Geometric basis of patterns

The construction of patterns for the purpose of decoration has been the subject of research and investigation for centuries. While I have witnessed the use of straight edges and string in the setting out of Islamic patterns, in watching craftsmen work it is evident that much is carried in their heads as they set out the details governing the patterns they are creating. Just as with this research, albeit over a much shorter time, the knowledge craftsmen have developed has accumulated over centuries.

Nowadays we have greater resources available to investigate pattern conceptualisation and construction; we also have the capability to examine non-Euclidian geometry and create designs which would not have been possible six hundred years ago, as has been effected in this study which has introduced the possibility of extending the beauty of traditional geometries into a third dimension. It is interesting to speculate on how an understanding of this mathematical approach might have altered the artefacts produced by Islamic craftsman in the past, particularly bearing in mind the use of pattern to cover large planar surfaces, and the manner in which this is said to have aided contemplation.

more to be written…

On a more prosaic level, execution of the work in the Gulf has, as a result of a number of factors, been relatively simple and has derived from the simple patterns based on four, five and six point geometries. I have never seen designs based on seven, nine or more complex geometries in Qatar. Although geometric patterns are found in nature it is likely that they would not have been observed by designers in the Gulf as there is little to see; rather they would have developed from the simple tools needed to create the pattern geometries, though there is the likely relationship of designers working coevally on the other side of the Gulf, and it is true that many craftsmen worked on both sides of the Gulf, some of them moving up to Qatar from Dubai.

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

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The beginnings of Arabic geometry

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

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

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

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

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

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

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

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

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

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

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

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

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Tools used in the design and layout of patterns

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

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

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

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Basis of geometry

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

From the geometrical constructions mentioned above generally, patterns with a greater complexity can be constructed. Patterns with 12, 15, 16 and 20 are readily established though approximations of 14 and 18 can be made based on approximate constructions for 7 and 9. I don’t know how to construct 13 and 19 point patterns, though there is a construction for 17. However, I digress… The next few notes look at the construction of these more complicated shapes.

Shapes that are capable of being constructed with an unmarked straight edge and compasses are known as constructible polygons and begin with polygons with sides that are: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285…

Shapes that are not capable of being constructed with an unmarked straight edge and compasses begin with polygons with sides that are: 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75…

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Three point geometry

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

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Four point geometry

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

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√2 geometry

Related to this is the development of geometries based on the diagonal of a square – √2 where the side is 1 unit. The geometries of the Alhambra depend, to some extent, on this geometry rather than on the Golden Section – 1:1.414 compared with 1:1.618. I don’t know why this should be so, though there is the Greco-Roman tradition, as mentioned above in the note on cosmatesque designs, of the use of √2 geometries in two dimensional patterning.

The second diagram illustrates the difference between the basic two proportions. Note that the √2 proportions are those of the International ‘A’ paper sizes – A4, the most commonly used, being 210mm x 297mm.

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

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

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

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

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Triangle and octagon approximations

One of the benefits of being able to construct geometries readily, is the ability to keep within tolerable limits of accuracy with relatively simple tools. Much of the work on these pages relates to the search for geometries that can be constructed with a compass and straight edge. But certain constructions became possible and suggested dimensions that, in turn, became generators for setting out. In effect, these measurements decided both shapes and proportions.

Such a system was employed by the Mughals who used the relationship between the numbers 7, 12, 17 and 24 to set out developments. There are three points to remember:

• An octagon whose width – not radius – is 17, will have a side with a length of 7,
• a right angled triangle with a side length of 12 will have a hypotenuse length of 17, and
• a right angled triangle with a side length of 17 will have a hypotenuse length of 24.

But if you take a closer look at the geometry involved you will see that these are very much approximations.

This first illustration shows the dimensions as described above. The red triangle has right-angled sides of 17 and a hypotenuse of 24, the blue triangle right-angled sides of 12 and a hypotenuse of 17, while the black octagon has sides of 7 and a width of 17. The geometry fits together accurately enough at this scale, so it is understandable that those using it must have been reasonably sure that it was accurate enough for their purposes.

Examining the diagrams in a little more detail, this second illustration reveals the mathematics behind the approximations. If you look at the two triangles of right-angled sides 5 and 7, with their respective hypotenuses of 7 and 10, you will understand that there is a small discrepancy in each of them.

These are drawn as 5² + 5² = 7² or 49, and 7² + 7² = 10² or 100.

But mathematically these should be 5² + 5² = √50, or 7.07106781², and 7² + 7² = √98 or 9.89949494² respectively. This is an error of just over 0.1% – one centimetre in ten metres – perhaps not that significant when setting out buildings in those days.

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Five point geometry

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

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

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

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

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

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

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Six point geometry

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

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

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

An important characteristic of six point geometry is its relationship with √3. This first illustration sets out the simplest construction that establishes the basic √3 development. This would have been a simple geometry to establish with a piece of string and a straight edge.

A circle, when sub-divided by the superscription of six circles located equally along it circumference, establishes a right angled triangle with its hypoteneuse the diameter of the circle, and its shortest side equal to half the circle’s diameter, or the radius. The remaining side is, therefore, √3 times the radius of the circle.

In the lower illustration the rectangle with its short side coincident with two opposite sides of the hexagon has its long side in the proportion of √3:1 to the side of the hexagon.

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

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Seven point geometry

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

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

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

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

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Eight point geometry

Obviously related to four-point geometry, eight-point geometry is extremely easy to construct, requiring just a pair of compasses and a straight edge.

The initial construction is exactly the same as with four point geometry. Using compasses with a fixed diameter, draw a circle in the centre of a horizontal line. From the two points that circle cuts the horizontal line, draw two more circles with the same diameter as the original circle. Through the points where the outer two circles intersect, raise a perpendicular through the centre of the original circle. Joining the four points where the horizontal and vertical lines cut the central circle will produce a square set on the diagonal.

With centres at the points where the vertical line intersects with the central circle, draw two more circles of the same diameter. From the points where these two circles intersect with the original two outer circles, draw two diagonal lines. The central circle will now be divided into eight equal lengths along its circumference, creating an octagon.

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A development of eight point geometry

There are only two regular two-dimensional shapes that will stack closely with no space between them – the square and the hexagon.

Much of Islamic geometry depends upon patterns composed of more than a single type of geometry, the linking elements being developed in relationship to the basic shapes used. These are usually based on the lines related to the construction of the simpler geometrical shapes, hidden within them, so to speak.

But some of the shapes have a simple link possible – in many cases a triangle and rhombus. The octagon will not stack as do the square and hexagon, but there are two types of pattern formed by relatively simple linkages. In the upper of these two examples I have turned the octagon through 22.5° and then added squares which have sides the same length as those of the octagon. Although there is a symmetry to this pattern on a large area of repetition, the pattern radiates from the central octagon outwards. Compare that with the second pattern below.

In this case the octagon has not been turned but balances on a point as in the construction above. This creates a more dynamic shape than in the first pattern above. The difference between this and the pattern above is that this is symmetrical in two dimensions. Its other characteristic is that it brings in an implied circular pattern that the eye reads, introducing a flow to the pattern. Both patterns are found commonly in Islamic design but it is interesting to note that the lower pattern is being incorporated into the façade of one of the buildings in the New District of Doha.

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Nine point geometry

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

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

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

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

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Ten point geometry

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

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

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

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A development of ten point geometry

Here are two other ways of using the simple ten-point pattern illustrated above. The pattern on the left is a simple ribbon development showing the geometry as a single plane. That on the right has both an applied shadow to give it something of a three-dimensional feel to it, but also has the ribbon inter-leaving itself as it moves around the geometry, a form that is sometimes a feature of two-dimensional geometry when there is a wish to elevate the design.

Although it was intended that the above two patterns should be read separately, and that they represent two different concepts, in putting them together in a single diagram it is noticeable that there is an implied relationship in their almost touching. Despite the two patterns’ conceptual differences, there is a strong feeling of natural attraction that suggests they should touch or even be linked. This is a natural consequence of the manner in which our brains react to visual stimuli and attempt to form more understandable patterns. It is a characteristic of our brain that operates all the time as it processes what our eyes feed to it.

With regard to the joining of patterns it is evident that the different types of geometry have the ability to make natural junctions, continuing the geometry to create an infinite pattern, bounded only by the borders of the object in which the geometry has been established. These three illustrations demonstrate how a simple horizontal shift of the basic rosette shown above, enables different geometric patterns to be brought into effect. There are other ways of joining the patterns, but these three examples are the main ways in which ten point geometries join naturally, as can be seen for two of them in the diagram below.

Patterns based on geometries that are not symmetrical about both their horizontal and vertical axes will not be able to create regular patterns in both directions. Essentially this means that only four, eight and twelve point geometries have the property of two-directional pattern symmetry within them, whereas six and ten point geometries will have different joining properties with regard to horizontal and vertical directions. In this diagram you can see two of the natural joining relationships for ten point geometry, and how the vertical relationships differ.

And here the relationship between the three rosettes is not based on the coincidence of the main lines of the pattern as is true for the diagrams immediately above, but of the centres of the rosettes with those lines. Note that the relationship between the corners of the two top rosettes with those of the lower one are not the same. This is one of the characteristics that allows for a degree of diversity and invention on the part of the designer, and leads to the apparently infinite number of patterns found in Islamic designs. Let’s look for a minute at the construction of just one of these relationships.

The pattern on the left of these two illustrations is the same as above and to the right. The additional lines show how the lower of the three rosettes is positioned.

In order to position the third rosette, extend the two, angled, oblique lines from the outer edges of the two upper rosettes to meet and then further extend them to cut perpendiculars dropped from the two centres of the upper rosettes. Where they cut, a horizontal line is drawn between them. A line dropped at right angles from the centre of the two upper rosettes will bisect the horizontal line, this point being the centre of the third rosette.

Note that, in the illustration on the right, the relationship between pairs of lozenges – one in the lower and one in either of the upper rosettes – a line drawn between them has the respective lozenge lying on each side of that line. This brings up the issue of the manner in which underlying geometry is used in delineating patterns.

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Eleven point geometry

First, be aware that there seems to be considerable confusion about the name of the eleven-sided polygon; I understand it to be the hendecagon.

Eleven point geometry relates to one of the polygons that are not constructible; it is not possible to form such a polygon using only an unmarked straight edge and compasses. There is, however, a rather complicated construction that will produce a relatively accurate eleven pointed polygon, for which I should refer you to Professor Bodner and her work not only on this figure, but also on other geometries relating to Islamic geometry.

This first construction is based on a piece of origami work suggested by Professor Dutch and illustrates how to produce a basic hendecagon. Sub-dividing a square three times consecutively along its diagonal produces a point on the edge, ‘A’, to which a line is drawn from the centre of the square, ‘B’. Bisect ∠ABC to obtain ‘D’. Next, bisect ∠EBD to obtain ‘F’ and then bisect ∠EBF to obtain ‘G’. The angle ∠EBG is a very close approximation to 32.7272°, which is the internal angle of a hendecagon. Repeating this angle around the centre, ‘B’, and superimposing a circle, enables you to draw the points that create the hendecagon.

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A development of eleven point geometry

The reason I have chosen to construct the hendecagon in this manner, following the work of Bodner, is that it is part of a process by which she recreated a pattern from the important Topkapi scroll that incorporates eleven point and nine point geometries, both of them unconstructible. A second point to note is that the rectangle upon which the two geometries are constructed does not locate the centres of the eleven and nine point geometries at its corners, as is common with Islamic patterns, but upon its edges.

Subdivide each of the sides of the hendecagon. From these eleven points, together with the eleven points of the hendecagon, connect to every ninth point around the polygon. In effect this draws a continuous line from a point of the hendecagon to a point on a side, to the next, ninth, point which will again be a point on the hendecagon, and so on – a,b,c,d,e… Following the lines that connect only the points on the sides of the hendecagon and linking them to every other central point, you will arrive at the star shape illustrated on the right.

In order to develop the pattern, the eleven-sided polygon has to be repeated and, in order to locate it, a rectangle has to be established on which it will be positioned. To do this, extend a vertical line through the centre of the right-hand polygon and draw a circle equal in radius and congruent with the circle describing the right-hand polygon. This forms the right hand side of the rectangle. From the centre of the top circle, extend a line left which will form the top side of the rectangle.

Next, establish the left and bottom sides of the rectangle by extending three lines from the centre of the right-hand polygon, the top one of which will intersect the top side of the rectangle. From this point of intersection, drop a line at right angles to the central of the three lines previously drawn from the right-hand polygon, and extend it to meet the lowest of those three lines. That point of intersection lies on the bottom line of the required rectangle which is formed by constructing a line at right angles from the vertical, right hand side of the rectangle. From the point where the central line and dropped line meet draw a circle with radius set to the centre of the right hand eleven-sided polygon. At the point of intersection of the central line and circle, construct a line to form the left hand side of the rectangle. At the point of intersection a second, eleven-sided polygon should be constructed.

As you can see from these sketches, the construction of this pattern becomes a little more complex now. Should you be interested in step-by-step instruction of the design, I suggest you look at the original article from which I developed my illustrations. The sketches I have placed here are not sufficiently detailed for them to be followed here but are placed in order to show not only something of the difficulty involved, but also to suggest that the method works, and that it would have been possible for the Topkapi illustration to have been constructed.

This last sketch in the series is as far as I will take the illustration of this construction as I have to admit I found it difficult to obtain the degree of accuracy to which I usually draw. However, that is not a criticism of the method, rather a comment on my drafting techniques.

So, here we have the reconstructed pattern from the Topkapi manuscript or, more accurately, a version of it drawn to produce a reasonable understanding of the basic rectangle with its associated eleven and nine point geometries.

The reason I have taken the trouble to set out this design in some detail above is both for the difficult nature of its construction as well as for the lively character of the pattern produced by using the two geometries together, most probably a function of combining the two odd-numbered geometries. Note that the two eleven-point stars on the left and right are mirrored with respect to each other on a horizontal axis, as are the nine-point stars at the top and bottom which are mirrored on a vertical axis.

Finally, here is a pattern created by extending the basic rectangle. This has been effected by mirroring both horizontally and vertically in order to have the pattern run through coherently. The result is a gently undulating pattern, perhaps dominated by the larger eleven-point star due to there being more white space within it compared with the nine-point star. But that is one of the characteristics of Islamic patterns, where balance is sought within an apparently infinitely extendable field, but where interest is created by point and counterpoint.

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Fifteen point geometry

It may seem counter-intuitive that it is not possible to construct a fifteen-sided figure easily. After all, five- and six-point geometries are relatively easy to construct, particularly the latter, requiring no more than a pair of compasses to produce six sub-divisions of a circle. Five-point geometry is also relatively easy to construct and, with it, the setting for golden mean geometry, mentioned below. However, if you combine the two geometries, it is possible to create both 12° and 24° angles, the latter being the internal angle of a circle subtended into fifteen equal parts. The diagram to the right shows the construction of a pentagon in red and, in blue, a hexagon both constructed on the same circle. With a pair of compasses set at the distance between the blue and red lines’ radial intersections with the circle, it is possible to step off fifteen divisions along the circumference of the circle. Incidentally, a fifteen-sided, two-dimensional figure is usually known as a pentadecagon, but you may also see them referred to as a quindecagon or pentakaidecagon.

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Seventeen point geometry

On first consideration it may seem impossible to construct a seventeen sided figure with compass and straight edge. However Gauss, in 1796 and at the age of nineteen, proved this not to be the case, and a few years later, Erchinger demonstrated how it might be constructed.

The construction of a seventeen-point figure, or heptadecagon is shown here, which I shall repeat below on this page. My diagrams tend to be over-complex, so I would strongly advise those with an interest in the construction to visit that site which has far more extensive and accurate information both on the heptadecagon as well as dealing with the underlying mathematics of shapes, together with other related matters.

Here, above and to the side, the first sketch is of a heptadecagon, or seventeen-pointed geometrical construction, the circles having been created in the laying out of the points of the heptadecagon on its containing circle. The method of construction establishes two points of the heptadecagon that are its first and fourth points. As there are an irregular number of points on the heptadecagon, compasses set at the distance between points one and four will enable all seventeen points to be established.

The second sketch shows some of the geometry relating to the construction of seventeen point geometry. These are the construction lines created in discovering the position of the fourth point on the circumference of a circle, shown in blue, that is to be developed into the heptadecagon illustrated in the first sketch above.

In order to construct the heptadecagon on the given circle, begin by drawing two diameters at right angles to each other.

On the vertical diameter, AB, locate a point, C, so that AC is a quarter of AB.

Join P1 to C, then locate point D so that ∠ ACD is a quarter of ∠ ACP1.

Locate point E so that ∠ DCE is 45° and, with diameter EP1 construct a semi-circle. This semi-circle will intersect with AB at F.

With radius DF, draw a semi-circle centred on D. Where this intersects with AP1 at G, raise a perpendicular to intersect with the circle at P4.

Now, as described at the beginning of this section, locate the rest of the points along the circumference of the circle by moving compasses around it with radius P1P4. This will create the points in the order shown here in red.

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Nineteen point geometry

The majority of divisions commonly seen are dealt with on these pages and can be seen all around us. Those that are based on simple numbers, or their multiples, are easy to spot, but as larger numbers are reached, patterns based on them are more difficult to construct, and to find. One of the reasons for that is there is not necessarily a reason to make unusual sub-divisions for the very fact that they are not easily recognised. Here, for instance, is a tile which, while not based on an Islamic design, caught my eye and is constructed with nineteen divisions – also known as a nonadecagon or enneadecagon. The basis for sub-division is, therefore, 19°, an angle that is not possible to construct simply. Note that the pattern is symmetrical about its vertical axis but not its horizontal axis.

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Using Neusis to trisect an angle

As you can see from the various studies above, it is possible to construct a number of geometrical shapes using just a pair of compasses and an unmarked straight edge. Some geometrical constructions, however, are not similarly possible, though Archimedes is said to have used a system known as neusis to resolve some of the difficulties faced by those attempting to draw what might be thought to be simple geometries. One of Archimedes’ original works, consisting of fifteen propositions, was lost but re-appeared in ‘The Book of Lemmas’, translated into Latin from the ninth century Arabic manuscript of al-Saabi’ Thabit ibn Qurra, in the mid-seventeenth century. A lemma, incidentally, is similar to a theorem, and is a subsidiary proposition that is assumed to be true in order to prove another proposition. There are fifteen propositions in the document, the eighth of them relating to the division of an angle by three. It is possible that this might have been a common method for assisting in laying out certain patterns.

While the theorem holds true, the method of construction does not have the precision of the regular geometries, but is likely to have been accurate enough for craftsmen to use in practice, and is shown here for interest.

The angle to be trisected is ∠ ABC. With centre, B, describe a circle of radius AB. Extend the horizontal line AB. On a ruler mark the same distance as the radius, in this case at points, D and E. Keeping the ruler attached to position C, slide it along until points D and E meet respectively the circle and the horizontal line. At that point, the lines DE and DB – which are both equal – will form an isosceles triangle with the angles ∠ DEB and ∠ DBE a third of angle ∠ ABC. Neusis can also be used in the construction of a regular heptagon, though it is not the method I have illustrated above.

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Trisecting a line

While the trisection of an angle requires a marked ruler, as is shown above, trisecting a line is a relatively simple process.

With the centre of a circle located at one end, point A, of the line AB and its radius the length of the line, draw a circle and then draw a second circle with its centre placed at the other end of the line, point B. With the same radius, draw a third circle at one of the points of intersection of the two circles, point C. Draw a line from A through C to intersect the third circle at point D. Draw a line from point D to the intersection of the first two circles at point E. Where the line DE intersects the line AB, at F, that point will divide line AB into AF and FB in the proportions of 2:1. If the latter exercise is repeated by extending the line BC, then the line AB will have been divided into thirds.

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The Golden mean or section

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

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

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

The proportions of the square to the rectangle are:

1:(√5+1)/2, or 1:1·618.

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

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

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Fibonacci series

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

a : b is as b : a+b

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

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

In order to clarify the above construction, here is the method I used. Create a square and extend its sides unidirectionally on one side only. Within the square construct a circle. With compass at position ‘b’ draw an arc anticlockwise from position ‘a’ to meet the vertically extended line from the square. With compass at point ‘c’ extend the arc anticlockwise to the horizontally extended line. From point ‘d’ extend the arc to the horizontally extended line. From point ‘f’ extend the arc to the vertically extended line. From point ‘c’…etc.

Although this construction is often used to produce spirals or volutes, the accretion of quadrants does not produce an accurate mathematical description of a spiral. A spiral is described by a line that increases its distance from a point by a constant factor as it moves around that point on a plane. That particular spiral, illustrated above, is a logarithmic spiral, and not an Archimedean spiral, illustrated here, which is distinguished by its lines being set parallel to each other, rather than increasing in distance as does the logarithmic spiral.

You can see in the above diagram of the logarithmic spiral, one of the many forms commonly to be discovered in the natural world. Shells, fruit and flowers are often found with this type of geometry driving the arrangement of their forms, the patterns being exhibited in a number of ways, all relating to their growth structures sometimes in their external form, often in cross-sections made across them.

Both the pineapple to the right, as well as the pine cone below it, have grown with eight divisions spiraling clockwise and thirteen anti-clockwise. These numbers, of course, are both in accord with the Fibonacci series and seem to be one of the most commonly found bases for the growth of plants – 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 etc.

Yet while arrangements based on eight and thirteen divisions seem to be the most commonly discovered in nature, those based on thirteen and twenty-one divisions seem also to be found everywhere you look. In this yellow flower head, for instance, it can clearly be seen that there are thirteen spirals rotating in an anti-clockwise direction, and twenty-one clockwise. There is a design coherence in the movement created in this static arrangement that contributes greatly to their attraction.

I should add that similar forms are used in the creation of designs expressed in work such as architecture and painting, and have been for thousands of years, the designers taking their geometrical structures from those which they have witnessed in the natural world around them. Designers have taken their lessons from nature and have attempted to develop their observations into a codified series of proportions which, they have argued, may improve our relationship, resonance or psychological comfort with the product of their designs, particularly architecture.

This has led to a number of proportions suggested as being suitable if not necessary for the basis of design work, perhaps the two best known being Leonardo da Vinci’s ‘Vitruvian Man’ and Le Corbusier’s modular systems which he utilised in the design of his buildings.

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

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

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

Before I end this note I would just like to add two more images as they are common examples, should be well known to most people, and are striking examples of this form of natural geometry. The first photograph shows a cross section of the shell of a nautilus pompilius where a logarithmic spiral governs the growth pattern of the buoyancy chambers the cephalopod uses in order to maintain its buoyancy in the water.

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

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

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

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

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

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

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

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Other systems of proportion

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

more to be written…

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Relationship of Arabic calligraphy to geometry

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

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

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

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

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Design and ‘e’

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

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

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

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

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

more to be written…

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Application of traditional patterns

In Qatar, traditional patterns are only applied in four areas:

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

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

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Platonic solids

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

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

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

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

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

The five figures are composed as follows:

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

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

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Seventeen different patterns

This note and the accompanying diagrams follow on from a note made on the preceding page, entitled symmetry. Its mention there has more to do with its meaning relating to balance and design aesthetics. But in the area of patterns, it has a more complex meaning, one to do with the way in which symmetrical patterns are produced and categorised, a complexity governed by the mathematics of group theory. The notation used in the accompanying illustrations is the IUC notation adopted by the International Union of Crystallography in 1952. Incidentally, you may come across other systems of notation such as the orbifold notation of John Horton Conway. This is based not on crystallography, as is the IUC system, but on topology.

Although there may appear to be an infinite number of patterns to be seen in the world around us, in reality there are only seventeen different ways in which patterns can be repeated two-dimensionally. This is sometimes referred to as ‘wallpaper’ patterning and commonly describes patterns created as covering on walls, floors and ceilings.

Examples of all or nearly all of these patterns can be found in the Alhambra, Spain. It has been stated by Grünbaum that there are four patterns missing from the Alhambra – pg, pgg, p2 and p3m1 – though the latter two have been found to have been constructed coaevally in Toledo. He also stated that the former two – pg and pgg – are not found in Islamic art at all, though this has been disputed. As you will see in the first sentence of this paragraph, there is even dispute about the number of symmetries found in the Alhambra. It is not my intent to set out the arguments for and against here; those who might have an interest in this issue should look elsewhere.

There is certainly a lesser number of designs than this to be found in the Gulf, if for no other reason than there are not that many examples to analyse and, of these examples, few cover areas of the size required to see the repeats effectively. More than this, the patterns seen in Qatar tend to be iconic and are not designed specifically to cover large areas as might be found in, for instance, the glazed tilework on many buildings in Iran and further afield in the Indian sub-continent, Egypt and Morocco among others. Tiling, of course, lends itself to mass production, the tiles enabling the formation of a wide variety of different arrangements and patterns.

It is possible that, if there had been a tradition in Qatar of tiling, there might have been a very different situation as craftsmen from Iran would most likely have made their influence apparent. But the fact is that the materials from which the architecture and its detailing were constructed in Qatar were relatively soft. Patterns were confined mainly to doors and wall panels, both of relatively transient materials, particularly the juss from which wall panels and patterns were constructed. The character of the juss, with its need to work relatively quickly, led to few of the patterns on the walls of Qatari buildings incorporating symmetrical geometries. Having said that, it is perhaps more accurate to state that where symmetrical geometries were employed, they were relatively simple.

There are two final points I should make. In writing about the seventeen different forms of symmetry, I should emphasise that this relates to a single colour and to flat patterns.

In the many examples of patterns to be found both in Islamic and other art, colour is often a major element of the design. If you include permutations of two colours, there are a possibility of an additional forty-six symmetries that may be found.

There are also complexities brought about when the patterns are not flat. A number of designs are enlivened by their designers by creating the appearance of three-dimensional work by simulating lines running over and under each other. Generally this will break the possibility of simple reflectional symmetry.

more to be written…

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Four basic pattern types of symmetry

The seventeen different ways for patterns to be formed, and illustrated in the five diagrams located above, have been established mathematically but in the diagrams are described notationally. Symmetry, in this sense, is created with the movement of a shape by one of four methods that preserve the relationships of distance, size, angles and shape. Each basic pattern type is constructed by taking an element and then repeating it by

• rotation, which turns a shape around a fixed point,
• translation, which simply moves a shape a specific distance without rotating or reflecting it, and
• reflection, which mirrors a shape across a fixed line.

In addition to the three types above, there is a fourth type,

• glide reflection, which reflects a shape across a fixed line, but then translates or moves it in a direction parallel with the line of reflection. It is the only type of the four that is established in two movements.

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

• the first group of four are made without rotation and are known as: p1, pm, pg and cm,
• the second group are constructed using rotations of 180°, without rotations of 60° or 90°: p2, pgg, pmm, cmm and pmg,
• the third group are constructed with rotations of 90°: p4, p4g and p4m,
• the fourth group use rotations of 120°: p3, p3ml and p3lm, and
• the fifth group are constructed with rotations of 60°: p6 and p6m.

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

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

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Determining a pattern type

This diagram relies on information from two different sources, although there is a third and larger version of it here where there is also more to be read on the subject of seventeen patterns. Reading the relevant sections of the sources should give a far wider understanding of the subject, though it is fair to say that elsewhere there is a considerable amount of literature dealing with the mathematics behind wallpaper patterns. This is a specialised area of mathematics but one that might be investigated further with benefit.

The diagram sets out a method for determining the category into which a particular symmetrical pattern might fall. The four categories of pattern illustrated in the diagram above –

• translation,
• reflection,
• rotation and
• glide reflection

– govern the process that starts with the number of rotations there might be, ranging in order from none to six – 0, 2, 3, 4 and 6. The orders relate to the number of rotations that can occur within 360°.

• Order 2 – 180°
• Order 3 – 120°
• Order 4 –   90°
• Order 6 –   60°

Starting with the rectangle on the left marked ‘Smallest rotation’, follow the line that defines the smallest number of rotations to be found in the pattern. The questions asked relate to whether or not there is reflection, rotation or glide reflection with regard to mirror lines or axes. The selection based on the decision or decisions will lead to the appropriate definition of the pattern type. I hope that’s clear…

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An eighteenth geometry

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

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

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

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

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

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

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

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

Prior to this, in 1974, Penrose established the basic units that first created what came to be known as Penrose patterns or tilings. The two shapes were nicknamed darts and kites and were derived from a pentagon as is shown on the left diagram of these illustrations.

The dart is produced by adding two of the central triangles together, and the kite by the addition of the two side triangles. When used to form a Penrose tiling, with the increase in the numbers used in the pattern, the ratio of kites to darts approaches the Golden Section, φ or 1.618. Note in this lower illustration, that the dart and kite shapes, when added together, form the rhombi that are sketched just above. These are known as Penrose rhombs.

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Decagonal and quasi-crystalline tiling

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

Here are the five tiles:

Termed ‘quasi-crystalline’ patterns, they are patterns which fill all of a space, but without the translational symmetry characteristic of true crystals. Essentially this means that attempting to match an exact copy of the pattern over itself will never produce a precise match. In this they are similar to the tiles to which Dr. Penrose put his name in the 1970s. The thesis is that Islamic designers developed these geometries between the thirteenth and sixteenth centuries, and that the geometries were not familiar to Western mathematicians for a further five hundred years.

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

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

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

At present this thesis is unproved but, as I have written elsewhere, considerable advances were made in the Islamic world in science and the arts, based on earlier Greek and other work. Little of this has been taught in Western schools and universities as evolving from Islamic study and it seems there is yet more research to be carried out.

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Artisans and geometry

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

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

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

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

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

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

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The appearance of the circle

Very rarely in Islamic geometric design is the circle apparent as such. There appear to be circles in many of the designs, and approximations of the circle can be seen in geometric patterns, but these circles are formed from the straight sides of polygons. As is illustrated here, for a given diameter, the greater the number of sides to the polygon, the more likely a circle is to be approximated. Yet the circle is the generator for most if not all geometric forms. These polygons have sides of, reading from left to right, 3, 4, 5, 6, 7, 8 and, on the lower level, 9, 10, 11, 12, 15 and 20. You can see that the last one is very nearly a circle.

There are some who claim that a circle is not needed in deriving or forming certain patterns, but others argue this not to be so. To some extent this is a semantic argument as a circle, or at least a pair of compasses, is necessary to construct any shape in order to form right angles and to sub-divide angles.

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A simple pattern

Here is a relatively common pattern found in Islamic designs, often in the form of a background. It has similarities with patterns found in Qatari naqsh. Based on a lozenge shape that repeats infinitely in two directions, it is constrained only, as in this case, by a relatively arbitrary selection of frame. It can be seen as a linear pattern or, here coloured, as a slightly more interesting pattern.

Yet the pattern is based on, or derived from, a square. Here I have shown three adjacent squares at a different scale in order to make the construction a little more apparent. But I have also drawn the contained circle as this form is argued to create the base planar surface for the design of the unit pattern.

In order to construct the lozenges, first take a square, here tinted blue, and draw diagonals from its corners. Through its centre, a horizontal and vertical line are taken to its perimeter, subdividing the square into four smaller squares. These squares are further divided by diagonals. Next, lines are taken from the corners to the central points of each of the sides of the square, creating the square illustrated in the centre. Each of the small squares is then subdivided horizontally and vertically into three by joining and extending lines through the intersections of the diagonals, creating the square at the right of the diagram.

The first part of this illustration shows, rather than the square, the circle on which the lozenge has been established. The second part shows how the lozenge has been created from the diagonal construction lines. Obviously the lozenge might be arranged either horizontally or vertically. In the third part of the illustration I have added four stars that bound the lozenge, though these stars would be parts of the surrounding lozenges in an extension of the pattern.

This first diagram illustrates the extended pattern showing how the lozenges fit together, alternating their direction through 90° on an orthogonal grid or matrix. As happens with many geometric patterns formed of two shapes, the eye tends to see one or the other shape, rather than the overall two-dimensional patterning. Here the eye moves between seeing the lozenges and stars.

The next two diagrams use the same geometric construction as the lozenges above, but the pattern is orthogonal. I have included the two patterns as they are one of the patterns found in Qatari naqsh work, usually as background or within a complete pattern. The two patterns are essentially the same, the only difference being the points at which the corners of the rectangles are selected.

This lowest sketch, based on the exact geometry of the study above, illustrates something of the character of a naqsh panel with an attempt to give a three-dimensional effect with a drop shadow. A simple effect such as this would have been relatively easy to carry out on either wet or dry naqsh. The pattern is found on the other side of the Gulf where the workers are likely to have originated and is a relatively common pattern in traditional buildings in Qatar.

At first glance these two photographs appear to illustrate an application of the pattern, but will not have required any use of the circle to lay it out. You can see that the pattern has the addition of a small square to create more of an even balance between solid and void, the recessed carvings having the proportion of 5:1. The photograph from which these two details were taken was made in an old majlis at Wakra in the late nineteen-seventies. In the upper photograph you can see a typical naqsh panel with the pattern taking up the whole of the panel as a ground. In the lower photograph you can see how the setting out lines were scratched on the dry or drying plaster in order to define the areas for carving out.

There are two points to note in the above photograph. The first is that the setting out lines are made along the centre of the area to be left untouched, when you might have anticipated that the craftsman would mark out the edges of the pattern, leaving much of the setting out lines hidden. The second is that the carving is relatively deep and has straight sides. On external work it was not necessary to have such deep carving in order for the sun or light to make its effect apparent.

This note began with a suggestion that the argument shapes might or might not be based on circles, was semantic. I have gone through this brief exercise in order to illustrate how a shape is formed by following lines of a pre-constructed pattern. The note on decagonal and quasi-crystalline tiling above showed that artisans used a number of standardised shapes with which to produce geometrical patterns. While only a compass and straight edge are necessary to produce many of the shapes used by designers, it is apparent that they would have used a simplistic system rather than construct all patterns from scratch. My own experience is in seeing artisans using primitive compasses, wooden set squares and chalk. The ancient Egyptians commonly used knotted ropes to form right angles; knots set at three, four and five units apart along a rope would enable a right angle to be readily formed, this being the basis of the simplest Pythagorean right angled triangle.

I have illustrated the construction of a square above, but there are a number of other ways of constructing squares, some of which might be more suited to the production of a lozenge.

Here, to the right, is an alternative way of constructing a square upon a given line. All the necessary construction lines are there but I have added the adjacent squares, circles and diagonals in order to show the relationship with the diagrams above.

The lower illustration has been added as a reminder of the relationship between the lozenge patterns and the square and circle. The illustration clearly shows the importance of the square in establishing the pattern of the lozenges. It is clear that there is a different relationship between the four lozenges in the corners of the squares and the four on the periphery of the circle.

So, the argument about the generator of patterns being square or circle may be semantic, but it is apparent that the circle, or at least compasses, are necessary in order to generate patterns found in Islamic geometric designs. Nevertheless, it would be a rational progression for artisans to use formers constructed to a variety of angles and lengths with which to construct those patterns.

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Numbers in Islam

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

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

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

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Seven

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

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

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

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

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

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

And verily We have beautified this lowest heaven with lamps…

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

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

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

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

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

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Four

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

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

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

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Nineteen

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

Over it are nineteen.

Some translations have this as:

Over it are nineteen angels.

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

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

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

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

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

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

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

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

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

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

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

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

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Colour

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

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

The system of three colours comprises:

• black,
• white, and
• sandalwood.

The system of four colours comprises:

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

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

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

more to be written…