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.

While many of the designs illustrated here, including my attempts to deconstruct them, are based on two-dimensional designs, there are many examples of three-dimensional design work in the Islamic world. Here to the side is the top of a fifteenth century wooden Egyptian minbar that has been articulated with pendentives, a form of cantilever that is commonly used in masonry constructions, though here is more decorative than structural due to the inherent character of timber which has both compressive and tensile qualities which stone lacks.

These next two examples of artefacts exhibiting Arabic geometry have been included even though they are less complicated in their underlying geometry and far more crudely assembled than that above. I have shown them because they are, perhaps, more the type of example with which we are familiar in our daily lives. This standard and character of design is commonly found all over the Middle East. They represent the character of inlaid work that is displayed in many decorative pieces. Sold to tourists many such items are brought back from the Arab world. The lower example, in particular, is extremely poorly set out. Nevertheless, it has sufficient geometrical integrity for the pattern to be easily seen and readily comprehended. Incidentally, all of these first three examples are based on eight point geometry, a common and relatively easy framework to establish.

While many regard Islamic designs as being based on strict geometrical constructions, there are also Islamic designs that are formed with floral devices and where the governing geometry might not be immediately obvious. This first example is placed here as a reminder that there are other forms of decoration that were, and are still, produced by Islamic designers, this piece being made in Turkey of hardwood inlaid with white shell pieces. As yet I have made no effort to analyse this design as it doesn’t really fall within the ambit of these studies.

Like the previous example, this photograph is here to remind and illustrate that not all Islamic design has a geometrical basis to it. In fact it appears to be completely free-form in its design and is a modern piece of work. The photograph shows the junction of the wall and floor within the Grand Mosque at Abu Dhabi. It may seem out of place in the context of Gulf architecture, and perhaps has more resonance with Mughal design. Significantly, and another reason for the illustration being here, the flowers in the wall are raised, a technique that encourages the the sense of touch, introducing another sense to the enjoyment of the space and its containing surfaces.

This next example, that of an embossed and inlaid brass dish in three finish colours, is perhaps a more familiar example of Islamic design work. Geometrically it is based on an eight-point geometry for setting out its outer cursive decoration, and a six-point geometry within the central area which incorporates and surrounds a simple calligraphic motif. The design has a very strong cursive feel to it.

Here is a heavily embossed silver dish in a kufic calligraphic style. The calligraphy creates the character of the design. though there can be seen to be a strong geometrical control to the design. There is a small amount of cursive work that helps to embellish the dish. The design contrasts with the dish above and is placed here deliberately for contrast, these first examples indicating something of the variety there is within Islamic design.

Perhaps more than the above, mosaic tilework is considered by many to be typical of Islamic geometry. Its scale and the degree of accuracy found in shaping and cutting the tiles differs considerably with region and time. Found all around the Mediterranean basin from Moorish work in the west, to Syrian and Turkish work in the east and Egyptian work in the south, there is also tilework to be found in Persia, Iraq and the Indian sub-continent. As the geometry that underlies these patterns is universal, it is sometimes difficult to determine where an individual design might come from. This first photograph, for example, is of Moorish work from Andalusia, the detail below it, of a fountain based on twelve and eight point geometries, is Syrian.

These next two examples are of different types of tilework. The first is an example of what is known as cross and star tiling, a simple form of interlocking, and comes from thirteenth century Kashan in Persia. While the shapes of the tiles have a strict geometry in order to fit together, their internal designs have a more free structure, though still within a framework of four and eight point geometry. Note that one of the patterns has its geometry rotated through 22.5° to the normal.

The last of these three tiled examples is of a single, large carved and glazed earthenware tile from the tomb of Buyanquli Khan, Uzbekistan, and dated 1358. In three colourways – brown, beige and turquoise – it has a heavy three-dimensional modelling created by carving into an earthenware panel up to 20mm in depth. Despite a number of eccentricities in its geometrical layout, it demonstrates considerable mastery in its design, construction and firing.

This photograph illustrates 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.

One of the points to bear in mind when looking at designs is the relationship between the setting out lines and the pattern lines. Many designs are constructed along the setting out lines, but pattern lines generally have width to them and, depending upon the designers’ decisions, there will often be the need to have the pattern lines centred on, or adjacent to, the setting out lines creating both problems and opportunities for designers. The design process has been likened to the setting out lines being the dry bones upon which the flesh is artistically draped, with the designers making a series of decisions in creating their individual patterns.

These decisions are also what are looked for in the investigation of patterns as they need to be understood in order to establish how the geometry was set out. The earthenware tile above is a case in point. It is obviously based on five-point geometry though, as you can see at a casual glance from the two sketches on the right, there may be a hint of a six-point pattern in it. To make matters worse for investigation, there is a significant amount of layout inaccuracy as well but, despite this, the pattern is strong and the layout looks accurate. It is a good example of its sort, a tribute to the strength of Islamic geometrical patterns.

The pattern consists of beaded lines which divide the plane into a series of simple shapes based on five-point geometry. These first two sketches show, firstly, the pattern created by the lines and, secondly, the small infill shapes used to bring character to the pattern. Both these sketches are approximations of the line and shapes and are used only to illustrate the basic pattern; examined in detail, there are a small number of inaccuracies in the execution of the tile which make it difficult to overlay the analysis lines and shapes precisely.

This third sketch, a detail of the panel above, has been drawn to illustrate how the geometry is not constructed as simply as you might anticipate. Given that there is a little distortion in the photograph which accounts for some inaccuracies in the setting out lines, you should be able to see the setting out lines drawn along the main elements of the pattern are not generated from common centres as are most of the others on this page. I have deliberately omitted many of the setting out lines in order to keep the diagram relatively legible, but there should be enough to illustrate the point.

The shapes used in the pattern can be seen in the fourth illustration to the right, and consist of a half pentagon, pentagon, rhombus, triangle and kite plus, not shown here, the make-up pieces at the periphery used to complete the rectangular frame. The basic shapes are used as shown here, but there are some that are mirrored and rotated in order to fit to the pattern. Note that the governing angles in the pattern are 0°, 36°, 72° and 90° from the horizontal, the angles associated with five-point geometry.

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Carpets and rugs

In addition to discovering Islamic geometries on metalwork, tilework, plasterwork and woodwork, we also associate them 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 of carpets, the carpet illustrated here in its setting within the Victoria and Albert Museum is one of the most famous Persian carpets, the Aradbil, named after the area in modern Iran from which it originates.

The Ardabil carpet is large, being approximately 1040 x 530 cms, and 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 features based on the same geometries in the corner or the surrounding frame. There is an inscription at the top of the carpet, shown on the right in the second of these three photographs.

Looking at the Ardabil carpet, even in these extremely small photographs, 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 and is not confined to naqsh but also to wood, metal, tiles and a variety of other materials.

But 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.

These structures underly much if not all Islamic patterns, though sometimes this may not be immediately discernible or, more likely, not immediately spring to mind. One of the great achievements of Islamic design is the manner in which the geometries or underlying patterns are subservient and not immediately apparent.

The first two examples to the side are from different areas of the Middle East and separated by a hundred hears. They are of floral or foliate designs which, while free flowing have a very strong geometrical basis in the setting out of their pattern, geometries enlivened by the cursive nature of the applied design.

You will see in the upper photograph, on a tile from the 1378 tomb of Buyanquli Khan of Uzbekistan, the use of two colours to bring definition to the design whereas, in the second photograph, of a 1277 Syrian marble bowl from an ablutions fountain, there is a single colour. Despite this, the complicated design is still legible. Note also that there is calligraphic script running round the basin just below its rim, though wear and tear has affected the top of it a little.

This third photograph is an example of calligraphic Arabic script, another of the Islamic arts that are applied as embellishments or integral elements of an Islamic design. In this example, a page from a Holy Quran, great skill has been used in the application of a set of rules to the creation of the page that has considerable meaning to both the artist and subsequent readers. While calligraphic, neither this nor the example below were constructed with single strokes of a pen. They are both artistically embellished.

The last of this group of photographs is of a plate from Eastern Iran or Uzbekistan, from around the tenth century, this time decorated with a highly stylised calligraphic phrase. Not only is there directional geometry centred on the plate, but the letterforms also have considerable attention paid to the geometry generating their individual and collective forms. At the bottom of the photograph you can glimpse part of a smaller plate where the outside rim is the baseline for the calligraphy whereas, in the larger plate, the baseline of the calligraphy is based on the centre of the plate.

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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A Western geometry

While this may not be the place to discuss Western geometries, it would be useful to illustrate something of the character of Western geometrical designs in order that it might be compared with the Arabic geometries being carried out around the same time. This is not intended to be anything more than a note on a single design.

This first illustration is part of the east window of an English cathedral photographed from inside the choir. It is easily recognisable as an element of a religious building and can be considered typical of this type of architectural detailing. Although much restored in the nineteenth century, this particular window was constructed, as were many others in England, in the fourteenth century and can be considered contemporary with many Islamic designs. What is significant, and the reason for its being shown here, is the difference in its constructional patterning, here based on six-point geometry.

Despite this there can be no doubt that while these features are a characteristic of Christian religious buildings, they owe something not just to their development from the Roman oculi that brought air and light into buildings, but to Islamic designs that would have been seen by travellers and those engaged on the Crusades.

It is not my intention to write notes about the conceptual issues underlying these designs but, in common with other similar constructions it should be borne in mind that they are considered to have a hierarchy of geometries relating to the

• visual form,
• symbolic meanings, and
• hidden structures or geometries.

The visual form consists of the interplay between the containing physical frameworks – usually the carved stone mouldings – together with the contained plain or coloured glass and the structural lead cames that are also design elements. In patterns such as these there is a three-dimensional character to the design which creates additional interest to the two-dimensional form.

Symbolic meanings are common to many of the art forms associated with religions. Numbers, colours and shapes play a significant part in these windows relating them to a variety of concepts, natural and supernatural elements as well as a variety of religious constructs and resonances. Much has been published on these issues and it is not my intention to repeat them here, but they are thought to inform a considerable range of designs all over the world and can give added levels for contemplation in those understanding them and, perhaps, those who might not.

The hidden structure or geometry is that which is mostly the subject of these notes: it is the basic geometry on which the different shapes are established and set out in relationship to each other.

Looking solely at the rosette at the top of the window, you can see that it contains three round trefoil rosettes alternating with three pointed trefoil rosettes. The round trefoils shown here are termed as lying, the pointed trefoils as standing. This first sketch illustrates the construction lines for a simple lying trefoil and can be made with only a pair of compasses set at the same radius and created around a central circle.

The illustration below it illustrates the way in which a standing pointed trefoil may also be constructed. What is evident with these simple constructions is that there are a number of choices that can be made when setting out the circles that underlie the the shape of the trefoil. Generally their centres are always located on the junction of two circles but may also lie on lines that join centres of circles or that radiate from the centre of the central circle. These are not shown on the two sketches of the round and pointed trefoils.

This type of construction can be extended to form more complex forms such as quadrefoils and cinquefoils. Here the same basic construction has been developed in order to produce a shape with six pointed foils – a hexafoil or sexfoil. In this case a second series of circles have been drawn, their centres on the circumference of the central circle, and with a radius illustrated by the horizontal line on the top circle.

While the sketches above illustrate simple constructions, the medieval stonemasons varied their geometries by moving the centres of arcs to positions off the main construction lines. Significant variations can be formed this way, and it suggests that the manner in which this was calculated may have been based on empiric or non-rational rules.

These four sketches are intended to illustrate how the centres for a pointed sexafoil can be constructed, and how its pattern can be varied by moving the centres of the arcs forming it.

In the first sketch, seven small circles are drawn that are mutually coincident, and six of whose centres fall along the circumference of a larger circle having twice the diameter of that of the small circles.

Where the outer six circles touch, a circle can be drawn passing through these points and having its centre at the centre of the central small circle.

Lines can be drawn from the points where this inner circle cuts the junction of pairs of the small circles and projected through the centre of the small circles.

In the third of these sketches above those lines are continued to meet the other side of the circle, making them the diameter of those circles. In the last of the sketches the line is projected further to a point representing a factor of 1.15 of the diameter of the circle, a point that does not fall on any construction line. From these two points, arcs can be drawn having their radii from those points to the junction of the two circles, and creating the shape of the sexfoil.

Note the difference between the two bottom sexfoils shown above – and here extracted for simplicity – where the establishing arc centres are 1.0 and 1.15 respectively but where, due to the necessities of its construction, the inner diameter of the foils remains the same. Obviously, the longer the line extends the greater the diameter of the sexfoil will be, and the flatter the shape of each of the foils.

In most of their respects, the geometry of these shapes are common to both Arabic and Western geometries and the designs produced from them, though the materials commonly used to form them might differ. As the geometries are so simple, this is not really a surprise. However, the method of determining centres outside the lines of geometry seems to be more common to medieval Europe.

Returning to the eastern window of the cathedral above, it is evident that the layout is very different in concept from an Arabic design, the geometry combining several different radii in order to produce what appears to be regular shapes, particularly in the six lines that radiate from the centre of the rosette, establishing the alternate round and pointed trefoil rosettes. To some extent this is due to the manner in which the masons set up the three-dimensional stone mullions. Note how, in this photograph, the heads of the adjacent pointed arches meet the ring describing the central rose.

One of the considerations of the masons was that the windows present very different effects when seen from the outside and the inside. The outside emphasises the two- and three-dimensional forms of the stone structure while, from the inside, the coloured glass is the main feature – though at night there will be a similar effect to that of viewing from the outside when there is no light behind the glass to illuminate it. Compare this photograph with that at the head of this section.

Here, then, is an illustration setting out the construction lines behind the central feature of the cathedral window. This is what is referred to above as the hidden geometry, though in blue I have emphasised the six trefoils. This may not be as accurate a representation of the layout of the window as I would like, but it is sufficient to see how this compares with the patterns illustrated on this and the other Arabic geometry page. In particular, you will notice that the majority of the Arabic patterns are running patterns; they extend beyond the frame in which they exist. This is a characteristic that has much to do with contemplation in Islam. Here the frame contains the pattern and this is a common feature of such Western geometries. The window forms a focus for those in the body of the church; it is a different form of contemplation that has much to do with the structure and hierarchy of the Church, and the sense of awe or wonder these buildings were designed to induce in their congregation.

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Geometric design

Returning to Arabic geometry, 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.

Related to the above grid is this one which is often used as the basis for a simple pattern known as a hammerhead or axehead design, and which can be clearly seen below in the detail photograph of part of a modern tile executed with fine craftmanship using materials having a strong contrast.

The construction of these geometries is easily established with only simple tools. The basic geometry of the pattern above was created by moving circles one radius length apart, both horizontally and vertically to form the underlying grid. In the case of the lower pattern, the establishing grid is made on the diagonal, and rows of circles, touching on their circumferences, are located on those diagonals, alternative rows slipping by one radius with respect to the next.

The pattern is very common and can be seen all over the Islamic world as well as other parts of the world in both of which many commercial products have been produced on its geometrical basis. Here is an example of its use as paving in front of a hotel in Qatar, the basic pavior being enlivened by traditional Islamic decoration which has the effect of creating a texture to the otherwise simple shape. In this sense it can be compared with the pattern above where the use of shells creates the secondary interest.

Returning to the first of the four-point patterns shown above, 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 previously, 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 between them, and the way in which basic shapes may be used.

In a slightly different arrangement from the eight-point geometry above, the illustrative sketch below has been assembled in order to illustrate the way in which geometric designs may be assembled. In this example I have not included the construction lines for the guideline geometry in order to simplify the graphic. The importance lies in the selection of the points that are joined in order to set out, first, the central rosette, then the linking elements and, from those, the positioning of the next placement of rosettes that surround the first one.

The circle is important in establishing the squares as well as the location of the parallel lines adjacent to the centre of the circle that will form the central rosette of this pattern.

Their width is set by running through the points where the extended diameters cut the circle. These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle. This creates a basic geometry allowing eight lozenges to be formed equidistant from the centre of the circle.

With centres on the corners of the larger squares, circles are described with a radius set to the points where the smaller squares meet, respectively, the horizontal and vertical lines drawn through the centre of the first circle. With a centre on the point where these second circles meet the horizontal line, a third, smaller, circle is drawn which intersects with the horizontal line as well as the extended lines of the larger squares. These intersections create the octagon which establishes the distance between the first rosette and those that are located around it.

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A twelve and fifteen point geometrical pattern

The sources mentioned above that deal with the ways in which geometries are brought together, tend not to help with the more artistic decisions that govern the designs of patterns. These can depend on a number of factors relating to traditions, the location of the finished work and the time available to design and execute the works. I have to admit that I have not seen a designer making one of these patterns other than artisan craftsmen in Iran and Qatar who were producing relatively simple designs, and who were able to explain something of what they believed governed their work. In those cases tradition was the main generator, together with a number of rules that governed what was – and what was not – permitted within that tradition and the designer’s understanding of the part the panels were to play in the overall design of the buildings in which they would be located. What is interesting is the degree of freedom they have to make design decisions. Yet a number of designs are evidently copies of other work and can be seen to be the continuation of traditional work in a newer context.

I have to admit to a fascination in the geometry behind such patterns that is as interesting to me as the patterns themselves. Here, for instance, is a working sketch behind a pattern that comprises a mixture of twelve-point and fifteen-point geometries. The basic elements of the geometry are far simpler than set out in this illustration, but I have added the more complete construction lines in order to make their underlying patterns more apparent. You should also be aware that the panel occupies only a small part of this pattern and, like many patterns, should be understood to be an infinite pattern of which a part has been selected for display. This is part of the charm or beauty of such designs. While the eye moves over them, there appears to be an understanding of their extending beyond the frame. This is one of the effects produce in the tilework and plasterwork of masaajid enabling the faithful to relax while immersing themselves in the infinite.

As you can see from many of the diagrams in this section, it is not difficult to construct a rosette with a variety of points on its compass. Generally rosettes can be constructed with a simple compass and straight edge, though it is evident that in many cases it would have been practical to operate with pre-cut formers that would establish shapes and relationships more rapidly. Rarely would a design be drawn from scratch although there may be be a need to experiment before establishing a final design.

However, when designs are complex it requires more than a set of formers, at least in the initial stages when there is a need to test potential designs and investigate the manner in which the patterns might best fit together. In particular, the designer must establish the points from which the overall setting out of the design are made. Conversely, when investigating the geometry of patterns, it is understanding this framework that is the analyser’s goal.

So, here is the basic setting out of the twelve-point rosettes for the pattern of rosettes illustrated above. The key generators in this layout are the circles, shown blue, of the same diameter as that in which the first, central, rosette is configured. These establish not only the positioning of the six rosettes that surround the central rosette, but also the centres on which the fifteen point rosettes will be constructed. In geometries of this sort there is a considerable degree of coincidence in the geometries relating each rosette to the others.

The construction of a twelve point rosette is shown elsewhere, but here is a detail of this particular rosette. Generally all these rosettes are constructed the same way: the circumference is divided into the required number of points and those points joined, each to all the others. The possibility also exists for lines to be drawn between the points of intersection of these lines, and that enables shapes such as the elongated lozenges, here shown in blue, to be drawn. Also shown in blue on the right is the circle construction that locates the centre of the three rosettes as is shown in the diagram above.

Here is a detail of the fifteen-point rosette. The greater the number of divisions of the circle, the more intersections there will be, and the more opportunities there are for establishing lozenges and other shapes on those points of intersection. In the case of odd numbered rosettes, it is not possible to construct the rosette as a single lozenge, rotated as you can see in the above illustration. In this case a continuous line creates the rosette giving the appearance of a number of rosettes, but not permitting the same kinds of treatment that even numbered rosettes allow the designer.

Although this note has to do with the construction of patterns, I have not shown all the construction lines for this part of the pattern, the extension or linking of the rosettes. Nevertheless, these lines are formed in a similar way to the rest of the pattern, by the extension of lines governing the rosettes. The skill of the designer lies in selecting the clearest elements to follow.

Here is the completed pattern as it appeared in an article a number of years ago relating the twelve and fifteen point geometries to a complex Saracenic pattern. There are many designs similar to it, some simpler, some more complex, but they all have a similar theoretical basis in their constructions; they rely on the skill of a designer to understand the underlying geometry of each rosette, and to develop and extend the controlling geometry along reciprocal lines, so linking them rationally and artistically. All the designs on the page will have been constructed in a similar manner, most of them less complicated than this.

There is a relatively small selection of design parameters within which the designer has very sophisticated decisions to make, most of them dependent upon the inherent geometric characteristics of the circle. The most important, perhaps, is the need to keep the design readily legible and comprehendible. Aesthetically, the more points on the circle, the more the design is likely to block up, or lack clarity due to the density of lines. In achieving this, one of the methods available to the designer is to increase the circle diameter as the number of points increase. The second significant opportunity lies is in keeping the circles at a distance from each other. Within this area the linking lines are organised, usually as extensions of the lines within the circles. The final decision, or perhaps the first, is the size and, particularly, the proportion of the frame within which the design is to be made, and the size of the circles that are to be located within it.

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An eighteen and twelve point geometrical pattern

This photograph, a detail of an old pair of doors to the Grand Mosque in Mecca, Saudi Arabia is a beautiful example of Islamic geometric design. It demonstrates perfectly the attraction of geometric adornment with what appears at first sight to be a simple application of a linear pattern selectively raised to produce three-dimensional relief. In this case the underlying geometry has been developed with considerable modeling added to enhance the door, creating a heavily modeled design that benefits from cast shadows.

In order to study the panels I have taken the liberty of amending the photograph slightly in order to make it more accurate geometrically and, therefore, easier to investigate its underlying structure, but essentially this is how it appears. The door has a similar layout to many other panels in that there is a grouping of large rosettes with smaller rosettes between them. The design structure is also similar in that the larger and smaller rosettes are both distributed to fall on the centres of a six-pointed framework, the two frameworks being set at 30° to each other.

The photograph above shows, like many of the panel patterns seen in Islamic design, only a part of a regular geometric patterning that extends, in theory, to infinity. In the illustration below it you can see how the pattern selected to enliven the door relates to this infinitely patterned ground, as well as the relationship between the underlying frameworks of the different rosettes.

Although I have had to construct the geometry of the rosettes from scratch, I have generally omitted much of the constructions from these sketches in order to keep them relatively simple. Having said that I have obviously included much that is not easily seen as I have to admit to finding the density of lines an added attraction to the studies. This sketch is such an example, the larger 18-point rosettes have been laid out with red lines, the smaller, 12-point rosettes with blue lines. The illustration shows, on the left, the setting out of the two different types of rosettes and, on the right, the basic relationships of those rosettes. The linking geometries have been omitted. The sketch shows the relationship with the pattern on the door and can be compares with the diagram above.

These next two sketches illustrate the point made relating to the incorporation of detail. This first sketch shows the basic geometry created by the two rosettes but to which I have added the linking geometry that fills the spaces between the two rosettes. Below it is a sketch showing most of the geometry used to produce the rosettes, though omitting a little of it for clarity. There are two points to make here; the first is that there is a degree of choice allowed the designer in selecting which linking geometries may best be used. Secondly, there are always complications introduced when thin construction lines are developed into wider channels as can be seen in the photograph of the doors. This introduces another series of possibilities for constructing the design. What is apparent is that there are eccentricities in the setting out of the door, but I find this part of the attraction of craftsmen’s work.

In order to clarify the design, here I have simplified the geometry again and given a degree of colour to the different elements – the two rosettes and the linking geometry. As I mentioned above, the linking geometry has a degree of selection to it which is usually resolved through linkages that produce irregular four- and five-point stars. The stars I have produced here are not as accurate as they should be, but I hope they are good enough to understand.

Below it I have placed the same photograph shown at the top of this note, but this time with the geometric pattern superimposed on it. There is a small lack of coincidence, but I think it is good enough to see how the designer took a panel design and applied it to a pair of doors. He might have elected to produce two panels, one for each door leaf, but I believe the use of a single panel, simply divided and heavily modeled, has produced a powerful design, and one well suited to introducing those who came to worship at the Grand Mosque.

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Other studies

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.

I should add that the above original three studies were made using traditional drawing instruments on A4 paper and are all I have left of a number of studies, the others now being in private collections. The majority of sketches on these pages were made using computer software.

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 on the next page, 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 three based on six-pointed geometry below them. While the first two photographs here show balance and a calmness created by their even regularity, the flowers based on five-pointed geometry do look to me more interesting in the irregularity of that geometry. I should add that by using photographs which have the six-point geometry balanced on a point, they are a little more lively than when rotated through 30°. 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.

Having said that, the lowest photograph here, illustrating six-point geometry in a tulip, shows a more lively form by the uneven opening of its alternate petals suggesting three-point geometry, reinforced by the three-point form of its stigma. However, the six anthers correctly reflect the six-point geometry reflected in its petals. 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.

Thirteen point

It seems that wherever you look at plants, there is something that catches your attention. Here are two photographs, the first taken a year after the first of the fourteen-petalled plants below, this having thirteen petals. I wasn’t looking for this, but the slight eccentricity caught my eye in both of them. I must go back and look at more of the flowers to see how many other variations in petals there are. What is interesting is how easily the eye spots these oddities, and yet how normal they seem, perhaps as they are not perfectly symmetrical in their shape and size of leaves. As a visual exercise, look at these thirteen-petalled flowers and then at the fourteen-petalled ones below and you should be able to see almost instantly how one has an odd number of petals the other an even number, without knowing the exact numbers of petals.

Fourteen point

These two photographs of large daisies show fourteen-petalled flowers. Looking at the top one 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. The lower photograph shows two fourteen-petalled flowers, these being part of the same plant which had two thirteen-petalled flowers illustrated immediately above. It seems this flower is prone to many variations in the number of petals it displays, yet casual viewing suggests there is no variation.

Other divisioning

The first of these photographs show yet another plant that exhibits a different number of petals on the same plant. In this case there are, from left to right, nineteen and twenty-one petals on the two heads, though they may appear on casual inspection, to have the same number. With this number of petals it is unlikely that the argument suggesting that variation creates interest, will hold good, though it may well be significant with a smaller number of petals, exhibiting a lack of balance that will keep the eye more interested when close to the flower heads.

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. I had anticipated that, even with eccentricity, the number of petals would fall on specific numbers, but this appears not to be the case.

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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Perfection

I intend to write a little more about this sometime, but it might be useful to place a marker here on the issues relating geometry and divisions to the concept and desire for perfection. Islam clearly holds that there is only a single source of perfection and that, although perfection may be sought in all things, tradition appears to be that it is a reflection of an inner perfection, ihsaan, but may not be attained by man. It is a commonplace that tilings, carpets and other areas of artistic endeavour have built into them eccentricities or obvious mistakes in order that it will not be thought their authors believe they can produce perfect examples of their different arts. In a sense this practice replicates nature with its wide variety of variations from the norm or from perfection. Yet modern production methods strive for a degree of accuracy that might be thought to represent perfection.

A diverse collection of authors have held that perfection, though it may be an end in itself, leads only to sterility and mediocrity. This seems to be particularly true when we look at the increasing ability of modern equipment and techniques to produce accurate products. The Victorian writer and critic, John Ruskin, John D. Sedding of the Arts and Crafts movement, Christopher Alexander and, even, Shakespeare, for instance have all held that the search for perfection in art misunderstands its meaning, and is likely to mar what is good.

But the reason I make this note here is that while geometry has an obvious mathematical precision to it, the artistic work that uses it as a basis for design can not hope to achieve the same degree of accuracy by virtue of the materials that are used. In a sense this is reflected in nature, particularly when you look at the example of the flowers above; none is perfect, but all have a beauty in their essence and variation. This is true, too, of work such as Qatari naqsh, Moroccan tilework and Persian carpets. Works such as these compare and contrast favourably with, for instance, the modern tilework that is being used to decorate new buildings and can appear sterile, if not soul-destroying, in its accuracy of production and execution.

more to be written…

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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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Symmetry

There are many ways of understanding symmetry. In these notes it is likely that I will mix some of them, for which I apologise if it is found confusing; those needing to know more about the subject should look elsewhere. Here I wish to write about it in both its general and specific geometrical meanings.

At its most basic, symmetry refers to a geometric quality where there is a precise balance that can be described mathematically. Commonly this is taken to be a correspondence of shape about an axis, central point or plane. This diagram illustrates the four basic methods by which symmetry is achieved. There is more written about this on the following geometry page where the seventeen basic patterns relating to symmetry are discussed in a little more detail.

In the natural world there is argued to be a tendency towards the achieving of symmetry in all its aspects. Symmetry is thought to be beautiful with a character illustrative of completion, an end to be sought. Though we can observe it visually, symmetry is best described in mathematical terms. More than this it is thought that the nearer a mathematical theory can be described by a simple declaration, the more likely it is to be correct – an area of study that continues to focus the attention of mathematicians and physical scientists.

But it is in asymmetry that we often notice the beauty of symmetrical designs, for while the eye is drawn to eccentricities, it tends to prefer perfection to imperfection. The issues here also relate to a state of mind; symmetry and perfection lead to a relaxed, peaceful and contemplative state of mind whereas asymmetry heightens observation, leading to a more interested state of awareness, one more focussed on what it is looking at and, by definition, less relaxed.

As I have mentioned elsewhere, there is an extension to the arguments about perfection in Islam where it is considered that man is unable to produce a design that is perfect, the consequence of this being a degree of imperfection that traditionally is built into designs. But this is not a concept that is peculiar to Islam; the same concept exists in other parts of the world, for instance, in Japan where it was written, in a 14th century essay on idleness:

In everything…uniformity is undesirable. Leaving something incomplete makes it interesting and gives one the feeling that there is room for growth… Even when building the Imperial palace, they always leave one place unfinished.

As you can see, this is not a consideration brought about by religion, but one having to do with aesthetic appreciation, a phenomenon that resonates in many areas that govern the way in which we perceive the world about us.

Symmetry can be defined in terms of grid and detailed design, and although there are only seventeen basic pattern arrangements, 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.

more to be written…