During the workshop, you may discover how to teach your students to work with patterns. It is about how to build complex architecture with a limited set of basic figures. They 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. In the Turkish city of Beyşehir is the Eşrefoğlu mosque, built in the thirteenth century, with beautiful muqarnas. Eric Broug has posted inspiring videos on YouTube. During the workshop, we bring in the 3D printed units. Yaghan: An online course to Muqarnas Harmsen, Kröker: Seljuk Muqarnas along the Silk Road Eric Broug: A Practical Introduction to Muqarnas Abdelghani Nhari: Constructing a Muqarnas Composition Shiro Takahashi: muqarnas database Yaghan: Decoding ... TakhtI Sulayman Presentatie Vaknetwerk Amsterdam The Topkapi Scroll: Geometry and Ornament in Islamic Architecture Gülru Necipoğlu (PDF) Karbandi and muqarnasMuqarnas 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.
SourcesEarlier workshops on muqarnas have been held by Bart Post, Maartje Elsingavan 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 DoldSamplonius. Her thesis covers both systems. NRC Handelsblad: Een zevenhoek te Isfahan Hogendijk: Sources relating to ArabicIslamic science Nieuw Archief voor Wiskunde: De Sleutel tot de Rekenkunde van alKashi U Talent: Islamic Patterns and Muqarnas Publication in Euclides: Creatief 'DrieDee': Muqarnas NWD 2016: Powerpoint and handout The Topkapi Scroll: Geometry and Ornament in Islamic Architecture Gülru Necipoğlu (PDF)
Examples of the twelve basic unitsWith a limited number of standardized building units, all different kinds of muqarnas can be built. Below are examples of the twelve 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.
CodingFor indepth 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.
ShapesShapes 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 224 because its angles are 45°, 45°, and 90°. Rhombus B is therefore 332 and large biped J is 521.
Student work piecesAt 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.
They quickly realized the system of papers with red arrows and green lines and they quickly discovered which 3D piece belonged to which paper.
Within half an hour I received the photos below.


Student assignment animation


Starting assignment young studentsYoung students start with a simple assignment where the rules are quickly discovered. This twolayer muqarnas follows the rule "back on front". Students were handed over a construction plate for a quick start.


Variations for the same construction plateWhile the printer is serviced for regular maintenance, Max has ideas for variations which all fit at the same construction plate. An extension to the easy assignment is the task to generate alternatives.


Bursa Abdal Mehmet CamiiIn Bursa, you will find the Abdal Mehmet mosque. This muqarnas follows the rule "back on curve".




Bursa Yildirim Bayezid 

Nigde Ak MedresseThe 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.


Nigde Ak Medresse animation
Nigde Ak Medresse (back on front)The first attempt was entirely based on the rule "back on front". It contains four layers. The result looks like a collapsed wedding cake. So this is not the correct interpretation. The computer reconstruction pinpoints the cause. According to the rule "back on front" the blue units should be "intermediate units". As a consequence, the first layer does gain height, but rather depth, creating the effect of the collapsed cake.


NatanzNatanz 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 (Masjidi 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 nonstandard 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 nonstandardhalf 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


Natanz animation


KayseriAccording to Shiro Takahashi, in the city of Kayseri there are a number of muqarnas with the same basic shape with different interpretations, all of the type "back on curve". Below is an overview of those different versions. A special variant is above a door in the Cifte Madrasah Gevher Nesibe hospital. Shiro Takahashi: muqarnas database
Kayseri Cifte Madrasah Gevher NesibeAbove a door of the Cifte Madrasah Gevher Nesibe hospital, a muqarnas with a special set of two nonstandard 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. Researchgate: Seljuk Muqarnas along the Silk Road Shiro Takahashi: muqarnas database
KayseriThis muqarnas has a number of dedicated units, flat surfaces in the middle and triangular units at the end of the first layer. Shiro Takahashi: muqarnas database


ÇorumIn 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.


A lot of variation is possible on the same rectangular construction plate with a straight back wall on the first layer. Below are examples of alternative muqarnas. Half of a standard intermediate unit is required on the first and third layers. In the second and fourth floors, half of a unit with a back wall is required. Alternatives are the result of permutations of combinations of full and intermediate units. Below, a series of alternatives for the square unit are presented. Each of these combinations has a square footage, the square itself, the orange jug or diamond and the orange biped, the red half squares, two yellow half rhombus and a large biped, the pink small biped, but not the purple almond. The pink one is a kind of inversion of the purple one. The purple almmond is a full unit and the corresponding small biped is the intermediate unit, but in pink, the small biped is the full unit. For the pink one, an intermediate unit complementing to the square does exist, but there is not such an intermediate unit which complements the purple almond.


AmasyaThe 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.

