For example, consider this pattern in a window in the external courtyard wall at Baku’s Taza Pir mosque.
This beautiful pattern, with its eight-pointed stars set within octagons, turns up on plate 67 in Jules Bourgoin’s 1879 Les Éléments de L’Art Arabe (which you can download from archive.org).
It’s wallpaper group is the fairly common *442 (p4m) and it is generated by tessellating a square cell.
Construction of this pattern is straightforward. The eight-pointed star in the centre is inscribed in a circle whose radius is one quarter the side of the square. The vertices of the octagon are found by extending the sides of the star. The rest of the construction lines are extensions of the octagon sides, and lines connecting star dimples that are three apart.
But one enters the Taza Pir compound via a stairway from the street. The panels in the stairwell are related, but subtly different from the window design!
What did they do here? There is the same eight-pointed star in the centre, and the same enclosing octagon, but in this case they’ve trimmed back, to the borders of the octagon, the square that one repeats.
As a result, the square tile borders stand out strongly as lines, and around each point where four tiles meet we get a big diamond holding four small diamonds.
(This also belongs to the *442 wallpaper group.)
Now, in the Old City, I came across a piece of octagon-based decoration that illustrates what happens if one doesn’t follow best practices, as explained by Eric Broug. This pattern involves starting with the same pattern as in the Taza Pir windows (above; a.k.a. Bourgoin’s Plate 67), but then repeating a somewhat random subset of it. In other words, incorrect tessellation.
It would appear that the manufacturer of these pre-cast concrete blocks selected a piece out of the overall pattern that was not the all-important basic square, but rather a rectangle.
Hence each of the concrete blocks looks like this:
When you put them together, lines match up, but the effect of the original design is lost.
The wallpaper group of this pattern would be*2222 (pmm).
Elsewhere in the Old City, there were pre-cast patterns that did tessellate pleasingly, again with octagons.
But back at the Taza Pir mosque, I spotted this on an adjacent building, which I believe is Baku Islamic University:
The grill pattern is octagons packed together, with squares in between; and eight radial lines emanating from the centre of each octagon. It’s basically the central column of this pattern:
But look at what they did in the point of the arch. It’s beyond my knowledge to know whether this is best practice or not, but it is definitely creative.
Eric Broug, from the School of Islamic Geometric Design, writes that sometimes while travelling he sees a piece of contemporary Islamic geometric design and recognizes it as, well, let’s say less than best practice. (He sometimes posts images of this sort of thing on Instagram under the hashtag #cpigd, which stands for Common Problems in Islamic Geometric Design.)
What sort of mistakes are they? He explains on his page on best practices, but to me the most common one is incorrect tessellation, where a block of pattern is repeated in ways that cause lines to abruptly stop instead of continuing on.
I thought it would be interesting to take a look at the designs I found in Azerbaijan, in the cases where they were identifiably Islamic, and ask the same question: are they examples of good, traditional design.
So, bearing that in mind, let’s look at a grill window I found in the Divankhana of the Palace of the Shirvanshahs in Baku, Azerbaijan.
The Palace of the Shirvanshahs is the premier piece of historical architecture in the Old City of Baku, or, as it’s called in Azerbaijani, İçərişəhər or Icheri Sheher. The original buildings have been deduced to date from the 15th century, but most of the palace was heavily renovated/restored in the 20th century so it’s not immediately clear whether the details one sees are original or the work of a restorer.
The Divankhana (which is also variously called the Divan-Khane or Divanhane) is a structure in its own courtyard just off the outer courtyard of the palace. It holds a pleasing octagonal pavilion. whose original function is unknown (there are many theories). The pavilion is domed and consequently two stories in height, so it stands above the courtyard wall and can be seen from the palace courtyard. In one corner of the Divankhana, there is a staircase leading up to a locked door on the upper storey of the pavilion.
This window is at the top of that staircase, and looks out into the palace’s outer courtyard. Here it is seen from inside.
What I thought when I saw this is There’s no way this can be a best practices design. There were so many wacky elements that I had never seen in an Islamic geometric design before. For one, I couldn’t find a single axis of reflection in it, anywhere. For another, it contained a number of strange, three-way intersections.
Is it a bad design, perhaps a modern artist not working within traditional lines, or could this be authentic traditional design?
Let’s look at how the pattern works.
The whole pattern begins with a pair of adjacent large octagons.
These fill the window, as shown.
Centred on the vertices of each octagon are eight smaller octagons, sized so that when they overlap they bisect each other’s sides.
These circles of smaller octagons define an empty space in the centre of each of the larger octagons, a space which is an eight-pointed star.
So far, so good, and very symmetrical and, in fact, infinitely tile-able. The cleverness comes with what they did inside each 8-pointed star.
They divided this space with a four-armed pattern which has rotational symmetry but no mirror symmetry.
They ran it in opposite directions in the top and bottom halves of the window. So, looking from inside the window, there is a counter-clockwise star in the top half and a clockwise star in the bottom half.
This is fairly mind-boggling for traditional Islamic design — I think. (I’m no expert.) The little pattern of four “hammerhead” shapes that circles within each 8-pointed star looks more M. C. Escher than standard geometric design.
Mathematically, we might ask: does this pattern at least pass the test of being able to be continued in all directions?
Well, yes. One would simply add more big octagons above, below and to the sides, and then add the smaller octagons, etc. This could go on forever.
Each big octagon hosts an eight-pointed star in its centre. If you alternate big octagons that host clockwise stars with big octagons that host counter-clockwise stars, following a checkerboard-like pattern, the centre of each star would be a four-fold centre of rotation. The corners where four octagon tiles come together are two-fold centres of rotation. And there are axes of reflection along the lines where adjacent big octagons touch.
Patterns with these axes of reflection and pattern of rotational centres belong (mathematically) to the “wallpaper group” known in orbifold notation as 4*2, and in IUC notation as p4g. (There are seventeen possible wallpaper groups.) Patterns that are 4*2 have a “twist” to them so that the basic square unit of the pattern is not the same as its mirror image.
In fact, the fundamental tile for this pattern, a tile from which the entire pattern can be generated through reflection and rotation, is triangular.
4*2 is an unusual wallpaper group for an Islamic geometric design, most of which are *442 (p4m) or *632 (p6m). But it’s not unheard of, and would not lead us to conclude that this pattern doesn’t use best practices.
And, then it gets a little more complicated.
At the Palace of the Shirvanshahs I never shot a picture of the outside of the window, so when I got home I found that, while there’s no Google Street view in Azerbaijan, someone had conveniently taken a photosphere in the main courtyard of the palace four months before I was there.
Here, in the photosphere, is the facade of the Divankhana as seen from the outer courtyard. The Divankhana is the building with the whitish dome.
Here’s a close-up of the three window openings on its second floor:
Only the central one is full-sized; the outer ones are reduced in height, and the left one is completely blocked. I was inside the rightmost window, and in this image we see it as the reverse of my original picture.
But look at the centre window.
The stars in this window turn the same way. They’re both counter-clockwise (as seen from outside).
This pattern group, incidentally, would be 442 (orbifold) or p4 (IUC); it has no axes of reflection and two kinds of centres of four-fold rotation. I have not seen an Islamic geometric design before in the 442 wallpaper group.
So the windows are not the same. Prompting some contemplation.
Design problems in Cairo. *All* the restored, replaced geometrical windows on *all* the restored buildings in this street in Darb al-Ahmar are identical. Originally, there would have been a great diversity of design.
This post prompted some pretty hot discussion in the comments about whether window patterns in a single building were traditionally uniform or not. Broug was of the opinion that
conventionally in Islamic Architecture, different patterns would be next to each other. possibly to encourage contemplation and reflection by looking for differences and similarities
At the Divankhana, the blocked and un-restored left-hand window suggests that this second storey of the Divankhana has not been aggressively restored, and that we’re not looking at recently restored windows here.
And I wonder if this subtle difference between the centre and right windows was an example of encouraging contemplation and reflection (no pun intended). It certainly got me contemplating.