Building patterns is one thing. Taking one apart is quite another. This section of the site looks at some of the ways of approaching that procedure.
We begin with an illumination from the Quran (Figure 1, which shows one half of a mirror image double page. The image is a scan of Plate 71 from 'The Quranic Art of Calligraphy and Illumination' by Martin Lings). The Quran was written by Ali Ibn Muhammad al-Mukattib al-Ashrafi and illuminated by Ibrahim al-Amidi in 775/1372. The original is in the National Library of Cairo (10,ff.iv-2r).
The Illumination consists of a rectangular section surrounded by an unusually broad border. The two parts are separated by an interior border applied in gold leaf. The whole is further unusual in that it contains no text from the Quran. In the central section the design is dominated by a series of ten-sided polygons, each with a pentangular form at it's centre. The border consists of a complex repeated floral motif which appears to be minimally influenced by the pattern in the centre.
The Rectangular Section
In attempting to determine the native design for this panel, we have a number of alternative approaches.
Firstly, we could simply turn to one of the numerous pattern books on the market and try to find one that matches. It is unlikely that we would find an identical design since each artist tends to add their own variations on a standard model. Finding an approximation would of course be useful but in this case I could not find anything close enough. This is perhaps not too surprising since designs based on ten-sided polygons occur infrequently.
Secondly, we could trace the original and scale it up to the desired size. This method would enable us to reproduce the original, but it would tell us little of the underlying structure.
Thirdly, we could attempt to find the main construction lines using our knowledge of geometry. In this instance the arrangement of the decagonal forms clearly indicates that both the x and y axes can be divided into four equal sections such that the centre of each decagon coincides with an intersection on the grid so formed (see Figure 2).
By careful examination of the design we further note that certain lines allow us to determine the horizontal and vertical spacing of the decagons. They are not perfect lines but are close enough and have enough geometric purity to them that we may assume any errors can be attributed to graphical misplacement or artistic interpretation. In Figure 2 the red lines, which are coincident with one side of two adjacent decagons and pass through the mid point between them, enable us to calculate the horizontal displacements. The blue lines, which pass through the centre of the decagon in the middle are also coincident with sides of decagons diametrically opposite each other which provide the information needed to calculate the vertical spacing.
Construction - Step 1
Using the red line feature in Figure 2 we can calculate the grid spacing in terms of the radius of the circumscribing circles around the decagons - See Figure 3.
Calculation of 'X'
OB = r (where r is the radius of the circumscribing circle)
Angle BOA = 72 degrees (ie 2/10ths of 360 degrees at O)
Angle OBA = 72 degrees (since segments of the decagon are isosceles)
Therefore triangle OBA is isosceles
Drop a perpendicular from A to line OB to form a right angled triangle OAC where OC = r/2
The Cosine of 72 degrees = r/2 divided by X
Therefore X = r/2Cos72 = 1.618r
We now have the horizontal scaling needed to build our rectangle. But we also have something else. The number 1.618 has another significance in that it is the ratio of the Golden Section. It should not surprise us that this relationship occurs since pentagonal forms have a close association with it. (For readers who wish to know more, go to Phi: The Golden Number). We move now to the vertical spacing.
Calculation of 'Y'
Angle sar = angle qpr = 36 degrees
Distance sa = distance qp
Therefore triangles sar and qpr are congruent reflections of each other,
and therefore sr = rq = 0.5Y
The tangent of 36 degrees = 0.727 = sr/sa
sr/sa = 0.5Y/1.618r
Y = 1.618r x 1.454 = 2.352r
Ratio X:Y = 1:1.454
Dimensions of the Rectangle
The dimensions of the original page in the National Library of Cairo are said to be 74 x 52 cm. Unfortunately the copy in the book does not show the top and bottom edges so we would have to guess their position. In this case however, we are not attempting to recreate the original page exactly and so we will work on a size of rectangle suitable for a web page - 300px by 436px.
And now on to the design of the decagons.
Before we look at the construction of the decagons, we need to resolve an incongruity.
At the centre of each decagon is a pentangular structure. As these are polygons with an odd number of sides they can drawn with a vertice pointing left, right, up or down. Now one would expect that whatever orientation was chosen by the designer it would be consistent throughout the pattern, but it isn't. Of the five complete decagons two have their pentagular centres pointing up whilst three are pointing down - there is no rationale to this layout. Quite simply it is a mistake.
So what are we to make of this error? Do we recreate it in our reconstructed design or do we correct it, and if we did the latter, which direction is correct?
We will deal with the reconstruction below, but it may be instructive to consider the circumstances whereby the error originated. My opinion follows this argument; pentagons are one of the most difficult structures to build geometrically; a logical short cut would be to construct the pentagonal part of the design as a template which could be repeated as often as required (possibly as a tracing, but more likely using pin pricks). Provided the original is accurate and the orientation of the template is consistent this is a perfectly acceptable modus operandi. Unfortunately, in this case neither is true.
That the orientation of the template was inconsistent is self-evident, but further evidence can be brought to bear to show that the accuracy of the original left something to be desired. If we take a careful tracing of one of the decagons with it's central pentagonal form pointing upwards we can show that this fits neatly over the others with a similar orientation but that for those with a downward pointing pentagon the template has to be rotated by 36 degrees in order for it to attain as tight a fit. Whilst not being conclusive evidence this does suggest that a template was used and that it became displaced during the transfer exercise. An alternative scenario is that the draughtsman, wary of defacing the original template, turned it upside down at some point without appreciating the consequences.
Whilst this analysis is interesting, it doesn't help determine the 'correct' orientation. We have no clear indication of which direction the pentagonal structures should be pointing, but we note that an upwards pointing form has by definition a flat base and is therefore inherently stable. With a vertice pointing downwards the whole device is pivoted on that point which suggests instability. We will opt for the more stable structure and adopt a design with vertices pointing upwards.
Deconstructing the decagons
The design within the decagons is constructed from a simple arrangement of chords to a circle somewhat larger than that which contains the decagon (see Fig 4). The relative sizes of these circles is not easy to determine accurately, but by striking off further chords of the outer circle which connect every fifth vertice it is possible to gain tangents to the circle in a fairly elegant manner. The intermediate circle is formed using chords connecting every fourth point.
Deconstructing the inter-decagonal spaces
We note that a rectangle with opposing corners at the centres of two diagonally adjacent decagons is proportionally the same as the whole rectangular panel but one quarter the size of it (see Fig 5). This smaller rectangle contains the remainder of the panel design which can be duplicated by reflection across the whole surface.
In the original panel, all of the lines in this section are approximately parallel to, or perpendicular to, one or another side of the decagon. There are significant departures from this norm, but the design can be completed by maintaining this principle with no breakdown in the integrity of the pattern. The reason for these departures is thought to lie in the use of 'ribbons' rather than dimensionless lines between the elements of the design. As I have mentioned elsewhere, these ribbons often lie such that the construction line lies along one side of the ribbon rather than along the axis. This is done to conserve the shape of the smaller units but has the knock on effect of displacing all connected lines which are thereby thrown off their true bearing.
As we have not yet reached the stage of laying the ribbons, but are concerned with the accuracy of the pattern we shall ignore the departures and define the pattern using lines that are all parallel to sides of the decagons. This can be achieved by simply extending all salient lines and picking out those sections which accord with the final pattern. It is at this point that a secondary usage of the ribbons is revealed.
The pattern derived in Fig 5 works geometrically but it does generate some odd looking shapes that are without symmetry through either of their major axes. In the finished illumination however, this is disguised by allowing the construction lines to fall diagonally across lengths of ribbon. This also explains why some of the finished ribbons are not quite parallel to sides of the decagons.
This section of the design illustrates the difficulties of working with 5/10 sided polygons.
To be continued..........