ibn Ahmad al-Biruni around the turn of the eleventh century, spherical geometry and, from this, latitude, longitude and, of course, the science relating to astrolabes.

In summary Arab mathematicians, in common with Arabs operating in scientific areas, produced work based heavily on Hellenistic principles but, in doing so both preserved and expanded this opus as well as propagating this knowledge widely through the medium of the Arab language and the influence of Islam. This enabled later European mathematicians and scientists to benefit from a significant body of work in developing these disciplines.

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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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Basis of geometry

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. Constructions 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 the geometrical constructions mentioned above generally, 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… The next few notes look at the construction of these more complicated shapes.

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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 unit. 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. I don’t know why this should be so, though there is the Greco-Roman tradition, as mentioned above in the note on cosmatesque designs, of the use of √2 geometries in two dimensional patterning.

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.

Here are illustrations of the paper standards adopted by the International Organisation for Standardisation from the German standard which has been in use there for over eighty years and has now been adopted in most of the world except the United States and Canada. The ‘A’ series, on the right, is based on a sheet of paper having a total area of one square metre and its sides, as mentioned above, in the proportion of 1:1.414. Each time the paper is halved the proportions of the new sheet remain the same. The ‘B’ series, on the left, is mainly used for posters and in the book industry and has sides in the same proportion, but with the sides of the basic sheet being 1.000 x 1.414 metres.

Incidentally, the area relationship between a square of side one unit, and a rectangle of sides 1.618 to 0.618 – the golden section, is the same – 1. There is an interesting note to be read here on some of the characteristics of φ – phi – the 21st letter of the Greek alphabet, and which is used to represent the golden number, mean or section.

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.

The issue of proportions introduced by this note on √2 geometry, is a subject I want to come back to later. It relates not just to the Golden Section mentioned below, nor to classical architecture, but also to the search for perfect proportions that has been continuing for centuries.

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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. The pineapple to the right, as well as the pine cone below it, both have eight divisions spiraling clockwise and thirteen anticlockwise both, of course, in resonance with the Fibonacci series. 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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Other systems of proportion

Further up the page I have noted briefly systems of proportions, particularly relating to the Golden Section, the Fibonacci series and Vitruvian man. Here, although it may seem as if it has little or nothing to do with Arabic geometry, I should note other systems of proportions as they have also been used in the design of buildings. The best known of these is, perhaps, the system developed by the Swiss architect, Le Corbusier, a system he termed ‘Le Modulor’, and which he based on the Golden Section.

more to be written…

 

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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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Design and ‘e’

While it may be thought to have little or nothing to do with geometry, I should also mention here the relevance of the mathematical constant ‘e’. I hope to expand on this relevance later. Generally known as the base of natural logarithm, ‘e’ has a significant importance in mathematics, as great as that of pi. ‘e’ is an irrational number which, to compare with other numbers given on this page, can be taken as 2.718 to three decimal places. You might note the following square roots:

• √5 = 2.236
• √6 = 2.449
• √7 = 2.646
• √8 = 2.828

You can see that ‘e’ fits between √7 and √8, though has a very different mathematical significance. I have placed these numbers here as the theory noted in the above paragraph has a theoretically wide possible variation; with ‘e’ being valued from two to seven, in fact.

The reason I have introduced ‘e’ on this page is that it is held to be a factor relating scales of elements within an overall design. It is argued that a design may have significant internal coherence when its elements have that relationship at different scales. Moreover it is held that this relationship also applies to the structure of biological forms. The paper referred to here is argued on the basis of the design of traditional carpets but, by its very nature, there should be no reason why these relationships might not hold good for other areas of design. In fact, in another paper this is explicitly stated, relating the comments to the work of Christopher Alexander, who based his work on the study of carpets but extrapolated the concept to design in general.

Its relevance on this page is in an aspect of Islamic design alluded to on both this page as well as others in this section of the site: the complexity in apparent simplicity that is so often a characteristic of geometrically based Islamic design. Above I have perhaps concentrated on the complexity achieved from simple geometries, but here I wish to introduce another way in which designs might be seen.

more to be written…

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Application of traditional patterns

In Qatar, 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 sixteenth centuries, and that the geometries were not familiar to Western mathematicians for a further five hundred years.

Here are the same five tiles set against their ten-point constructional geometry.

The illustrations of the five basic tiles include their concomitant strapwork lines. It is these lines that are most associated with Islamic design and for which the geometries I have described above form the basis. Using these five tiles as templates, it is argued that it would not have been difficult for artisans to assemble the complex patterns that we associate with Islamic design. In fact it is argued that this might be far simpler than organising the more complex strapwork geometries and tiling or decorating within them. Having spent a long time working on this kind of geometry I have to say I have a feeling that this might well be the case.

If this was the case – that Islamic artisans designed and used these tiles – then it explains how they were able to accomplish the complex setting out of their designs without making the mistakes that would have been likely if they had to create each line segment separately. It is not thought that they assembled the tiles individually, but that the five tiles were used as templates to trace patterns for fixing mosaics.

At present this thesis is unproved but, as I have written elsewhere, considerable advances were made in the Islamic world in science and the arts, and little of it has been taught in Western schools and universities.

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Artisans and geometry

Here, below, is evidence of such activity in this page taken from a Persian document, written between the eleventh and thirteenth centuries, whose author is not known, and entitled ‘Fi tadakhul al–ashkal al–mutashabiha aw al–mutawafiqa’ – ‘On interlocking similar or congruent figures’.

There are two points I would like to make here relating to the diagram. The first is that it appears to support the argument that geometric studies such as this might have been made, or might have been used, to record or investigate geometric tiling patterns. The rationale for this has been suggested by a study of the translated document which holds that the document, in its lack of structure and differing quality of work, appears to be the work of a scribe noting conversations held between mathematicians and artisans, rather than a geometric treatise or artisans’ handbook.

The second point has perhaps more to do with my possible lack of understanding of the diagram. The diagram on the page appears to show a single quadrant each of a twelve-point – lower left, and a ten-point – upper right, geometrically divided circle, each of which touch at a point. To its right is a simplified illustration of the diagram.

Here I have crudely assembled a number of the full patterns to illustrate how the diagram might have been used in a tiled wall or floor. It shows horizontal runs of ten-point and twelve-point stars though it is possible, of course, that the stars might have been assembled in different ways. But referring back to the initial page and the simplified diagram I have made to its right, there appears to be an anomaly. This can also be seen in this, second, simplification of the pattern. In it you can see that the points of the twelve-point stars at three and nine o’clock don’t meet as they should, and that the stars are irregularly shaped. There is a reason for this.

Elements A and D belong to the ten-point star, elements B and C to the twelve-point star. Elements C and D have regular points in that the two lines of the star pointing outward have equal lengths. But you will see that the lines forming the points of elements A and B are of unequal length and, although you can’t see it, the construction appears to have been forced in order to have the ten- and twelve-point stars touch. So far I have not been able to work out exactly how the geometrical construction has been derived and will re-write this when and if I do.

To reiterate the main point I wish to make here, it appears that there was significant study made of tile patterns and their underlying geometry, and it seems highly probable that there is more to learn about the methods of tiling as suggested by the work above on girih tiles.

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Numbers in Islam

In discussing Arabic geometry I have concentrated so far on the basic geometries relating to the construction of two-dimensional shapes – the elements from which patterns are assembled in Arabic tilework and similar decorative materials. But there are related areas that receive little attention in the West, those of

• numbers – in their relationship with pattern and geometry, and
• numerology – the divining of meaning from numbers.

There is considerable discussion about the rationale behind numerology, with many claims made for it as a science – and as a pseudo-science by its detractors both in the West and those who have come to it from an Islamic perspective. Just as there are those who argue for the significance of certain numbers in the Bible, particularly seven, twelve and forty, there are Islamic scholars who argue the significance of four, seven and nineteen in the Quran. I should add here that my understanding of the number ‘forty’ is that it used to signify a large number, perhaps as we might today say ‘thousands’ or ‘millions’, but not meaning it in an accurate or literal sense. Bear in mind that, in those days, many people could not count, shepherds customarily using stones to ascertain the number of their flocks.

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Seven

There are many references to the number seven in the Quran. It is a number which has, because of this, not only its usual numeric value but also important symbolic meanings.

According to the Quran, God created seven heavens – 65.012:

Allah it is who hath created seven heavens, and of the earth the like thereof…

To those who observed the heavens there appeared to be seven bodies which were important to them and could be seen with the naked eye:

• Saturn,
• Jupiter,
• Mars,
• the Sun,
• Venus,
• Mercury, and the
• Moon.

These they perceived to sit and move each on their own ceiling above the earth. They might be envisaged as a nested set of geocentric ceilings each supported on invisible columns, with the stars attached to the lowest of these ceilings - 67.005:

And verily We have beautified this lowest heaven with lamps…

According to a hadith there were also seven phases through which the Prophet, guided by Gabriel, made an ascent to Heaven meeting, at each stage respectively, Adam, John the Baptist and Jesus, Joseph, Idrees, Aaron, Moses and, finally, Abraham before moving to the last heaven where there were four rivers, two within – the rivers of Paradise – and two outside – the Nile and Euphrates. It is not my intention to discuss these in any detail. My only purpose is to note that these numbers exist in the Quran and hadith, and must be considered significant.

The number seven is found in many Islamic buildings including, for instance, the Hall of the Ambassadors in the Alhambra, Granada where, with the number four it appears all over the room. This space, like many of those in the Alhambra, was designed for quiet and contemplation. It is a space where the observer may bring himself closer to his God through spiritual reflection. It is significant, then, that the number seven was considered important enough to be integrated so strongly into its decoration.

As obtains with the number nineteen, mentioned below, there are a significant number of coincidences relating to the number seven in the Quran, and which have been the source of interest by Islamic commentators:

• the number of words in the first and last verses of the Quran is 7 each,
• the number of words in the first and last suwar of the Quran is 49, or 7 x 7,
• the number of suwar which have verses being a multiple of 7, is 14, or 7 x 2,
• the number of letters used as initials are 14, or 7 x 2,
• from sura 1 to sura 112, where the name for God appears first and last, there are 112 suwar,
• the number seven is the first number mentioned in the Quran,
• the first chapter of the Quran is composed of 7 verses, and the
• number of times the letters of God’s name occur in this opening sura of the Quran is 49, or 7 x 7,
• and so on…

There is a large body of study into the mathematics of coincidence, or theory of probability, and I do not intend to go into it here. The purpose of setting out some of the above coincidences is solely to establish the interest there is in the number seven.

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Four

The number four is also one which regarded as being important in the Islamic world. As with the numbers seven and nineteen, it’s difficult to know where to start. In no particular order:

• the creation of the earth took four days,
• the name, Muhammad, appears in the Quran four times,
• the name of Muhammad in Arabic has four consonants,
• there are four Archangels, Jibril – who revealed the Quran to Muhammad, Mikail – the Angel of Mercy, Israfil – who will signal the Day of Judgement, and Izra’l – the Angel of Death,
• there are four sacred months in the Islamic calendar. They are the twelfth to the third months of the Islamic year – Dhuw al-Hijjah, Muharram, Safar and Rabi’ al-Awwal. The name, Rabi’ is derived from the word for four. Curiously to a non-Muslim, Rabi’ al-Awwal, or Rabi’ the First is named as such as it is the fourth of the sacred months. The month which follows it is Rabi’ al-Thani, or Rabi’ the Second, so named as it is the fourth month of the calendar,
• the creation of the earth took four days,
• there are four rivers in Paradise; they are of water, milk, wine and honey,
• relecting the four rivers of Paradise, Islamic gardens are traditionally divided into four parts by water channels,
• and so on…

As I have mentioned before, the point of this is solely to demonstrate some of the sources which give rise to the importance of the number four in Islamic thinking.

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Nineteen

With regard to the importance of the number 19, sura 74:30 of the Quran reads, simply:

Over it are nineteen.

Some translations have this as:

Over it are nineteen angels.

though the word ‘angels’ doesn’t appear in the Arabic and appears to have been introduced in order to make better sense of the sura.

There seems to be uncertainty as to what the quotation means as the suwar of the Quran are not set down in the order in which they were received. This gives rise to speculation about how this verse might read in relationship to their original reception. As the ordering of the verses is ascribed to Khalif Uthman and not Muhammad, then arguments relating to the numerology based on the present order of verses are unlikely to be sound.

For instance, it is known that the first revelations received by Muhammad were the first five verses of Sura 96 followed by verses 17 to 30 of Sura 74 which ends with the quotation given above – ‘over it are nineteen’. It is argued that 5 verses plus 14 verses add up to 19 verses; hence the verse ‘over it are nineteen’.

This has led scholars to look for instances and relationships in the Quran having nineteen as a basis. There are a wide variety of combinations of counts in the Quran which, it is claimed, are divisible by nineteen. For instance, the

• Quran is composed of 114 suwar – 19 x 6,
• the Quran is also composed of 6,346 ayyat – 19 x 334,
• the first revelation made to Muhammad comprised 19 words, and
• these 19 words contained 76 letters – 19 x 4,
• the number of suwar in which the word ‘God’ appears in the Quran is 118,123 – 19 x 6,217,
• the phrase that begins every verse of the Quran has nineteen different letters in it, excluding repetitions – in English this is usually translated as:

In the name of Allah, the most gracious, the most merciful

• the word ‘name’ appears in the Quran 19 times,
• the word ‘God’ appears in the Quran 2,698 times – 19 x 142 times,
• the phrase ‘the most gracious’ appears in the Quran 57 times – 19 x 3 times – half the number of suwar in the Quran,
• the phrase ‘the most merciful’ appears in the Quran 114 times – 19 x 6 times, the same number of suwar in the Quran,
• 19 associates the first and last cardinal numbers, 1 and 9, and
• 19 is the sum of the cardinals, 9 and 10, as well as the difference between the squares of those two numbers – 100-81.

These relationships or coincidences have led to claims of conclusive proof that the Quran can only be the work of God.

I don’t intend this list to be fanciful but a search on the Internet will turn up even more examples of the perceived importance of the number, nineteen:

• Halley’s comet appears every 76 years – 19 x 4,
• the human body contains 211 bones – 19 x 9,
• a full term foetus develops 266 days or 38 weeks after fertilisation – 19 x 14 and 19 x 2 respectively,
• and so on…

It is not my intention to try to promote or disprove the importance of nineteen, solely to mention that it is deemed important by many Islamic scholars, and reinforces the importance that numbers have in Islam.

Having said that I should also add that it appears that all religions place significance on certain numbers, though the numbers differ with the different religions. One, three, five, six, seven, nine, twelve, seventeen, twenty-six, twenty-eight, thirty-six, forty-nine, one hundred and twenty-eight, one hundred and forty-four, one hundred and fifty-three, six hundred and sixty-six, seven hundred and eighty-six and so on are all numbers believed to have a meaning beyond their functions as elements of counting.

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Colour

It might also be useful to mention the theory of Islamic colour dealt with here where it is related to the numbers three, four and seven as well as to other factors. Essentially it holds that the Islamic system of colour arranges colour on three levels:

• a system of three colours, and a system of four colours,
• a system, based on the first, of seven colours, and a
• system of twenty-eight colours based on four times seven.

The system of three colours comprises:

• black,
• white, and
• sandalwood.

The system of four colours comprises:

• red – representing fire,
• yellow – representing air,
• green – representing water, and
• blue – representing earth.

These seven colours are also associated with the seven heavenly bodies. Based on an Egyptian system, it holds that:

• Saturn is associated with black,
• Jupiter, sandalwood,
• Mars, red,
• the Sun, yellow,
• Venus, white,
• Mercury, blue, and the
• Moon, green.

more to be written…