www.fransvanschooten.nl

Work in progress

This webpage is in development. A lot of research has to be done. My aim is to develop rules for drawing and analysing muqarnas. Visualisation is a major issue. The actual coloring scheme is just a try out and needs improvement.

3D printing

The teaching material consists of six boxes for six groups of three tot five students, each with nearly 200 muqarnas pieces of two by two centimeters and four centimeters high. Everything is printed on a Flashforge Adventurer 3 using PLA from Kexcelled. This PLA can be processed better on the Flashforge than other popular brands. Printer and PLA supplier is Badgers Lair based in Hoevelaken. They also did the consultancy and printed large demonstration objects.

BadgersLair.nl

Chris Cambré

The Belgium master of Geogebra created a workbook with animations and pictures to explain the basic ideas of muqarnas. Chris Cambré has travelled through many countries and created geogebra workbooks on islamic geometry.

Muqarnas

Islamic Geometric Patterns

Ali Reza Sarvdalir

Muqarnas

In Islamic mosques you often see abstract decorations in an interplay of regularity and variation. Muqarnas are decorative elements that, for example, fill the space above the entrance to a portal. It looks like a large niche being filled with all kinds of small niches. For the craftsman, it is a technical challenge to design new creations within the available space with a limited number of standardized building blocks or units. You can find beautiful examples in Turkey and Iran, but also in Moorish Spain or Morocco. The formal language is a fine application of mathematics and lends itself well to an alternative class lesson.

 

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Takht i Sulayman Southern Vault
© M. Yaghan

Design Yaghan

Alternative Design

The design in the middle matches the drawing at the left, but the right one is a full layer higher. Printing designs in 3D helpts to investigate the true mature of these muqarnas.

Student work pieces

Student activity: Bursa Abdal Mehmet Camii

Student activity: Nigde Ak Medresse

Inquiry: Natanz

Inquiry: Kayseri

Inquiry: Çorum

Inquiry: Amasya

 

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.

Chris Cambré: Muqarnas

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

Eric Broug: Muqarnas

Dan Owen: Muqarnas

Shiro Takahashi: muqarnas database

Cristˇbal Vila: Isfahan

Dan Owen: interactive course

Yaghan: Decoding ... Takht-I Sulayman

Beyşehir: Eşrefoğlu

Video Beyşehir: Eşrefoğlu

Presentatie Vaknetwerk Amsterdam

Presentatie Studiedag NVvW

The Topkapi Scroll: Geometry and Ornament in Islamic Architecture Gülru Necipoğlu (PDF)

 

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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.


Drawing in SketchUp


Final Product

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Sources

Earlier workshops on muqarnas have been held 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.

Dissertatation: Algorithmic Computer Reconstructions of Stalactite Vaults - Muqarnas - in Islamic Architecture, Harmsen, Silvia

NRC Handelsblad: Een zevenhoek te Isfahan

Practical Geometry: Muqarnas

Hogendijk: Sources relating to Arabic-Islamic science

Nieuw Archief voor Wiskunde: De Sleutel tot de Rekenkunde van al-Kashi

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)

 

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Examples of the twelve basic units

With 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.

 

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Coding

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.
During my inquiries, I found out that I needed more, dedicated units for special purposes, for example at the end of a layer or in the middle of a layer to gain extra height. These new units are often half the size of a standard unit, but can also be something surprisingly new. These units are also coded. For example, the codes A1 and A2 represent the left and right half of the blue square building block that got the letter A. An example of a new building block is the flat wall, coded with the letter Z. The software accepts all triangular and quadrangular shapes.

A kind of inversion of the pink almond, being the full unit, and the small biped, being the intermediate unit, does exist: a pink small biped, being the full unit, and an almond, being the intermediate unit. This new unit has nice properties. See the alternatives at the inquiry of Çorum.

Inquiry: Çorum

For students, assignments have been made with increasing complexity, depending on the level of the student and the available lesson time. The easy assignment has only two layers and is coded as:

D L C C L D
A J A I E I A J A

 
easy assignment

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The starting assignment is coded this way:

C C C C
F I D I D I D I F
J C C C J C C C J C C C J C C C J
I E I A I I A I E I I E I A I I A I E I

 
Starting assignment

 

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Shapes

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.

 

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Student work pieces

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. 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.
Math teachers, on the other hand, work much more thoughtfully, go for complete understanding, and needed more time.


Analysis


Design

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Teachers in action


Final product

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Coding

C C C C
F I D I D I D I F
J C C C J C C C J C C C J C C C J
I E I A I I A I E I I E I A I I A I E I

 

 

 

 

 

 

 

 

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Student assignment animation

 

 

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Starting assignment young students

Young students start with a simple 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.


Analysis


Student Task

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Design

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Construction Plate


Final product

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3D printer maintenance

BadgersLair.nl

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Variations for the same construction plate

While 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.


Basics

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Variation on the same Construction Plate


Front View

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Variation on the same Construction Plate


Front View

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Variation on the same Construction Plate


Front View

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Variation on the same Construction Plate


Variation on the same Construction Plate

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Variation on the same Construction Plate


Variation on the same Construction Plate

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Bursa

Bursa is the fourth most populous city in Turkey with large industries. In the fourteenth century, it was the capital of the Ottoman State. Bursa has a modern Science Center. Every year, they organize the Turkish Airlines Science Expo, a four day experience with more than hundredthousand visitors. Mountain Uldağ is a lovely place, in winter there are ski resort, in summer it hosts the Bursa AstroFest.

Wikipedia: Bursa

Bursa Science Expo

Uludag Astrofest

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Bursa Abdal Mehmet Camii

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


Bursa


Muqarnas

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Analysis


Design

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Construction Plate


Final product

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Coding

M M M M
F N D N D N D N F
J M M M J M M M J M M M J M M M J
N D N D N N D N D N N D N D N N D N D N
F J B J D J B J D J B J D J B J F
J E C C E J E C C E J E C C E J E C C E J

 

 

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Bursa Yildirim Bayezid

 

Nigde

Nigde is a small city with some tourist attractions including an archeology museum and the Alaaddin mosque with its beautiful muqarnas.

Wikipedia: Nigde

Wikipedia: Nigde Alaaddin Mosque

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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.
Below, the five-layer reconstruction is shown which is based on the rule "back on curve".


Nigde


Muqarnas

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Analysis M. Fatin Uluengin


Design

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Construction Plate


Student Task

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Computer reconstructie (STL) binnenkant van onderen


Computer reconstructie (STL) binnenkant links voor

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Coding

C C C C
F I D I D I D I F
J B J B J B J B J
A A A A A A A A
F C C C E C C C E C C C E C C C F

 

 

 

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Nigde Ak Medresse animation

 

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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.


Computer reconstruction (STL) inside


Computer reconstruction (STL) outside


Final Product

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Natanz

Natanz is a smalll city with a long history. Its monuments deserve a visit.

Wikipedia: Natanz

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Natanz

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
Blair, S., Bloom, J.M, The Art And Architecture Of Islam (1250- 1800), New York, Yale University Press, 1994
Harmsen, S., Algorithmic Computer Reconstructions of Stalactite Vaults - Muqarnas - in Islamic Architecture, 2006, PhD thesis
Yaghan, M., Decoding the Two-Dimensional Pattern Found at Takht-i Sulayman into Three-Dimensional Muqarnas Forms, Iran, Vol. 38 (2000), pp. 77-95


Natanz


Foto tour
© SmugMug

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Muqarnas
© Margi Rothengatter-Lake


Muqarnas

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Mosque


My paper analysis

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Analysis Harmsen


My interpretation

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Analysis Harmsen


Analysis Yaghan

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Analysis Yaghan


My analysis

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Design


Final Product

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Dedicated Unit
at the end of the third layer


Construction plate

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Coding

D J A J A L A L A J A J D
A L A L C D B B D C L A L A
B A I L A L A A A A L A L I A B
Z B G C E C G C E C G C C C E C C C G C E C G C E C G B Z

 

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Natanz animation

 

 

Kayseri

Kayseri is a city with sturdy Byzantine city walls and Ottoman mosques decorated with attractive muqarnas.

Wikipedia: Kayseri

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Kayseri

According 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

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Cifte Madrasah Gevher Nesibe hospital

In the design of this muqarnas squares are drawn in the bottom center, top right and top left with a special diagonal notch. An elaboration of this muqarnas is below.

analysis and examples

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Alaca Kumbet

This muqarnas can be created using the standard building units.

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Dner tomb

This muqarnas can be created using the standard building units.

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1793

In the middle, it is intriguing how the third layer rests on the bottom layer together with the second layer. The analysis indicates that these three layers come together. It looks like that the muqarnas runs straight up with a vertical unit. Furthermore, the second layer is closed at the corners with the special triangular building unit that occurs in all designs with this basic shape.

Analysis and examples

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1755

window

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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.

Researchgate: Seljuk Muqarnas along the Silk Road

Shiro Takahashi: muqarnas database

 


Kayseri


Analysis
by Shiro Takahashi

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Detail


Overview door and muqarnas

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Construction Plate


Design

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Dedicated Unit
in the second and third layer


Final product

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Coding

M M M M
A N A N A N A N A
B B B B
Q M M M M Q Q M M M M Q
K P K N A N A N A N K P K K P K N A N A N A N K P K
B B B B B B B B

 

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Kayseri

This 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

 


My analysis


Analysis
by Shiro Takahashi

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Dedicated Unit
at the end of the first layer


Final Product

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Construction plate

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Coding

M M M M
F N D N D N D N F
J M M M J B J B J M M M J
N D N D N D Z Z D N D N D N
E J B J B J B B J B J B J E

 

Çorum - Osmancık

In het Turkse district Çorum ligt de stad Osmancık.

Wikipedia: Çorum

Wikipedia: Osmancık

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Çorum

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.


Çorum


Analysis
by Shiro Takahashi

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Muqarnas


Student Task

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Construction Plate


Design

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Dedicated Units
at the end of the first and third layer


Final product

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Coding

D D
J D D J D D J
E J D D J D J D D J E
K E J E E J E J E J E E J E K

 

 

 

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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.

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Alternative


Final product

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Design

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Alternative


Final product

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Design

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Alternative Design


Alternative Design

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Alternative Design


Alternative Design

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Alternative Design


Alternative Design

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Alternative Design


Alternative Design

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Alternative Design

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Amasya

Amasya is an ancient city with a long history that starts before the Roman times. It was also an important city in Ottoman times. Sultan Selim I was born there, for example.

Wikipedia: Amasya

Archnet: Amasya Bimarhanesi

Yavuz Selim Mosque in Istanbul

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Amasya

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.


Amasya


Analysis
by Shiro Takahashi

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Muqarnas


Final product

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©Dick Osseman

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Construction Plate


Design

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Dedicated Unit
at the end of the second layer


Student Task

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Coding

C C C C
F I D I D I D I F
J C C C J C C C J C C C J C C C J
I E I E I I E I E I I E I E I I E I E I

 

 

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https://www.todocoleccion.net/libros-segunda-mano/primera-sigunda-parte-reglas-carpinteria-por-d-lopez-arenas-1619-facsimil-1966~x42835811 http://www.alhambra-patronato.es/mocarabes-versus-muqarnas-2
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