Mirosław Majewski
INTRODUCTION
It was in 2010 when I first time visited the Patriarchal Church of St. George in Istanbul. The church located in the Phanar district, facing the Golden Horn, is full of beautiful Byzantine icons and valuable historical objects. Between them is the very famous patriarchal throne that is attributed to St. John Chrysostom (398404). At this day I have spent there about half an hour admiring beautiful icons and the very ornate interior of the church. While leaving it I saw near the entrance a mysterious box covered with a very curious pattern. I made a photograph of it and I went out to visit other Greek churches in the Phanar. I did not pay any attention to this box, as at this time I was more interested in Byzantine icons.
Fig. 1 The Patriarchal church of St. George is the fifth church in Constantinople to house the Ecumenical Patriarchate since the 15th century. Formerly a convent for Orthodox nuns, it was converted to the patriarchal offices by Patriarch Matthew II (15981601). Patriarch Timothy II refurbished the church in 1614, and Patriarch Jeremiah III rebuilt it after a fire in 1720). It was repaired in 1836 by Patriarch Gregory VI and restored recently under Patriarch Bartholomew. 
Fig. 2 Dazzling geometric ornament on the 17^{th} century candle stand. The glass covering the candle stand reflects lights from the church. Photo year 2010. 
In September 2014, I visited the St. George Church again. This time I was paying more attention to the geometric patterns and especially Islamic geometric patterns. The intriguing box was still there, protected with glass, and the pattern on it was teasing my curiosity. The pattern looked very Islamic, but it had a small inscription written in old Greek characters. I started looking for the function of the box, its history and meaning of the inscription. Unfortunately no one was able to give me a truthful translation of the inscription and even a brief history of the box.
When I came back home I started exchanging letters with the Patriarchate and finally I got answers to my questions: the box is a candle stand made of walnut and inlaid with ivory petals in the shape of pentagons, the 17thcentury candle stand in the patriarchal church is a replica of early Egyptian craftsmanship. It is hard to say what it means “early Egyptian craftsmanship”? How early? Egyptian culture is one of the oldest in the world. There was some hint in the letter: the inscription records that it was a gift of “Manuel, son of Peter, from Kastoria, Greece, donated in the year 1669.” A furrier by trade, Manuel was a great benefactor of Constantinopolitan education (information obtained from George E. Sarraf).
This really means that the candle stand is definitely older than from 1669. In this moment I could also distinguish in the inscription words ‘γιός’ (son). ‘Petro’ and ‘Kastorias’. The remaining text was still unreadable for me, but I guess, Greek characters changed significantly since 1669.
However, I am still not convinced about Egyptian origins of the pattern. None of the books in my collection, including the famous Islamic Art in Cairo by E. Prisse d’Avennes, does not show a similar pattern.
Fig. 3 The inscription records that it was a gift of Manuel, son of Peter, from Kastoria, Greece, donated in the year 1669. A furrier by trade, Manuel was a great benefactor of Constantinopolitan education” 
I found a slightly similar pattern in J. Bourgoin book “Arabic Geometrical Pattern & Design”. It is well known that Bourgoin has been many years in Egypt and he depicted in his book many patterns from this area. However, later I found also information that this particular pattern Bourgoin probably copied from somewhere in Turkey. Thus I was back with my search in Turkey. At the time when the mysterious pattern was created Istanbul was full of architectural developments. The most famous Ottoman mosque Sultan Ahmed Mosque was opened in 1616. Here we can find a number of very complex patterns created using the same designing approach. In many other buildings from this period of time in Istanbul we can find similar patterns, but none of them has such unusual features.
Thus we can conclude that the origins of this design are here in Istanbul? Judging from the quality of design and its complexity, as well as the excellent craftsmanship the candle box was designed by one of the top artists of this period of time and created in one of the best workshops in the town. Designer of the pattern on the box used concepts that we can find in many Islamic patterns in Istanbul, but the pattern has specific symbolism that place it outside of Islamic geometric patterns. It is definitely Christian art but designed using the same methods that were used while creating patterns for Ottoman mosques and Topkapi Palace.
In the next pages I will show a hypothetical way how the pattern was constructed. This is my personal interpretation and probably we will never be sure if the method described here was really used. However, the pattern is so complex that there must be a systematic method, let us say – an algorithmic technique, to construct it. There is no way that one having a collection of pentagonal, rhomboidal, and other tiles, could create such pattern by simply assembling these tiles using a tryanderror method.
Geometry of the pattern
On the first sight the pattern on the candle box looks very complex and chaotic. One has to spend some time looking at it from a distance of 2, 3 meters in order to see some repeating motives. The very first thing we are able to distinguish is the big rosette in the middle and a quarter of such rosette in each corner of the pattern. Then after some more time we may see the 24 shapes that are marked in the picture using a red regular decagons. We will call them rubies. There are 10 rubies around the central rosette and then 7 of them on each side of the rosette. Later we will see that the size of the ruby, and precisely – the length of its side, determines sizes of all remaining elements of this design, and the pattern on the ruby determines patterns on all remaining shapes.
The number of rubies – 24 and one central rosette suggests a symbolic meaning of this part of the design. If we look at a typical iconostas in a Greek orthodox church we can easily identify on top of it two rows of icons. One of them, called Deisis, contains Jesus Christ in the middle and twelve apostles next to Him. Another row of icons contains the Old Testament prophets and patriarchs – the latter sometimes including the twelve sons of Jacob. In a limited space in this row there are usually twelve icons only. Thus we have together 24 icons representing apostles, prophets and patriarchs. Is this really the reason why we have here the 24 round shapes, that I call rubies, around the central big rosette? Can we assume that these shapes in the pattern represent apostles, prophets and patriarchs? The two numbers 12 and 24 we will find again in this pattern in other places. This specific organization of rubies it is another feature that, according to my experience, is not present in any Islamic pattern.
Fig. 4. The 24 decagons marking rubies on the design 
Starting from this point the artist had two, sometimes contradicting tasks. Having the limited space he could not expand this design much further and at the same time he had to fill empty spaces between rubies as well as to conclude this design in such a way that in each corner of the box will be a big rosette. In fact he could design only ¼ of the whole design, e.g. righttop rectangle located between the central big rosette and the rosette that is located in the topright of the design. The remaining three quarters of the design are similar to the topright one. However, we have to notice that the design has only one line of the mirror symmetry. The bottom part is a mirror reflection of the top part. This design has no vertical mirror symmetry. The left side is not a mirror reflection of the right side. The two central rubies do not have a mirror symmetry line passing through centers of their sides.
Let us again look at the figure 4. Inside of each ring of 5 rubies on the sides of the design there is a starlike shape that is worth filling with something. For this purpose we can use small rhombi, each with its side equal to the side of a ruby. We will call them diamonds. Our diamonds will fit also in a number of other places. It is easy to find out that the smaller angle of a diamond is equal to 72 degrees. Construction of a diamond can be easily obtained from the shape of a ruby. We will talk about their construction later.
Although the figure 5 shows large quantity of diamonds filling space in our design there are still more places where, in the future, we will put them. Now we can easily notice that between diamonds and rubies there are still gaps that can be filled by a shape that looks like a trapezium with shorter of the parallel sides equal to the side of the ruby. These shapes we will call emeralds and we use a green color to distinguish them from the remaining shapes.
Fig. 5. The 24 decagons marking rubies on the design with diamonds filling gaps between rubies. 
Fig. 6. Further extension of the geometry of the pattern. Emeralds were added. 
Fig. 7. Next step in reconstructing geometry of the pattern. 
Fig. 8. Next step in reconstructing geometry of the pattern. 
The main problem is the area marked on the figure 7 by yellow circles. Here he should close the two small rings of rubies. However, there is not enough space there to put two rubies, one for each ring. They will overlap. Therefore, he had to fill this space by something that will look like a ruby, it will not overlap with anything else and it will preserve the symmetry of top and bottom parts. He could not use diamonds as this could destroy the nice shape of pentagonal blue stars. Thus the only option is the hexagonal shape – citrine. This will determine a few further steps of the design. We can easily see it on the enclosed picture (fig. 8).
At the very moment it is evident that one more shape is needed. This is a kite like shape that can fit into the empty space between emeralds and diamonds (see the right side of the figure 8). This new shape we will call amethyst and we will use a pink color to distinguish it from the other shapes.
Due to a perspective distortion of the camera the far right side of the photograph does not match well with the pattern of polygons. Thus we will do the further reconstruction of the pattern without the photograph in the background. The further reconstruction is quite simple. However, we may see there some errors and peculiarities of the pattern near the right edge of it. The first thing is the unnecessary gap (see number 1 on the figure 9) that was left between the two rubies and amethyst. The artist had to invent one more shape to fill this gap. However, this place in the pattern I could easily fill with the shapes used already (see number 2 on top of the figure 9). Is this really an error of the artist? Let us look again on the geometry of the pattern (still figure 9). We can easily calculate the blue stars made out of diamonds? We have 6 of them on the right side and there will be another 6 on the left side of the pattern. Thus we have 12 of them together. Again number 12 comes into our analysis. Now, let us calculate the orange shapes, we named them citrines. With the designer’s “error” we have 12 of them on the right side of the pattern and 12 on the left side. Perhaps it was an objective to have again number 24 in this design?
Two more problems we can see on the right edge of the pattern (see numbers 3 and 4). The diamonds as well as the amethysts are cut in a very unusual way. According to the way how hundreds of similar patterns were created the cutting line should go along a symmetry line of a polygonal shape. This means, in case of diamonds, the cutting line should go from one vertex to the opposite vertex. In fact, the shape that I marked here using number 3 in the original ornament is very irregular. Therefore using an amethyst here is just my guess only. In the left side of the ornament in this very place the craftsmen used an emerald shape but the shape above it is again irregular.
Thus we should ask the question – is there any other, or better, way to position all shapes used in the peripheral sides of this pattern so the right edge will not cross a polygonal shape or go through its mirror symmetry line?
In this whole design we will deal with regular decagons and objects derived from them. Thus we will need a bit of geometry in order to feel comfortable with all designing steps described in this text.Constructions of regular decagon and shapes derived from it
Angles in a regular decagon
One can easily calculate that the internal angle of a decagon is 36 degrees (360/10=36). It is easy to prove that the segment CB is parallel to the segment AE’. Thus the external angle of a decagon must be also 36 degrees. Therefore, if we want to construct a regular decagon and all remaining shapes we have to know how to construct a decagon having given the length of its side. now how to construct a decagon having given the length of its side. 
Our first task will be constructing angles 72 and 36 degrees, and then constructing a regular decagon if its side is given.
Construction of the 72 degrees angle

Construction of the 36 degrees angle

Construction of a decagon from a given side

Construct bisectors of AB and BH. Their intersection creates point O that is a center of the decagon.

Now, we can try to construct all shapes used in reconstruction of geometry of this pattern. The ruby is simply a decagon and we constructed it already. Below I show construction of the four remaining shapes.
Constructions of other shapes
Here are four more shapes necessary to fill gaps in our design. We call them

We will start by constructing the pattern of the ruby. We already know that it is based on a regular decagon.
Construction of patterns for components of the ornament
Construction of ruby pattern

As we did mention before, all remaining polygons and patterns on them depend of the ruby pattern. Thus while developing all remaining patterns we should always compare them with this what we created a while ago.
We should also notice one important thing. If we look at the original pattern on the candle stand, or its photograph, we may easily notice that edges of the polygons often form vertical or slant lines going through large parts of the whole design. This effect was achieved by precisely matching patterns on the neighboring polygons.
The next pattern we will create is the amethyst. It is just part of the ruby.
Construction of the pattern of the amethyst
This pattern is 1/5 of the pattern of the ruby. We created it in the first step while developing the ruby. See figures (a) and (b) in the previous construction. 
Construction of the pattern of the diamond
A construction of the diamond is also related to the ruby, or if one prefers to the regular decagon. The next figure shows this relation.
In order to match the pattern of the diamond we have to place two regular decagons next to it in such a way, that the sides of the diamond are also the sides of the decagon. Then we construct the shape of the pattern for diamond by simply following edges of the pattern on the ruby.
The picture shows how the diamond will fit between two rubies. Note also that the pentagon crated between the pattern on the rubies and the pattern on the diamond is a regular pentagon. 
Below are shown a configuration of diamonds with a ruby and the fivediamond star that we often see on the original pattern. 

Now is time to create the pattern on the emerald. We need it to fill the space between rubies and diamonds in the central part of the whole design as well as in a few other places.
Construction of the pattern on the Emerald
We start by creating a shape of emerald inside of the regular decagon. Then we place two diamonds next to it on its left and right side. 
By extending the pattern on the diamonds and following the same procedure that we used to create the pattern on the ruby we construct a grid of lines (dashed) that we will use to produce the pattern on the emerald. 
The image below shows how the pattern on the emerald and on ruby fit together. 
Next missing piece for this big ornament is the pattern on the citrine.
Construction of the pattern on the citrine
This construction is straightforward. We create shape for the citrine, we mark centers of its sides and then from the centers of the two parallel sides we construct lines perpendicular to the opposite sides. This way we obtain lines AC, AD, BE, and BF. Then we draw segments HI, GJ, GH and IJ connecting centers of the sides around the acute angles. 
Now, we are ready to create the pattern on the citrine. Above it is shown how it looks. The picture below on the right shows a simple pattern created using only the citrine shapes. 
Construction of the pattern of the chrysolite
We already constructed most of the shapes needed for the candle box pattern. One of them is the minor shape filling the two gaps near the top and bottom border of the pattern. We will create it in a while. At the very moment it is necessary to create the major object from the center of the pattern – the large rosette. We will call it the chrysolite. Its construction is the most complex part of this design.
By drawing lines through points AE and BD, and extending the side BC we create a shape that we developed while constructing the emerald.
We find centers of the sides of emerald. Here these are the points H and I. This gives us opportunity to obtain the two points J and K. These are they key points necessary to create the whole pattern on the chrysolite. Now, in any convenient way we should create similar points on each side of the decagon. We can do this by drawing an appropriate small circles for each vertex of the decagon, or by using parallel lines, or by measuring distances of these points from the vertices. Any logical and reasonably simple way will be good enough for this task. 
This way we will obtain a chain of the points on each side of the chrysolite. By connecting these points in a way shown here we will produce a small grid for the external part of the chrysolite pattern. 
By constructing two additional dashed lines parallel to the edges of the pattern we obtain a framework for a small triangles finishing the external part of the pattern .
By repeating this triangle for each side of the decagon we finish the external part of the rosette. 
This picture shows how the petals of the rosette can be set up. First we draw the large circle passing through the point P. Then by drawing diagonals of the decagon we create the intersection point of the large circle with one of the diagonals (point R).
Now we draw a small circle with the radius PR and center in R. This way we will obtain point S as an intersection of one of the diagonals with the small circle. Finally the medium circle passing through S will be a convenient tool to replicate point S around the center of the decagon. Lines passing through these replicas of the point S will allow us to create the petals of the rosette. This is shown on the next two pictures. 
The image below shows the chrysolite surrounded by the other shapes.
Reconstruction of the complete ornament from the candle box
At the very moment we have all components needed to restore the whole ornament from the candle box. We start by completing its geometry.
In this picture the polygons play the role of placeholders for each element of the ornament. The irregular shapes on the left and right edges of the original ornament were replaced by tiles that should be there if we follow the logic of this design.
Now we can place decorations on these placeholders, remove placeholders and finally convert the obtained result into white and brown colors in order to imitate the original materials used from the ornament.
Bibliography
Prisse d’Avennes E., Islamic Art in Cairo (reprint of the book Arabic Decoration published in 1885), A Ziitouna Book, The American University in Cairo Press, Cairo, 1999.
Bourgoin J., Arabic Geometrical Pattern & Design (reprint of the book published in 1879), Dover Publications, Inc., New York, 1973.
Copyright notes
All photographs, illustrations and text were created by the author. All sketches were created using Geometer’s Sketchpad^{®}, a computer program by KCP Technologies, now part of the McGrawHill Education. More about Geometer’s Sketchpad can be found at Geometer’s Sketchpad Resource Center at http://www.dynamicgeometry.com/.
Miroslaw Majewski
New York Institute of Technology
College of Arts & Sciences
Abu Dhabi, UAE
15 April, 2015, Zlotoria, POLAND
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