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Algorithmic Computer Reconstructions of Stalactite Vaults

1. o d sin a 2 cos a 2 With the same arguments we state that the normals of the vertices on the right curve are given by 0 0 1 and the normals of the vertices on the left curved side are given by Bocconi For the front sides we can verify that the normals of the right front side are given by Rab atb o tn 2 normals and the normals of the left front side are given by Ra a 0 0 7 2 normals 4 3 4 Complexity Analysis The structure information of the muqarnas is already contained in the muqarnas graph G The task of the program graphtomuq is to translate this structure into a three dimensional reconstruction We find the faces by looking for each node at his neighbors We conclude that the computing time to set the faces is linear in the number of nodes M of the graph G To set the element type we use operations with computing time not depending on the size of the graph For the position only the calculation of the height depends on the size of the plan For this we use Algorithm A 3 3 We have seen in Section 4 2 2 that its computing time depends on M this cubic computing time is then crucial The program graphtomug needs at most M operations Chapter 5 Results of Algorithmic Muqarnas Reconstructions In this chapter we present the results of some examples of muqarnas plans on which we applied our algorithms to create computer reconstructions from the corresponding muqarnas vaults We
2. 2 2 1 Elements as Abstract Objects 222 22 none nenn 2 2 2 MugarnasPlam 24 02 25 ea rule DEE XS 2 2 3 Structure of Il Khanid Muqarnas 3 Algorithm for Reconstructing Muqarnas 3 1 Representation of the Muqarnas Structure ina Graph 3 1 1 Definition of the Muqarnas Graph 3 1 2 Properties of the Muqarnas Graph 3 2 Construction of Mugarnas Graphs from the Plan 3 3 Conversion of Muqarnas Graph into Muqarnas Structure 3 3 1 Mugarnas Reconstruction Process 2 2 2 22m nennen 3 3 2 Determination of Faces from the Graph 3 3 3 Conversion of Faces into Mugarnas Elements 3 3 4 Reconstruction from a Simplified Plan V iii n 11 11 11 16 19 19 21 26 vi Contents 3 4 Preparation of the Plats zs scs uns E XAVIER te ent 51 3 5 Uniqueness of the Reconstructions leen 55 4 Software Tools for Reconstructing Muqarnas 57 4 1 The Program plantograph 2 223 WAREN De ware so als 57 4 1 1 Conversion of the Plan into the Graph 58 4 1 2 Representation of the Muqarnas Plan 60 4 1 3 Conversion of the Input into a Muqarnas Plan 61 4 1 4 Boundary of aMuqarnasPlan 2 2 64 4 1 5 Determination of the Direction of the Edges 67 4 1 6 Complexity Analysis 5 3 Goa Gy asia en ae end 69 4 2 The Program removeli
3. Apart from globally different muqarnas we also consider the locally different ones Sometimes it is possible to exchange a muqarnas element by others without changing other parts of the muqarnas The mugarnas graphs are then the same and this decision need to be considered during the three dimensional reconstruction process executed by the program graphtomuq We applied our algorithms on the oldest known muqarnas design which is found at the Takht i Sulayman The last decennial different researchers studied the design and gave their interpretation of the design This work shows how new methods give an overview of all possible interpretations of a design By applying a more structural method of studying the mugarnas designs we are able to propose suggestions differently from the interpretations found in the literature but more realistic with the style of muqarnas from the same time and region Research about muqarnas could develop in different directions One direction could be to extend our software tools so that they can handle other kind of muqarnas Our al gorithm is designed for muqarnas fitting into domes and niches and therefore demands rectangular plane projections in the input Muqarnas which appear as a decoration on minarets have a plane projection with another geometrical form Those cannot be han dled by our algorithm Another definition of the boundary of a muqarnas plan is required to analyze such muqarnas In other styles of mu
4. Definition A 5 incident Two edges e f are incident if they have exactly one common end node An edge e is incident to node v if v is an end node of e Definition A 6 predecessor Let G G N C be a directed graph and v w N We say that node w is a predecessor of v if there is an arrow c C pointing from w to v Definition A 7 successor Let G G N C a directed graph and v w N We say that node w is a successor of v if there is an arrow c C pointing from v to w The notion of path is used to denote connections in the graph The length of a path indicates distances between nodes If the initial point and the end node of a path are the same we call this path a circuit The circuits which are of most interest of us are those which do not coincide themselves the so called cycles Definition A 8 path A path p of length l p n in an undirected graph G N E is an ordered sequence of nodes vi v N such that there exist edges edge v viii in E for i l n 1 We write p v for the initial point of the path p and p v for the end point of p Similar an ordered sequence of nodes vi v N defines a path p of length l p n in a directed graph G N C if there exist arrows arw v vj41 in C for bebes Definition A 9 connected graph An undirected graph G is connected if for all nodes v w there is a path with initial point v and end point w A directed graph G is connected if its underl
5. R given by its coordinates read from the xfig file divided by the unit u This method first observes the actual node list of the plan P to check whether their are already indices ip v and ip w given If they are not available in the node list we create index numbers ip v respectively ip w not existing yet in Ip The connection ip v ip w is then added to the linked list by adding ip v to the neighbors of ip w and adding ip w to the neighbors of ip v 4 1 4 Boundary of a Muqarnas Plan From the subplan Q M C given in the input we create the object geoPlan which main task it is to store the boundary information of the plan This object works with plans corresponding to a muqarnas in a niche or dome For reconstructing more general mugarnas forms the methods in this object need to be adapted It uses the information which section of the muqarnas is represented in the plan as given by the command line option p in calling plantograph It stores the number given behind p in the command line and uses it to calculate the boundary center and bottom boundary nodes of the plan They are stored as vectors of index numbers referring to the corresponding nodes We restrict to describe the situation dealing with a quarter muqarnas plan in the input as we also gave the boundary definitions in that context If a larger part of the muqarnas plan is given we can use symmetry arguments to calculate the complete boundary The boundary nodes a
6. The node w v is therefore given by rotating 1 0 or 1 0 over the origin by k7 4 Let the other edge of length one corresponding with a curved side be given by edge v u 2 2 Muqarnas as an Abstract Geometrical Structure 29 then the position of the polygon p is given by translating with v and rotating over r or over kn 4 a v u w As a v u w is the angle of the element a v u w km 4 for a k Z oO We use the property that the grid Z gt o x Z gt o is countable to represent each node in L with one integer We define the function f Z gt 0 x Z gt o Z gt o by FAN 3 8G 3 j 1 2 2 we denote r for the largest integer smaller than or equal to r Proposition 2 2 12 The function f is invertible with the inverse g Z gt o Zso X Zo given by gi ke n kd in n4 1 k in n 4 1 2 3 with n 3 V 8k 1 PROOF We define g as in 2 3 and have g Zso C Zso x Zso and f Zso X Z gt o C Z gt o It suffices to prove that g f i j i j for all i j Z gt o x Z gt o and g f k k for all ke Z9 Let i j Z0 x Z0 then sw sgG iG 3 3j 1 with k j i j i j 1 and with n 3 3 8 j 6 t 3 4 1 4 is We note that i 8G c i0 jJGc j 1 1 24 j 1 i j 3 and itj h lt G f 41 i4 j 3 lt i454 which can be seen by writing i j 1 as i j 4 3 j li j 3 3 for the left estimate and i j 1 as i j
7. 0 we conclude that We proceed the proof by showing that for all nodes v there is ak lt M such that h v h v Because h v is increasing in k 3 3 and smaller than or equal to h v 3 4 we can then conclude that h v h v There is at least one node v M with h v 0 For this node already ho v 0 and thus h v h v If the height of a node v is set well v h v then in the next iteration step the height of its neighbors are set well Algorithm A 3 1 will care about that hy w h w for all successors w of v and Algorithm A 3 2 will force hpp w h w for all predecessors w of v We define the plan P M E given by the graph G M C as the undirected graph for which edge v w E if arw v w C or arw w v C Let v be such that h v A v Because the graph G is connected we find that for all nodes w M there exist a path in the plan P from v to w Let this path be given by the sequence v vo 11 Un W Then arw v vj 4 or arw v 1 v exist in With induction we can conclude that after at most k n iteration steps also h v h v As the shortest path has length smaller than M the height of all nodes are given in less then M steps oO The next feature which we will formulate in Lemma 3 1 7 concerns about the direc tion of the edges in the muqarnas graph To formulate this lemma we need the notion of opposite edges in a graph In the definition of opposite edges
8. Figure 4 8 Element model as used for our virtual reconstructions m m IN NK JEN va N y N C A N Figure 4 9 Elements and half elements approximated by triangles In Figure 4 9 we see how we divide the surfaces of a cell and intermediate element in different triangles These pictures show which coordinates we need to calculate for modeling the geometry of the elements The curved side which represents the curved side of the elements is drawn in Figure 4 10 The curved side as it is described by al Kashi is given by straight lines and a circle arc On the circle arc we calculate n vertices where n depends on the smoothness and velocity of the reconstruction which we require for the drawing By increasing n the reconstruction will be more realistic but slower This means that it takes longer until the reconstruction appears on the screen and it influences the speed with which we can move the muqarnas interactively The curve is described by al Kashi as according with the method of the masons indicating that it is taken from practice A detailed explanation of his calculation of the curve can be found in Dold Samplonius 1996 p 66 An animation to illustrate his calculations can be found in Dold Samplonius et al 2002 We use the symbols as given in Figure 4 10 to describe the curve Our origin is chosen so that vertex G 0 0 Setting our unit equal to the module we draw a straigh
9. SZ N a pl_tis_plate_yagh fig Inn er x DL BEEN MAP E POP CN CCCo BOE BOYD b gr_tis_plate_yagh fig c WRL_yaghan main wrl Figure 5 21 Reconstruction proposal of Yaghan Chapter 6 Conclusions Reviewing our research questions which we formulated in the introduction see Section 1 2 1 we present the following answer There is an algorithm for reconstructing muqarnas There are different muqar nas reconstructions with the same plane projection possible The possibilities are reduced by the proportions of the vault into which the muqarnas should fit We started this work with a problem taken from the practice concerning reconstruc tions of muqarnas an architectonic decoration appearing in the Islamic architecture By describing the problem in a more mathematical context we created a framework in which we are able to analyze it For this aim we defined in Chapter 2 the three dimensional muqarnas structure and muqarnas plans motivated by existing muqarnas vaults and properties of muqarnas designs In Chapter 3 we have used this framework to analyze muqarnas from the Seljuk and Il Khanid periods We developed methods to study the structure of muqarnas directly from their designs These methods were formulated so that it was possible to write software tools which are able to analyze muqarnas designs see Chapter 4 In this context we have written the computer programs removelines plantog
10. standard position 20 subgraph 118 Takht i Sulayman Plate 105 South Octagon 103 tier 17 Topkap Scroll 5 Index Acknowledgements This thesis never could have been realized without the help of different people I am grateful to Professor G Reinelt who gave me the possibility to write this thesis about my research on muqarnas This study on muqarnas started with the project Mathe matische Grundlagen und computergraphische Rekonstruktion von Stalaktitengew lben Mugarnas in der Islamischen Architektur financed by the German Bundesministerium f r Bildung und Forschung BMBF under Grant 03WNX2HD I am deeply indebted to Yvonne Dold Samplonius for initiating this project and sharing her experience with me I would like to thank Professor W J ger who supported the initiative of this project and Professor H G Bock who enabled me to finish it Professor Jan Hogendijk motivated me to start the project and was very helpful during the process I would like to thank him for that and for his faith in me I also want to thank Susanne Kr mker and Michael Winckler for coaching me during my research and giving me the possibility to work in the Computer Graphics group where I had the possibility to develop my knowledge in different areas Bernhard Keil and Daniel Jungblut assisted with a lot of work In particular I want to mention the preparation of the input plans so that my algorithms were able to read them for creating recons
11. 2 at a 2 i j 2 4 for the right estimate We find n 3 1 86 6 G4 5 1 41 i By setting n i j in 2 4 we can verify that g f i 7 i j To prove that f g k k we calculate f i j with i n k in n4 1 and j k n n 1 We can verify that i j n and therefore i j k n n 1 n n 1 k oO 1 2 30 Chapter 2 Structure of the Muqarnas The function f is only invertible on Z gt and not on Z we translate the muqarnas plan so that all nodes can be given by non negative integers a b c d Zso We first translate the plan so that the smallest x value and the smallest y value both equals 0 Then we can represent the coordinates v a b V2 c d V2 by a b c d Zo X Za X ZZ X Z gt We need the 1 to be able to represent all coordinates of the biped In standard position the node opposite of the central node of the small biped is given by v 1 V 2 1 1 4 2 so we need d 1 to represent its coordinates The node opposite of the central node of the large biped in standard position is given by v 1 1 V2 1 1 V2 and so b d 1 To be sure that we do not need to use negative integers we add 1 to all a b c and d 0 0 0 0 0 0 4 4 40 1 4 2 0 0 1 0 0 14 4 77 0 1 0 0 1 0 4 7 73 Table 2 2 Example of indices v for nodes v given in coordinates a b v2 c d V2 In Table 2 2 some examples of representation
12. E the xfig file is scanned and every connection in the plan is inserted as an edge in our data structure of the plan Therefore this scanning process needs computing time linearly dependent of the number of edges E We find 70 Chapter 4 Software Tools for Reconstructing Muqarnas that the computing time equals to O E O N To set the boundary of the plan we iterate over the nodes N of the plan We survey for each node v N whether it is surrounded by figures or islands The maximal size of an island does not depend on the size of the plan The validation process whether a given combinations of nodes defines a figure does also not depend on the size of the plan Therefore checking whether a fixed node belongs to the boundary depends only on the neighbors of the node v As the maximal number of neighboring nodes does not depend on N we conclude that computing time for setting the boundary is given by O N To set the center of a muqarnas graph we calculate R with R L D the path defined by the boundary nodes and k the number of center nodes For calculating R we look for each edge to neighbors if its end point We use this argument to conclude that calculation of R costs O k D operations As D lt E and the center has alway less than N nodes we can estimate computing time with N E O N For calculating the outer boundary we iterate over the set of nodes on the boundary and decide for each nod
13. ZHE ESNE WA gt Liz 5 0 RI ERS ER BEIN ER 22x NAT A ee Ls ES fe ug an 4r Mv d L6 Lb 2 VEX VEX NIU Su V RN p a Plane projection of Harb s interpreta b gr tis plate harb fig tion of the design c WRL_harb main wrl Figure 5 18 Reconstruction of Harb s interpretation of the design on the plate found at the Takht i Sulayman 110 Chapter 5 Results of Algorithmic Muqarnas Reconstructions er EN Y lt i UIN H 1 P DIE NEN EN y 7i A Low ae A Qo oc oe a pl tis plate regularcenter fig b gr tis plate regularcenter fig c WRL regularcenter main wrl Figure 5 19 Reconstruction proposal of a mugarnas with regular center correspond ing to the design on the plate 5 2 Computer Reconstructions of Il Khanid Muqarnas 111 NSZ p ee SDA AN un I SR N sa S EN gt on x Ya K S YNSZYNSZY cS N A d 7 Se OO NS of Q D a NO IHRE e KDA SRR X x SES 5 N RVA a pl_tis_plate_our fig b gr_tis_plate_our fig AZAN c WRL_our main wrl Figure 5 20 Reconstruction process if specified nodes on the diagonal are removed 112 Chapter 5 Results of Algorithmic Muqarnas Reconstructions N Zy pm AV ENS f CN Z YN x m AY SA N a a AO SOLO RO EISEN FARIAKD wy wa Ww 7Y Y BEN ya WwW PDD VNC x
14. is the set of nodes n N which are not surrounded by figures or islands In this way the boundary is the collection of nodes not belonging to the inner part of the plan The boundary of a muqarnas plan as defined here consists of nodes corre sponding to vertices at the bottom the side or the center of the muqarnas We therefore split the boundary of a plan also in the center and the bottom boundary x mirrorline y mirrorline Figure 2 20 The bold nodes represent the boundary of the plan corresponding to the vault over the east portal of the shrine in Bistam The blue part defines the bottom boundary and the red part the center To separate the different parts of the boundary we assume that the plan is a quarter plan with the usual orientation We consider the plan in the zy plane and represent its nodes with x y coordinates Let zmax R be the maximal x value appearing in the plan and ymax R be the maximal y value We define the x mirrorline parallel to the z axis by the line y ymax and the y mirrorline parallel to the y axis by x ry The boundary of the z mirrorline is given by the nodes on the z mirrorline with maximal or minimal x value The interior of the x mirrorline is what remains after re moving the boundary of the z mirrorline The boundary and interior of the y mirrorline is defined analogously Definition 2 2 9 center The center W P of the plan P is the shortest path in W P betw
15. 2 1 1 Our description of the elements is based on the definitions given by al Kashi After presenting the elements we give an overview about the way they can be combined for constructing a muqarnas vault see Section 2 1 2 2 1 1 Mugarnas Elements In al Kashr s definition of mugarnas see citation in Section 1 1 2 it is explained that a muqarnas is constructed from different cells These cells are the main building blocks of which the muqarnas is built Beside the cells some space need to be filled with another kind of building blocks the so called intermediate elements Al Kashi already roof roof facet Int Figure 2 1 Examples of curved muqarnas elements On the left a cell On the right an intermediate element formulated a definition of a cell In this definition a cell is compared with a house see Dold Samplonius 1992 p 226 11 12 Chapter 2 Structure of the Muqarnas Definition 2 1 1 cell al Kashi The two facets can be thought of as standing on a plane parallel to the horizon Both facets together with their roof are called one cell bayt We adopt from Dold Samplonius 1992 p 226 the word cell as translation for the Arabic word bayt The word bayt could also be translated with house Like a house a cell is then split in its facets which correspond to the straight part of the elements and its roof which correspond to the curved part see Figure 2 1 In the context of al Kashi s manus
16. 2 13 d Considering the complete muqarnas structure we define the bottom of the muqarnas as the set of elements which do not stand on elements of a lower tier This bottom is called regular if all elements of the bottom are contained in the first tier of the muqarnas 2 2 Muqarnas as an Abstract Geometrical Structure 19 Similar the center of the muqarnas consists of the set of elements which do not join to elements in a upper tier The center is called regular if all elements of the center belong to the most upper tier of the muqarnas The front part of a muqarnas fitting into a niche is formed by the elements at the border of the tiers 2 2 Mugarnas as an Abstract Geometrical Structure To define a muqarnas in a more mathematical context we define it as a finite set of mugarnas elements together with their position and orientation We will start this sec tion with a more abstract definition of muqarnas elements and try to formulate some properties for the set of muqarnas elements to make clear what kind of sets are al lowed see Section 2 2 1 In this more abstract characterization of muqarnas only the structure of the muqarnas is considered no information about the exact appearance ma terial color measurements of the muqarnas is given We continue with an definition of the muqarnas plan which will substitute the muqarnas design see Section 2 2 2 In the last part of this chapter see Section 2 2 3 we calculate the proport
17. 5 1 1 In the center elements appear of smaller size In Figure 5 7 c the output after running plantograph r Qrmlines pl arslanhane fig Ppl arslanhane fig is given This output is not satisfying as too less directions of edges are set If we look to the reason we conclude that a lot of orbits in this plan only contain a few edges due to the fact that most rhombi are divided in two triangles In the definition of opposite edges see Definition 3 1 6 we excluded edges in polygons with cross edges as being opposite This was done for the case that the triangles should be interpreted as two half intermediate elements joining at their front like we have seen in the reconstruction of the entrance portal of the enclosure of the Sultan Han near Kayseri see Section 5 1 3 In the real muqarnas see Figure 5 8 b such element combinations do not appear The divided rhombi are the plane projections of a half cell standing on a half intermediate element both elements with plane projection a half rhombus It is a similar situation of a rhombus dividing in an almond and a biped see Section 3 3 4 By running plantograph r Qrmlines_pl_arslanhane fig the complete plan is not given in the input and the information of the cross edges dividing the rhombi is not included Therefore the edges of the rhombi are interpreted as opposite edges In this way we do not regard element combinations consisting of half intermediate elements joining at their fron
18. Find the cell faces corresponding to full cells 2 Find the int faces 46 Chapter3 Algorithm for Reconstructing Muqarnas ge Figure 3 13 Set c v w defines a face On the left cell face On the right int face 3 Find the cell faces corresponding to half cells 4 Translate cell faces to cells 5 Translate int faces to intermediate elements 4 M NL L f Cos AN m he ER b EX A3 A3 N APA 77 PER N a Initial situ b After set c After set d After set ation ting cell faces ting int faces ting cell faces corresponding to half cells Figure 3 14 Finding the faces from the muqarnas graph corresponding to the muqar nas in the basement of the north iwan of the Friday Mosque in Natanz Faces already set are colored gray Black arrows are twice available for setting faces blue arrows are once available and red arrows are not available any more In Figure 3 14 the initial situation and the first three steps of the algorithm are vi sualized We mark in each step how often the arrows are used so that after setting the cell faces we can decide whether there is place for an intermediate element left To find the set of cell faces corresponding to half elements another method is used as for finding the cell faces corresponding to full elements Therefore this is done in a separate step In the next section we explain more detailed how the faces can be recognized from the muqarnas grap
19. Muqarnas 3 1 1 Definition of the Muqarnas Graph In the left part of Figure 3 1 a cell and an intermediate element are drawn with red arrows pointing to the apex of the curved side In the plan P N E the edges which are projections of curved sides appear as red arrows and define a directed subgraph G M C of P N E The blue arrows in the cell indicate the required direction of curved sides of elements in the previous tier on which the cell can be arranged ZN Figure 3 1 On the left cell and intermediate element with arrows on the curve in upward directions On the right corresponding plane projections Definition 3 1 1 muqarnas graph A muqarnas graph is a directed subgraph G M C C P N E such that an edge e C E appears in C iff it is the projection of a curved muqarnas side or the projection of a backside of a full cell at the bottom of the muqarnas If edge v w C correspond to the projection of a curved side with w the projection of the apex of the curve then the edge v w is directed so that it points from v to w If edge v w is the projection of a backside of a cell then edge v w points to w with w the projection of the vertex joining to the curved side of the cell We write arw v w to refer to an arrow in the muqarnas graph pointing from v to w The muqarnas graph mainly consists of the projection of the curved element sides Beside that the projection of the backsides of bottom cells are included in the muqa
20. Sort its successors so that a c vo Vi lt a c vo vj M yes As we see in Figure 3 16 the int faces are now given by F c v v 1 for i 0 n 1 Un e vo Figure 3 16 The successors vo Un of c are ordered in counter clockwise direction During the step of setting the int faces we do not search for a possible fourth node k in the plan P In this step we find the int faces corresponding to full intermediate elements as well as the int faces corresponding to half intermediate elements We need to mark the arrows arw c vo and arw c vn corresponding to the outer curved sides of the intermediate elements as used The arrows arw c v for i 1 n 1 which correspond to the inner curved sides of the intermediate elements are used twice The front sides of the intermediate elements do not appear in the graph G because they cannot touch curved sides see Section 2 1 2 Therefore they are not considered during the administration We set the cell faces corresponding to half cells at similar way as the int faces In this step we also iterate over the nodes c C of the graph G M defined by the available edges We scan the predecessors vp v of the node c and find the cell faces by sorting these predecessors by increasing angle a C vo Vi lt a c vo vj ifi lt j like in the situation for the int faces We mark the arrows arw c vj which corresponds to curved sides as used 3 3 3 Conversi
21. a half cell or a gap is preferred It is decision which leads to locally different muqarnas Chapter 4 Software Tools for Reconstructing Muqarnas In this chapter we describe three computer programs which are developed to reconstruct muqarnas The first program plantograph is written for converting a muqarnas sub plan into a muqarnas graph see Section 4 1 This subplan consists only of the edges which appear in the muqarnas graph The task of plantograph is to give the edges a direction A second program removelines is used to find the subplan from the complete muqarnas plan see Section 4 2 The third program graphtomug calculates a three dimensional muqarnas reconstruction from the muqarnas graph with the initial plan as plane projection see Section 4 3 For this aim a list of muqarnas elements is calculated together with a list containing the positions of the elements For the visualization of the two dimensional objects that are the plans and graphs we use the xfig file format which is designed for the open source program xfig The three dimensional data are given in the VRML 2 0 file format The xfig files are then used as input and the computer reconstructions are given in VRML 2 0 As the calcu lations are all done in our own data structures the muqarnas reconstruction algorithm does not depend of these input and output formats A short description of the xfig file format can be found in Section 4 1 2 and the VRML 2 0 form
22. a survey of the different reconstruction possibilities is realizable with this software The reconstruction process is worked out for muqarnas vaults from the Seljuk and Il Khanid time fitting into domes or niches Knowledge of art history concerning the common muqarnas styles corresponding to a certain region time period or building materials is required to select the muqarnas reconstruction which fits best in the historical context The Il Khanid muqarnas as we know them from Iran were often built from pre fabricated elements They mainly exist of a finite set of different elements We will consider this property in the reconstructions For the Seljuk muqarnas a wider variety of different elements is possible This work is organized as follows besides of the introduction Chapter 1 it contains three main parts each represented by one chapter The first part which is presented in Chapter 2 is concerned with a formalization of the terms muqarnas vault and muqar nas design by defining these in a more mathematical context Having clear definitions available makes it possible to analyze the muqarnas structure This analysis is done in the second part presented in Chapter 3 In that chapter we present the steps executed to calculate the muqarnas structure from its design The reconstructing process is split in two steps in the first step we construct a directed subgraph from the design of a muqarnas vault see Section 3 1 and in the second st
23. an example of a muqarnas in a niche it shows the basement vault of the north iwan from the Friday Mosque in Natanz Examples of muqarnas as a decorative frieze can be found at balconies of minarets see Figure 1 1 a towers which are attached to a mosque often used for the call to prayer Cornices the uppermost section of moldings along the top of a wall or just below a roof are also often decorated with muqarnas see Figure 1 1 b The fifteenth century mathematician Ghiyath al Din Mas d al Kashi died in 1429 defines the muqarnas in his Key of Arithmetic see Dold Samplonius 1992 p 226 from a practical point of view The muqarnas is a roofed musaqqaf vault like a staircase madraj with facets dil and a flat roof sath Every facet intersects the adjacent one at either a right angle or half a right angle or their sum or another combination of these two The two facets can be thought of as standing on a plane parallel to the horizon Above them is built either a flat surface not parallel to the horizon or two surfaces either flat or curved that constitute their roof Both facets together with their roof are called one cell bayt Adjacent cells which have their bases on one and the same surface parallel to the horizon are 4 Chapter 1 Introduction called one tier tabaqa The measure of the base of the largest facet is called the module miqyas of the muqarnas In addition there are intermediate elements
24. boundary is the projection of vertices above the vertices with projection the end node not in the bottom boundary We distinguish two kinds of bottom boundary nodes the singular and the non singular ones They are so defined that the height of the non singular nodes is always smaller than the height of its neighboring nodes not on the bottom boundary For the singular nodes we cannot do any statement about this We define the line u w from u to w by l u w tu 1 t w t R and the line section from u w from u to w by l u w tu 1 t w t 0 1 We say that the line u w separates v from the plan P N E if and only if there is a line section l v v with v N_ v Ul u w so that I u w N Lw v Z 0 see Figure 3 9 Figure 3 9 Node v is non singular as it is separated from P by I u w Node 9 is singular the line i does not separate it from the plan P Definition 3 2 3 non singular Let v Wy P with neighbors u w in Wy P If l u w does not separate v from P and v I u w then we call v a singular bottom boundary node If v l u w or l u w separates v from P then v is non singular The set of non singular boundary nodes of the plan P is denoted by Wu P The first rule can now be applied to set the directions of edges pointing from the non singular bottom boundary nodes to other nodes of the plan Rule 1 We consider a muqarnas structure with plan P N E
25. bring over the directions of these arrows to the to the orbits in the plan For an arrow arw v w in the island we look for the orbit in OrbitsValued containing the corresponding edge edge v w or 4 1 The Program plantograph 69 edge w v The direction of the corresponding orbit can then be set For applying Rule 4 the graph has no niches we iterate over the orbits For the orbits of which the direction is undetermined the direction is set arbitrarily If this result in a mugarnas graph containing a niche we know that the direction should be swapped For both possible directions we execute this test if both resulting graphs have no niches the direction of the orbit is set back to unknown Finally the contents of the object OrbitsValued is written in a xfig file For each orbit we first a comment line in the xfig file is written to denounce the begin of a new orbit This is done so that the orbit information is not lost After that all edges of the orbit are inserted as arrows according to its direction If the direction is not known we insert an xfig object representing a dotted line The program planedit see Jungblut 2005 is written for interactively changing the graph This program offers the possibility to change the directions of the orbits manually The directions of the undetermined orbits can also be set 4 1 6 Complexity Analysis We finish this section with some remarks regarding the computing time of the methods applied
26. can be found in Chapter 5 At first we consider the situation that removing nodes on the diagonal results in different muqarnas reconstructions By removing these nodes an island in the plan emerges and a certain element combination on the diagonal is forced in the muqarnas If we fix locally the elements we need to ensure that the adjacent elements still fit to this fixed part This is in general not the case and we need to change more elements to create an approved muqarnas structure Therefore after removing nodes not only the elements on the diagonal vary We have seen in Section 3 4 that this can influence the number of tiers in the muqarnas Because more elements are involved and the shape of the mugarnas is changed we call these muqarnas being globally different These variations need to be considered in the algorithm by finding the subplan from the complete plan Secondly while converting the subplan into a subgraph some directions of orbits may be undetermined Giving these orbits different directions results in different muqarnas reconstructions In this case also more elements are involved and the different recon structions can have different shapes consisting of a distinct amount of tiers Therefore we say that the different possible directions for the orbits which are undetermined by the constructing of the subgraph from the subplan lead also to globally different computer reconstructions In Section 3 3 4 we have seen some exampl
27. computer reconstructions of the muqarnas directly from their designs Before going into details we will use this introductory chapter to explain what muqarnas are We focus on muqarnas from the Seljuk and Il Khanid style In Sec tion 1 1 a short overview of the history of the Seljuk and Il Khanid time periods is given After that muqarnas and their designs are explained In Section 1 2 the motivation of this work and problem formulation is presented Furthermore a very short summary of previous research works concerning muqarnas is included We finish this chapter with an outline of the research contained in this work 1 1 Mugarnas in the Islamic Architecture In this work we study muqarnas built in Seljuk and Il Khanid times the periods in which they got important in Islamic architecture We therefore start in Section 1 1 1 with a historical overview of these times After that we will use in Section 1 1 2 the definitions from al K shi a mathematician from Timurid time to explain what muqarnas are Fi nally we discuss in Section 1 1 3 the aspects of muqarnas designs illustrated by a famous design found at the Takht i Sulayman 1 1 1 Overview about Seljuk and Il Khanid Architecture The word Seljuk is used to denote a mighty group of nomadic Turks who invaded a large part of Asia see e g Aslanapa 1971 Seljuk was the name of one of their first supervisors Under his supervision a tribe of the Oghuz Turks converted to Islam around 96
28. computing time linear depends on E the number of edges in the muqarnas plan Computing time is then given by E O N The method setHeight unsigned int n calculates for each diagonal node a possible muqarnas reconstruction The muqarnas structure is calculated by applying plantograph We have seen in Section 4 1 6 that this program needs N opera tions After the graphs are calculated the Algorithm A 3 3 is used to calculate the height of the nodes This Algorithm will call to the Algorithms A 3 1 and A 3 2 at most N times For executing both the Algorithm A 3 1 and A 3 2 we iterate at most N times over the nodes in N We find that the computing time of Algorithm A 3 3 can be given by O N Summarizing iteration over diagonals running plantograph and running Algorithm A 3 3 gives computing time of N N N O N For the methods DeleteDivisionLines and DeleteTriangleLines we remark that we iterate over the nodes in the plan to decide whether these nodes divide a polygon or define a triangle respectively The decision does a given node define a division line or a triangle depends on the neighbors of the nodes Therefore the computing time of this method can be given by 4 3 The Program graphtomuq The task of the program graphtomug is to create three dimensional computer recon structions from the muqarnas graph It reads the structure information contained in the gr
29. determined by using similar arguments as we did by reconstructing the Alay Han see Section 5 1 1 In Figure 5 5 d the complete muqarnas graph is given In this figure we colored one square gray The direction of the arrows incident to this square show the motivation for excluding the edges of a polygon with cross edge from being opposite see Definition 3 1 6 This part corresponds in the mugarnas to two half intermediate elements with plane projection half squares which join in their front as can be seen by comparing this part to the right picture of Figure 5 6 b The computer reconstruction is given by graphtomuq r Ppl slthan ksi fig Gcrc gr slthan ksi fig We see that there is a small part in the muqarnas not consisting of muqarnas elements In our reconstruction a gap is left in the real muqarnas this space is filled with a stone ornamented with a kind of rosette see left of Figure 5 6 b Our computer reconstruc tion contains 10 tiers but the real muqarnas vault contains only 9 tiers The eight tier of the computer reconstruction only contains cells with projection rhombi In the real 90 Chapter 5 Results of Algorithmic Muqarnas Reconstructions muqarnas the rhombi are the projection of intermediate elements of the seventh tier It is an an example of a local different muqarnas as described Section 3 5 If a tier only consists of cells standing on curved sides of elements in the previous tier we can push down the complete tie
30. directed graph containing the structure infor mation of the muqarnas from the muqarnas plan see Sections 3 2 and 3 4 Secondly we extend this graph into a three dimensional muqarnas structure see Section 3 3 We discover that different steps of the algorithm contain some freedom This result in different muqarnas reconstructions with the same plane projections The differences between such mugarnas reconstructions are discussed in Section 3 5 3 1 Representation of the Muqarnas Structure in a Graph For converting a muqarnas plan into a three dimensional mugarnas vault we want to identify for each polygon in the muqarnas plan the corresponding element together with the position and orientation of the element We recall from Section 2 2 that an element can be specified by its diagonal angle and its type cell or intermediate element To determine its type we decode which edges of the polygon correspond to the curved sides of the element If we are acquainted with the curved sides we can specify the central node see Definition 2 2 1 The direction of the curves according to the central node fixes then the type of the element We will see that also the height information can be assigned from knowing all directions of the curved sides in the plan As the plane position is already included in the coordinates of the central node the position and orientation of the element can then be determined 31 32 Chapter3 Algorithm for Reconstructing
31. find situation III in muqarnas graphs The lower four pictures show the situation the left polygon corresponds to an inter mediate element In that case the edges e and e join in their initial points and present the curved sides of this intermediate element black arrows The first situation I may appear in a muqarnas graph G In this situation a cell stands with its backsides on the front of the intermediate element The projection of the front of the intermediate does not appear in the graph The edges e and f have in this case the same direction The situation II is in contradiction with Lemma 3 1 4 For the third situation III we re mark that this situation only appears when e and e define a half intermediate element and f and f also define a half intermediate element These half intermediate elements appear on the same tier and touch in their front parts This can only happen in case of half intermediate elements In that case the projection of the front is a cross edge joining 3 2 Construction of Muqarnas Graphs from the Plan 39 09 Bd a a a Jo eg v eee Figure 3 6 Possible directions of the edges of the polygon given by e e f and f If edges e and f are opposite edges in a muqarnas graph then only cases I and I are allowed the end points of the edges e and e in the plan P This is in contradiction with the assumption of e and f being opposite edges We conclude that the only allowed situations are I an
32. into the vault In the center of the upper most tier we often find ele Nam NEA NG K EB Kan EB zv B ER OU EX Y a X b FASSUS LI IS RER Figure 2 8 Plane projection of a muqarnas vault in the entrance portal of the Friday Mosque 1315 1316 in Ashtarjan Iran We find figures corresponding to non basic elements gray at the front of the vault The center is filled with barley kernels dark gray 16 Chapter 2 Structure of the Muqarnas ments with barley kernels as plane projection A barley kernel has two sides with length of the module The length of the diameter varies and needs to be adapted to the vault in which the muqarnas is built In II Khanid muqarnas it only appears as plane projection of intermediate elements of the upper tier It is a special property of Il Khanid muqarnas that they are formed from only a few different kind of elements In muqarnas of other styles we find more variation in the elements For example in the Seljuk muqarnas we see that there is a larger freedom for varying the angles see e g Figure 2 9 In the designs found in the Topkap Scroll see Necipoglu 1995 this is also the case but the structure of the muqarnas remains the KIS IRF SALI SACRA eg BAS ALZ AN PASOS Figure 2 9 Plane projection of the mugarnas in the entrance portal of the Hunat Hatun Camii 1238 in Kayseri Turkey In other kinds of muqarnas for example the ones in Morocco besides of the cells a
33. is long then k is large and a lot of space is needed to store the plans R for i 1 k This means that the above described method for calculating the shortest path between two nodes needs a lot of computer capacity To calculate the center of a muqarnas plan this is not a problem as the center is mostly small The path defining the bottom boundary see Definition 2 2 10 can be larger Therefore we have chosen to use a different method in that case From the set of boundary nodes we delete the center nodes and the nodes on the x mirrorline and the y mirrorline From the remaining set of nodes L we calculate the maximal subplan R L D of Q M C given by these nodes That means that if edge v w C and both v w L then v w L and edge v w D We add the nodes from the x mirrorline and the y mirrorline with minimal x and minimal y value respectively 4 1 5 Determination of the Direction of the Edges The main object during the program flow is the object OrbitsValued In this object the orbits of the muqarnas plan are stored and we apply the rules as formulated in Section 3 2 on this object to set the directions of the orbits After setting the direction we return the information of this object as a graph An edge edge v w is stored as set of two nodes v w In this context we do not identify edge v w with edge w v An orbit is a vector of edges and the object OrbitsValued contains a vector of orbits O along with for eac
34. not only the muqarnas graph but also the plan is involved The definition is illustrated in Figure 3 5 Definition 3 1 6 opposite edges We consider a muqarnas with plan P N E and muqarnas graph G M C If e edge e e and f edge fs f are edges in the graph G M C such that there exist edges e edge es fs f edge e f in G M C but no cross edges edge e f or edge f e in P N E then we say that e and f are opposite edges in G according to P In other words edges e and f are opposite if both end nodes of e are connected in the graph G as well as in the plan P with exactly one and not the same end node of f In this definition of opposite the end nodes of the edges need to be connected in the graph G for being opposite To exclude the possibility to be opposite if these end nodes have a second connection we look at the plan P In Figure 3 5 the edges which appear in P but not in G are dotted The edges drawn by continuous lines appear both in the plan P as well as in the subgraph G We see three different kind of structures which may appear 38 Chapter 3 Algorithm for Reconstructing Muqarnas fr ff f f n je f es e et Figure 3 5 On the left edges e and f are opposite In the middle and on the right edges e and f are not opposite in the graph In the first picture edges e and f are opposite because the end nodes of e are connected in the graph G to the end nodes of f In the second picture e an
35. regular center but the bottom boundary is not regular anymore We find that the elements corresponding to the orbits of which we changed the direction are pushed upwards This idea can be used to create several globally different muqarnas of different heights Changing the direction of orbits makes it possible to push parts upwards or downwards OPO a pl_farumad fig b gr_farumad fig Figure 5 13 Plan analysis for reconstructing the muqarnas in the sanctuary iwan of the Friday Mosque in Far mad 102 Chapter 5 Results of Algorithmic Muqarnas Reconstructions a grl_farumad fig b reci farumad fig i c gr2 farumad fig d rec2_farumad fig Figure 5 14 Two different computer reconstructions corresponding to the simplified plan of the sanctuary iwan of the Friday Mosque in Far mad 5 2 Computer Reconstructions of Il Khanid Muqarnas 103 5 2 4 Takht i Sulayman South Octagon Location Takht i Sulayman Iran Building Il Khanid palace Vault South octagon Height unknown Years of construction 1271 1274 References Harb 1978 pp 43 46 Discussion In the Il Khanid palace at the Takht i Sulayman parts of a muqarnas were found in a room with octagonal ground plan The palace contained two rooms with octagonal ground plan and the muqarnas remains were found in the most south one Adopted from Harb we will refer to this room as the south octagon In Figure 5 15 b a computer reconstruction of the c
36. same plane projection can be local or global Local means that by exchanging several elements the muqarnas are equal Muqar nas are globally different if more elements are involved and the shape or number of tiers of the muqarnas differ We can restrict the number of reconstructions if we know the pro portions of the vault into which the muqarnas needs to fit This gives a restriction on the amount of tiers that can be used This work focuses on the muqarnas vaults fitting into domes or niches from the Seljuk and Il Khanid periods Different tests of such muqarnas from Anatolia and Iran are given Additionally a new interpretation for the oldest known design which is found at Takht i Sulayman is presented iv Contents Zusammenfassung Abstract 1 Introduction 1 1 Mugarnas in the Islamic Architecture 4 1 1 1 Overview about Seljuk and Il Khanid Architecture 1 1 3 M garnas Designs gt 4 2 52 ra P cto a lhe ee RN 1 2 Purpose and Contribution of our Research 1 2 1 Motivation A 2228 dob RN near Beers Oe S de 1 2 2 Literature Overview 2 bie owe on go REUS RAP NUS EURO ae eS 1 2 3 Outline of this Work ins s Gx 2 0 oR ee we a XD Structure of the Muqarnas 2 1 Muqarnas in the Architecture llle 2 1 1 Muqarnas Elements 6 4 2 4 se ek A C OX oe oso xU 2 1 2 Three Dimensional Mugarnas Structure 2 2 Mugarnas as an Abstract Geometrical Structure
37. side of an element in the next tier Because of this property we say that a muqarnas is oriented in upwards direction following the curved sides gives a possibility to go up in the muqarnas Further we see in Figure 2 13 a that the backsides of a cell can be put on curved sides of elements below The Figures 2 13 b and 2 13 c show that backsides of a cell also can be put on front sides of intermediate elements An intermediate element stands at the front of a cell from the previous tier a Back on curve b Back on front c Back on front d Back on front Figure 2 13 Different ways of building cells upon elements in a lower tier We summarize the possibilities how we can combine the different elements curve to curve A curved side of an element joins a curved side of an element on the same tier see Figure 2 11 front to front The front part of an intermediate element can touch the front part of an intermediate element on the same tier see Figure 2 12 back on curve The backside of a cell can stand on the curved side of an element of the tier below see Figure 2 13 a The apex of the curved side of the element in the lower tier then joins the curved sides of the cell in the upper tier curve points to curve back on front The backside of a cell can stand on the front of an intermediate element of the tier below see Figures 2 13 b and 2 13 c An intermediate element stands on the front of a cell see Figure
38. true Stop If change false or if k M The main algorithm is A 3 3 This algorithm calls the algorithms A 3 1 and A 3 2 in alternating order The Algorithm A 3 1 forces that all successors of the nodes at height h get height h 1 and the Algorithm A 3 2 forces the predecessors at height h having height h 1 In Figure 3 4 we see a part of the muqarnas graph from the interpretation of Harb of the design on the plate found at the Takht i Sulayman On the left we see the result of calculating the heights of the nodes by only applying Algorithm A 3 1 and on the right in red the changes after applying Algorithm A 3 2 By applying the Algorithms A 3 1 and 36 Chapter 3 Algorithm for Reconstructing Muqarnas Se SED Qoxuy GRY 0 Figure 3 4 Part of a muqarnas graph On the left calculated heights of the nodes after running A 3 1 On the right calculated heights of the nodes after running additionally A 3 2 A 3 2 in alternating order we correct the height information needed for the nodes not direct reachable by a path in the graph starting at height 0 We force the algorithm to stop if hmaz gt M This is needed for the situation we apply the algorithm on a graph which is not a muqarnas graph In that case it may contain a cycle and applying the algorithm could result in an infinite loop Applying the algorithm on graphs different from muqarnas graph can be useful to test whether a given graph may correspond to a muqarnas or not The fol
39. v w is then given by c h 9 with h h c and a c c eo v with eg 1 0 The central node c of an intermediate element in tier r corresponds to the bottom of the curved sides of this element This bottom stands on elements from the previous tier or corresponds to the bottom of the muqarnas Therefore the height c equals to r 1 The position of an intermediate element corresponding to the int face F c v w is given by c h 9 with h h c 1 and a c c eo v with eg 1 0 3 3 4 Reconstruction from a Simplified Plan The plate found at the Takht i Sulayman shows that a muqarnas design is not always composed of the projection of all muqarnas elements see Figure 1 3 The drawing on this plate mainly consists of squares and rhombi We call such a plan a simplified plan Some rhombi need to be split into an almond and a biped to get a proper interpretation of the design For this reason we extend our reconstruction algorithm so that it can also handle designs only consisting of squares and rhombi This extension is only done for muqarnas consisting of basic elements like common in the Il Khanid architecture In a muqarnas where a wider variety of elements is used we cannot determine the place the polygons can be split MN Mer WU Figure 3 17 On the left square and rhombi of the plan interpreted as one element On the right rhombi and center square in the plan interpreted as two elements In Figure
40. we create the subplan Q and the plan P containing the coordinates of the nodes in the muqarnas plan and their connections Besides the object Plan we create an object called geoPlan The object Plan is responsible for the structure of the plan which is given by the connections between nodes The object geoPlan cares about the features of the plan depending on the coordinates of the nodes It consists of the boundary information of the plan and stores which section of the muqarnas is represented by the plan given in the input This is given by the number behind the command line option p As the direction of parallel edges is the same we first analyze which edges are par allel We store parallel edges in an orbit which is represented by a vector of edges In 60 Chapter 4 Software Tools for Reconstructing Muqarnas this way we create a vector of the orbits with for each orbit a value to fix the direction of the orbit This vector of orbits together with a value for its direction is stored as an object OrbitsValued In this object our algorithm steps as presented in Section 3 2 are applied to set the directions of the orbits We then can directly convert this object into an xfig file where edges with unknown directions are drawn by dotted lines If the direction is known we use arrows to represent these directions In the remaining part of this section we give a more detailed description of the used data structures and explain the steps executed duri
41. which connect the roofs of adjacent cells or other intermediate elements A more comprehensive explanation about the muqarnas elements will be given in Chapter 2 Al Kashi describes the plane projection of an ele ment either cell or intermediate element as a basic geometrical form namely a square half square cut along the diagonal rhombus half rhombus isosceles triangle having as base the shorter diagonal of the rhombus almond deltoid jug quarter octagon large biped complement to a jug and small biped complement to an almond These elements are constructed according to the same unit of measure and fit together in a wide variety of combinations Al Kashi uses in his computation the module of the muqarnas defined as the base of the largest facet the side of the square as a basis for all propor tions Different materials can be used to construct muqarnas For example plaster brick or wood are common materials In Seljuk architecture brick is often used in Il Khanid architecture plaster is more common Dependent of the material used for the muqarnas elements there can be a wider variety in the elements If the muqarnas is not constructed from prefabricated elements we only need to care that two joining elements need to fit together In other parts of the muqarnas the elements may have other measurements When a mugarnas structure has to be inserted into an existing vault the height of the facets of the elements need to be adapt
42. 0 They ruled parts of central Asia and Middle East from the 11th to 14th century The Great Seljuk Empire reached its peak during the reigns in the time between 1038 and 1093 of the three sultans Tughrul Bek grandson of Seljuk Alp Arslan and Maliksh h Under Maliksh h and his two Persian viziers Nizam al Mulk and Taj al Mulk the Seljuk 1 2 Chapter 1 Introduction state expanded in various directions to Persian border so that it bordered China in the East and the Byzantine in the West After the death of Malikshah the empire split into smaller parts Ahmed Sanjar one of the sons of Maliksh h was the ruler of most of Persia This part the so called Seljuk Dynasty collapsed after his death in 1157 In Anatolia for example the Seljuks ruled longer They are called the Seljuks of R m and reigned until the Mongol invasion in 1307 From 1220 on there were different successful attacks on Central Asia by the Mon gols under supervision of Genghis Khan see e g Wilber 1955 In 1256 his grandson H lagu Khan invaded Baghdad and founded the Il Khanid state During the leadership of H lagu s great grandson Ghazan Khan the Il Khanids converted to Islam and the culture flourished After the death of the last Khan Abu Sa id in 1335 the Il Khanids lost their power and the state broke up into different dominions which developed separately By the diffusion of the Islamic religion during the Seljuk and Il Khanid periods a lot of religious bui
43. 3 17 we see two different muqarnas reconstructions corresponding to a small plan based on squares and rhombi The graphs of these interpretations are the 50 Chapter3 Algorithm for Reconstructing Muqarnas same only the plans differ These reconstructions are for illustrating they do not corre spond with real architecture In the left picture we see a cell with a rhombus as plane projection In this cell the curved sides join in the smaller angle Such elements do not appear in Il Khanid architecture Therefore we interpret these rhombi as the plane projection of two elements a cell with plane projection an almond and an intermediate element with plane projection a small biped see Figure 3 17 on the right The interme diate element corresponding to the biped appears one tier lower In similar way the square in the middle of the plan is in the right reconstruction interpreted as a combination of a cell with plane projection a jug and an intermediate element with plane projection a large biped The element corresponding to the biped appears one tier lower We say that we split the cell into a combination of a cell and an intermediate element We push down the part corresponding to the intermediate Figure 3 18 On the left part of the the plane projection of a mugarnas vault in a shrine at Bist m see Section 5 2 2 On the right the plane projection of a muqarnas vault in a niche at the Friday Mosque in Natanz see Section 5 2 1 Split
44. INAUGURAL DISSERTATION Zur Erlangung der Doktorw rde der Naturwissenschaftlich Mathematischen Gesamtfakult t der Ruprecht Karls Universit t Heidelberg vorgelegt von Dipl Math Silvia Harmsen aus Ermelo die Niederlande Tag der m ndlichen Pr fung 27 Oktober 2006 Algorithmic Computer Reconstructions of Stalactite Vaults Muqarnas in Islamic Architecture Gutachter Prof Dr Gerhard Reinelt Prof Dr Jan P Hogendijk For Cor Zusammenfassung Mugarnas oder Stalaktitengew lbe sind dreidimensionale Ornamente die in der isla mischen Architektur verbreitung finden Sie werden in Gew lben Kuppeln Nischen auf Bogenkonstruktionen oder als fast flache dekorative Friesen genutzt Ein Muqarnas hat die Aufgabe einen flie enden bergang von einer geraden Wand zu einem gekr mmten Teil zu gew hrleisten Ein Mugarnasgew lbe wird von verschiedenen nischeartigen Ele menten die in horizontalen Stockwerken angeordnet sind aufgebaut Ein Hauptmerk mal der Mugarnas ist dass die Form eine dreidimensionale Einheit darstellt welche in einem zweidimensionalen Grundriss repr sentiert werden kann Wir konzentrieren uns auf die Frage ob die zweidimensionale Projektion alle Strukturinformationen des dreidi mensionalen Gew lbes enth lt Um einen Rahmen zu schaffen in dem wir die Muqarnasstruktur beschreiben k n nen f hren wir explizite Definitionen ein Jedes Muqarnaselement wird parametrisiert um Typ Abmessungen und P
45. K u VN EN c d e rmlines2_ciftemed fig gr2 ciftemed fig crc gr2 ciftemed fig f g Photo of the vault and a de WRL main wrl tailed photo of the area in the muqarnas not consisting of mugar nas elements Figure 5 4 Reconstruction process and photos of the entrance portal of the hospital of the Gifte Medrese in Kayseri 5 1 Computer Reconstructions of Seljuk Muqarnas 89 5 1 3 Sultan Han near Kayseri Location On the road between Kayseri and S vas Turkey Building Sultan Han Vault Entrance portal to enclosure Height 9 tiers Years of construction 1232 1236 Date of Visit April 11th 2005 References Erdmann 1961 pp 90 97 Aslanapa 1971 p 150 Takahashi 2004 051 Discussion The Sultan Han near Kayseri is in a good condition and richly ornamented with muqar nas We study here the muqarnas in the entrance portal to the enclosure In Figure 5 5 a the plane projection of this muqarnas vault is given We start the reconstruction process by removing edges edges of the plan by applying removelines r Ppl slthan ksi fig Additionally the edges represented by dotted lines in Figure 5 5 b need to be removed to create the subplan consisting of the projection of the curved sides The muqarnas graph is calculated by running plantograph r Qrmlines pl slthan ksi fig Ppl slthan ksi fig There are eight orbits for which the direction is not set The direction of the undeter mined orbits can be
46. PEEK K y K D SER TES 2 gt SH ZX FAN Figure 1 3 On the left picture of the plate found at Takht i Sulayman with a mugar nas design incised in it On the right the lines of the plate as recognized by Harb after Harb 1978 The first study of the plate found at the Takht i Sulayman is done by Harb one of 6 Chapter 1 Introduction the members of the excavation team who found this plate see Harb 1978 More recent studies and also suggestions to decode the plate can be found in Yaghan 2000 and Dold Samplonius and Harmsen 2005 The design on the plate see Figure 1 3 on the left is a about 3 5 to 4 cm thick rectangular plate with height 47 cm and width 50 cm It contains a geometric grid in a quadratic field of 42 cm length A small part of the bottom left corner is broken off and did not survive The remaining plate is broken into seven parts which fit together except near the middle of the plate where there is a small hole Under the clearly drawn grid are poorly erased lines visible At some points these may have been auxiliary lines but at others seem to have no direct connection with the actual design The artisan seems to have reworked his design and it is not clear whether the present design was ever used or whether it was altogether abandoned In the right picture of Figure 1 3 the lines of the plate as recognized by Harb are shown As the
47. We use for the visualization of our virtual reconstructions the curved muqarnas elements as described by al Kashi see Dold Samplonius 1992 p 232 This means that our reconstructions give a representation of the structure of the muqarnas but do not need to resemble the original muqarnas To create a virtual muqarnas resembling the real muqarnas we may need to replace the curved elements by other kind of elements Our model for the curved muqarnas elements fit well to the elements found at the Takht i Sulayman see Harb 1978 Because we focus on the structure and not on the static part of the muqarnas in our model only the surfaces which are visible are drawn see Figure 4 8 This means that we restrict ourselves by modeling the curved sides the curved front surfaces and the bottoms of the elements which are visible if we watch the muqarnas from beneath The backside of an intermediate element only consists of the edge where its curved sides join and are therefore already reconstructed by drawing the curved sides The backside of a cell is not visible in the muqarnas and hence not included in our model The same holds for the top surfaces as we do not look at the muqarnas from above they are not visible and therefore not drawn 78 Chapter 4 Software Tools for Reconstructing Muqarnas right curved side left curved side right curved side left curved side right front right front left front Joe k left front bottom bottom
48. ain wrl b Photo of the muqarnas Figure 5 2 Computer reconstruction and photo of the muqarnas vault in the portal to the enclosure of the Alay Han 5 1 2 Kayseri Cifte Medrese Location Kayseri Turkey Building ifte Medrese Vault Entrance portal of hospital Height 7 tiers Year of construction 1205 Date of visit April 11th 2005 References Aslanapa 1971 p 129 Takahashi 2004 040 Discussion The ifte Medrese in Kayseri consists of a medical school and a hospital which are joined by a corridor It is the oldest hospital building in Anatolia In Figure 5 3 a a quarter muqarnas plan is given It represents half of the plane projection of the muqarnas vault in the entrance portal of the hospital The projection suggests that the muqarnas consists mainly of basic elements Running 5 1 Computer Reconstructions of Seljuk Muqarnas 87 removelines r Ppl ciftemed fig successfully removes edges which do not correspond to curved sides of muqarnas ele ments The removed edges correspond to the projection of backsides of cells standing on front sides of intermediate elements We need to remove additionally the edges which are given by dotted lines in Figure 5 3 b EN a pl_ciftemed fig b rmlines_pl_ciftemed fig Figure 5 3 Plan and subplan of the muqarnas in the entrance portal of the hospital of the Cifte Medrese in Kayseri By applying plantograph r Qrmlines pl ciftemed fig the direction of all orb
49. ams Books 2004 123 124 Bibliography Dold Samplonius and Harmsen 2005 Dold Samplonius Y and Harmsen S L 2005 The Muqarnas Plate found at Takht i Sulayman A New Interpretation Muqarnas An Annual on the Visual Culture of the Islamic World 22 82 91 Dold Samplonius et al 2002 Dold Samplonius Y Harmsen S L Kr mker S and Winckler M J 2002 Magic of Mugarnas Video about Muqarnas in the Islamic World IWR Preprint 2002 39 Erdmann 1961 Erdmann K 1961 Das anatolische Karavansaray des 13 Jahrhun derts Verlag gebr Mann Foley et al 1994 Foley J D van Dam A Feiner S K Hughes J E and Phillips R L 1994 Grundlagen der Computergraphik Addison Wesley Grabar 1992 Grabar O 1992 The Alhambra Harvard University Press Harary 1974 Harary E 1974 Graphentheorie R Oldenbourgh Verlag M nchen Harb 1978 Harb U 1978 Ilkhanidische Stalaktitengewolbe Beitr ge zu Entwurf und Bautechnik volume IV of Arch ologische Mitteilungen aus Iran Dietrich Reiner Berlin Herzfeld 1942 Herzfeld E 1942 Damascus Studies in Architecture Ars Islamica The Research Seminary in Islamic Art IX 1 52 University of Michigan Ibrahim 2002 Ibrahim S M A 2002 Mugarnas ein Stalaktitenf rmiges Architek turelement sein Struktureller Aufbau und seine Bildungsregeln PhD thesis Universit t Stuttgart Jones and Goury 2001 Jones O and Goury J 2001 Pl
50. and graph G M C Let v Wy P and w adjacent to v with w Wy P then arw v w C The second rule is motivated by the fact that paths over curved sides finish at the front or in the center of the muqarnas From the center it is not possible to go upwards without leaving the center 42 Chapter3 Algorithm for Reconstructing Muqarnas Rule 2 We consider a muqarnas structure with plan P N E and graph G M C If the corresponding muqarnas has a regular center and c C has end nodes c c such that cs W P and c W P then c arw c ci Rule 2 can only be applied if the center is regular that means all center nodes appear as projections of vertices of the upper most tier This is usually the case the only example we know where this is not true are several interpretations of the design on the plate found at the Takht i Sulayman see Section 5 2 5 In Figure 3 10 a part of the center of a possible muqarnas graph corresponding to the design on the plate is drawn If we ex bow AL Lz D Figure 3 10 Part of the center of a muqarnas graph inspired by the design found at Takht i Sulayman The dotted arrow has a direction which is in contradiction with Rule 2 observe the dotted arrow we see that for its end nodes c W P and c W P but arw c cs Z C which is in contradiction with Rule 2 In this special case node c belongs to the center but not to an element in the upper most tier Thi
51. and on the front of an element below back on front In both situations the bottom of the element touches elements a tier lower We find that v also appears as the projection of an element in the lower tier and h v r 1 If the curved side of which arw v w is the projection belongs to an intermediate element not at the bottom of the muqarnas then this element stands with its backside on the front of an element in the previous tier In that case we also find that h v r 1 For 3 2 we remark that this follows directly from Definition 3 1 2 The height h v cannot be negative and the projection of vertices of the bottom of the elements in the first tier appear as nodes v in the muqarnas graph with h v 0 El Lemma 3 1 4 If G is a directed graph corresponding to a muqarnas structure then paths p q in G with the same end nodes have the same length In other words if p q are paths in G with lengths l p l q respectively and end nodes p qs and pe qe then PROOK We remark that following an arrow corresponds in the muqarnas to going up wards to the next tier This means that if we start at node v and we are able to walk to w 3 1 Representation of the Muqarnas Structure in a Graph 35 we need to pass h w h v tiers with h v the height of node v Therefore the length of our walk equals h w h v It depends only of the end nodes of the path and not of the route oO From the previous property it follows that a
52. anos Alzados Secciones y Detalles de la Alhambra Mar a ngeles Campos Romero Jungblut 2005 Jungblut D 2005 Muqarnas Plan Editor http pille iwr uni heidelberg de muqtool Last Visit 19th November 2006 Lemmens and Springer 1992 Lemmens P and Springer T 1992 Hoofdstukken uit de Combinatoriek Epsilon Uitgaven Utrecht Necipoglu 1995 Necipo lu G 1995 The Topkapi Scroll Geometry and Ornament in Islamic Architecture Topkapi Palace Library MS H 1956 The Getty Center Santa Monica Noltemeier 1975 Noltemeier H 1975 Graphentheorie mit Algorithmen und Anwen dungen Walter de Gruyter amp Co Berlin Notkin 1995 Notkin I 1995 Decoding Sixteenth Century Muqarnas Drawings Mugarnas An Annual on the Visual Culture of the Islamic World 12 148 171 Bibliography 125 Pope 1939 Pope A U editor 1939 A Survey of Persian Art from prehistoric times to the present volume II Oxford University Press London and New York Sato and Smith 2002 Sato T and Smith B V 2002 Xfig User Manual http www epb lbl gov xfig Last Visit 19th November 2006 Sedgewick 2002 Sedgewick R 2002 Algorithms in C Graph Algorithms Addison Wesley third edition Senechal 1995 Senechal M 1995 Quasicrystals and Geometry Cambridge Univer sity Press Cambridge Takahashi 2004 Takahashi S 2004 http www tamabi ac jp idd shiro mugarnas Last Visit 19th Nove
53. aph and defines a set of muqarnas elements see Section 4 3 1 For visualizing the mugarnas this set is converted into VRML files containing the geometrical data of the mugarnas see Section 4 3 2 4 3 1 Calculation of the Muqarnas Structure The following command line options are available to call the program graphtomuq 74 P lt string gt G lt string gt p lt number gt v lt number gt o lt string gt t lt string gt Chapter 4 Software Tools for Reconstructing Muqarnas This option is followed by the name of the file repre senting the complete muqarnas plan If this option is not given a subplan derived from the input muqarnas graph is used instead This option is necessary It is followed by the name of the input file representing the muqarnas graph The option p can be followed by the integers 1 2 or 4 representing a full half or quarter muqarnas plan respectively By default the input plan is considered being a quarter plan This option can be followed by 1 or 2 representing whether a full or half vault should be reconstructed This option expects the name of the directory in which the output files containing the geometrical data are written By default the output directory gets the name WRL This option is followed by the name of the texture file By default the file texture jpg is used N EN setFaces i ElementList i VRML output Figure 4 6 Program fl
54. at is explained in Section 4 3 2 4 1 The Program plantograph To convert a muqarnas plan into a muqarnas graph we first determine the subplan con sisting of the plane projection of all the curved sides of the elements and of backsides of cells appearing at the bottom of the muqarnas We then give each edge in this sub plan a direction to set the muqarnas graph see Section 3 2 The determination of the directions is executed by the program plantograph In this section we first describe the program flow of the program plantograph see Section 4 1 1 After that we work out some parts more in detail In Section 4 1 2 the 57 58 Chapter 4 Software Tools for Reconstructing Muqarnas data structures used for the plans and graphs in the program is explained After that we explain in Section 4 1 3 how we can convert the input to our intern data structure To set the direction of the edges we use the rules which we formulated in Section 3 2 The first two rules depends on the boundary of the plan In Section 4 1 4 we explain how this boundary is calculated The main process of the program the execution of the rules is then given in Section 4 1 5 Finally in Section 4 1 6 some remarks about the computing time of the program are given 4 1 1 Conversion of the Plan into the Graph The program plantograph applies the rules as presented in Section 3 2 to set as many directions of the edges as possible With the additional tool planedit written b
55. axis with angle a so that it joins to edge 0 b Then we rotate over 7 2 so that the normal is perpendicular to edge 0 5 The normals of the vertices on the left curved side are thus given by Resna LOO The normals of the front sides are given by rotation of the matrix normals The nor mals in this matrix are given for three dimensional surfaces created by expanding the two dimensional curve in three dimensions as the left two pictures of Figure 4 11 The normals as given in the matrix normals correspond to the normals of three dimensional surfaces perpendicular to edge a 0 as drawn in the middle picture of Figure 4 11 with direction GI For the front side the projection of GI needs to join edge o a Therefore 82 Chapter 4 Software Tools for Reconstructing Muqarnas we rotate the matrix normals over 7 2 so that it joins to edge a 0 and then rotate over a a 0 o We find the normals of the vertices on the right front surfaces by Re a 0 0 n 2 normals Since edge a b b is parallel to edge a 0 we find with similar arguments that the nor mals of the vertices on the left front surface can be given by Ra b a b 0 7 2 normals For an intermediate element in standard position given given by a a 0 b b b R and with diameter d R we proceed in a similar way Let a a 0 a b be the angle of the element we write again o for the node opposite the central node its coordinates are given by
56. boundary on the heights is determined If the command line option r is given more variations of elements can be used in the reconstruction We have seen in Section 3 4 that the edges corresponding to the places where backsides of cells stand on front sides of intermediate element can often be recog nized in the muqarnas plans as they divide a polygon of four edges in two polygons of four edges or two polygons of three edges see Figure 3 23 To handle the first case the 4 3 The Program graphtomuq 73 method DeleteDivisionLines is written This method deletes all non boundary nodes which have only two neighbors The second case where a polygon of four edges is divided in two triangles is handled by the method DeleteTriangleLines It removes for each triangle the edge with most differing length This is done because a triangle corresponds to a half element and the sizes of the curved sides of such a half element have in general the same length Triangles are recognized in the plan as a node with two neighbors which are connected to each other 4 2 2 Complexity Analysis We finish this section with some remarks about the computing time of the methods used in the program removelines We use again that the number of edges F linearly depends on the number of nodes in a mugarnas plan see Section 4 1 6 The method FilterLinesLengthOne iterates over the edges in the plan to determinate for each edge its length Therefore the
57. c and vector w c join see Figure 2 14 Let c R be the central node of an element and v vz vy w Wz wy R such that the edge edge c v connecting c and v and the edge edge c w connecting c and w 20 Chapter 2 Structure of the Muqarnas Figure 2 14 To join v c with w c we have to rotate v over c by an angle of a c v w in counter clockwise direction correspond to the curved sides of a muqarnas element The nodes v and w are chosen in the order so that if we rotate v in counter clockwise direction to w we cross over the projection of the interior of the element Definition 2 2 2 standard position We say that the element is in standard position if its central node c lies in the origin and edge c v joins with the x axis so c 0 0 and Uy 0 The left two pictures of Figure 2 15 show the plane projection of muqarnas elements in standard position The curved sides are represented by arrows This motivates to classify the type of a basic muqarnas element as follows Definition 2 2 3 type The type of a muqarnas element is given by two nodes a a a b b b R representing the curved sides and in case of a full element a number d Rso representing the diameter so that a 0 and 0 lt a 0 a b The type as formulated in this definition is then given by the element in standard position it is given by the two edges edge 0 a and edge 0 b joining in the central nod
58. cos km 4 1 vV2 1 V2 Because all entries of R are equal to 1 2 Z x Z 2 we can confirm that R l L To prove that these are the only transformations we need we use induction on the num ber of elements in the muqarnas If a muqarnas consists of one element we choose the orientation of the grid such that this element is in standard position For a plan of n 1 elements we look at a polygon at the boundary of the plan The element corresponding to this polygon needs to join with its curved side to another element Therefore one of the edges of the polygon has length 1 Let this edge be edge v w As the other n poly gons have coordinates in the grid IL the end nodes v w of the edge edge v w are in the grid L One of the end nodes of edge v w corresponds to the central node We assume that this is the node v By translating over v the central node lies in the origin The coordinates of the other end node w v is then given by w v a b v2 c d V2 with a b c d Z As the length of the edge edge v w equals to 1 we find that 0 0 v3 c a v2 a 309 e 5d ab cd 2 1 which gives us that so we find that a ib pid 1Aab cd 0 The only integer solutions are now a b c d 1 0 0 0 a b c d 0 0 1 0 and a b c d 0 1 0 1 This means that a 6 v2 c d v2 STEKO i a 6 2 c a v2 0 1 or a b v2 c d v2 41 v2 41 Vv2
59. cript such a division makes sense because his aim was to calculate the surface of the muqarnas elements Therefore it was important to describe the exact curve on the side of the elements and calculate the surface of the roof and the facets separately We adopt the word intermediate element Zwischenelement from Harb 1978 p 34 to describe the other type of building blocks An intermediate element is an element which connects the roofs of two adjacent elements These adjacent elements can be a cell or an intermediate element as we see in Figure 2 2 The place between two elements is not necessarily filled with intermediate elements Figure 2 2 On the left an intermediate element joining two cells In the middle an intermediate element joining a cell and an intermediate element On the right place between the cells is not filled with intermediate elements We often refer to the horizontal plane projection of an element These projections are two dimensional polygons with three or four vertices which represent the geometric figures we recognize if we watch a muqarnas from beneath see Figure 2 3 We want Figure 2 3 Portal to the enclosure at Sultan Han near Aksaray Turkey On the left view from the front In the middle View from underneath On the right plane projection to emphasize that sides of elements become edges in the plane projection Nodes in the plane projection are then the projection of lines where si
60. ction with maximal 16 tiers We run 5 2 Computer Reconstructions of Il Khanid Muqarnas 107 removelines hl16 Ppl plate tis fig The textual part of the output contains that we need to remove one or more of the nodes indexed by 144 312 1200 3280 or 7320 for creating a computer reconstruction containing less than 17 tiers The corresponding nodes are marked red in Figure 5 17 a Experimenting with the program plantograph shows that to create a reconstruction with regular bottom boundary we need to remove two nodes We remove the nodes as marked in Figure 5 20 a The left node which we remove is the place on the diagonal where four squares join We have seen a similar situation by the reconstructions of the muqarnas in the north iwan of the Friday Mosque in Natanz see Section 5 2 1 and the muqarnas in the east portal of the shrine of Bayazid in Bistam see Section 5 2 2 In both cases we removed this node for creating the reconstruction which coincides with reality The other node we remove is the place where six rhombi join A similar situation can be found in the famous mugarnas vault over the tomb of Skaykh Abd as Samad see Harb 1978 pp 55 58 The muqarnas graph is now calculated by plantograph c Qpl tis plate new fig and by setting the direction of the remaining undetermined edges so that the bottom boundary is regular we calculate the computer reconstruction by running graphtomuq vl Ppl tis plate new fig Gcrc gr tis plate n
61. d I in those cases e and f have the same direction oO 3 2 Construction of Muqarnas Graphs from the Plan In Definition 3 1 1 we defined the muqarnas graph as the projection of the curved ele ment sides of the muqarnas To construct the muqarnas from its design we should find from the plan the directed subgraphs with the same properties as muqarnas graphs We facilitate the problem by considering first the situation where the plan only consists of edges appearing in the muqarnas graph The starting point is already a subplan and our task is to determine the directions of its edges We have seen in Lemma 3 1 7 that opposite edges of the muqarnas graph G according to the plan P have the same direction Therefore we bind opposite edges together in one orbit see Figure 3 7 and we determine for each orbit a direction instead of determining separately for each edge a direction An orbit is defined as an equivalence class of the equivalence relation generated by the opposite edges Edges of the same equivalence class are then called parallel edges Definition 3 2 1 parallel edges The edges e and f are parallel if e f or if e and f are opposite or if there exist an edge parallel to both e and f We can prove that this definition indeed defines an equivalence class by verifying the reflexive symmetric and transitive relations Definition 3 2 2 orbit An orbit is an equivalence class of the equivalence relation given by the parallel e
62. d Samplonius and Harmsen 2005 In the mugarnas designs of the Topkapi Scroll the designers mark where figures have to be split In these designs polygons often appear with more than four nodes which suggests that the projection of the sides of some elements is not always drawn We see for example in Figure 2 16 a detail from the left lower corner of the design nr 16 from the Topkap Scroll see Necipoglu 1995 p 294 In the right picture a dotted line is inserted to mark a side of an element which is not drawn in the design but necessary for the construction In the software we developed we use a muqarnas plan as an input A mugarnas plan is the set of lines corresponding to the plane projection of all sides of the elements The definition of mugarnas plan is inspired by the muqarnas designs we know but we need to be aware that we often have to adapt the design before it is a muqarnas plan containing 22 Chapter 2 Structure of the Muqarnas Figure 2 16 Part of a design from the Topkap Scroll from Necipoglu 1995 p 294 the plane projection of all sides of the elements We define a muqarnas plan as a graph see Definition A 1 Interpreting it as a graph gives us a way to store the structural information contained in the plan As a graph the edges E and nodes of intersection N are stored Definition 2 2 5 muqarnas plan A muqarnas plan P P N E is a two dimensional drawing existing of a finite set of edges E and a f
63. d f are not opposite as the end node e is connected in the plan P to both end nodes of f In the last picture e and f are not opposite as not both end nodes of e are connected in the graph G to the end nodes of f Lemma 3 1 7 We observe a muqarnas with graph G and plan P If the edges e and f are opposite in the graph G according to the plan P then they have the same direction in the graph G PROOK We look at the polygon at the left of Figure 3 6 We consider this polygon as a part of the graph G with opposite edges e and f As this polygon appears in the graph G we have to determine for all edges a direction The upper four pictures represent the situation that this polygon corresponds to a cell In that case the edges f and f join in their end points represented by black arrows in the upper four pictures The red arrows present all possible directions of the remaining edges e and e Situation I may appear in muqarnas graphs the backsides of the corresponding cell could stand on curved sides of elements in the lower tier In this situation the opposite edges e and f have indeed the same direction Picture II presents a situation which is in contradiction with Lemma 3 1 4 there is a path of length 1 and a path of length 3 both with the same end nodes Picture III implies that f and f as well as e and e define a cell We need to join these cells with there backsides This situation does not appear in muqarnas and therefore we do not
64. dent to the edges e E We use the notation P Po oP k times for applying k times the operator o For a plan P there is a path between v and w of length k if edge v w PF For computing the shortest path between v and w which defines the center we store the subplan given by the boundary nodes of the plan as a subplan R We determine for i 0 k the plans R where k is chosen so that this is the smallest integer for which edge v w R We construct the path of length k defining the center by finding a sequence of nodes s such that edge v s R and edge si w R fori 1 k 1 The process of calculating the center is illustrated in Figure 4 5 Here the plans R are drawn for the plan corresponding to the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz We see that edge v w R so the center is given by a 66 Chapter 4 Vv 51 w S5 a R y w c R vr 1 w 55 e R Software Tools for Reconstructing Muqarnas vU 52 54 w b R ye 52 54 w d R Figure 4 5 The center of the plan corresponding to the basement of the north iwan of the Friday Mosque in Natanz is calculated by multiplication of the subplan defined by the boundary nodes 4 1 The Program plantograph 67 path of length 6 It is defined by the sequence consisting of the common neighbors s of v in R and of w in R If the shortest path between nodes v and v
65. des join see Figure 2 4 2 1 Mugarnas in the Architecture 13 Figure 2 4 Elements with curved sides marked together with their projections We often refer to the curved sides of the mugarnas elements To describe the structure of the muqarnas the exact curve of the sides of the elements is not that important But to know which sides are curved gives us the orientation of the element and the place where the elements join In our drawings we represent these sides indeed by curved faces as they are common in Il Khanid muqarnas see Figure 2 4 Without changing the structure of the muqarnas we could change their form see e g Yaghan 2001a p 141 To clear terminology we define the apex of the curved side by the line at the front of the curved side The backside of the curved side is the line on the opposite side In the left picture of Figure 2 4 we marked the apex of the curved side with A and its backside with B Both apex and backside of the curved side are in the plane projection represented by a node Curved sides of a cell join in their apex the curved sides of an intermediate element join in their backsides Mugarnas of the Il Khanid period only consist of a small set of basic elements These basic elements were already described by al Kashi see Dold Samplonius 1992 In Dold Samplonius 1996 an extensive explanation of his description is given We give an overview of the plane projections of the basic elements of the Il Khan
66. dges 40 Chapter3 Algorithm for Reconstructing Muqarnas Figure 3 7 The colored lines represent the orbits in the muqarnas graph An arrow in the mugarnas graph corresponds to a curved side Following the direction of an arrow in the graph means in the muqarnas going to the next tier Our job is to set the direction of the edges in such a way that we can walk over the arrows from the bottom boundary to the center In the practise a muqarnas is built from the outside in see Ibrahim 2002 pp 77 78 Hence we read a mugarnas design in this direction We formulate the following rules 1 Arrows at the bottom boundary need to point inward 2 Arrows at the center need to point to the center Figure 3 8 The red nodes are the bottom boundary nodes of the plan Rule 1 should be able to set the directions of the blue arrows The direction of the dotted edge is undetermined In Figure 3 8 we see the desired result after applying the first rule Not for all edges incident to a bottom boundary node the direction can be determined For example the direction of the dotted edge cannot be set We can only set the direction of edges of which we can validate which end node is the projection of the lowest vertex Not all bottom boundary nodes necessarily appear as the projection of a more lower vertex than its neighbors For the dotted edge in Figure 3 8 the end node belonging to the bottom 3 2 Construction of Muqarnas Graphs from the Plan 41
67. direction The marked node then gives by using Rule 4 the direction of these two orbits c rmlines pl alayhan fig gr alayhan fig X FT d Set undetermined orbits e crc gr alayhan fig Figure 5 1 Reconstruction process for the muqarnas in the portal to the enclosure of the Alay Han For the blue marked orbits we remark that the quarter plan only represents half of the plane projection of the muqarnas If we copy the plan in the y mirrorline we find that the direction of the most right blue orbit can be determined as the corresponding orbit in the full plan contains an edge incident to the center node Rule 2 determines then its direction To set the direction of the other blue marked orbit we use the argument that the plan is mirror symmetric in its diagonal and therefore we expect the muqarnas graph being mirror symmetric in its diagonal The muqarnas can then be reconstructed see Figure 5 2 a by the command graphtomuq r Ppl alayhan fig Gcrc gr alayhan fig 86 Chapter 5 Results of Algorithmic Muqarnas Reconstructions We see in the reconstruction that there appear intermediate elements with almonds as plane projection These elements carry cells with a biped as plane projection These elements were not listed in our set of basic elements see Section 2 1 1 as in Il Khanid mugarnas only bipeds appear in the plan as plane projection of intermediate elements and almonds as the plane projection of cells a WRL m
68. during the runtime of the program plantograph We want to emphasize that the muqarnas plans are quite small and in practice computing time is not main point We favor a clearly implementation over finding the most efficient algorithm We restrict ourselves by presenting raw estimations showing that the algorithms are of polynomial time The computing time is formulated according to the size of a muqarnas plan The size of a muqarnas plan is given as the number of nodes N in the plan We announce the computing time by using Landau s symbol In a muqarnas plan a node can only have a few neighbors independently of the total amount of nodes in the plan Looking at the smallest angle in a muqarnas plan consisting of basic elements only we find that this is the angle of a rhombus and equals to 7 4 A node in such plan can thus have maximal eight neighbors independently of the size of the plan For more general muqarnas where a larger variation of angles is allowed the number of neighbors of a certain node is also restricted and does not depend on the amount of nodes in the muqarnas plan From this we can conclude that the number of edges in a plan depends linearly on the number of nodes in the plan For a muqarnas consisting of basic elements for example we can state that 8 N E lt a 4 3 as each node as at most eight neighbors Because of this relation we can always give the computing time in terms of N By reading a muqarnas plan P N
69. e The size of the diameter of the element equals d In case a 0 we are dealing with a cell and if a gt 0 the figure corresponds to an intermediate element The fourth node of the element can be found by drawing a line from 0 with end node k so that the distance from k to O is equal to d and a 0 a k a 0 a b If we are dealing with a half element there is no diameter The right two pictures of Figure 2 15 illustrate the definition of the position of an element We calculate c and from the plane projection The height h Zs of the element is the number of the tier to which the element belongs The tiers are counted from below so that the elements in the bottom tier have height h 1 The position c R will be the position in the plane projection given by the coordinates cx cy of the central node The orientation of the element is given by 0 27r Let c v w R as before so that the edges edge c v and edge c w are the projections of the curved sides of the muqarnas element and the element lies at the side we cross over by rotating v counter clockwise over c to w We define eo 1 0 In case of a cell the position of the element is given by c h 9 with a c c eo v so the element is rotated from standard position in counter clockwise direction over and translated over c Similar the position of an intermediate element is given by c h o with o a c c eg v 2 2 Muqarnas a
70. e length of the side of a square For the visualization we use a texture which is inspired by the appearance of a muqarnas vault in the dome of the Nur al Din Mausoleum in Damascus Syria We have chosen this texture to emphasize the curves on the elements In this way it is easier to recognize the different elements separately 5 2 1 Natanz Friday Mosque Location Natanz Iran Building Friday Mosque Vault Basement of the north iwan Height 5 tiers Years of construction 1304 1309 References Wilber 1955 pp 133 134 Harb 1978 pp 54 55 Takahashi 2004 120 Discussion The Friday Mosque in Natanz is in a pretty good state In the north iwan two muqarnas placed above each other can be found We study the lowest one and refer to it with the mugarnas in the basement vault The plane projection of this muqarnas vault is given in the file pl natanz fig see Figure 5 9 a By running the program removelines without the optional command line option h we remove the edges of length unequal to 1 not corresponding to element sides at the front of the vault The resulting subplan rmlines pl natanz figis given as output see Figure 5 9 b The possible muqarnas graph corresponding to this subplan is calculated by applying plantograph Qrmlines pl natanz fig the output is shown in Figure 5 9 c Almost all directions of the edges are set by the program plantograph There is only one direction undetermined dotted edge in F
71. e that they have been found in three different sizes namely with units of measurement 21 26 and 42 cm Only one element of 42 cm twice 21 has been found and its height is undeter mined as it is an intermediate element without facet The height of the cells with a unit of measurement of 21 is 42 and the height of the cells with a unit of measurement of 26 is 52 Hence the height of the cells is twice their unit of measurement The elements with a unit of measurement of 21 might originate from a muqarnas in the south octagon 1 1 3 Mugarnas Designs A mugarnas is usually designed by drawing the plane projection of the muqarnas ele ments The first known example of a muqarnas design is a stucco plate found at Takht i Sulaym n with a muqarnas design incised in it see Harb 1978 Another famous example of a collection of designs is the Topkap Scroll see Necipoglu 1995 Until Necipoglu s discovery of the Topkap Scroll the earliest known examples of such architec tural drawings were a collection of fragmentary post Timurid design scrolls on sixteenth century Samarkand paper retained at the Uzbek Academy of Sciences in Tashkent see Notkin 1995 These scrolls almost certainly reflect the ambitious Timurid drafting methods of the fifteenth century The Timurid and post Timurid scrolls show a deci sive switch to more complex muqarnas with an increasing variety of polygons and star polygons N d Q
72. e the plane projection of Yaghan s interpretation in the input line 5 1 If the original design is used cells with plane projection a square are used instead The other proposed reconstructions of Yaghan contains uncommon element combinations and will therefore not be considered here see Yaghan 2000 and Dold Samplonius and Harmsen 2005 Comparing the different reconstructions corresponding to the design we remark that the reconstruction of Harb contains 18 tiers the reconstruction with regular center con tains 17 tiers and Yaghan as our reconstruction contains 12 tiers The main difference between the interpretation given by Harb see Figure 5 18 c and our interpretation see Figure 5 20 c is the global geometric form of the muqarnas In Harb s interpreta tion the mugarnas starts in the four corners The first tier consists only of two elements in these corners The number of elements in subsequent tiers increases which means that the length of the subsequent tiers grows until the elements span the whole circum ference This happens for the first time at the eighth tier In our interpretation and also in that of Yaghan the elements span the whole circumference in each tier ANSP NYSE COE OO 4 A a pl tis plate fig b gr tis plate fig Figure 5 17 First steps in the analysis of the design found at the Takht i Sulayman 5 2 Computer Reconstructions of Il Khanid Muqarnas 109 SA SEN IN NX
73. e them by ordered triples u v w The corresponding polygon is then given by a sequence vo Un starting with u v w so vo u v v v9 w For being an island this sequence needs to define a counter clockwise oriented cycle in the plan such that the polygon given by this cycle does not contain other nodes of the plan P Because it defines a cycle v vo and being counter clockwise oriented means that by rotating Uj41 over vj to v _ for i 1 n 1 in counter clockwise direction we cross over the polygon given by vo Un Definition 2 2 7 island An ordered set u v w defines an island in P N E if there exist a counter clockwise oriented sequence vo v N with vo u v v v9 w and Un vg u such that tom fans Yo 1 0 lt t lt LE AN 2 1 i 0 We need the condition 2 1 so that polygons with nodes in the interior are excluded from being an island We use in general a quarter muqarnas plan for our input With a quarter plan we mean the part of which four copies are needed so that a dome can be filled with the muqarnas see Figure 2 18 We get these copies by mirroring the quarter plan twice If we mirror it once we get a plane projection of a muqarnas corresponding to a niche In that case a quarter plan is the projection of half the muqarnas We always choose the orientation of a quarter muqarnas plan such that the left lower corner of the plane projection corresponds to the right cor
74. e whether if belongs to the center or a mirrorline From the remaining nodes we calculate the maximal subplan defined by these nodes This is also done by iterating over the nodes We can verify that the computing time can be given by O NI In the recursive process which we use for setting the orbits for each edge we find parallel edges from the graph with the edges already used removed That means that the ith edge which we search is taken from a set of E i edges The number of operations is then not larger than E o YCIE i 0 le NP Setting the direction of orbits according to Rule 1 is done by iteration over the nodes on the boundary For each node v among the bottom boundary we look for the orbit to which the edges joining in the node v belongs The corresponding orbit is found by examining for each orbit its edges In the worst case we iterate over all edges until we find the corresponding orbit The number of boundary nodes is smaller than N and the computing time of this rule can be estimated by N E O N For Rule 2 we can use similar arguments to conclude that the computing time can be estimated by O ND During the execution of Rule 3 we iterate over the undetermined edges and validate whether these belong to an island Setting the directions of the edges of an island is an operation which does not depend in the size of the plan as the size of an island is bounded For the edges in the i
75. eans of angles and circle arches This makes it possible to give a quite exact calculation of the surface area In this thesis we work with the translation Measuring the Muqarnas by al Kash of Yvonne Dold Samplonius see Dold Samplonius 1992 Ulrich Harb was one of the members of the German excavation team who discovered the plate found at the Takht i Sulaym n The aim of his work is to understand this mugarnas design In his research he gives an exact description of the muqarnas elements and he studies several still existing Il Khanid muqarnas vaults He concludes with an interpretation of the design on the plate A quarter of his interpretation is presented as a pen drawing More recent research about muqarnas is done by Mohammad A J Yaghan In his thesis Yaghan gives a review of the terms used in the literature to describe muqarnas see Yaghan 2001a The origin of muqarnas is also discussed and a computer algorithm to generate mugarnas is introduced This muqarnas generation system is restricted to create one type of muqarnas only The generated muqarnas are constructed in a radial way and every tier consists of one type of repeated elements The system is not able to generate muqarnas which consists of varying element combinations like common in Seljuk and Il 1 2 Purpose and Contribution of our Research 9 Khanid architecture Besides this work Yaghan studies in Yaghan 2001b the muqarnas elements in order to design new elem
76. ed The height of the facets often decreases in the upper tiers as the curvature of the vault diminishes The part above the last tier can then be finished in several ways In some vaults the original brick work is left visible in others the ceiling is plastered and ornamented by painting or the upper part is filled with barley kernels see Section 2 1 1 Figure 1 2 Muqarnas elements as found at the Takht i Sulayman On the left cells On the right combinations of intermediate elements Examples of muqarnas fitting well in the definitions of al Kashi are the muqarnas re mains found at the Takht i Sulayman The Takht i Sulaym n is an Il Khanid seasonal palace in the Azerbaijan region of Iran two hundred miles south of Tabriz and south east of Lake Urmia This palace is built over a Sasanian fire temple that seems to have been in use from the late fifth to the early seventh century and continued to serve as a Zoroastrian sanctuary for two more centuries after Persia had been conquered by Islam 1 1 Mugarnas in the Islamic Architecture 5 Buried under the ruins of the palace several muqarnas remains were found con structed from probably prefabricated muqarnas elements see Figure 1 2 These could have been used to construct several different vaults within the palace but where such vaults might have been located can no longer be determined Concerning the excavated muqarnas elements described in detail by Harb see Harb 1978 we not
77. een the end node of the x mirrorline with maximal x value and the end node of the y mirrorline with maximal y value Definition 2 2 10 bottom boundary The bottom boundary Wy P of the plan P is given by the path in W P not containing center nodes between the end node of the x mirrorline with minimal x value and the end node of the y mirrorline with minimal y value 26 Chapter 2 Structure of the Muqarnas y mirrorline g mirrorline Figure 2 21 The bottom boundary nodes are given by the blue nodes in this plan They are found by removing the red nodes and then the path between v and w is calculated over the remaining boundary nodes In Figure 2 21 we see the motivation to define the bottom boundary in this way It is defined by the path between the node v on the the z mirrorline with smallest x value and the node w on the y mirrorline with smallest y value By excluding the nodes marked red in Figure 2 21 on the center and the mirrorlines we find the path at the right side We will see that often there are several reconstructions possible from a given plan These reconstructions differ in their global form The shape can often be fixed by restric tions on the boundary We say that a given muqarnas has a regular bottom boundary if and only if all nodes in the bottom boundary of the plan correspond to elements in the first two tiers of the muqarnas Similarly we define the center of a muqarnas as being regular if and only if a
78. einander berf hrt werden k nnen Muqarnas sind global unterschiedlich wenn mehrere Elemente miteinbezogen sind und dadurch die geometrische Form und die An zahl der Stockwerke variiert Die Rekonstruktionsm glichkeiten k nnen eingeschr nkt werden wenn die Abmessungen des Gew lbes in das ein Mugarnas eingebaut werden soll bekannt sind Die Anzahl der Stockwerke ist dadurch beschr nkt Wir konzentrieren uns auf Mugarnasgew lbe die in Kuppeln oder Nischen eingebaut sind und der seldschukischen oder ilkhanidischen Zeitperiode entstammen Der Algorith mus wurde an verschiedenen Mugarnasgrundrissen aus Anatolien und dem Iran getestet Zudem wird eine neue m gliche Intepretation f r den ltesten bekannten Muqarnasent wurf eine in Takht i Sulaym n gefundene Platte pr sentiert ii iii Abstract Mugqarnas or stalactite vaults are three dimensional ornaments common in the Islamic architecture They are used in vaults domes niches arches and as an almost flat deco rative frieze It is the function of a muqarnas to guarantee a smooth transition between straight walls and more curved parts A muqarnas vault is built from different niche like elements arranged in horizontal tiers One of the main characteristics of muqarnas is its form as a three dimensional unit that can be represented as a two dimensional outline In this study we focus on the question whether this two dimensional projection contains all structure information
79. ence a rhombus in the plan may correspond to two elements a cell with plane projection an almond standing on an intermediate element with plane projection a small biped For the squares in the plan a similar situation is valid A square may correspond to a cell with plane projection a jug standing on an intermediate element with plane projection a biped The program graphtomug should to be able to decide which rhombi need to be split It is not able to recognize all squares which need to be split but the combinations of squares where splitting is necessary are recognized As it is known that the original vault consists of 6 tiers we apply removelines h6 Ppl bistam fig This method removes the edges with length unequal to one and suggests to remove the node indexed by 612 which has coordinates 2 1 2 2 1 V2 In Figure 5 11 a the removed edges are drawn with dotted lines The directions of the edges is calculated by applying plantograph Qrmlines pl bistam fig and calculating the computer reconstruction of the vault can be done by running graphtomug Ggr_bistam fig We see that the rhombi incident to the bottom boundary of the plan are indeed in the computer reconstruction of the muqarnas interpreted as almonds and bipeds covering the first two tiers The squares are not split in the computer reconstruction In the real muqarnas they are split as we see in the plane projection of the existing vault as given in 5 2 Computer Reconstruc
80. ents for creating new kind of muqarnas Studies of Yaghan concerning the decoding of muqarnas designs contain a survey about corbelled mugarnas see Yaghan 2003 and also a reconstruction proposal for the plate found at Takht i Sulayman see Yaghan 2000 The thesis of Sayed M A Ibrahim documents the architectural features of muqarnas see Ibrahim 2002 This documentation serves to show how to implement repair work at historical buildings as well as to show how to integrate this traditional element into modern architecture It is a complete overview which is very useful to understand muqarnas Both Yaghan and Ibrahim give an overview about notices of muqarnas found in the literature Methods for a structural reading of muqarnas designs are not found Decoding designs is based on heuristic methods using experience and intuition leads to the proposal of interpretations 1 2 3 Outline of this Work In this work new methods are developed for doing research on muqarnas We bring dif ferent disciplines namely art history mathematics and computer science together for a better understanding of the muqarnas structure For the first time a more mathematical context is used to define the muqarnas vaults and the muqarnas designs This makes it possible to develop an algorithm that is able to analyze the muqarnas designs The result is a reconstruction algorithm which is given as a software package A faster under standing of a muqarnas design and
81. ents on the same tier join together at their curved sides The curved sides can join directly together like we see in the case of the elements A and P or they can join only at the backside of the curved side so that there remains an angle between the curved sides of two adjacent elements elements C and D Cells can only join other elements in the same tier at their curved sides Intermediate elements most often join other elements on the same tier at their curved sides but also their front parts can join front parts of intermediate elements at the same tier see Figure 2 12 In that case the curved sides still join elements on the same tier and the front sides touch each other The two front parts of a full intermediate Figure 2 12 Examples of structures where front sides of intermediate elements touch other intermediate elements in the same tier elements touch two different elements The situation as shown in Figure 2 12 where two intermediate elements lie opposite of each other and touch in their front only appears for half intermediate elements and not for full intermediate elements 18 Chapter 2 Structure of the Muqarnas The next tier will then be arranged on the previous one in such a way that in the projection no gaps will be left and the projection of the elements do not overlap In Figure 2 13 we demonstrate how cells are built upon elements a tier below We find that the apex of a curve always ends in the bottom of a curved
82. ep the muqarnas is reconstructed by reading the structure of this subgraph see Section 3 3 10 Chapter 1 Introduction The third part Chapter 4 focuses on the conversion of the reconstruction algorithm into a software tool This tool is able to do suggestions for reconstructions of muqarnas designs The program flows of the different parts the software consists of are explained Additionally a detailed description of the main data structures and methods used in the software is given In Chapter 5 we show the results of applying the software tool for reconstructing dif ferent muqarnas vaults from the Seljuk and Il Khanid periods The last chapter Chapter 6 contains a review in form of conclusions and proposals for further study For studying the muqarnas structures we use terms from graph theory The exact definitions can be found in the Appendix A Chapter 2 Structure of the Muqarnas In this chapter we formalize the structure of muqarnas In the first part see Section 2 1 we study muqarnas as they appear in the Islamic architecture and formulate the properties we need to consider in our reconstructions These properties are used in the second part see Section 2 2 to define a muqarnas in a more mathematical way 2 1 Mugarnas in the Architecture In order to understand the structure of a muqarnas we first introduce their building blocks or muqarnas elements as we find them in the Seljuk and Il Khanid architecture see Section
83. es of locally different muqarnas Here only a few elements are exchanged without changing the shape of the muqarnas We have discussed the situation where a polygon in the plan is interpreted as the plane projection of a cell or of a combination of a cell together with an intermediate element If the polygon was interpreted as the plane projection of a cell only the back sides of this cell stand on the curved sides of an element in the previous tier We can remove the cell and fill the emerged gap with a cell and an intermediate element The intermediate element appears in the previous tier The adjacent elements of the removed cell are not 56 Chapter3 Algorithm for Reconstructing Muqarnas changed If we reconstruct the muqarnas from a simplified plan we need to decide about splitting elements during the reconstruction from the subgraph to the muqarnas vault The muqarnas graphs of these mugarnas vaults are the same Figure 3 24 The cells in the second tier of the left reconstruction are pushed down in form of intermediate elements in the right picture In a similar way we can construct locally different muqarnas if it is possible to push down a complete tier If a tier only consists of cells with all backsides of the cells standing on curved sides of muqarnas elements we can exchange all these cells by intermediate elements and push them to the previous tier All the upper tiers then also need to be pushed down one tier In Figure 3 24 an i
84. es the type of the element Intermediate elements are required to fill in the gaps in the view from underneath Therefore two arrows starting from the same node only represent an intermediate element if there will appear a gap otherwise We use the word face to refer to a set of nodes which corresponds to an element More formal we define a face as follows Definition 3 3 1 face A face F c v w in a muqarnas graph G M C is a set of three nodes c v w M such that the edges edge v c edge w c correspond to the curved sides of the same element in the muqarnas structure A face defines a figure see Definition 2 2 6 in the plan P by ordering the nodes c v w adequate The node c is the central node of the corresponding element since this is the place where the edges edge v c and edge w c join In the case that the face F c v w corresponds to a cell we call it a cell face A cell face is represented in the mugarnas graph by arrows arw v c and arw w c see Figure 3 13 on the left Similar if it corresponds to an intermediate element we call it an int face In the muqarnas graph the int face F c v w is represented by the arrows arw c v and arw c w see Figure 3 13 on the right The algorithm for finding the muqarnas structure from a muqarnas graph is split in five steps where the first three steps concerns about setting the faces and the last two convert the faces into the elements defining the muqarnas structure 1
85. ew fig Removing another set of nodes from the suggested ones results in other computer re constructions These reconstructions contain element combinations we have not seen in other mugarnas from the Il Khanid period Yaghan proposed different computer reconstructions We discuss here his interpre tation with plane projection as given in Figure 5 21 a and computer reconstruction as given in Figure 5 21 c The corresponding muqarnas graph can be found by removing the nodes on the diagonal as marked in Figure 5 21 a The graph is then calculated by plantograph Cc Qrmlines pl tis plate yagh fig And the corresponding computer reconstruction can be found by graphtomuq vl Ppi tis plate yagh fig Ggr tis plate yagh fig 5 1 The result is shown in Figure 5 21 c The diagonal nodes which need to be removed for creating Yaghan s reconstruction proposal are not suggested by the program removelines The explanation for this can be found in the fact that Yaghan interprets the hexagons on the diagonal as cell with projection a square standing on two intermediate elements with plane projection rhombi In the muqarnas plan this is not drawn the hexagons are given by a jug a square and two half rhombi Yaghan has chosen to interpret most squares as an intermediate element with plane projection a large biped carrying a cell with plane projection a jug Therefore we need to 108 Chapter 5 Results of Algorithmic Muqarnas Reconstructions giv
86. f an element in the overhead tier therefore not appear in the graph G Both objects a muqarnas plan and a muqarnas graph are graphs if we consider them as mathematical objects A muqarnas plan is an undirected graph and a mugarnas graph is a directed graph To avoid confusion we will use the term graph only for the directed graphs and use the term plan to refer to an undirected graph P d 4 SLs Figure 3 3 Muqarnas plan and graph of the muqarnas vault in the basement vault of the north iwan in the Friday Mosque in Natanz see Figure 1 1 c 3 1 2 Properties of the Muqarnas Graph The muqarnas graph is defined in terms of the three dimensional muqarnas structure Our aim is to reconstruct the muqarnas directly from its design For this reason we need to recognize the muqarnas graph in the plan without using other information of the mugarnas vault as that contained in the plan In this section we formulate some properties of the muqarnas graph so that we can clear what kind of object we want to create From the property of a muqarnas structure that the elements are arranged in tiers it follows that the length of the paths in a muqarnas graph with same end nodes is the same To formalize this statement we first introduce the terminology of height for nodes in a muqarnas graph G M C 34 Chapter3 Algorithm for Reconstructing Muqarnas Definition 3 1 2 Let G M C be a muqarnas graph We define the height h v Z gt o of a node
87. f element combinations also needs an adaptation of definitions If the definitions cover more kind of element combinations reconstruction can be more difficult With the program graphtomug it is possible to create a muqarnas reconstruction directly from the graph We have seen that there may be different muqarnas reconstruc tions possible with same plane projection We distinguish locally and globally different muqarnas Concerning globally different muqarnas we have seen that the direction of orbits not determined by the program plantograph may result in different interpreta tions with different shapes see Section 5 2 3 Another operation during the analysis causing global different interpretations of a muqarnas plan is removing nodes on the diagonal of a plan We have seen that by removing a node from the plan an island is constructed In the three dimensional re construction this island expands to a set of intermediate elements By fixing locally the elements we use in the reconstruction the whole muqarnas can be influenced It is the task of the program removelines to select the edges of the plan which appear in the muqarnas graph and to find which nodes can be removed to create an island If we know into which vault a muqarnas needs to fit we know the height and therefore the num ber of tiers the muqarnas consists of The number of possible muqarnas reconstructions corresponding to a plane projection can then be restricted see Section 5 2 1
88. first consider examples of Seljuk muqarnas see Section 5 1 In these examples we use the data structure for the plans and graphs with the nodes stored as floating point numbers see Section 4 1 2 In the second part we consider examples of Il Khanid muqarnas see Section 5 2 Here we represent the nodes of the plans and graphs with integers see Section 4 1 2 For each mugarnas we first present a compact overview of properties of the muqar nas vault This overview includes information about the location and building where the mugarnas vault can be found The number of tiers the muqarnas consists of is also given Considering this number during the reconstruction process may result in a restriction on the possible muqarnas reconstructions Furthermore the overview contains information about the time period in which the muqarnas vault probably was constructed These years of construction are taken from the references presented in the overview In the reference list for most examples a reference to the website of Professor Shiro Takahashi see Takahashi 2004 is given This refers to a big data base of muqarnas plans avail able on the web The muqarnas plans contained in this data base are represented by a number denoted with n We adopt this notation in our reference list After this overview we discuss the results of applying the programs plantograph and graphtomuq on the muqarnas designs in order to reconstruct muqarnas vaults In the exa
89. g that the integer representation of the node is not known A node v will get status 1 if its integer representation is known but not of all its neighboring nodes the integer representation is known If for a node both its integer representation is known as well as the integer representation of its neighboring nodes then it gets status 2 We start the converting process by giving the origin status 1 For each node v with status 1 we look for w with status 0 such that v w lt 1 1 v2 and determine its integer representation The node w gets status 1 and if for all w with v w lt 14 1 2 the integer representation is determined the node v gets status 2 If the set of nodes with status 1 is empty but there are still nodes v with status 0 we have to look for the nearest node w of v with status 2 and estimate its integer representation by choosing a dy from the set Zr j w 1 jv w 41 Fhe converting process will stop if there are no nodes of status 0 anymore After converting all floating point numbers to integers the edge list is used to link each v N to its neighbors w by linking w to v The object Plan contains a method insert i v w which task it is to add w to the neighbor list of v If we work with a data structure based on floating point numbers we can directly add the edges which we find in the xfig file to the plan P The object Plan contains a method insert v w for nodes v w
90. g the z coordinates by 1 1 For the same reason we divide the y coordinates by 2 1 that is the length of KL As a consequence of this rescaling the proportions of the curve change a little The curve does not exactly coincides with the curve of al Kashi but still serves as a good approximation We list all coordinates of this curved side into a matrix curve For a cell given by a a 0 b b b R with diameter d R see Definition 2 2 3 and Figure 2 15 we calculate the coordinates of the right curved side by trans lating over 1 0 0 so that the projection of the central node lies in the origin To control the length of the curved side we rescale by multiplying the x coordinates with a The coordinates on the left curved side are given by translating the matrix curve over 1 0 0 and we rescale in the x direction by multiplying the x coordinate by b Additionally we multiply the matrix by the rotation matrix Ra 0 a with cosy 0 sing R 0 1 0 4 4 siny 0 cosy 80 Chapter 4 Software Tools for Reconstructing Muqarnas gives the coordinates of the vertices on the left curved side The matrix R defines a counter clockwise rotation by an angle y over the y axis To calculate the coordinates on the diagonal again we translate the coordinates as stored in the matrix curve over 1 0 0 then we multiply the z coordinates of the matrix by the diameter d and rotate over Ra 0 6 2 For an inter
91. h For the administration how often the edges are used we remark that an edge in the interior of the plan corresponds to the sides of two elements If this edge is represented by an arrow in the graph then it can correspond to two curved sides curve to curve or to one curved and to one backside of an element back on curve During the administration process we therefore also need to consider the situation that edges are used as backsides 3 3 Conversion of Muqarnas Graph into Muqarnas Structure 47 At the boundary the edges are only one time available as curved side of an element or as backside of a cell We start our administration by setting all edges at the boundary as used once 3 3 2 Determination of Faces from the Graph In Definition 3 3 1 a face is defined as the projection of parts of elements in the muqarnas structure Because our task is to reconstruct the muqarnas this information from the muqarnas structure is not available We need to recognize the faces by using the plan P N E and the muqarnas graph G M C We find the cell faces corresponding to full cells by looking for combinations of ar rows joining in their end nodes For each node c M we search predecessors v w with a common neighbor k Z c in the plan P This fourth node k verifies that c v w is part of a figure and we can set the cell face F c v w We look for the fourth vertex k in the plan P and not in the muqarnas graph G They do not necessarily exist
92. h orbit an integer r Q 1 0 1 If r Q 1 the edges in the orbit point from w to v The value r 9 0 means that the direction of the edges in Q are undetermined If the edges in the orbit point from v to w then r Q 1 Let P N E be the complete muqarnas plan and Q M C the subplan of which we need to determine the direction of the edges To set the orbits in Q according to P we work with a copy D of the set of edges C from the subplan Q M C We calculate the orbit O to which an edge e D belongs with the definition of parallel edges see Definition 3 2 1 Let O C be the orbit in C to which the edge e belongs To find the orbit 2 C is done by using a recursive process split in three steps i Insert e to the orbit O C ii Delete e from D iii If there is an opposite edge f of e then insert O D to the orbit O C Else stop and return the orbit O C We insert for the edge e its orbit 2 C to the object OrbitsValued its direction is marked by r Q C 0 We set the next orbit by choosing an arbitrarily edge f of the actual set of remaining edges D and insert Q C in the object OrbitsValued with 68 Chapter 4 Software Tools for Reconstructing Muqarnas r Q C 0 This process repeats till the set D is empty in that case all orbits are set After setting all orbits 7 Q 0 for all orbits Q After all preparation work that consisted of calculating the boundary of the plan and sett
93. he word arrow to refer to the directed edges The direction is in its visual representation given by an arrow pointing form v to w Definition A 2 directed graph A directed graph G G N C is an ordered pair of disjoint sets N C such that C is a subset of the set of ordered pairs of N The set N is the set of nodes and the set C is the set of arrows 117 118 Appendix A Graph Theoretical Terminology We say that an arrow c given by the ordered set v w points from v to w we write c arw v w The node v is called the initial point of the arrow c and the node w is called its end point Both v and w are called end nodes of the arrow c We denote c for the initial point of an arrow c and c for its end point Definition A 3 subgraph A graph H M F is a subgraph of G N E if M C N and F C E We write H C G for the subgraph H of G The subgraph of G N E induced by the set of nodes M C N is defined by the subgraph H M F such that edge v w E belongs to F if and only if both end nodes v w M To refer to parts in the graph connected to nodes or edges we use the terms adjacent or neighbor and incident In the case of a directed graph a neighbor of a node can be a predecessor or a successor depending on the direction of the arrows incident to these nodes Definition A 4 adjacent neighbor We say that two nodes v w N of a graph G N E are adjacent or neighbors if there is an edge e E with end nodes v and w
94. id period in Figure 2 5 The names of the geometric figures are adopted from al K shi We see that the plane projection of the Il Khanid muqarnas elements are figures based on a square left column or on a rhombus right column We choose our unit according to the edges of the square The length of this side will be our measure unit and we adopt from al Kashi the word module miqyas for its size see Dold Samplonius 1992 p 226 Concerning square based elements we find a square with all sides equal to the mod ule Elements with a square as plane projection can appear as a cell or as an intermediate element Furthermore we find a jug with smaller diagonal as the square In Il Khanid muqarnas a jug only appears as plane projection of a cell The longer sides of the jug have the module as their length these are the sides which correspond to the curved sides of the element The length of the smaller diagonal of a jug equals the module The large biped is what remains if we subtract the jug from the square In Il Khanid muqarnas we only find a large biped as plane projection of an intermediate element in general in connection with a jug The remaining square based figure is the half square where the square is cut in its diagonal Elements with a half square as plane projection in general appear in the first tier as a cell but may also appear as an intermediate element Based on a rhombus we first consider the rhombus itself The rhombus has fou
95. ig ure 5 9 c By copying the plan in the y mirrorline this edge belongs to an orbit of which the direction is set It is therefore obvious what the direction of this edge should be and we set it manual see Figure 5 9 d By calling to graphtomuq a computer reconstruction is given containing 7 tiers see Figure 5 9 e The muqarnas vault in Natanz contains 5 tiers and not 7 therefore we call removelines h5 Ppl natanz fig 96 Chapter 5 Results of Algorithmic Muqarnas Reconstructions a b c pl_natanz fig rmlines_pl_natanz fig gr_natanz fig E N P a Y r a in f 4 Ana i vas d e WRL main wrl crc gr natanz fig Figure 5 9 Plan analysis for reconstructing the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz without specifications concerning the height used for reconstruction This program states that reconstructing a muqarnas containing 5 tiers is possible by re moving the node indexed by 312 which has coordinates 1 1 2 1 1 v2 We remove this node by hand We examine the effect of our rules see Section 3 2 applied on this subplan by calling plantograph Qrmlines2 pl natanz fig In Figure 5 10 a the result of applying Rule 1 the arrows at the bottom boundary point 5 2 Computer Reconstructions of Il Khanid Muqarnas 97 inwards in Figure 5 10 b the result of applying Rule 2 the arrows point to the center and in Figure 5 10 c the result of applying Rule 3 di
96. in the G because edge k v and edge k w could correspond to the backside of the cell These backsides do not appear in the graph in the situation that the corresponding cell stands with its backsides on the front sides of intermediate elements No curved sides at this place are involved then To mark which arrows of the graph G are used we first select the real backsides of the cell For setting the cell face we were satisfied by knowing that v and w have a common neighbor k but this common neighbor does not need to be unique see Figure 3 15 For the backsides we choose for the common neighbor k of v and w nearest to c k argmin k c k N k Z c edge k v edge k w E If the arrows arw k v arw k w exist in the muqarnas graph G they need to be marked as used In addition the arrows arw v c and arw w c which correspond to the curved sides of the cell need of course also be marked as used C Ne ko w Figure 3 15 Backsides of a cell are given by the edges connecting the initial points of joining arrows To set the int faces we look at the subgraph G M C of the muqarnas graph G M C consisting of the arrows still available for defining a face That is we remove all arrows from C which are used twice to define C The set M consists of the nodes incident to the remaining arrows C For each node c C with two or more successors vo Un we 48 Chapter3 Algorithm for Reconstructing Muqarnas
97. ing all orbits we apply our rules on the object OrbitsValued to set the directions of the orbits For applying Rule 1 arrows point from non singular bottom boundary nodes inward we use the bottom boundary information from the object geoP1an First the vector of bottom boundary nodes need to be filtered as we can only apply Rule 1 on the non singular bottom boundary nodes Non singular bottom boundary nodes are defined in terms of their neighbor nodes We therefore convert the vector of bottom boundary nodes into a path so that we have fast access to the neighbors u w of a node v The path representing the bottom boundary is stored as a plan R W Q E with Wy Q the set of bottom boundary nodes in Q and E C E such that e E if its end nodes e Wy Q Then for each v Wy Q we can calculate whether v is a singular or non singular bottom boundary node see Definition 3 2 3 by calculating the line l u w and check if u w separates v from the plan Q The set of non singular bottom boundary nodes is stored as a vector For each non singular bottom boundary node v we iterate over its neighbors in the plan Q If we find a neighbor w which is not a non singular bottom direction then v points to w in the muqarnas graph We iterate over all orbits in the object OrbitsValued to find the orbit Q containing edge v w or edge w v If Q contains edge v w we set r 9 1 otherwise if Q contains edge w v we set r Q 1 T
98. inite set of nodes N such that if l m E then m N To understand the meaning of the edges in the muqarnas plan we first remark that the edges in the projection of a cell correspond to the two curved sides and the two backsides of the cell The edges in the projection of an intermediate element correspond to the projection of the two curved sides and the two front parts of the intermediate element In case of a half element only one edge in the projection corresponds to the backside respectively the front part of the element An edge in the interior of a plan corresponds to a place where two elements of the muqarnas join We already listed all cases in which sides of elements join to sides of other elements in Section 2 1 2 Each edge in the interior of the muqarnas plan then corresponds with one of the situation as listed there curve to curve front to front back on curve or back on front The geometrical information is included in the information of the coordinates of the nodes N The structure information is presented by the connection between these nodes which is given by the set of edges In this way the plane projections of the elements are not explicitly stored in the plan but can be calculated from the coordinates of the nodes and their connections We call the plane projection of a muqarnas element a figure As each muqarnas element has three or four nodes in its plane projection the figures are the cycles in P of length three o
99. intermediate element with angle 7 4 but diagonal d too larger for a biped We set this diagonal equal to the length of the diagonal of a biped d 1 v 2cos 1 8 76 Chapter 4 Software Tools for Reconstructing Muqarnas We recall that we interpret a square as the plane projection of a biped and a jug only if squares do not fit otherwise see Figure 3 19 We recognize this situation by working during setting the faces with a plan UsedBacksides In this plan we store the arrows used as backsides of the elements instead of curved sides We then recognize the structure as given in Figure 3 19 as putting two squares would result in using arrow c two times as backsides of an element and not as a curved side 4 3 2 Visualization of the Muqarnas in VRML To visualize the geometry of the muqarnas we have chosen to use the VRML 2 0 format This data format gives a possibility to define three dimensional geometrical data Dif ferent browsers supporting VRML 2 0 are available to visualize the data We present here a short overview of the attributes of the VRML 2 0 format which we use to vi sualize the muqarnas a detailed explanation of the VRML 2 0 format can be found in Ames et al 1997 a b Figure 4 7 On the left the xyz coordinate system as defined in VRML On the right smooth surface dotted lines approximated by triangles the arrows represent the normals vectors at the vertices on the surface In Figure 4 7 a we see an illu
100. ional muqarnas vault takes place This conversion is done by iterating over the nodes in the muqarnas graph An extensive description is already given in Section 3 3 We recall that first the faces are recognized from the graph see Section 3 3 2 and then these faces are converted into a list of elements containing information about the type and the position see Section 3 3 3 When the muqarnas only consists of basic elements it is possible to work with a simplified plan see Section 3 3 4 In that case only the muqarnas graph is necessarily in the input During the conversion to the three dimensional muqarnas vault we need to decide which rhombi and squares in the plane projection may need to be interpreted as two elements A rhombus will be interpreted as an almond and a biped if otherwise we need to insert a cell with diameter the longer diagonal of the rhombus We take this into account during the conversion of the faces to elements In step 4 of the algorithm translate cell faces to cells we calculate the diameter d of the element If this diameter is too large namely larger than that of an almond whereas the angle equals to 7 4 we set this diameter equal to that of an almond d 1 cos 7 8 We insert the nodes belonging to the backsides given by this cell face as nodes of an int face From this int face a biped is created during step 5 translate int faces to intermediate elements We recognize this biped in the same way We find an
101. ions of the ele ments appearing in Il Khanid muqarnas and formulate special properties of an Il Khanid muqarnas plan 2 2 1 Elements as Abstract Objects The crucial sides of the elements for describing the structure of the muqarnas are the curved ones In the previous section we have seen that these are the surfaces where elements on the same tier are connected to each other We see in Figure 2 1 that the two curved sides of a cell join at the front of the element in the apex of the curved sides The two curved sides of an intermediate element join at the backsides of the curved sides In Figure 2 4 we see that the plane projection of the place where the curved sides join is a node and we call this node the central node of the element Definition 2 2 1 central node The central node of an element is the plane projection of the place where the curved sides join To classify the elements by means of parameters such as an angle and some typical length we use the angle between the curved sides of the element and the diameter In the case of a full element the diameter is defined as being the distance between the central node and its opposite node In the case of a half element where the projection only consists of three vertices there is no diameter Before formalizing these definitions we introduce the notation a c v w for the size of the angle in radians over which we can rotate v in counter clockwise direction over c so that vector v
102. is plane For calculating normals for the front sides of the muqarnas elements we rotate normals of the curve adequately over the y axis The normal of the vertices on the line Y A are given by 1 0 0 and the normals of the vertices on the line AZ are given by 1 14 3 0 For the vertices v on the circle arc ZH 2 2 the normals n are given by v T E TT lv TI 4 3 The Program graphtomuq 81 as they point to the circle center T The normals on the line GH are then given by 1 0 0 We list our normals in a matrix normals and will use them for the front sides of the muqarnas elements We first consider the situation of a cell in standard position given by a a 0 b bz by and diameter d Let a a 0 a b be the angle of the cell We denote o for the vertex opposite of the central node Its coordinates can be given by o d sin a 2 cos o 2 We use again the notation R to denote the rotation over the y axis see Equation 4 4 il i N GN a 0 b oO 0 a Figure 4 12 Calculation of normals on the surfaces in our element model In the left picture of Figure 4 12 a cell is drawn with its projection given in standard position The projection of the right curved side is given by edge a 0 The normals of the vertices on this surface are then given by 0 0 1 The projection of the left curved side is given by edge 0 b To decide its normals we rotate 1 0 0 over the y
103. its can be determined see Figure 5 4 a Calculating the cor responding muqarnas reconstruction results in the reconstruction as shown in Figure 5 4 b Comparing this reconstruction with the mugarnas vault as given in Figure 5 4 g we find that our virtual reconstruction does not fit to reality This can for example be seen in the middle of the first tier our virtual reconstruction has no element in the mid dle of the first tier but the real muqarnas has This is caused by the stone as shown in Figure 5 4 g which is found in the middle of the muqarnas It covers two tiers and does not correspond to a muqarnas element If we remove by hand the lines from the plan corresponding to this area see Figure 5 4 c we are able to correct the corresponding muqarnas graph The reconstruction calculated by graphtomuq r Prmlines2 ciftemed fig Gcrc gr2 ciftemed fig results in the reconstruction of the muqarnas with a gap on the place of the area not containing muqarnas elements The effect of removing these edges is that the elements on tier 2 and 3 are pushed down so that place for the stone is created Like we have seen in the reconstruction of the Alay Han see Section 5 1 1 also in this muqarnas cells with plane projection a biped standing on intermediate elements with plane projection an almond appear 88 Chapter 5 Results of Algorithmic Muqarnas Reconstructions NA 4 a b grl ciftemed fig WRL main wrl AY INA JN At
104. lan 51 L Figure 3 19 The projection of two squares needs to be interpreted as two jugs com bined with a biped if the squares join at their backsides otherwise necessarily to split a square This will be the situation as given in Figure 3 19 we are not allowed to join squares with there backsides together as drawn on the left We exchange the squares by a combination of a jug and a large biped as shown on the right 3 4 Preparation of the Plan To determine the muqarnas graph G from the plan P we first have to determine the subplan Q of P only consisting of the projection of the curved sides of the muqarnas elements The muqarnas graph is then given by a directed graph based on the subplan Q We start this section with an example to show the effect of removing edges in the plan on the corresponding reconstructed muqarnas Then we formulate these observation so that we can use them in our software tool to find which edges we need to remove for reconstructing a muqarnas with desired properties We consider the muqarnas in the basement of he north iwan of the Friday Mosque in Natanz to observe the consequence of removing edges from the plan In Figure 3 20 the complete plan corresponding to this muqarnas is drawn Aim is to find a subplan so that there exist a muqarnas reconstruction with muqarnas graph based on this subplan and with plane projection the plan of Figure 3 20 The muqarnas consists except at the front boundary of the vault on
105. ldings like mosques and medreses arose The mosques are the places where Muslims come together for their Friday prayer They became the social centers of the city The first mosques from the seventh century were open air spaces During Seljuk and Il Khanid time they mainly consist of a quadratic room covered by a dome Medreses are educational establishments nowadays also known as Koran Schools The vizier Nizam al Mulk developed a political system where jurisprudence became very important Besides of theology in the medreses law political science languages litera ture and science was taught The medreses transformed into an extensive state organi zation of institutions An example of non religious buildings which raised during Seljuk and Il Khanid times were the caravanserais see e g Erdmann 1961 A caravanseray is a roadside inn which was important as a place for the merchants and the animals to rest and recover from their journey Often the caravanserais contained shops where the merchants could dispose some of their goods Caravanserais supported the flow of commerce information and people across a network of trade routes of Asia North Africa and South Eastern Eu rope Most typically it was a building with a square or rectangular walled exterior with a single portal wide enough to permit large or heavily laden animals to enter The court yard is almost always open to the sky and along the inside walls of the enclosure we find se
106. ll nodes in the center of the muqarnas plan correspond to elements in the top tier 2 2 3 Structure of Il Khanid Muqarnas We finish this chapter by applying our definitions of the muqarnas elements and the mugarnas plan to Il Khanid muqarnas From the plane projections see Figure 2 5 of the basic muqarnas elements we can calculate their angles and diameters In Table 2 1 an overview of these angles and diameters is given by using the module as our unit of measurement Il Khanid muqarnas are designed by mainly using the basic elements This has as con sequence that the coordinates of the nodes in their plane projection can be represented by integers In Chapter 4 we will see how we can use this integer representation in our software as a data structure to represent a muqarnas plan An advantage of this integer representation is that it is possible to verify whether nodes are the same by comparing 2 2 Mugarnas as an Abstract Geometrical Structure 27 name angle diameter square large biped half square long rhombus short rhombus almond small biped half rhombus Table 2 1 Overview of the measures of the basic muqarnas elements integers instead of floating point numbers Because floating point numbers are rounded to integers we can work with drawings which are not very exact the nodes are rounded to the nodes fitting in a muqarnas plan We can verify the following properties for the basic elements by checking these s
107. llustration of this situation is given The second tier of the left reconstruction only consists of cells with plane projection a rhombus In the right reconstruction these rhombi are interpreted as intermediate elements appearing in the first tier The right reconstruction has then one tier less as the left reconstruction If we look at the corresponding muqarnas graph of the right situation we see that the we need to remove the dotted edges from the plan before calculating the graph The di rection of the remaining edges does not change Our algorithm is not able to reconstruct this right muqarnas correct from its muqarnas graph as this graph is disconnected The Algorithm A 3 3 for calculating the height of the nodes only works for connected graphs We therefore only reconstruct the left situation In our muqarnas reconstruction we need to be aware that maybe tiers only consisting of cells with their backsides standing on curved sides need to be removed and intermediate elements in the previous tier inserted We call this a local difference although the amount of tiers changes This is because we only exchange the elements of one tier in such a way that the new elements still fit in the muqarnas Adaption of adjacent elements is not required Finally we want to mention the situation of half cells at the bottom of the muqarnas If place for such half cells is available we can insert them but also leave them out The designer needs to declare whether
108. lowing lemma affirms that the height is calculated well by Algorithm A 3 1 Lemma 3 1 5 If G M C is a connected muqarnas graph then after running the Algorithm A 3 3 for each node v M the calculated height h v equals the height h v of the node v as defined in Definition 3 1 2 PROOF We write h v for the calculated height h v after the iteration step k in Algo rithm A 3 3 The Algorithms A 3 1 and A 3 2 cause only a change of the actual value h v for a node v if this results in an increase We conclude that hya v gt hy v 3 3 We define the sequence v vg given by the history of h v With this we mean that v is the node which determines the value of h v in the iteration step i 1 The following situations can appear 1 igi vidi hi vi then v v there was no change in the iteration step i 1 2 hipi Vi h v 1 there is an arrow arw v v the Algorithm A 3 1 causes a change in the iteration step 7 1 3 Pasa ind h v 1 there is an arrow arw v 1 vi the Algorithm A 3 2 causes a change in the iteration step i 1 These situations compared with Lemma 3 1 3 show that in all three cases hisa visr hi vi h vii h v 3 1 Representation of the Muqarnas Structure in a Graph 37 From this it follows that gt m ev 1 hx vr ho vo hiss vi41 ha vi h vi 1 h vi h vx h vo i ll o ll o i As ho vo
109. ly found by trial and error Here we present our more methodical way for understanding the plate and creating computer reconstructions We first want to emphasize that it is unknown whether the design was correct On the plate different lines emerge in the center of the design It could be that the designer was uncertain about the design for the upper most tiers and tried different possibilities To take in account these problems during the reconstruction process we use the alternative Rule 2 by applying the command line option c for calculating the muqarnas graphs In Figure 5 17 a the muqarnas plan based on the design as recognized by Harb is given The only difference between the plan and the design is that we changed the inner lines of the hexagons on the diagonal see Dold Samplonius and Harmsen 2005 so that the design only consists of projections of basic elements Comparing the diagonal of the plan in Figure 1 3 to the diagonal of the plan in Figure 5 17 a shows the distinction It is only a local difference and does not influence the global reconstruction We could replace the corresponding elements in the reconstruction by elements according to the design on the plate without changing other elements in the muqarnas reconstruction By applying plantograph Cc Qpl tis plate fig almost all directions can be determined see Figure 5 17 b The resulting graph looks quite similar to the muqarnas graph corresponding to the proposal for reco
110. ly of basic elements In Section 2 1 1 we have seen that the curved sides of the basic elements always have length one Thus all edges corresponding to curved sides of elements which are not located at the front of the vault have sizes equal to the module We therefore remove the edges of sizes not equal to the module which not correspond to sides of elements at the front of the muqarnas This are the edges which are marked in Figure 3 20 by dotted lines We apply the Rules 1 4 as formulated in Section 3 2 to calculate a possible muqarnas graph In Figure 3 21 this graph and corresponding computer reconstruction are visual ized Comparing to the real muqarnas corresponding to this plan see Figure 3 22 we find the following differences i The virtual computer reconstructed muqarnas consists of 7 tiers The real muqarnas consists of 5 tiers 52 Chapter 3 Algorithm for Reconstructing Muqarnas Figure 3 20 Plane projection of the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz the dotted edges have length unequal to one ii The virtual computer reconstructed muqarnas has a non regular bottom boundary some nodes in the bottom boundary of the plan correspond to vertices of elements of tier 3 and 4 The real muqarnas has a regular bottom boundary all nodes of the bottom boundary correspond to vertices on the first two tiers We compare the muqarnas graph of our virtual reconstructed muqarnas with the mugar
111. mber 2006 Wilber 1955 Wilber D N 1955 The Architecture of Islamic Iran The Il Khanid Period Princeton University Press Princeton New Jersey Yaghan 2000 Yaghan M A J 2000 Decoding the two dimensional Pattern found at Takht i Sulayman into three dimensional Muqarnas Forms Iran XXXVIII 77 95 Yaghan 2001a Yaghan M A J 2001a The Islamic Architectural Element Mugar nas Phoibos Verlag Vienna Yaghan 2001b Yaghan M A J 2001b The Mugarnas Pre designed Erecting Units Analysis Definition of the Generic Set of Units and a System of Unit Creation as a New Evolutionary Step Architectural Science Review 44 3 297 309 Yaghan 2003 Yaghan M A J 2003 Gadrooned Dome s Muqarnas Corbel Analysis and Decoding Historical Drawings Architectural Science Review 46 1 69 88
112. mediate element given by a a 0 b bz by R and with diameter d R we calculate the coordinates similar In this case the projection of K corresponds to the central node and a translation is not needed The vertices on the right curved side are then calculated by multiplying the z coordinates of the matrix curve by a The vertices on the left curved side are calculated by multiplying the x coordinates of the matrix curve by b and rotating over the y axis by a 0 a b The coordinates of the vertices on the diagonal are calculated by multiplying the x coordinates by the diameter d and by multiplying the resulting matrix by Ra 0 a b 2 for rotating purpose Calculation of the Normals To give the viewer of our computer reconstructions the impression that the muqarnas elements are built with smooth surfaces instead of joined triangles we explicitly write normals in the Geometric Nodes of the VRML 2 0 files For each triangle we deter mine the normal vectors for its vertices where not the normals of the triangle is given but the normals of the surface we are approximating Y D JT G Figure 4 11 Calculation of the normals on the front surfaces of the elements In the calculation of the normals of the elements we will use the normals on the curve as drawn in the most left picture of Figure 4 11 We consider the curved side as two dimensional surface in the zy plane and calculate its normals as vectors in th
113. mples we always use a quarter muqarnas plan for the input The plans are oriented as presented in Section 2 2 2 The origin is chosen so that the smallest z value and y value of the nodes appearing in the plan have coordinate 0 Most graphs are drawn with blue and red colors If not stated otherwise these colors are alternating for easier recognizing the tier information from the graphs 83 84 Chapter 5 Results of Algorithmic Muqarnas Reconstructions 5 1 Computer Reconstructions of Seljuk Muqarnas The specialties of Seljuk muqarnas are the shell shaped ornaments which we represent by textures We use for our reconstructions of Seljuk muqarnas curved elements without facets which means that we remove the lower parts of the elements The combination of using elements without facets and applying textures gives our reconstructions a similar appearance as the still existing Seljuk muqarnas We visited all sites of the examples of Seljuk muqarnas presented here 5 1 1 Alay Han Location On the road between Aksaray and Nevsehir Turkey Building Alay Han Vault Entrance portal to enclosure Height 7 tiers Years of construction 1180 1200 Date of visit April 9th 2005 References Erdmann 1961 pp 81 83 Aslanapa 1971 pp 147 148 Takahashi 2004 043 Discussion The road from Aksaray to Nevsehir crosses the area where once the courtyard of the Alay Han the oldest known caravanseray in Anatolia has been From the courtya
114. mplete plan P is required in the input We recall from Section 3 2 that for giving the edges a direction opposite edges are bound in orbits For each orbit a direction is given To decide whether edges are opposite we 4 1 The Program plantograph 59 need the information of the subplan Q as well as the information of the complete plan P For setting the orbits the complete plan is only scanned to check whether a polygon has a cross edge connecting the edges which are candidates for being opposite see second picture in Figure 3 5 The edges of a polygon with cross edge are not considered in our definition of opposite because they could correspond to the projection of half intermedi ate elements joining at their front In that case the edges which are candidates for being opposite do not necessarily have the same direction As this situation rarely appears in most cases the muqarnas graph can still be calculated correctly without information from the complete plan P In Figure 4 1 the program flow of the program plantograph is visualized The thick arrows designate the main stream of the program Dotted arrows are used to indicate that the objects are not changed but serve as information The first step in our algorithm is to seortan tue OrbitsValued ee N i L Ps Figure 4 1 Program flow of the program plantograph read the input plan and convert it into our own data structure represented by the object Plan In this way
115. muqarnas graph does not contain cycles A path starting in a node v and ending in the same node v has always length 0 After formulating properties of the height h v we now concern on the calculation of h v The process of calculating the height is based on determining the distance of the nodes to the boundary Let M be the number of nodes in the graph G M C The notation M is used for the set of nodes v M with h v h For an arrow c C we denote c for its initial point and c for its end point We use the following algorithms to calculate the height of the nodes v M A 3 1 Initial situation For all v M we set h v 0 h 0 and change false Iteration step For all c C with c My if h c lt h 1 we set h c h 1 and mark the situation as changed by setting the boolean change true If this is done for all c C we seth h 1 Stop condition If h gt hmaz or h gt M with hmar maxyey h v we return the value of change A 3 2 Initial situation We set h hmar change false Iteration step For all c C with c My if h c lt h 1 we set h c h 1 and mark the situation as being changed by setting change true If done for all c C we set h h 1 Stop condition We stop if h 0 and return the value of change A 3 3 Initial situation We set k 0 change false Iteration step Apply A 3 1 and A 3 2 and set k k 1 If A 3 1 or A 3 2 returns true we set change
116. nas graph of the real muqarnas In the graph in Figure 3 21 the direction of the lower edge of the orbit marked in this graph is given by the direction of the upper edge from this orbit as they are parallel The direction of the upper edge is fixed by Rule 3 as this edge is a member of an island By removing the edges marked red in Figure 3 20 we can split this orbit and the direction of the lower edges can be swapped so that we are able to reconstruct a muqarnas with a regular boundary see the graph in Figure 3 22 We find that removing edges makes it possible to split orbits This makes it possible to change the direction of some edges The edges which we need to remove are projections of non curved sides joining to other non curved sides In Section 2 1 2 we have seen that non curved sides can only join to other non curved sides in the following situations 1 Front sides of intermediate elements touch see Figure 2 12 2 Backsides of cells stand on front sides of intermediate elements see Figures 2 13 b and 2 13 c In our example we removed the edges corresponding to situation 2 already from the plan by removing the edges with sizes unequal to the module If the size of the backsides of the cells stand on front sides of intermediate elements equals to one it can be more difficult to recognize such edges We will see an example of such a situation in the next section We use the observations i and ii for trying to remove edges in o
117. nd intermediate elements other types of elements appear In Cast ra 1996 p 316 319 an overview of the elements common in muqarnas used in Morocco is given As example we mention the rectangular element as given in Figure 2 10 We see that this is neither a cell nor an intermediate element In this element the curved sides do not join but they lie opposite to each other As we did not find such elements in Il Khanid and Seljuk mugarnas it will not be considered in our reconstructions Figure 2 10 Rectangular element as used in muqarnas in Morocco 2 1 2 Three Dimensional Mugarnas Structure In this section we study the structure of muqarnas by discussing how the different muqar nas elements of a muqarnas can be arranged to form a muqarnas structure In al Kashr s 2 1 Mugarnas in the Architecture 17 definition a muqarnas is compared with a staircase We have to think about a structure constructed from building blocks arranged in different levels which are the stairs Al K shi calls such a level a tier tabaga and we adopt this terminology He defines a tier as follows see Dold Samplonius 1992 p 226 Definition 2 1 2 tier Adjacent cells which have their bases on one and the same surface parallel to the horizon are called one tier AB C D Figure 2 11 Part of a tier of the muqarnas niche in the basement of the north iwan in the Friday Mosque of Natanz In Figure 2 11 we see a part of a tier We observe that the elem
118. ner at the back of the muqarnas if we stand in front of a muqarnas niche see Figure 2 19 The diagonal nodes of a quarter muqarnas plan are then the nodes on the line from the left lower corner to the right upper corner of 24 Chapter 2 Structure of the Muqarnas Figure 2 18 The thick lines represent a quarter of a muqarnas plan The complete plan could be the projection of a muqarnas vault of the south octagon at the Takht i Sulayman see Harb 1978 p 46 Figure 2 19 Plan and photo of muqarnas in the basement of the north iwan from the Friday Mosque in Natanz In red the diagonal nodes the plan If the muqarnas plan corresponds to a muqarnas vault in a dome this diagonal gives a symmetry line of the plan For niches this is not the case In the next chapter we will see that for converting a muqarnas plan into a muqarnas structure it seems to be important to know how a certain node in a muqarnas plan is situated according to the boundary of the plan The definitions concerning the boundary are visualized in Figure 2 20 The boundary of a plan consists of the nodes which are not surrounded by figures or islands This means that v is in the boundary if it has neighbors u w such that u v w and w v u do not define a figure or island We denote the set of boundary nodes of a plan P by W P 2 2 Muqarnas as an Abstract Geometrical Structure 25 Definition 2 2 8 boundary The boundary W P C N of a muqarnas plan P N E
119. nes eu fun wre 71 4 2 1 Removal of Edges from the Plan 2 71 4 2 2 Complexity Analysis 4 4 4 ko omo xe wR nee 73 4 3 The Program Grapntomuc c ews Er res SS seque ms 73 4 3 1 Calculation of the Mugarnas Structure 73 4 3 2 Visualization ofthe MugarnasinVRML 76 4 3 3 Visualization ofthe Elements 77 4 3 4 Complexity Analysis 2 2o xS ER REG Rs 82 5 Results of Algorithmic Muqarnas Reconstructions 83 5 1 Computer Reconstructions of Seljuk Mugarnas 84 Bll Alay Han 54 ans Ee EROR Ra RE RE 84 b 1 27 Kayseri ifte Medrese iro ece ne CROIRE CR RC BOR c 86 5 1 3 Sultan Han near Kayseri Ya ore x XR Ex RAE SR 89 5 1 4 Ankara Arslanhane Camii a e 92 5 2 Computer Reconstructions of Il Khanid Mugamas 95 5 2 1 Natanz Friday Mosque 6 5 5405 EN URUEACAON E XS 95 5 2 2 Bist m Shrine of Bayaz d 4 u cepe cx se gus e Rw d etn 98 5 2 3 Far mad Friday Mosque wins Sed lt 2 dO fec uM WOO aa 100 5 2 4 Takht i Sulaym n South Octagon 103 5 2 5 Takhi i Sulayman Plate 42e xe e x X5 105 6 Conclusions 113 A Graph Theoretical Terminology 117 Acknowledgements 121 Chapter 1 Introduction In this work we study muqarnas a special three dimensional ornamentation common in the Islamic architecture The aim is to analyze the two dimensional designs of muqarnas and to create three dimensional
120. ng the program flow 4 1 2 Representation of the Muqarnas Plan The open source program xfig written by Supoj Sustanthavibul and improved by Brian V Smith and Paul King is a program for drawing pictures in vector format The xfig file format is well documented in the xfig manual see Sato and Smith 2002 We use this file format for the input plans and the output graphs This makes it possible to visualize the graphs In the xfig file format different drawing objects like polygons splines or circles are defined Most objects are given by two or more lines containing numbers see Figure 4 3 The first line starts with a number which is a code for the kind of object defined For example a line starting with the number 2 defines a polygon The remaining num bers of this first line decodes some properties of the object These properties can for example consist of information about the line thickness or color of the object The sec ond line consists of coordinates of nodes defining the object In a possible third line extra properties of the object are defined For example in the case of a polygon drawn with arrows these extra properties contain information about the drawing style of the arrows To represent muqarnas plans and graphs in the xfig format only the object polygon is of interest for us An open polygon in the xfig format is represented by a sequence of coordinates of the nodes connected to each other These coordinates are given in
121. nstruction given by Harb We first set the direction of the remaining edges of this graph according to Harb s reconstruction see Figure 5 18 b and run graphtomug v1 G gr tis plate harb fig Ppl tis plate fig to create a computer reconstruction of Harb s interpretation of the design The plane projection of this reconstruction is given in Figure 5 18 a where the tiers are colored alternately Figure 5 18 c shows the three dimensional reconstruction which exists of 18 tiers For creating a reconstruction with regular center the directions of the remaining edges are set by running plantograph without the command line option c They are then directed as shown in Figure 5 19 b It results in another computer reconstruction containing 17 tiers see Figure 5 19 c We prefer a muqarnas with a regular bottom boundary Without removing edges this is not possible independent of how we set the direction of the undetermined edges in Figure 5 17 b the corresponding reconstruction has a non regular bottom boundary We try to find other reconstruction possibilities by applying the program removelines A reconstruction with regular boundary contains supposably less tiers as a path from the corner of the muqarnas plan to the center needs to have the same length as a path starting at another location of the bottom boundary to the center see Section 3 4 We therefore watch what happens if we try to remove edges in order to create a muqarnas reconstru
122. nstruction is available An important initiation is made in order to study muqarnas in a more structural way To have an algorithm that is able to analyze the structure of a muqarnas leads to a better understanding of muqarnas designs The integration of the computer in our research provides a fast analysis which is an important advantage compared with older methods based on trial and error 116 Appendix A Graph Theoretical Terminology In this appendix we give an overview of the main definitions from graph theory which we use in our research The definitions are taken from Lemmens and Springer 1992 Chapter 7 and 10 and Bollob s 1979 Definition A 1 undirected graph An undirected graph G G N E is an ordered pair of disjoint sets N E such that E is a subset of the set of unordered pairs of N The set N is the set of nodes and the set E is the set of edges If the edge e is the unordered pair of nodes v and w we call v and w the end nodes of the edge e and write e edge v w Because such an edge is defined as an unordered pair we identify edge v w with edge w v Often graphs are visualized by drawings The nodes are represented in the plane as points an edge edge v w is represented as a line connecting the points v and w see Figure A 1 Figure A 1 Visual representation of an undirected graph We create a directed graph by giving each edge a direction Then the pairs v w are ordered pairs and we use t
123. ntermediate elements with blue arrows Remark that the arrows of all these figures have AN GUT XM Yd AAS X UA A Figure 3 11 Overview of different islands with their common directions in the graph the same structure at the most right node on the diagonal two arrows come together The other arrows have an alternating direction so that the last arrow of the alternating ones points to the arrow pointing to the most right diagonal node In formulating this rule we use the term sink for a node in a directed graph which is an initial point for all arrows incident to this node Similar a source is a node which is an end point for all arrows incident to it We remark that an island always consist of an odd number of nodes as each element is represented by two arrows in the graph Rule 3 Let vo Un vo v be a counter clockwise oriented cycle in the subplan Q representing an island in a muqarnas graph G Let v be the diagonal node of this cycle in G with largest x coordinate then arw v _ v arw v 41 0 G and for i Z 2 r and j 2 n r the nodes v _ v are sources in this cycle for i j even and sinks in the cycle for i j odd We formulate a fourth rule which exclude some forbidden element combinations In the left picture of Figure 3 12 we see a situation where node u is the central node for all intermediate elements containing u in its plane projection In this example the only possibility to add new element
124. o apply Rule 2 arrows with one end node incident to the center points to the center we work similar In this case for each center node v we look for its adjacent nodes w not in the center Then there is an orbit Q with edge v w Q or edge w v Q and the direction of this orbit is set so that w points to v If the command line option c is given by calling to plantograph an irregular center is allowed an we need to apply the alternative Rule 2 Before setting the direction of the orbit to which edge v w or edge w v belongs we need to check if this orbit does not contain an edge edge v w with both v w W P If such an edge exist the direction of Q cannot be set For applying Rule 3 set the direction of the islands we iterate over the orbits Q for which the direction is unknown r Q 0 All the edges of an undetermined orbit are tested on being a member of an island To test whether an edge edge v w belongs to an island is done by iterating over the neighbors k of v We check for each neighbor the sets v k w and v w k for being an island If an island is found it is stored as a sequence of nodes v v The next step is then to determine the direction of the edges edge v vj 41 0 n 1 For this we find the node v fora r 0 n with largest z coordinate by comparing the coordinates of the nodes vo un The island is stored as a graph with arrows as given by Rule 3 Finally we
125. of intermediate elements touch front to front Two methods are designed to handle these situations The first method FilterLinesLengthOne calculates for each edge in the input plan not corresponding to the front border of the muqarnas its length If this length is unequal to one the edge is removed from the plan The second method setHeight unsigned int m is only executed if the com mand line option h is given in the input It uses the number n which follows after the option h in the input line The effect of removing nodes on the diagonal to create a muqarnas graph with maximal height n is analyzed by this method For each diagonal node v the minimal height of a possible muqarnas reconstruction corresponding to the plan P N vj E is calculated The notation N v is here used to denote the set N with the node v removed The program removelines produces some text stating whether there is a reconstruction possible and what the minimal height of such a reconstruction is Special attention is taken on the nodes which results in a reconstruction of height n The possibility of reconstruction and the height of the reconstruction is approximated as follows we first calculate a graph corresponding to the plan P N_ v E by applying the Rules 1 4 as formulated in Section 3 2 If the command line option c is given the alternative Rule 2 is used In the case that the rules are in a conflict situation that means that different rules try to
126. of the three dimensional muqarnas vault Explicit definitions are given to create a framework in which we are able to describe the muqarnas structure Each muqarnas element is described by means of parameters representing its type and measurements together with parameters that describe the po sition of the element in the muqarnas vault A graph theoretical approach makes it possible to include the structure information of the three dimensional muqarnas in the two dimensional outline This is done by constructing a directed subgraph from the muqarnas design The main task is then to find all directed subgraphs corresponding to a mugarnas design for which a three dimensional muqarnas representation is pos sible We can construct three dimensional computer reconstructions directly from the directed subgraphs An algorithm is developed for constructing the directed subgraphs and reconstructing the three dimensional muqarnas Two software tools are designed to execute this process The program plantograph finds the directed subgraphs corresponding to a muqarnas structure from a muqarnas de sign The program graphtomuq converts a subgraph into a three dimensional computer reconstruction of the muqarnas The various subgraphs result in different muqarnas re constructions with the same plane projection Some interaction is necessary to select the required reconstruction for this art historical expertise is essential The differences between muqarnas with the
127. omplete muqarnas as proposed by Harb is given together with its plane projection in Figure 5 15 a This reconstruction consists of 6 tiers The corresponding muqarnas graph is a disconnected graph see Figure 5 15 c Because the graph is disconnected we are not able to reconstruct the muqarnas by applying graphtomuq Ppl tis southoctagon Ggr tis southoctagon Our tier algorithm see Section 3 1 Algorithm A 3 3 cannot be applied We need to reconstruct the center and the outer part separately see Figure 5 15 d and include the half rhombi in the fourth tier manual If we calculate a muqarnas reconstruction with the program plantograph from the plan by calculating the muqarnas graph without removing edges see Figure 5 16 b we get as corresponding muqarnas almost the same reconstruction This reconstruction contains 7 tiers The rhombus which Harb interpreted as intermediate elements appear ing in the fourth tier appears in this new reconstruction as cells in the fifth tier The other elements are not changed We see that this means only a local change and the global mugarnas is not influenced see Section 3 5 Chapter 5 Results of Algorithmic Muqarnas Reconstructions 104 b Reconstruction proposal of Harb a southoctagon fig pl tis by using Reconstruction d c Muqarnas graph of Harb s reconstruction proposal graphtomuq Figure 5 15 Reconstruction proposal of Harb for the muqarnas in the so
128. on 2 5 in Section 2 2 3 Like in Section 2 2 3 we use again the notation r to denote the largest integer smaller than or equal to r To write a given floating point number r by integers a b such that r a b 2 we need to find a b such that b e V2 r a 4 1 The Program plantograph 63 There does not necessarily exist an optimal solution such that the error which is given by V2 r a v2 r a is minimal We choose a of a finite set Zi jj k k 1 k 1 k fora certain k such that V2 r a 3 V2 r a 4 1 is minimal for a Z j This means that for the a for which Equation 4 1 is minimal the corresponding b V2 r a 1 2 The representation by integers is only possible if the muqarnas consists of basic muqarnas elements We will use that in such a mugarnas plan there is always a cer tain distance between the nodes in the plan If a node v appears in a plan and a node w appears in the plan such that v w lt 1 1 2 then they need to be connected by edges belonging to a square or rhombus based figure see Figure 4 4 We find that in that case las aul lt 1 Il lt 1 4 2 by buw lt 2 Id dull lt 2 hence a 1 0 1 b 2 1 0 1 2 This means that if we know the integer Figure 4 4 Adjacent nodes in a mugarnas plan consisting only of basic elements are connected by an edge belonging to a rhombus
129. on of Faces into Muqarnas Elements To convert the faces into elements we have to determine for each face its type and position In this section F c v w is a face in a muqarnas graph G M C so that the ordered set v c w defines a figure in the corresponding muqarnas plan P N E w so To determine the type of the element we remark that F c v w is a cell face if arw v c C and an int face if arw c v C The diameter d of the corresponding element is given by d min k c k N k Z c edge k v edge k w E If the set k c k Z c edge k v edge k w P is empty there is no diagonal and the face corresponds to a half element we set d 0 The type of the element is given by 3 3 Conversion of Muqarnas Graph into Muqarnas Structure 49 the nodes a v c b w c and the diameter d The angle of the element is given by a c v w which equals to a 0 a b The element has curved sides of sizes a and b The position of the element corresponding to the face F c v w can also be calculated from the muqarnas graph G M C We determine the height h c of the central node c in the graph G by applying Algorithm A 3 3 In the situation that the face represents a cell the central node c corresponds to the apex of the curved sides It does not touch elements of a lower tier Therefore the height of the cell is given by h c The position of a cell corresponding to an cell face F c
130. onal these paths are interrupted and the maximal length of the paths decreases which results in muqarnas reconstructions containing less tiers the bottom is nearer to the center Therefore we remove a diagonal node and consider the heights of the different possible muqarnas reconstructions We will see in Chapter 5 that by trying to remove nodes on the diagonal we are indeed often able to locate the islands in Il Khanid muqarnas Figure 3 23 Plan of portal to enclosure at Sultan Han near Kayseri Removing the right edges in a plan not only consisting of basic elements as in Seljuk mugarnas appear can be more complicated For the muqarnas only consisting of basic elements most edges we need to remove are recognized as they have size unequal to the module In Seljuk muqarnas the curved sides can have arbitrarily size and therefore different measures of edges in the plan may appear In Figure 3 23 we see an example of a plan corresponding to a Seljuk muqarnas The edges not corresponding to curved sides are red marked The black edges corresponding to curved sides do not all have the same length the polygons near to the bottom boundary have larger sizes as the polygons in the center To recognize the edges which correspond to back sides of elements standing on front sides of intermediate elements we remark that these can be recognized as they divide a polygon of four edges e g a parallelogram or a square into two triangles or into two
131. or a square based figure representation ay bu Cv dy of a node v we can calculate the integer representation of neighboring nodes w by setting r v w and calculating the minimum of Equation 4 1 with a 1 0 1 Then in the integer representation aw bw Cw dw of w the a is given by aw a a Analogously we can calculate c by setting r v wy in Equation 4 1 In general if v w we choose a a a from the set Zr 1 3 ii We set the integer representation of the nodes by comparing their coordinates with the coordinates of the nodes in their neighborhoods of which the integer representation is already known To apply this idea we first read all coordinates of the xfig file and list them as a vector of edges where edges are given by two nodes with floating point coordinates This vector is the set of floating point numbers we need to convert As we would like to have a b c d gt 1 so that we can calculate a b 2 c d 2 we first translate the plan so that all coordinates are positive We search the corner 64 Chapter 4 Software Tools for Reconstructing Muqarnas z y such that v gt z v gt y for all nodes v N in the muqarnas plan P N E By subtracting x y from all coordinates v all coordinates can be represented with positive coordinates and x y is represented with integers a b c d 0 0 0 0 The nodes get all a status 0 which has the meanin
132. osition in dem Gew lbe beschreiben zu k nnen Ein gra phentheoretischer Ansatz erm glicht es die Strukturinformationen des dreidimensiona len Muqarnas in einem zweidimensionalen Grundriss zu integrieren Dies wird durch die Konstruktion gerichteter Teilgraphen aus dem Muqarnasentwurf realisiert Die Haupt aufgabe ist es alle gerichteten Teilgraphen eines Muqarnasentwurfs zu finden f r die eine dreidimensionale Mugarnasrepr sentation m glich ist Die dreidimensionalen Com puterrekonstruktionen k nnen dann direkt von den gerichteten Teilgraphen ausgehend aufgebaut werden Wir haben einen Algorithmus eintwickelt zur Konstruktion der ge richteten Graphen und zur dreidimmensionalen Rekonstruktion des Muqarnasgewolbes Der Algorithmus wurde in zwei Computerprogrammen implementiert Das Programm plantograph findet alle gerichteten Teilgraphen die mit einer Muqarnasstruktur eines gegebenen Muqarnasgrundrisses korrespondieren Das Programm graphtomuq ber setzt einen Teilgraph in eine dreidimensionale Computerrekonstruktion Die verschiede nen Teilgraphen ergeben verschiedene Muqarnasrekonstruktionen mit gleicher Projek tion Eine Interaktion mit dem Anwender ist n tig um die gew nschte Rekonstruktion auszuw hlen wof r Fachwissen in Kunstgeschichte erforderlich ist Die Unterschiede zwischen Muqarnasgewo lben mit gleicher Projektion k nnen lokal oder global sein Lokal bedeutet dass die Muqarnas durch Austauschen einzelner Ele mente in
133. ow of the program graphtomuq 4 3 The Program graphtomuq 75 r With this option we work with a data structure where nodes in the plan are stored as floating point num bers By default we work with integers to represent the nodes in the graphs The program graphtomug expects in the input the complete muqarnas plan as well as the muqarnas graph If the muqarnas plan consists only of basic elements we may use the subplan derived from the muqarnas graph instead of the complete plan This subplan contains the same edges as the muqarnas graph but without directions In Figure 4 6 the program flow of the program graphtomug is visualized The input consists of xfig files which we convert into our own data structure like we also did in the program plantograph We use the numbers behind the command line options p and v to mirror the input plan and graph so that they correspond to the complete projection of the muqarnas vault which we want to reconstruct For this mirroring we first compare the coordinates of the nodes to determine the maximal x and y coordinate of all nodes appearing in the plan We iterate over the edges and calculate for each edge the mirror image of its end nodes after mirroring in the x mirrorline or y mirrorline respectively The edge connecting the mirror images of the end nodes is then inserted in the plan After mirroring the graph and plan are stored in the object setFaces in which the conversion to the three dimens
134. polygons of again four edges In the examples in Chapter 5 we will see that we are often able to remove most edges not corresponding to curved sides automatically by using the remarks before Some edges still need to be removed by hand but these edges are mostly easy recognized as edges not corresponding to curved sides as they have uncommon length To locate the 3 5 Uniqueness of the Reconstructions 55 islands is still a process of trial and error Our software is able to do suggestions which diagonal nodes to remove but cannot decide by its own 3 5 Uniqueness of the Reconstructions During the reconstruction process the muqarnas plan is converted into a directed sub graph and this subgraph is extended to a three dimensional muqarnas structure In both steps there may be some freedom which lead to different muqarnas reconstructions In the first part the conversion from mugarnas plan to a directed subgraph these differ ences can be caused by removing nodes in the plan see Section 3 4 or by changing the direction of undetermined edges see Section 3 2 In the second part the extension from the directed graph to a three dimensional muqarnas structure sometimes polygons can be interpreted as different element combinations see Section 3 3 4 The answer to our question whether muqarnas are uniquely reconstructible will therefore be nega tive In this section we study how muqarnas having the same plane projection can differ Examples
135. projection of a muqarnas If we are able to write such a program we can confirm our first question The final software product should show all possible reconstructions corresponding to a certain plane projection so that we can answer the second research question This software then can be useful during restoration and for designing new muqarnas It gives the possibility to test different alternatives for a muqarnas reconstruction with a given plane projection 1 2 2 Literature Overview Our work is mainly inspired by a chapter of the manuscript Key of Arithmetic writ ten by the Timurid astronomer and mathematician al Kashi see al Kashi 1558 and a research about Il Khanid muqarnas of Harb see Harb 1978 Al Kashi ranks among the greatest mathematicians and astronomers in the Islamic world Two years before he died in 1429 he had finished the Key of Arithmetic Miftah al Hisab one of his major works which he intended for everyday use By far the most extensive part of the work is Book IV On Measurements The last section of Chapter 9 Measuring Structures and Buildings is about muqarnas The aim of al Kashi is to mea sure the muqarnas He does not give us information about the construction of muqarnas and his work does not contain information about how we need to interpret muqarnas designs For measuring the muqarnas al Kashi gives an overview of the different muqar nas elements He describes the surfaces of the elements by m
136. put consists of three kind of VRML 2 0 files files representing elements files for the tiers and a main file binding the tiers to a muqarnas For each muqarnas element we create two files In one of the files the roof of the element is stored and in the other the facets are stored see Definition 2 1 1 This is done to have the freedom to rescale these parts of the elements separately see Section 1 1 2 The files have names like type d a right left part wr1 where type can be cell or int to denote whether the file contains a cell or an intermediate element d is a floating point number which represents the diameter of the element and a is a floating point number to represent the angle of the element The variables right and left contain the sizes of the right and left curved sides The variable part can be equal to u or 1 for denoting if the file contains the upper or the lower part of the element For example the roof a cell with a square as plane projection is then given in the file ce11 1 41 1 57 1 0 1 0 u wrl and its facets are defined in cell 1 41 1 57 1 0 1 0 l wrl The main file main wrl consists of Transform Nodes putting the different tiers on each other Each tier i is given by two files tieri_u wrl and tieri l wrl rep resenting the roof and the facet of the tier In the tier files the elements are included by Transform Nodes containing the location and orientation of the elements 4 3 3 Visualization of the Elements
137. qarnas but also give an algorithmic way to build them directly from their design We give in Section 1 2 1 the motivation of this study and the problem formulation Then we summarize in Section 1 2 2 the contents of previous research works concerning muqarnas The final part Section 1 2 3 contains an outline of this work 1 2 Purpose and Contribution of our Research 7 1 2 1 Motivation This research focuses on the connection between a three dimensional muqarnas struc ture and its two dimensional plane projection In the plane projection the muqarnas elements do not overlap which makes it possible to design a muqarnas by drawing its plane projection Nevertheless such two dimensional designs do not include explicitly all the three dimensional information A muqarnas consists of different niche like ele ments arranged in tiers In the drawing we can distinguish the different elements but we do not have information to which tiers the elements belong It is also not directly clear which side of a certain polygon in the projection corresponds to the front part of the element and thus the orientation of the elements cannot be recognized We have furthermore to take into account that there exist different elements with the same plane projection Figure 1 4 On the left plane projection of the muqarnas vault in the basement of the north iwan of the Friday Mosque in Natanz On the right computer reconstruction of the muqarnas vault In this
138. qarnas other elements and other combi 115 nation rules may appear which influences the mugarnas structure As we now possess software tools to analyze muqarnas it is possible to test new designed muqarnas on mistakes For renovation purposes we can also first view the muqarnas reconstruction on the computer If only a part of the muqarnas survived and therefore only a part of its plane projection is known we need to fill the plane projection with polygons such that there is a muqarnas representation possible This can be done by using studies about tilings to find the possibilities to fill the two dimensional space with polygons If the muqarnas consists only of basic elements the muqarnas plan can be considered as an octagonal tiling see e g Senechal 1995 p 213 218 Studying what kind of assumptions are needed so that a finite part of a tiling has a muqarnas repre sentation can be useful for reconstructing partly destroyed muqarnas or for constructing new muqarnas designs Our software tools can then be used to validate whether there is a three dimensional reconstruction possible We already created a data structure based on integers in which we can represent tilings given by the plane projection of muqarnas only consisting of basic elements We could imagine that a better algebraic description of the structure of the plans is possible and useful Our data structure could then be more specified to exclude plans for which no muqarnas reco
139. r a pl slthan ksi fig b MNT NAZI SATIS BREN ONIN a ey VA AS c gr slthan ksi fig d Gro gr slthan ksi fig Figure 5 5 Plan analysis for reconstructing the muqarnas vault in the entrance portal to the enclosure of the Sultan Han near Kayseri 5 1 Computer Reconstructions of Seljuk Muqarnas a Photo of front view b Photos of some details 91 c Bottom view of the computer reconstruction d Photo of bottom view Figure 5 6 Pictures and reconstruction of the entrance portal to the enclosure of the Sultan Han near Kayseri 92 Chapter 5 Results of Algorithmic Muqarnas Reconstructions 5 1 4 Ankara Arslanhane Camii Location Ankara Turkey Building Arslanhane Camii Mosque Vault Entrance portal Height 9 tiers Year of construction 1290 Date of visit April 7th 2005 References Aslanapa 1971 pp 121 122 Takahashi 2004 882 Discussion The last example we study of a muqarnas in Seljuk style is the vault in the entrance portal of the Arslanhane Mosque in Ankara In Figure 5 7 a the plan of this muqarnas is shown By removing edges not corresponding to curved sides of elements we find that the subplan mainly consists of squares and rhombi see Figure 5 7 b Comparing to the plans from muqarnas of the Il Khanid period we see that the plan is contained in the lattice L rotated over 7 8 like we also have seen in the reconstruction of the vault at the Alay Han see Section
140. r sides with length equal to the module two opposite angles of 45 and two opposite 14 Chapter 2 Structure of the Muqarnas EN Square rhombus jug a iq m 3 A large small biped II IS A half square half rhombus Figure 2 5 Overview of the basic mugarnas elements angles of 135 The rhombus appears as plane projection of a cell and of an intermediate element In Figure 2 6 three different elements with a rhombus as plane projection are visualized In the first picture we see a cell with plane projection a rhombus the curved sides join in the apex with an angle of 135 In the right two pictures of Figure 2 6 we see two different intermediate elements In the first the shorter diagonal is used for the orientation of the element and the curved sides join in an angle of 135 In the second picture the larger diagonal is used and the curved sides join in an angle of 45 In Il Khanid muqarnas we do not find cells with a rhombus as plane projection where the larger diagonal is used for the orientation The almond and the small biped represent a Figure 2 6 Three different elements with a rhombus as plane projection possibility to split the rhombus like a jug with a large biped for the square In Il Khanid muqarnas the almond appears only as plane projection of a cell This cell has two sides 2 1 Mugarnas in the Architecture 15
141. r four They will be given by ordered sets u v w such that if we rotate edge v w over v to edge v u in counter clockwise direction we cross over the figure defined by v w u In case of a figure consisting of three nodes this means that edge w u must exist and a v w u lt 7 In case of a figure corresponding to a polygon of four nodes there exists a node k connected to w and u with a v w k lt a v w u as we need to cross the polygon given by v w k and u during rotating w to u We say that the figure is oriented in counter clockwise direction as a walk over u v w k u defines a counter clockwise cycle see Figure 2 17 2 2 Muqarnas as an Abstract Geometrical Structure 23 Un 1 u gt Der UC XE C v4 w w Dane U3 Figure 2 17 On the left two examples of ordered triples u v w defining a figure On the right example of an ordered triple u v w defining an island Definition 2 2 6 figure The ordered triple u v w defines a figure if nodes v and w are connected and a v w u m or if nodes v and w have a common adjacent k v and a v w k lt alv w u In the next chapter we also work with a subgraph of the plan P by removing certain edges from the plan Thereby polygons arise which consists of more than four nodes and do not fit in our definition of a figure We call these polygons islands see right picture of Figure 2 17 In this context an island is a generalization of a figure and we also defin
142. rallel with different copies of a plan and switch between them we have to look up the coordinates corresponding to a certain index and compare the coordinates directly instead of comparing the indices We have chosen to use an adjacency list as data structure for our graphs so that we have a fast access to neighboring nodes For the directed graphs this means that we have only fast access to the successors of a certain node Therefore we often work parallel with the reverse graph Gev of Graph Grey has the same set of nodes and edges as the graph G but the edges are directed in the reverse order compared to G The graph Grev gives then the possibility to have also fast access to the predecessor of a node in G 4 1 3 Conversion of the Input into a Muqarnas Plan The first task of the program plantograph is to convert a xfig file into our intern data structure of a plan P To do this we start with an empty plan P read the information of the xfig file line by line and put this information in the plan P While scanning the xfig file a polygon can be recognized by the first line of the object starting with the number 2 see Figure 4 3 This first line also contains the informations whether this polygon is drawn with lines or with arrows In the context of reading the plan we only consider the lines and if we read a graph from a xfig file we only store the arrow information We scan the second line for the coordinates of the nodes connected to each o
143. raph and graphtomuq From the examples of Chapter 5 we conclude that the created software is able to give a fast analysis of the designs and results in a better understanding of the structure of the muqarnas We defined the mugarnas structure by a set of muqarnas elements together with rules how they can be arranged see Section 2 1 The combinations of elements covered by our definitions are those common in Seljuk and Il Khanid architecture A new way of representing the three dimensional structure of the muqarnas in the design is given by drawing arrows These arrows form a directed subgraph of the muqarnas design Properties of the muqarnas structure lead to properties of this graph This makes it possible to formulate different rules to find the graphs from the design with an allowed mugarnas realization The software tool plantograph applies these rules to analyze a muqarnas design In the examples taken from the Seljuk and Il Khanid periods we have seen that most directions are fixed by these rules and all possible muqarnas reconstructions cor 113 114 Chapter 6 Conclusions responding to a given plan can be calculated If we consider more general muqarnas the properties can change and the rules need to be adapted We may need to take into account more different element combinations and less directions in the graph can be set We could state that the definitions depend on the style of muqarnas we want to reconstruct Allowing other kind o
144. rd less remained but the entrance portal to the enclosure is in good state of preservation see Figure 5 2 b It contains the oldest muqarnas vault we study in this work In Figure 5 1 a the plan pl_alayhan fig representing the plane projection of the muqarnas is shown The muqarnas plan is mirror symmetric in its diagonal Further more we find that the muqarnas is built mainly from basic elements The elements are rotated over 7 8 kn 4 k 0 7 compared to standard position Because of this property we can represent the nodes by integers by remarking that the nodes in the plan are in the lattice IL rotated over 7 8 Instead of rotating the lattice we use here for our reconstruction a data structure based on floating point numbers In Figure 5 1 b the resulting subplan after running removelines r Ppl alayhan fig is shown We find that indeed most lines not corresponding to curved sides are removed The dotted lines in this figure represent edges we need to remove manually to create the subplan consisting only of the projection of the curved sides Running plantograph r Ppl alayhan fig Qrmlines pl alayhan fig 5 1 Computer Reconstructions of Seljuk Muqarnas 85 calculates the direction of several orbits of the muqarnas graph There are four orbits of which the direction is undetermined see Figure 5 1 c As the mugarnas plan is mirror symmetric in its diagonal we expect the two orbits marked red in Figure 5 1 d having the same
145. rder to calculate a subplan so that after applying Rules 1 4 there is a reconstruction possible with the desired number of tiers Looking in our example at the edges marked red in Figure 3 20 which are the edges we need to remove we find that removing these edges would results in an island Islands can be created by removing all edges incident to the node in the 3 4 Preparation of the Plan 53 J D Figure 3 21 Muqarnas graph and computer reconstruction of a muqarnas with same plane projection as the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz Figure 3 22 Muqarnas graph and picture of the of the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz Photo by Mohammed Bagheri 54 Chapter 3 Algorithm for Reconstructing Muqarnas center of the island Islands generally appear on the diagonal of a muqarnas graph The reason for this can be explained by looking at the paths in a muqarnas graph The longest path usually corresponds to a walk in the three dimensional muqarnas from the lowest tier to the upper most tier This lowest tier is found in the muqarnas graphs by arrows in cident to bottom boundary nodes of the plan The largest distance of a bottom boundary node to the center can be found at the diagonal of the plan Therefore most probably the longest path in the muqarnas graph follows a path approximating the diagonal of the plan By putting islands on the diag
146. re determined directly from Definition 2 2 8 We iterate over 4 1 The Program plantograph 65 allv M to decide for each node whether it belongs to the boundary or not For this we select for each node v M its neighbors 1 ao a 1 and order them such that a v ag a lt a v ag a 1 We apply the definitions of figure see Definition 2 2 6 and island see Definition 2 2 7 to decide for alli 0 n 1 j i 1 mod n whether v a a defines a figure or an island The definition of the boundary shows that v is in the boundary if there are i j with i 0 n 1 j i 1 mod n such that v a a neither defines a figure nor an island To determine the center nodes see Definition 2 2 9 we select the nodes on the boundary between the end node VU ex max Ug Me v M vy ymax of the x mirrorline and the end node uU eere ip cup of the y mirrorline The nodes v w can be found by comparing the coordinates of all nodes in the boundary and select from the nodes with maximal x value the one with largest y value and similar from the nodes with maximal y value the node with largest x value is selected The shortest path between v and u is set with assistance of a multiplication operator o defined on plans We define for a plans P N E and Q N E the product P o Q N E by if edge u v E and edge v w E gt edge u w E The set of nodes C N U N is given by the nodes inci
147. re remain only a few undetermined orbits and we can calculate for all possible graphs the corresponding muqarnas reconstruction Often the different directions of these undetermined orbits result in different valid muqarnas computer reconstructions 3 3 Conversion of Muqarnas Graph into Muqarnas Structure 45 3 3 Conversion of Muqarnas Graph into Muqarnas Struc ture In the previous chapter we interpreted a muqarnas structure as a set of elements These elements are given by their type and their position In this section we explain how the mugarnas structure can be found from the muqarnas graph In Section 3 3 1 we give an overview of the whole conversion process This process is split in two parts First we determine which nodes together define a muqarnas element see Section 3 3 2 After that these combinations of nodes are converted into a set of muqarnas elements combined with their position see Section 3 3 3 In Section 3 3 4 we show that it is in some situations possible to find the muqarnas structure directly from the muqarnas graph without using the muqarnas plan 3 3 1 Mugqarnas Reconstruction Process We recall from Section 2 2 1 that if two arrows in the muqarnas graph join in their end points these two arrows define a cell Intermediate elements are given in the graph by two arrows joining in their initial points Therefore we are interested in joining arrows to find the elements of the muqarnas The way these arrows join decod
148. re were more muqarnas in the palace at the Takht i Sulayman and because we do not know whether the design was ever used the prefabricated elements found at the Takht i Sulayman do not necessarily correspond to the plate Nowadays the plate is kept in the Islamic Department of the Iran Bastan Museum in Tehran The design consists mainly of squares and rhombi with isosceles right triangles along the frame of the field The sides of the squares and rhombi as well as the legs of the triangles are all 3 5 cm in length Their areas have been symmetrically arranged around a diagonal axis The construction is completed in the upper right corner by an irregular quarter octagon The angles of the various figures shapes are all multiples of 45 with the exception of some semi regular quadrangles and isosceles triangles along the diagonal The Topkap Scroll is a late fifteenth or early sixteenth century scroll containing patterns for ornamentations and patterns to be used as designs for muqarnas This scroll is preserved at the Topkap Museum in Istanbul and published by G lru Necipoglu see Necipoglu 1995 It includes several rough designs of which the artist has shown a small part worked out in detail probably to avoid confusion It ranks among the oldest extant designs for muqarnas 1 2 Purpose and Contribution of our Research Our research differs from previous works about muqarnas We do not only concern about a better understanding of the mu
149. rections of arrows in an islands are known is shown Rule 4 there are no niches allowed has no effect in this example The computer reconstruction corresponding to the new graph see Figure 5 10 e indeed agrees with the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz VA te Cr AS I a b c afterrulel fig afterrule2 fig afterrule3 fig o D d e WRL main wrl crc gr natanz fig Figure 5 10 Plan analysis of the muqarnas in the basement of the north iwan of the Friday Mosque in Natanz with node removal on the diagonal allowed to get desired height 98 Chapter 5 Results of Algorithmic Muqarnas Reconstructions 5 2 2 Bistam Shrine of Bayazid Location Bistam Iran Building Shrine of Bayazid Vault East portal Height 6 tiers Year of construction 1313 References Wilber 1955 pp 127 Harb 1978 pp 44 45 Takahashi 2004 058 Discussion In the small village of Bistam which lies between Tehran and Mashad two architectural elements dated in the Il Khanid period can be found These are the shrine of Bayazid and the Friday Mosque We study here a muqarnas vault of the shrine of Bayazid Inspired by the design on the plate found at the Takht i Sulaym n we use in this ex ample a simplified plan see Section 3 3 4 This means that lines which split a rhombus into an almond and small biped and lines which split a square into a jug and large biped are removed As a consequ
150. res a xfig file representing the complete muqar nas plan The output consists of a xfig file with some edges removed and some text with suggestions of additional nodes to remove Removing a node v means that we remove not only the node v but also all edges incident to v The following command line options are available to call the program removelines P lt string gt This necessary option is followed by the name of the file representing the complete muqarnas plan o lt string gt This option is followed by the name of the output file representing the subplan Q Without this option the name of the input plan with the prefix rmlines_ placed before is used C With this option an irregular center is allowed in the reconstruction h lt integer gt This option is followed by a number representing how many tiers the corresponding muqarnas contains r With this option we work with a data structure where nodes in the plan are stored as floating point numbers Without this option we work with integers to repre sent the nodes in the graphs If the option r is not given we assume that the muqarnas only consists of basic el ements We have seen in Section 3 4 that in that case we need to remove two kind of 72 Chapter 4 Software Tools for Reconstructing Muqarnas edges Those which correspond to backsides of cells standing on front sides of intermedi ate elements back on front and the edges corresponding to places where front sides
151. rnas graph In this way the structure information in which way the muqarnas can be continued downwards is obtained If we add elements at the bottom of the muqarnas the backsides of the bottom cells can only stand on the curved sides of the new elements in the direction as fixed by the mugarnas graph The arrows in the interior of the graph G can correspond to one or two curved sides At this place elements on the same tier or of different tiers can join At the left of Figure 3 2 the red arrow is the projection of two curved sides curve to curve in the right figure the red arrows are projections of a backside of an element and a curve side of an element in the lower tier back on curve The projection of the place where a backside of a cell stands on the front of an intermediate element back on front and the projection of the place where the front of an intermediate element touches the front of an intermediate element front to front do not appear in the muqarnas graph In Figure 3 3 we see an example of the relationship between a muqarnas plan P and the corresponding muqarnas graph G In this example the graph G is a proper subgraph of the plan P as there are lines in the plan that do not correspond to curved sides and 3 1 Representation of the Muqarnas Structure in a Graph 33 ial Figure 3 2 Projections of curved sides may correspond to two joining curved sides of the same tier left or to one curved side of an element and a backside o
152. s an Abstract Geometrical Structure 21 1 Vv I P 2 x AS e Elm CI eo E a Tr Figure 2 15 On the left plane projection of a cell and an intermediate element in standard position On the right the projection of a cell and an intermediate element with position c h Definition 2 2 4 position Consider an element in tier h with central node c Let edge c v be the projection of a curved side of the element such that if we rotate edge c v in the plane in counter clockwise direction then we cross over the projection of the element The position of the element is then given by c h 9 with a c c e o v for a cell and a c c eo v for an intermediate element 2 2 2 Mugarnas Plan We have seen in Section 1 1 3 that a muqarnas design is a drawing which consists of polygons representing the plane projections of the muqarnas elements A muqarnas de sign therefore mainly consists of geometrical figures as given in Figure 2 5 the plane projections of the building blocks of the muqarnas In practice a muqarnas design does not correspond to the exact projection of all the elements If we study the plate found at the Takht i Sulaym n we observe that only square and rhombus figures are drawn During the analysis of this design we find that it is not possible to construct a muqarnas only consisting of squares and rhombi we have to interpret some rhombi as a combina tion of an almond and a biped see e g Dol
153. s in the next tier is to put cells with their backsides on the fronts of the intermediate elements see Section 2 1 2 The next tier will then consists of cells spread over the whole circumference of the structure as given in this picture In this way a column appears a structure not fitting in our niche or dome We see that node u cannot be a source in the muqarnas graph In the second picture node v is the central node of all cells which have v in its plane projection In the mugarnas this implies that all curved sides join already to other curved sides in this structure and no free curved sides are left In Section 2 1 2 we have seen 44 Chapter3 Algorithm for Reconstructing Muqarnas eO EK Figure 3 12 First two pictures show element combinations not allowed in a muqar nas fitting in a dome or niche The right picture shows a similar structure with a combination of elements which is allowed that the only way cells can join to other elements is by joining curved sides at the same tier together curve to curve backsides of the cells stand on front sides or curved sides of elements in the previous tier back on front or back on curve This means for the situation as given in the second picture that it is not possible to join elements in the same tier or to join a new tier above Such a structure therefore can only appear in the center of a muqarnas We exclude this situation by Rule 4 by avoiding sinks in the graph In the right pict
154. s makes it possible to walk from c further upwards to node c leaving the center Because we doubt whether the center is designed well see Dold Samplonius and Harmsen 2005 and this is the only example we found of muqarnas without regular center we do not consider this situation as being representative In the case we allow a non regular center we replace this rule by an alternative one which is a generalization of Rule 2 Rule 2 alternative We consider a muqarnas structure with plan P N E and graph G M C If c C belongs to an orbit which does not contain an edge e with end nodes s W P and for the end nodes c c of c it applies that c W P c W P then arw cs amp G M C A third rule for setting the direction of the edges can be formulated by recognizing some known structures This are the figures in the plan appearing after removing edges in the plan By removing edged there arise islands as defined in Definition 2 2 7 In Figure 3 11 an overview of such kind of figures is given together with the direction of the edges They only appear in the orientation as given in Figure 3 11 These figures correspond to more than one element in the muqarnas and will consist of one cell and different intermediate elements The projection of the curved sides of the 3 2 Construction of Muqarnas Graphs from the Plan 43 cell is in Figure 3 11 given by red arrows and the projection of the curved sides of the i
155. s of nodes are given We first add 1 to a b c and d to be sure that we apply f to non negative integers Then we apply f two times the first time on a 1 5 1 and on c 1 d 1 and the second time on the results of the first calculations The node v a 0 42 c d 4 2 is then represented by an integer given by uv f f a 1 5 4 1 f c 1 d 1 2 5 As f is invertible for each non negative integer i Zs we can calculate the coordinates L i by applying g two times on i see Table 2 3 0 0 0 0 0 0 0 1 1 42 1 1 4 2 10 4 0 1 1 0 0 0 1 1 2 20 0 5 0 0 0 2 1 1 2 V2 Table 2 3 Integers i represent nodes in the grid L Chapter 3 Algorithm for Reconstructing Muqarnas We focus on the analysis of the muqarnas structure directly from its design Our task is to find the set of elements the muqarnas consists of together with the position of these elements The different steps we perform to achieve this are described in this chapter The conversion of these steps into a computer program and the visualization of the set of elements is treated in the next chapter Chapter 4 In Section 3 1 we define a directed graph derived from the muqarnas plan which contains the main structure information of the muqarnas From this graph we can recog nize the mugarnas elements and their positions The process of calculating the muqarnas is then split in two steps first we find the
156. seen in Section 2 2 3 that if a node v IR can be represented by integers a b c d Z gt _ so that v a b V2 c d 2 then we can represent it by one integer dv f f a 1 5 4 1 f c 1 d 1 with f as given in Equation 2 2 We use these integers given by the grid as our in dex numbers to refer to a certain node in our adjacency list The integer then directly includes the information of the coordinates of the elements In the case the command line option r is given by calling the program plantograph the coordinates are stored as floating point numbers We store additionally a list of the floating point numbers In this list each floating point number is linked to an integer which is the index of then node Let vp i R the coordinates of the node corresponding to the index i Ip We denote ip v for the index of the node v N in the adjacent list representation of the plan P If we represent nodes v with integers v and use these integers for the indices then for the subplan Q of P Ig C Ip and vg i vp i for i Ig as the coordinates of the nodes are determined by there integer representation If we represent the coordinates with floating point numbers this is not the case The same index number in different plans in general refers to different nodes that means for i Ig N Ip in general vo i Z vp i they depend of the order the nodes are inserted in the node list This means that if we work pa
157. set the direction of the same orbit differently there is no reconstruction possible Otherwise we calculate the height of the nodes in the subgraph H M C given by the edges for which the direction is determined If there is a muqarnas reconstruction possible with muqarnas graph G M C then the graph H is a subgraph of this muqarnas graph G This subgraph H does not need to be connected as some edges of the muqarnas plan can be undetermined and therefore do not appear in the graph H Albeit the graph H does not need to be connected we approximate the height of the nodes H by applying Algorithm A 3 3 The algorithm calculates for each component in H the height of its nodes If this calculation fails that is it stops because hmar gt M there is a cycle in the graph and H cannot be a subgraph of a muqarnas graph Otherwise the Algorithm A 3 3 calculates for each component the height of the nodes well If there is a reconstruction possible these components are projections of smaller parts of the complete muqarnas We need to add for each component a constant to the calculated heights referring to the height of the smallest tier in which this muqarnas component appears The calculated heights h v of the nodes v M are then equal to or smaller than the height h v of the nodes We therefore can conclude that the number of tiers of the corresponding muqarnas is equal to or larger than the maximal height of the nodes in the subgraph and a lower
158. sland we need to find the corresponding orbit by iterating over the edges in the orbit we find that computing time can be given by O E O ND In Rule 4 we iterate over the orbits The number of orbits can be maximal E We give an orbit a direction and iterate over all nodes to check whether the resulting graph 4 2 The Program removelines 71 contains a niche The amount of operations depends then on E N and computing time can be given by O E N O N Adding the computing times of the different methods executed in the program plantograph we conclude that computing time of the program equals to O N 4 2 The Program removelines The program removelines is written to do the first analyzes on the muqarnas plan Its task is to calculate the subplan Q from a muqarnas plan P This subplan Q consists of the edges which appear in the mugarnas graph G The program removelines is not able to find all edges in the plan P which not correspond to a curved side but it is able to find most of them and it proposes suggestions to remove nodes to create islands in the plan Creating islands can make it possible to create reconstructions containing less tiers The user needs to decide whether these nodes indeed should be removed In Section 4 2 1 we present the methods executed in the program Some remarks about its computing time are given in Section 4 2 2 4 2 1 Removal of Edges from the Plan As input the program removelines requi
159. stration of the zyz coordinate system in VRML The x axis points to the front the y axis points upwards and the z axis points to the right Surfaces in VRML are approximated by triangles see Figure 4 7 b Besides of a list of coordinates z y z of vertices on the surface a list of triangles is given Each triangle is given by three numbers which refer to the selected vertices on the surface In this way a triangulation of the surface is defined To give a smooth appearance to a surface constructed by joining triangles we may also give a list of normal vectors explicitly see e g Foley et al 1994 p 541 These normals are then used by the VRML browser to calculate the amount of light to reflect in the different directions This geometric information coordinates triangulation and normals is stored in a so called Geometric Node 4 3 The Program graphtomuq 77 To translate rotate or rescale surfaces a Transform Node is available in VRML 2 0 It is used to change previous defined surfaces or objects containing more surfaces and place them on the required position in our model These surfaces can be defined in the same file or we call to another file containing the geometrical data of the object we want to transform For visualizing the muqarnas we create separate VRML 2 0 files containing only the geometrical information of the muqarnas elements By using Transform Nodes these elements are then placed in their right position Our out
160. t In the output graph see Figure 5 7 d much more directions are determined The remaining directions are given manually see Figure 5 8 a and the reconstruction is created by running 5 1 Computer Reconstructions of Seljuk Muqarnas 93 graphtomuq r Ppl arslanhane fig Gcrc gr arslanhane fig In Figure 5 8 c the reconstruction is shown after manual completion of the vault the muqarnas fits in a pl_arslanhane fig In Paz AE i Y EY Vi ny I e 4 27 n lt V Nu vtm uU Xo 4 4 NS L7 nn N A N b 1 4 X bd I U 4 gt N ru oe 25 V4 7 aqu MM LN l L I v2077 E VV NN gt c d grl arslanhane fig gr2 arslanhane fig Figure 5 7 Plan analysis of the muqarnas vault in the entrance portal of the Arslan hane Mosque in Ankara 94 Chapter 5 Results of Algorithmic Muqarnas Reconstructions A AN i a crc gr arslanhane fig AHI A Srt CAML 1230 b Photo of the entrance portal c WRL main wrl Figure 5 8 Reconstruction of the entrance portal of the Arslanhane Mosque in Ankara 5 2 Computer Reconstructions of Il Khanid Muqarnas 95 5 2 Computer Reconstructions of Il Khanid Muqarnas In the reconstructions of Il Khanid muqarnas we take into account that these muqarnas exist mainly of basic elements We represent the plans and graphs with the data structure based on integers In the examples we set our unit equal to the module th
161. t line BG of length 2 The perpendicular line BA which has size the module has then length 1 The oblique line AF intersects the opposite vertical BG with angle BAE equal to 30 This line AE is divided in five equal parts and on three fifth of this line we mark the vertex Z As ZBAE 30 we find that E has coordinates 0 2 1 4 3 and it is then easy to verify that Z 2 2 2 3 The line EZ is rotated down over E until it joins the vertical line We find the end point H 0 2 3 The arc ZH is described by 4 3 The Program graphtomuq 79 Figure 4 10 Construction of the curve according to al Kashi a circle with radius equal to the length ZH To describe this arc we look for the center of the corresponding circle which has distance ZH to H and to Z This center can be found by intersecting the circle with center Z and radius ZH by the circle with center H and radius ZH We find the center T 2 2 2 3 and describe the arc ZH by Qo 5 005 9 2 sine with 30 lt y lt 7 The lengths of LG and AY vary in our model we choose LG 0 1 and AY 0 2 The coordinates of the curved side are embedded in the three dimensional space by adding a z coordinate equal to 0 We translate by x 0 1 as we want to have vertex L in the origin instead of G In the muqarnas reconstruction the module will be the length of KY and not of BA as an edge in the muqarnas plan corresponds to this length Therefore we rescale by dividin
162. tate ments for all nodes in the projections drawn in Figure 2 5 e The plane projection of the elements are polygons of which all angles have sizes kn 8 k 2 e Nodes of the plane projections of the basic elements in standard position can be represented by coordinates in the grid L Z 2 v2 x Z z v2 We prove that nodes of a plan consisting only of basic elements are all in the grid IL Proposition 2 2 11 The set of nodes N of a plan P N E corresponding to a muqarnas only consisting of basic elements is a subset of the grid L Z Z v2 x Z Z v2 PROOF We give the proof by showing that we only need two transformations for putting the polygons from their standard position to their places in the muqarnas plan These two standard transformations are given by i Translation over n L ii Rotation by kz 4 k Z 28 Chapter 2 Structure of the Muqarnas We first prove that these two transformations are homomorphisms on L if we consider IL as a Z module Then we show that these transformations are sufficient to bring the polygons to their position in the plan Transformation i is an operation in L let n IL we can translate over n by adding n to l It can be confirmed that n l IL For transformation ii let L and let R be the rotation matrix for rotating over an angle of kz 4 0 k lt 8 hence wea cos km 4 sin km 4 1 V2 1 2 sin km 4
163. the second line of the object polygon The first line contains beside of the information that we are dealing with a polygon also information whether the nodes are connected by arrows or by lines In case of arrows this line also contains their direction In this way a plan or graph is in the xfig file format represented by a list of edges where each edge is given by an open polygon defined by the two connected nodes In the case of a plan the connections are given by lines and in case of a graph they are given by arrows In our software the structure of a muqarnas plan and a muqarnas graph are rep resented by an adjacency list each node in a plan is linked to a list of nodes rep resenting its neighbors A detailed description of this data structure can be found in Sedgewick 2002 pp 31 35 In the case of a graph only the successors of a node are listed In Figure 4 2 we see examples of such adjacency lists corresponding to a plan and a graph In the adjacency list we refer to the nodes by using indices These indices consist of a set of non negative integers so that each node uniquely corresponds to an index taken from this set For a plan P N E we denote with Jp C Z gt o the set of index 4 1 The Program plantograph 61 numbers of the nodes N If the command line option r is not given we store the coordinates of the nodes as integers This is possible if all nodes in the plan P N E are in the grid L Z Z 2 x Z Z 2 We have
164. ther The 62 Chapter 4 Software Tools for Reconstructing Muqarnas 222 312 70 112 218 509 73 112 222 73 112 70 73 312 218 218 70 257 312 509 257 222 73 312 70 257 218 312 112 218 222 509 218 222 312 70 112 218 Fa 509 73 112 222 73 A 112 312 218 218 312 509 Ness 222 312 Figure 4 2 On the left visualization of a plan and graph On the right connections between the nodes stored in an adjacency list 21040 7 50 1 1 0 000 00 1002 900 1536 1536 1800 Figure 4 3 Part of a xfig file which defines a polygon line 1 given by two connected nodes line 2 two lines illustrated in Figure 4 3 define an open polygon In the second line we see that the polygon is given by the connection between the node v 900 1536 and the node 1536 1800 In this example the unit is chosen to be equal to 900 so that we will refer to these coordinates with floating point numbers v 1 1 71 w 1 71 2 The method we use to convert the information of the xfig file into our own data structure depends on whether we represent our node with integers or with floating point numbers In the situation that we use an integer representation for the nodes v Vp vy R we need to be able to find a b c d Z gt _ such that v a b V2 vy c d 2 then we can convert a b c d into one integer by calculating a b V2 c d V2 f f a 4 1 b 4 1 f c 4 1 d 4 1 as defined in Equati
165. thesis we want to analyze the plane projection of muqarnas and study which part of the lost information can be calculated from the plane projection The aim of this work is to answer the following questions e Is there a direct algorithm for constructing a three dimensional muqarnas from its plane projection e Are these muqarnas uniquely reconstructible We restrict ourselves to muqarnas used in domes and niches which are built dur ing Seljuk or Il Khanid times This delivers an insight in the development of the oldest muqarnas forms The purpose of our reconstruction algorithm is to analyze the structure of the muqarnas It answers the question to what kind of element a polygon in the design corresponds and it determines its position location and orientation However it does not tell us what style colors material of elements need to be used The only known design of an Il Khanid muqarnas is the plate found at Takht i Sulayman We do not have a real muqarnas corresponding to this plate nor do we have 8 Chapter 1 Introduction measures of the dome the muqarnas could fit in Therefore we can not verify our re constructions For Seljuk muqarnas we do not even have an example of a design In this study we therefore work with plane projections of existing muqarnas instead of working with original designs Our goal is to write a computer program which is able to propose a three dimensional computer reconstruction for a given plane
166. ting a cell and pushing down a part in form of an intermediate element is pos sible if the backsides of the cell stand on curved sides of elements one tier below see Figure 2 13 a After splitting the backsides of the cell are replaced by the curved sides of the intermediate element Because this intermediate element is pushed down one tier its curved sides need to join to the curved sides of the element on which the cell stood before splitting Because splitting the elements does not influence other elements in the muqarnas structure we call it a local change Splitting the elements does not change the muqarnas graph as the places of curved sides do not change This means that the deci sion to split some elements needs in our algorithm be executed during the reconstruction of the mugarnas structure from its muqarnas graph It does not influence the algorithm of finding the muqarnas graph from the plan We cannot always automatize the decision of splitting a square If we compare in Figure 3 18 the plane projection of a vault in Bistam and of that in Natanz we see that the gray marked square in the left picture is split in the muqarnas structure In the right picture the gray marked square appears in a similar situation but here this square is not split These kind of decisions therefore cannot be automatized in real life the designer need to decide about it We restrict ourselves by automating the situation where it is 3 4 Preparation of the P
167. tions of Il Khanid Muqarnas 99 Figure 5 12 Another difference between the original vault and our reconstruction are the half squares which appear in the second tier of our reconstruction In the real muqarnas there are no elements on these places The differences between the virtual muqarnas and the real one are only local We need can exchange some cells with projection a square by a combination of a cell with projection a jug and an intermediate element with projection a large biped The half elements at the second tier with projections half squares need also to be removed for creating a more realistic muqarnas reconstruction D o A gt V b gr bistam fig Na c WRL main wrl Figure 5 11 Plan analysis for reconstructing the muqarnas in the east portal of the shrine of Bayazid in Bistam 100 Chapter 5 Results of Algorithmic Muqarnas Reconstructions Jy RN Figure 5 12 Plane projection with tiers colored alternately of the muqarnas in the east portal of the shrine of Bayazid in Bistam 5 2 3 Farumad Friday Mosque Location Farumad Iran Building Friday Mosque Vault Sanctuary iwan Height 6 tiers Year of construction 1320 References Wilber 1955 pp 156 157 Takahashi 2004 059 Discussion In this example we explain the effect on the muqarnas if we set the directions of unde termined edges in different ways We observe a mugarnas from the Friday Mosque in Farumad This mosque is si
168. tructions They also made it possible to convert the VRML data into other data formats which enabled them to revise the reconstructions with different programs Beautiful animations were one of the results Daniel has written a user interface planedit which makes it much easier to work with the algorithms Bernhard supported in installing and testing programs for me I thank them both for their assistance and interest in my work I am grateful to Professor Shiro Takahashi for sharing data from his large database of mugarnas plans By being allowed to use his plans I saved a lot of time I also would like to thank Mohammed Yaghan for his visit to Heidelberg for discussing about muqarnas with me I want to acknowledge the people who gave me the permission to use their pictures in this thesis Pictures not made by myself stem from Susanne Kr mker Mohammed Bagheri Yvonne Dold Samplonius and Michael Winckler Finally I would like to thank my family friends and colleagues who made an enjoying stay in Heidelberg possible In particular I want to thank Mariya Pau and Frank for listening and discussing not only about my thesis Frank I want to thank you for your many useful suggestions and your non exhausting patience during the preparation of this thesis 121 122 Bibliography al Asad 1994 al Asad M 1994 Applications of Geometry In Frishman M and Khan H U editors The Mosque History Architectural Development amp Regional Di
169. tuated on the outskirts of the village The earliest construc tion work at the mosque stems from Seljuk times the damaged north and south iwan are from Il Khanid time The plan of the muqarnas vault is given in Figure 5 13 a We delete edges by apply ing removelines h6 Ppl_farumad fig It removes the edges with length unequal to 1 and proposes to remove the node on the diagonal with integer representation 364 which has coordinates 1 3 2 1 3 2 The edges which we remove are represented in Figure 5 13 a with dotted lines The mugarnas graph is calculated by first applying plantograph Qpl_farumad fig 5 2 Computer Reconstructions of Il Khanid Muqarnas 101 we find that most directions can be determined The edges of the undetermined orbits can be set by using the fact that in the plan some squares are divided in jugs and bipeds For these squares the direction of its edges is fixed by the orientation of the jug This forces the direction of the undetermined edge and we get the graph as shown in Figure 5 14 a with reconstruction calculated by applying graphtomuq Ppl farumad fig Ggrl farumad fig given in Figure 5 14 b We study what would happen if we set the direction of the undetermined orbits of Figure 5 13 b in the reverse order This graph is shown in Figure 5 14 c and the corre sponding reconstruction is created by applying graphtomuq Ppl farumad fig Ggr farumad fig We see that this second reconstruction has a
170. ure node w is also a sink but here it is possible to add new elements In node w not only cells join but also half intermediate elements There are still free curved sides available to which we can join new elements We summarize our observations in Rule 4 Rule 4 Let v be a node in a muqarnas plan P N E then v cannot be a source in the muqarnas graph If all edges e E incident to v appears as arrows in the corresponding muqarnas graph G then node v cannot be a sink in the corresponding muqarnas graph We have now a set of rules available which make it possible to determine the direc tions of most edges in the muqarnas graph by using only the plane projection of the mugarnas and not the muqarnas vault itself The definition of opposite edges and the rules given in this section are formulated in accordance with the different element com binations appearing in Il Khanid and Seljuk muqarnas fitting in a niche or dome If we restrict ourselves to the reconstruction of Il Khanid muqarnas we can simplify our defi nition of opposite In Il Khanid muqarnas two half intermediate elements fitting at their front do not appear We therefore do not need to exclude the situation as presented in the middle picture of Figure 3 5 Developing reconstruction algorithms for other types of muqarnas may need a reformulation of the rules formulated in this section In Chapter 5 we will see that with this set of rules most directions of orbits can be set The
171. uthoctagon of the Il Khanid palace at Takht i Sulayman 5 2 Computer Reconstructions of Il Khanid Muqarnas 105 a b WRL main wrl gr southoctagon tis fig Figure 5 16 Reconstruction proposal of the muqarnas in the south octagon by using plantograph and graphtomug 5 2 5 Takht i Sulayman Plate Location Takht i Sulayman Iran Building Il Khanid palace Vault Unknown Height Unknown Years of construction 1271 1276 References Harb 1978 pp 60 66 Yaghan 2000 Dold Samplonius and Harmsen 2005 Discussion The most important example we consider in this work is the reconstruction of a muqarnas corresponding to the design on the plate found at the Takht i Sulayman We do not know how the muqarnas corresponding to this plate looked like not even the proportions of the room for which the muqarnas was designed It could be that it was only a sketch and the designed muqarnas was never built We will see that our reconstruction methods succeed for understanding the design and results in different suggestions for reconstruction In the last decades different interpretations of the plate were published The first interpretation was done by Harb see Harb 1978 Yaghan has published four different suggestions for reconstruction see Yaghan 2000 and we published another proposal 106 Chapter 5 Results of Algorithmic Muqarnas Reconstructions see Dold Samplonius and Harmsen 2005 These reconstructions were main
172. v M as follows if v appears as the projection of a vertex at the backside of a muqarnas element in tier r which belongs to the bottom of the muqarnas then h v r 1 Otherwise h v is given by the number of the minimal tier containing elements with node v in their projection We recall from Section 2 1 2 that following the direction of a curve in the muqarnas corresponds to going upwards This means that every arrow c C of the graph G M C corresponding to a curved side in tier r points to tier r 4 1 This property will later be used for calculating the heights of the nodes directly from the muqarnas graph Lemma 3 1 3 Let G M C be a muqarnas graph For the height h v of the nodes v M the following properties are valid h w h v Lif arw v w C 3 1 and min h v 0 3 2 veM PROOF Let the arrow arw v w be the projection of the curved side of an element in tier r Then w is the projection of the apex of this curve It cannot touch curved sides of elements in a lower tier and therefore h w r The node v corresponds to a vertex on the bottom of the curve This vertex belongs also to the backside of the element If the element belongs to the bottom of the muqarnas it follows directly from Definition 3 1 2 that h v r 1 Otherwise the element stands on an element of the previous tier If the element is a cell its backsides stand on curved sides of an element below back on curve or its backsides st
173. ver sity pages 55 70 Thames and Hudson London al K shi 1558 al K shi G a D n 1427 1558 Miftah al Hisab Key of Arithmetic Ms Or 185 Leiden Ames et al 1997 Ames A L Nadeau D R and Moreland J L 1997 VRML 2 0 Sourcebook John Wiley amp Sons Inc New York Aslanapa 1971 Aslanapa O 1971 Turkish Art and Architecture Fabe and Faber Limited London Bianca 2000 Bianca S 2000 Urban Form in the Arab World Hochschulverlag AG an der ETH Z rich Institut f r Orts Regional und Landesplanung ORL Institut ETH Z rich Bollob s 1979 Bollob s B 1979 Graph Theory an Introductory Course Springer Verlag New York Cast ra 1996 Cast ra J M 1996 Arabesque Art d coratif au Maroc ACR dition Internationale Paris Dold Samplonius 1992 Dold Samplonius Y 1992 Practical Arabic Mathematics Measuring the Muqarnas by al Kashi volume 35 of Centaurus pages 193 242 Munks gaard Dold Samplonius 1996 Dold Samplonius Y 1996 How al Kashi Measures the Muqarnas A Second Look Mathematische Probleme im Mittelalter Der lateinische und arabische Sprachbereich Wolfenb tteler Mittelalter Studien Vol 10 56 90 Dold Samplonius and Harmsen 2004 Dold Samplonius Y and Harmsen S 2004 Mugqarnas Construction and Reconstruction In Williams K and Cepeda E D edi tors Nexus V Architecture and Mathematics pages 69 77 Fucecchio Florence Kim Willi
174. veral identical stalls niches or chambers to accommodate merchants their servants animals and merchandise 1 1 2 What is a Muqarnas Mugarnas is the Arabic word for stalactite vault an architectural ornament consisting of niche like elements arranged in tiers It can be used in domes niches or on arches as an almost decorative frieze In Figure 1 1 different examples of muqarnas are shown A dome is a half sphere or an oval usually placed on a rectangular building A muqarnas can be fit into the dome to provide a smooth transition between the rectangular part and the vaulted part see e g al Asad 1994 p 65 or Bianca 2000 p 45 Figure 1 1 d 1 1 Mugarnas in the Islamic Architecture 3 a c d Figure 1 1 a Minaret of the Selimiya Camii in Konya Turkey decorated with muqarnas b Cornice at the Sultan Han near Kayseri Turkey ornamented with muqarnas c Muqarnas vault above a niche in the Friday Mosque in Natanz Iran d Muqarnas in a dome in the Mevlana Teke in Konya Turkey shows an example of a muqarnas fit into a dome in the Mevlana Teke in Konya Other famous examples of muqarnas in domes can be found in the Alhambra in Granada see e g Grabar 1992 or Jones and Goury 2001 In a niche a muqarnas is also used to hide the transition between the straight and curved parts of the niche Niches are often found at entrance portals of Islamic buildings for example at mosques In Figure 1 1 c we find
175. with sizes equal to the module which are the sides corresponding to its curved sides These two sides join in an angle of 45 The opposite angle is 135 and the other two angles have both sizes of 90 A small biped is what remains if we subtract an almond from a rhombus The two sides with sizes equal to the module correspond to the curved sides of the element These curved sides join in an angle of 45 and the opposite angle equals 225 The remaining two angles have sizes of 45 In Il Khanid muqarnas a small biped only appears as plane projection of an intermediate element The last figure based on a rhombus is the half rhombus The half rhombus is given by dividing the rhombus over its smaller diagonal yielding an equilateral triangle In Il Khanid muqarnas it only appears as plane projection of an intermediate element Often we find two half rhombi in combination with a jug and a square forming a hexagon see Figure 2 7 Figure 2 7 Common combination where half rhombi are used to carry a jug In Il Khanid muqarnas other elements than the basic ones only appear at the bound ary In Figure 2 8 the plane projection of an Il Khanid muqarnas vault in a niche of the Friday Mosque in Ashtarjan is drawn We see that this projection indeed mostly consists of polygons as given in the overview in Figure 2 5 At the part corresponding to the front of the mugarnas vault other muqarnas elements appear They need to be adapted so that the muqarnas fits
176. y Daniel Jungblut see Jungblut 2005 it is possible to set the direction of the remaining edges manually The program plantograph uses the complete muqarnas plan P and the subplan Q which consists of the edges which appear in the muqarnas graph G The following command line options are available to call the program plantograph Q string This option is necessary It is followed by the name of the file representing the input muqarnas subplan Q P lt string gt The option P should be followed by the name of the file representing the complete muqarnas plan P If this option is not given the subplan Q is used instead G lt string gt This option is followed by the name of the output file representing the muqarnas graph G By default we use the name of the input file representing the subplan Q with the prefix graph placed before p integer The option p can be followed by the integers 1 2 or 4 representing a full half or quarter muqarnas plan respectively By default the input plan is considered being a quarter plan C If this option is given an irregular center is allowed in the reconstruction r With this option we work with a data structure where nodes in the plan are stored as floating point numbers Without this option we work with integers to repre sent the nodes in the graphs The task of the program plantograph is to set the direction of the edges of the sub plan Q Besides of the subplan Q the co
177. ying undirected graph is connected This means that for all nodes v w there is a sequence vj v C with v v and v w such that arw vi vjj1 C or arw v 1 v C fori 1 n 1 Definition A 10 cycle A cycle c in a graph is a path c vi v such that v v and v ze vj for ij 1 n 1 j i Index adjacent 118 Alay Han 84 almond 14 angle 19 Ankara Arslanhane Camii 92 apex 13 arrow 117 backside 13 barley kernel 16 biped large 13 small 14 Bistam Shrine of Bayazid 98 bottom 18 bottom boundary 25 boundary 25 cell 12 center muqarnas 19 plan 25 central node 19 circuit 118 connected graph 118 curved side 13 cycle 118 diameter 19 edge 117 element position 21 type 20 end node arrow 118 119 edge 117 end point 118 face 45 cell 45 int 45 Far mad Friday Mosque 100 figure 23 global different 55 graph directed 117 muqarnas 32 undirected 117 incident 118 induced 118 initial point 118 intermediate element 12 island 23 jug 13 Kayseri ifte Medrese 86 Sultan Han 89 local different 55 module 13 Natanz Friday Mosque 95 neighbor 118 node 117 non singular 41 opposite 37 orbit 39 120 parallel 39 path 118 plan 22 simplified 49 regular bottom 18 bottom boundary 26 center 19 26 rhombus 13 separate 41 singular 41 sink 43 size 69 source 43 square 13

Algorithmic Computer Reconstructions of Stalactite Vaults

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