The design in the middle matches the drawing at the left, but the right one is a full layer higher. Printing designs in 3D helps to investigate the true mature of these muqarnas.
During workshops, students discover how to investigate patterns. It is all about how to build a complex architecture with a limited set of basic figures. Student can start with design drawings in the flat plane by placing pieces of paper next to each other like a puzzle. Next, they can create the design drawing with 3D printed units. During the workshop, we bring in the 3D printed units. As an alternative, we created an online environment to develop muqarnas in GeoGebra.
Karbandi and muqarnas
Muqarnas are decorative elements. Karbandi are constructive elements. Karbandi are the arches that carry a dome. The pressure of the building mass of the round dome is directed through the karbandi to the walls of the underlying rectangular box.
Earlier workshops on muqarnas have been held in the Netherlands by Bart Post, Maartje Elsinga-van der Veen, Susanne Tak, Rogier Bos, and Philip van Egmond, among others. They teach the system "back on front", neglecting the system "back on curve". Common sources are the dissertation by Sylvia Harmsen and articles by Yvonne Dold-Samplonius. Her thesis covers both systems, but pays most attention to "back on front".
A more practical thesis is written by Hüdai Sirri Şenalp, working at for Hassa Architects in Istanbul.
Examples of the fourteen basic units
With a limited number of standardized building units, all different kinds of muqarnas can be built. Below are examples of the fourteen basic building units for the 45° system based on squares, parallelograms and triangles whose angles are multiples of 22.5° are, for example, 45°, 67.5°, 90°, 135°. The top row shows the units with walls, the bottom row shows the intermediate units. An other system is based on 30° angles.
Shapes can be defined by their angles.
Every shape is symmetrical. Its symmetry line is a diagonal of a quadrilateral or a perpendicular bisector of a triangle. We code units by counting the quarters of a right angle, starting at the point of the unit, along the diagonal, to the wall and round to the point. Square A is therefore 2-2-4 because its angles are 45°, 45°, and 90°. Rhombus B is therefore 3-3-2 and large biped J is 5-2-1.
For in-depth analysis of the structure of muqarnas, I develop software that generates drawings, tables and statistics for any muqarnas. Input is a series of coded strings. The software interprets these strings using the geometrical definitions of a collection of muqarnas units.
For the software, a muqarnas is a collection of strings of letters, where each letter represents a unit. The letters A to F have been chosen for the top row of units with rear wall and the letters G to L for the bottom row of intermediate units without rear wall. One of the aims is to identify for each unit its neighbours (left, right, above, and below) and to detect missing neighbours and other inconsistencies.
Student Basic Assignment
At the NVvW 2019 study day, three helpful children asked if I may need their help. I was curious how far they would come with a minimal explanation. Within half an hour, they assembled their muqarnas from 3D printed units.
Easy Starter Activity
Young students start with an easy assignment where the rules are quickly discovered. This two-layer muqarnas follows the rule "back on front". Students were handed over a construction plate for a quick start.
M. Fatin Uluengin
Ermenek Tol Medresse
In Ermenek, you will find the Tol Medresse. This muqarnas follows the rule "back on curve".
Bursa Abdal Mehmet Camii
In Bursa, you will find the Abdal Mehmet mosque. This muqarnas follows the rule "back on curve".
Nigde Ak Medresse
The mosque Ak Medresse is situated in Nigde. The Turkish architect M. Fatin Uluengin has written a valuable book about muqarnas.
My reconstruction may have been done according to the rules, namely "back on front", the result looks weird and does not match the photo.
The first layer is much too wide.
Moreover, the photo shows five layers and the reconstruction only four layers.
After implementing the additional rule "back on curve", the missing layer becomes visible.
Instead of the blue "intermediate units" there will be muqarnas units with a wall.
The result is a slimmer and higher muqarnas that matches both Uluengin's analysis and the photo.
Natanz is a city between Teheran, Kashan and Isfahan with a beautiful great mosque. Yaghan and Harmsen described an example of a muqarnas in a niche, which is situated in the basement vault of the north iwan in that mosque (Masjid-i Jami).
Blair has written in detail about the shrine complex of Natanz.
Harmsen investigated a muqarnas in a niche into great extent. She made a replica of carton. Interesting is what happens at the third layer. She did fill in the gap in the paper version, but not in her computer models. Her rules to generate and interpretet graphs do not help her to solve what she omits: sometimes, you need flat walls. As a result, the software does not understand how to connect tiers, especially the rhombus at the first and second layer. In my interpretation, I added flat walls and a new non-standard muqarnas unit: a half square with a wall at the short side (instead of its wall on the long side.
Yaghan also studied this muqarnas. His drawings are very precise, but do not tell which object is drawn. In my analysis, I labeled every unit. Four units could not be recognized. My assumption is a combination of flat walls and a non-standardhalf square. Questions arise about the need of drawing the the short diagonals of the rhombus and the meaning of the large rectangle. His analysis drawing is almost identical to Harmsen, but there are differences, small in detail, but major impact on the interpretation. That is why I developed the paper 2D analysis, bought a 3D printer, started printing 3D objects and developed dedicated software to find out whether these differences matter.
Blair, S., The Ilkhanid Shrine Complex at Natanz, Iran, Cambridge Harvard University Press, 1987
Kayseri Cifte Madrasah Gevher Nesibe
Above a door of the Cifte Madrasah Gevher Nesibe hospital, a muqarnas with a special set of two non-standard building units has been realized. The set creates space within the square surface of the underlying layer for a smaller building unit on the next layer. As a result, the surface of a standard square is doubled in height.
Harmsen and Krömker describe this extraordinary unit because their software did not recognize its geometry.
In the province of Çorum in the city of Osmancık there is a mosque with an interesting muqarnas. This muqarnas follows the line "back on front". The muqarnas can be built from the standard basic units, with a few exceptions. Half of a standard intermediate unit is required on the first building layer. In the second, half of a unit is needed. In the third building layer half of a standard intermediate unit is also needed.
Shiro Takahashi's drawing suggests some unique units in the top layer. This muqarnas needs further study at this point. The assignment for students is slightly different and solely based on standard building units.
Amasya / Bimarhane
The Bimarhane mosque is located in Amasya. In the entrance portal, two small muqarnas are incorporated in the wall on the left and right. This muqarnas follows the line " back on front " and has as exception only two half intermediate units. That is why the assembly of this muqarnas is a clear assignment for young students.