A comparative analysis of the digital re-constructions of muqarnas systems: The case study of Sultanhanı muqarnas in Central Anatolia

Abstract

This paper presents a comparative case study on the digital modeling workflows of a particular muqarnas system. After the literature review and the definition of the context, several digital modeling workflows were described as element-based, tessellation-based and block-based workflows by using computer-aided design and parametric modeling software. As the case study of this research, these workflows were tested on a muqarnas design located at the Sultanhanı Caravanserai in Central Anatolia. Then, workflows were compared according to three qualities: analytical, generative, and performative. The outcomes of element-based workflow has more analytical solutions for the study, where tessellation-based workflow has more generative potential and block-based workflow is more performative.

Keywords

Anatolian Seljuk muqarnas digital modeling parametric modeling architectural geometry Sultanhanı Caravanserai

Introduction

Muqarnas is a computational design system; widely used in vaults, domes, niches, arches, portals, and minarets. There are many examples of muqarnas from India to Western Asia, the Middle East, Anatolia, North Africa, and Andalusia. The main purpose of muqarnas is generally defined as a smooth transition among building parts. In portals; the prismatic elements of muqarnas are arranged with a specific order so that they overlap and rise by forming a protrusion, starting from the interior side of the wall surface, transforming the rectangular plan into a polygon, and then gradually gather at a point on the facade wall. In this way, a high entrance space that gradually opens out is created.¹ (Figure 1). The aesthetic sophistication presented by muqarnas compositions is generally associated with its sculptural visual quality, causing light, and shadow effects. Most of the sources indicate that the first known muqarnas examples are originated in Iran dated to the beginning of the first millennium, and spread from Central Asia to Andalusia.² Although it emerged in the Islamic world, later examples can be seen in other regions such as Cappella Palatina in Palermo (1132) and Chapel of the Assumption in the Monastery of Las Huelgas in Burgos (13th century).³

Figure 1.

Sahip Ata Mosque Portal (1258–1283), Konya, Turkey.

Some sources translate the word muqarnas as an Arabic word of stalactite vault.^4,5 However, there are different opinions about the roots of the term. One of the most accepted hypothesis is based on an Arabic word qurnas, meaning similar to pointed. The other argument is based on a Greek word κοrwνιs, meaning curved.^6,7 Ödekan² states another hypothesis, based on Arseven⁸ in which, the word might be originated to karnas, which means protrusion in Yakut, or Sakha Turkish. These arguments help us to re-paraphrase muqarnas from today’s perspective as; a “modular, stereotomic system that is segmented-discontinuous in the subscale, while gradient-continuous in superscale.”

Muqarnas has been a major research field for art historians and archaeologists, studying the art and architecture of the Middle East and Near East. The muqarnas studies revealed many different interpretations, most of which claim that they are unique and original.⁹ It is possible to divide these studies into two major branches. One of them is the research branch that focuses on the social, political, and theological meanings of muqarnas. Since the subject covers a wide geography and time period, it is difficult, or impossible to make solid generalizations in this research branch. For this reason, the importance of focusing these studies within a certain context and historical perspective becomes evident.¹⁰ The detailed historical, cultural, theological, and political backgrounds of muqarnas can be followed from the studies of Tabbaa¹⁰ and Carillo.^3,11

Another major branch of muqarnas research focuses on geometrical qualities; where mathematicians, architects, and computational design specialists are at the forefront. The aim of this study is to contribute to the current studies on the geometry of muqarnas. There are many precedents on this topic, each presenting different digital modeling workflows. This paper is intended to reveal the potentials of different digital modeling workflows. In this paper, three workflows were defined, and explained step-by-step by using computer-aided design and parametric modeling software. Then, these workflows were compared according to three major qualities: analytical, generative, and performative. One of these workflows is based on the generally-accepted approach, and used as the control workflow. The other two testing workflows presented in this paper are the original contributions to the field, which are derived from the control workflow. As indicated above, it is difficult or impossible to make generalizations for all muqarnas examples. Therefore, this study is limited with a specific spatio-temporal context of Anatolian Seljuk muqarnas systems. A case study was conducted to demonstrate the application and comparison of different workflows. In the case study, a typical example of the Anatolian Seljuk muqarnas system was modeled and the analytical, generative, and performative qualities are discussed.

Related works

Most of the studies on the geometric analysis of muqarnas are originated from the works of Ghiyâth al-Dîn Jamshîd Kâshânî (al-Kashi) (1380–1429). He is mostly known for his book titled: Miftah al-Hisab(The Key of Arithmetic) (1427). In this book, al-Kashi made the first known analytical definition of muqarnas. The al-Kashi’s work not only explained the practical methods for designing and constructing muqarnas¹² but also made it possible for mathematicians and architects to establish a common language.¹³ Although al-Kashi did not explain the actual construction methodology of muqarnas, he dealt extensively with the geometric features of its elements. Today al-Kashi is still referenced as a solution to the computational geometry problem of muqarnas. Özdural¹⁴ and Dold-Samplonius⁶ explained the fundamental principles of the muqarnas units, based on al-Kashi. Harb¹⁵ also studied al-Kashi’s calculations. Dold-Samplonius⁶ and Harmsen⁴ used Harb’s methods and al-Kashi’s calculations in their studies to understand the basic design principles of muqarnas systems. According to these principles, muqarnas compositions generally consist of several types of three-dimensional units called as elements⁴ or blocks.¹⁶ These elements are named rhombus, large biped, small biped, almond, and jug. There are certain rules on the combinations of these elements.

Shiro Takahashi¹⁷ collected muqarnas examples from all over the world, classified and shared them in an online database. In this database, muqarnas examples are categorized into three different types: the square lattice muqarnas, the pole table muqarnas, and others that do not belong to these two types. The square style contains rhombus and square-based tilings, pole table style differentiate into polygons and other styles are unique muqarnas design based on a region or a building.

The dissertation of Silvia Harmsen⁴ is about the mathematical underpinnings of muqarnas systems. Harmsen⁴ developed software tools that are used to determine muqarnas designs from their projection views. This software set proved that it is algorithmically possible to precisely translate the two-dimensional muqarnas patterns into the three-dimensional compositions. A computer scientist, Hamekasi¹⁸ developed another software tool, intended to be used to design new muqarnas compositions.^18,19 Senhaji and Benslimane²⁰ also studied the method of three-dimensional modeling of muqarnas by using its two-dimensional projection pattern. In addition to these analytical studies, there are also design-research studies that aim to stretch the rules and the limitations of muqarnas geometry and open new paths for mathematical creativity. For example, Yaghan^21,22 created new versions of classical muqarnas forms by developing new units. Another recent research is the master thesis of architect Imani,²³ who suggested an alternative workflow for the computer-aided reconstruction of muqarnas designs. Imani²³ suggested two alternative approaches of analysis; which are motif-based and layer-based analyses of muqarnas patterns. Also in 2017, Alaçam²⁴ led a team of researchers who studied paper folding techniques to analyze muqarnas tectonics at the conceptual level. Since the usual design of muqarnas depends on mathematics and logic; they tried an alternative form-finding experiment to understand the generative, structural, and performative aspects of muqarnas tectonics. Alaçam and Güzelci have also studied the use of parametric modeling software to reveal the tacit potentials of abstract muqarnas geometry and generate new designs.^25,26

This study is about the early phases of a specific muqarnas tradition found in Anatolia, in the period of Seljuks (11th–14th centuries AD). The foundations of many Islamic arts in Anatolia, which reached their peak points in the Ottoman period (1299–1923), were laid in its predecessor, Anatolian Seljuk period (1075–1308). Tuncer²⁷ argues that the Sultanhanı Caravanserai represents one of the peak points of the unique Anatolian Seljuk palace architecture. He claims that, if the invasion and devastation of İlkhanate Mongols from the east had not taken place, this architecture would have given many other examples as a unique style. With the decrease of the influence of İlkhanate in Anatolia, the power vacuum formed by the Anatolian Seljuks created a suitable environment for the development of the new Ottoman state and started a new phase of evolution in many arts and architecture, including muqarnas. Readers are referred to Ödekan²⁸ for the continuation of Anatolian Seljuk muqarnas style, and its evolution in the early Ottoman period. Ödekan’s² dissertation explains the roots, history, and geometry of the muqarnas examples in Anatolian Seljuk portals. She states that the stone masters who produced the portal muqarnas used some ready-made schemes prepared before the application. The use of the same units in different portals indicates that the whole system must be planned and calculated before the construction begins. Therefore, it is possible to foresee that muqarnas in Anatolian Seljuk portals should have a certain rule set.

Among the researches on this Spatio-temporal context; Ödekan,^1,2,28 Tuncer,²⁷ and Uluengin²⁹ studied many examples of muqarnas made by Anatolian Seljuks and Ottomans. Uluengin²⁹ proposed an alternative classification methodology based on the angle of the half-star at the peak point of the muqarnas. Based on this classification, he defined the similarities among the muqarnas compositions found in different regions of Anatolia. The classification is based on the points on it like 8/2 = 4 point star or 10/2 = 5 star, etc. He also classified the point stars as flat and turned. If the star shape starts with semi pieces of star shape, it is called as turned, if it has the semi star pieces on the design, it is called as flat. Mehmet Fatin Uluengin²⁹ is an influential figure in the tectonics of muqarnas. He had been a master muqarnas craftsman for 70 years. He built many muqarnas compositions and created wooden and stucco models to explain its geometry to his workers. In addition to the actual constructions of muqarnas, he created approximately 2000 drawings, mostly from the muqarnas examples in Turkey. In the book which is published by his son, Bulent Uluengin after his death; Fatin Uluengin used an alternative classification methodology based on the angle of the half-star at the peak point of the muqarnas. Based on this classification, he defined the similarities among the muqarnas compositions found in different regions of Anatolia.

Methodology

This research is based on a case study, aiming to define, interpret, and generalize several digital modeling workflows within the context explained above. In the digital modeling process, Rhinoceros software was used. The generative algorithms were created with the Grasshopper interface for Rhinoceros. Grasshopper is used to create data-flow diagrams; as it enables fast and efficient algorithms to be developed by non-programers.³⁰ In order to test different modeling workflows, first, a control item was defined. This is named as the element-based modeling workflow, and it represents the widely-accepted approach for the analysis and re-construction of muqarnas compositions. This workflow will be explained and used to compare the performances of the proposed workflows. The second and third workflows are the testing workflows, which are the original contributions of this study. They are based on a dialectic between the abstract/geometric/logical eye, and the real/tectonic/structural eye. One of the testing workflows presented in this paper is named tessellation-based modeling workflow, which focuses on the generative performance of muqarnas systems. The other unique approach is named block-based modeling workflow, focusing on the physical performance of muqarnas systems. Below are the brief explanations of these workflows. A more detailed explanation will be presented and discussed along with the case study.

Control workflow: Element-based modeling

The element-based modeling workflow has three main steps expla

Step 1: Calculating the 2D shapes of the muqarnas elements

These elements are the fundamental building blocks of a muqarnas, and they are generally named as jug, square, half square, large biped, small biped, almond, and rhombus. The geometric definitions of these elements are based on the related works found in the literature, briefly explained in the previous section. In the literature, these elements are generally analyzed in two main categories. The cells are regarded as the main elements of a muqarnas, while the intermediate pieces are defined as complementary elements to the cells. In the plan view, jug and large biped are complementary pieces, seen as the two parts of a square. Almond and large biped are positioned as the two parts of a rhombus. In the section view, these elements are geometrically composed of facets and roofs. Dold-Samplonius⁶ explains the method to construct the profile curve of these elements. This method originated from al-Kashi, who is believed to learn it from the masons. [AB] defined as one unit and drawn vertically. Then [BC] has double the length of [AB]. From A point to [BC], [AE] drawn where BAE angle is 30°. [AE] divided into five parts, F point is determined where 2[AF] = 3[FE]. On [BC], G is determined where [FE] = [EG]. The h-centered circle tangent to the point F and point G, where the curved side of the elements are constructed (Figure 2). Since these elements have strict geometric and proportional relationships, the generative potentials of the element-based algorithm are mostly based on the decisions made in this step.

Figure 2.

Construction method of profile curve explained by Al-Kashi.

In the literature of Anatolian Seljuk muqarnas, the elements are generally reduced into two main categories, named as yaprak (leaf) which is a general name of the elements almond, jug, and rhombus; and kaz ayağı (goose-foot), which is the general name of the small and large biped. According to Tayla,⁹ the main elements of muqarnas are the concave units called as yaprak (leaf). In this approach, kaz ayağı is regarded as the additional unit which is used to complete the yaprak.

Step 2: Arranging the elements to construct a 2D layout

In this step, the proportional features of the elements are utilized in combining them together, and ultimately, creating the 2D projection of the muqarnas that fill the desired area without gaps or overlaps. This also represents the 2D projection of the muqarnas and provides valuable information about its overall composition. At this stage, the relationships between the layout (Figure 3(Step 2)) and the elements (Figure 3(Step 1)) becomes important. The modeling process reveals feedback loops between these two steps.

Figure 3.

The overview of the element-based modeling workflow: (Step 1) the 2D shapes of the muqarnas elements, (Step 2) the arrangement of the shapes into a layout, (Step 3) the 3D transformation of the muqarnas via tiering system.

Step 3: Elevating the elements by using tiers

The elements on the same level, sharing the same horizontal base are called a tier. There is a generally accepted numbering system for the muqarnas tiers.⁴ According to this system, the arrows are drawn to indicate the vertical transition of the elements. The network of arrows creates a directed graph of the geometric configuration. Since the muqarnas is a gradually rising design, this directed graph usually finalizes at the apex of the muqarnas (Figure 3(Step 3)). A number is assigned to each arrow when determining the rise of the composition, eventually creating the groups of cells on the same tier. The modeling process using this workflow have feed-back loops between this step and the first two steps.

Testing workflow A: Tessellation-based modeling

The tessellation-based modeling workflow is based on an abstract and mathematical point of view. According to this method, the patterns forming the two-dimensional projection of the muqarnas constitute the starting point of the analysis and generative modeling. The tessellation-based modeling workflow is an original approach we developed in this study. It is based on the idea of creating an abstract geometric system by focusing on the tessellation found in the 2D projection of the muqarnas. The workflow has four main steps briefly explained below. An implementation of these steps will be demonstrated in the case study.

Step 1: Determining the underlying tessellation

The first step of this modeling workflow is to choose a tessellation for the muqarnas to be analyzed/created (Figure 4(Step 1)). Parametric modeling of such tessellations may create variations without breaking the geometric continuity. Therefore, this underlying system reveals generative design potentials. Most of the Anatolian Seljuk muqarnas compositions have underlying tessellations. Therefore, this step is also useful in the analysis of these examples.

Figure 4.

The overview of the tessellation-based modeling workflow: (Step 1) a tessellation was determined as the basis of the construction, (Step 2) the development of the components from the tessellation cells, (Step 3) The placement of the components via a special numbering system, and (Step 4) the transformation of the component references by morphing the component references.

Step 2: Deriving the muqarnas components from the tessellation cells

The components of the muqarnas are based on the bounding volumes, which are created by extruding the cells of the 2D tessellation (Figure 4 (Step 2)). At this stage, a feedback loop with Step 1 is necessary to tessellation perfectly to the referencing bounding volumes. The resulting components of this step are generally different from the classical muqarnas elements, since they are derived from the tessellation cells. A more detailed explanation of this process will be presented in the case study.

Step 3: Elevating the components by using a numbering sequence

Since the components are not necessarily the classical muqarnas elements, the numbering of this system is usually a different sequence compared to the classical tiering system. Considering the ascendent system of the classical muqarnas design, components are placed like a staircase, and each should fill the gaps in between (Figure 4 (Step 3)). Numbering system is determined according to elevation of the components and their placement. It is derived from the classical muqarnas system however it has a difference. The classical muqarnas elements level are addressed to define the components level, the numbers on components show the level directly. Feedback loops to step 1 and 2 are necessary to determine the correct numbering on the 2D tessellation. This numbering sequence will be explained in detail, along with the case study.

Step 4: Morphing a collection of 3D shapes to the referencing components

This step finalizes the modeling of a muqarnas (Figure 4 (Step 4)). It uses morphing transformation from one source volume, to a target volume (the components). This step reveals the generative potentials of the workflow, as the source shape can be easily replaced to create variations.

Testing workflow B: Block-based modeling

The Anatolian Seljuk muqarnas are generally found on portal vaults, and they are not only decorative elements but also structural systems; meaning that the design of the pieces and their combinations are affected by aesthetic and structural considerations. The block-based modeling workflow is an original approach we developed in this study. It is based on the idea of creating a structural and tectonic model, by focusing on the 3D shapes of the stone bricks of a muqarnas construction. The workflow has two main steps briefly explained below. An implementation of these steps will be demonstrated in the case study.

Step 1: Merging the muqarnas elements into blocks

The Anatolian Seljuk muqarnas is generally realized by constructing and arranging the stone blocks of the portal; while carving the muqarnas shapes on the visible sides of the blocks. This indicates a design and planning problem that needs to be addressed in two ways; superimposing a visual system with a structural system. Tuncer³¹ argues that it was obligatory to exchange views between the architect of the building and the master of muqarnas at the planning stage of the work, regarding the dimensions of the portal and the desired visual effect. The first step of the block-based modeling workflow is to determine the stone blocks of the muqarnas by considering the bricklaying pattern used in the construction. In most cases, this step reveals a set of blocks, which are different from the classical muqarnas elements, or the tessellation cells. This step could be regarded as an analysis, considering the modeling of an existing muqarnas. The case study presented in this paper revealed this potential. This geometric approximation is based on an assumption that the stone blocks were cut perfectly, fitting each other without the need for a filling material (such as rubble) in between.

Step 2: Implementing the bricklaying pattern to finalize the model

In the construction site, the muqarnas master had to decide on a sequence and pattern within the given volume. While the construction of the portal rises to the level of muqarnas, the chipping works of the muqarnas blocks should be finished. The rows of the portal walls have to be laid together with the muqarnas so that each row of stones can be placed securely, and the backward connections could be made. In order to work comfortably at the backside, the muqarnas craftsmen work on the obverse, while the regular bricklayers work on the backside.³¹ At this stage the model is finalized by using basic rules on the arrangement of the stones. These rules are; the interweaving the stones by breaking joint, creating a discontinuity on the vertical joints; keeping the corner angles of the stones as much as possible (preventing sharp corners); and the dimensions of the stones to be portable and affordable (avoiding a big difference between the sizes of the blocks). This workflow presents a cyclic process between the two steps; creating and testing the stone blocks with feedback loops (Figure 5).

Figure 5.

The overview of the block-based modeling workflow: (a) The 3D blocks of the muqarnas are determined, and (b) The bricklaying pattern used to construct the 3D model.

The case study

The Sultanhanı Caravanserai was built in 1229 by Anatolian Seljuk sultan Alâeddin Keykubad I., located at 42 km. away from the center of Aksaray Province of Turkey (Figure 6). The caravanserai was made by cutting stone, covering an area of 4500 m², being one of the largest caravanserais of its period. It has a typical plan scheme of its period, with the only original exception of the small masjid located at the center of the complex. This caravanserai is regarded as an original historical document of the classic scheme of Anatolian Seljuk caravanserais. It is the work of architect Mohammed bin Havlan (of Damascus). It has been listed on the UNESCO World Heritage tentative list since 2014. In the caravanserai complex, there are two portals; one at the main entrance and another at the entrance of the inner part (winter section) accessed from the courtyard. These portals are made of marble, decorated with two muqarnas compositions.

Figure 6.

(a) The map of muqarnas instances, showing the location of the Sultanhanı Caravanserai, (b) Plan and photo of the caravanserai showing the location of the inner portal (The photo is used with permission).

The muqarnas which is the subject of this study is located at the inner portal (Figure 7). It is covering a rectangular area of 323 cm long and 161 cm wide, close to the standard proportion of 1/2. The Sultanhanı inner portal muqarnas has nine tiers and reaches the height of 361 cm. The reason for investigating this particular structure is that it is a typical example of the muqarnas geometry found in Central Anatolia. Because of its craftsmanship and material quality, it is a well-preserved example of many similar muqarnas around the same geography and time period. Ödekan² describes this muqarnas as “a perfect combination of structure and decoration,” as the stone laying is neat and systematic, compared to similar examples. Tuncer³¹ quotes the muqarnas as “composed of a rich and mature sequence of units.” Thus, it is thought that the analyses about this muqarnas will shed light on many similar examples. Throughout this study, the Aksaray Sultanhanı Caravanserai Inner Portal muqarnas will be called as Sultanhanı muqarnas, in short.

Figure 7.

(a) Photo of the muqarnas, (b) projection drawing of the muqarnas, and (c) elevation drawing of the portal (The drawing is used with the permission of Konya Regional Directorate of Foundations).³²

The latest restoration of the Sultanhanı Caravanserai had started in 2017 and finished in 2019. Although many restoration works had been done over time, the muqarnas on the inner portal remained original and had never been restored. In the latest restoration project, the muqarnas was only sandblasted and cleaned (Figure 7(a)). In 2007, Harmsen³³ visited the region and produced a research-film, named Seljuk Muqarnas Along the Silk Road. This film narrates and analyses several muqarnases located on the route of the Silk Road, including both of the Sultanhanı portals. Harmsen et al.³³ developed computer re-constructions of similar muqarnas examples in the region including Çifte Medrese (Kayseri, 1206), Alay Han (Aksaray 1180–1200), Ağzıkara Han (Nevşehir 1242–43), Arslanhane Mosque (Ankara, 1290).

Element-based model

Step 1

According to Shiro Takahashi’s¹⁷ categorization, the style of Sultanhanı inner muqarnas is square. According to Uluengin’s²⁹ categorization, this style is called dönük (rotated) around an eight-pointed star. These categorizations help to create the fundamental elements found in muqarnas compositions. The elements of the Sultanhanı muqarnas were generated by the 2D parametric evaluation of the edges of a square (ABCD) (Figure 8). In the Sultanhanı muqarnas, there are six parametric points generated in this way; five of them being on the square edges (P₁ and P₅), and one (P₆) on a diagonal line drawn between P₂ and P₅. These points are then combined to define the edges of the 2D shapes (jug (P₁P₂P₃P₆), large biped (P₁P₅P₃P₅), small biped (P₅P₃P₂P₄), almond (P₁P₂P₄P₅), and rhombus (P₂P₃P₂P₁). The major analytical and generative performances of this step are based on these parametric evaluations. The evaluation parameter for the P₁, P₂, P₃, and P₅ the Sultanhanı case is found as 0.721. The evaluation parameter of P₄ between C and D is 0.704. Finally the evaluation parameter of P₆ between P₂ and P₅ is 0.811. Changing these parameters creates new element shapes keeping the proportional relationships.

Figure 8.

(a) The fundamental unit square, used to create the elements, and (b) The 2D elements generated by the unit square.

Step 2

The overall arrangement of the shapes generated in the first step are applied to a custom grid of squares which includes overlaps. This grid is based on the diagonal half of a 4 × 4 square grid (Table 1-CS1a), mirrored two times (Table 1-CS1b). The parametric evaluation introduced in step 1 to this reference grid creates the 2D projection of the Sultanhanı muqarnas. The result is the set of muqarnas elements; Jug (Table 1-CS2a, CS2b), large biped (Table 1-CS2a, CS2c), small biped (Table 1-CS3a, CS3c, CS4b), almond (Table 1-CS3a, CS3b, CS4c), and rhombus (Table 1-CS5b).

Table 1.

The construction steps (CS) which were used to create the 2D projection of the Sultanhanı muqarnas.

	a	b	c	result
CS1
CS2
CS3
CS4
CS5

Step 3

The numbering algorithm of the tiers presented by Harmsen⁴ was used to define the elevations of the edges (Figure 9). The arrows are added to the curved sides of the elements, from the bottom to the top.The sequence follows a path with the help of the arrows. A number is assigned to each arrow when determining the rise of the composition, eventually creating the groups of cells on the same tier.

Figure 9.

The sequential order of tiers of the Sultanhanı muqarnas.

The finalized element-based model of the Sultanhanı muqarnas is shown in Figure 10. The element-based modeling workflow is found useful in analyzing the existing muqarnas compositions. Table 2 shows various muqarnas compositions with similar elements found in Anatolia.

Figure 10.

The final result of the element-based modeling. (a) the elements of the Sultanhanı muqarnas, and their proportional relationships; (b) the resulting model of the Sultanhanı muqarnas, perspective and projection views.

Table 2.

Examples of 2D muqarnas projections of similar spatial and historical period with the Sultanhanı muqarnas. The red squares represent the fundamental square unit used to generate the elements.

Building and location	Projection of the muqarnas
Arslanhane Mosque, Ankara
Hacı Kılıç Mosque, Kayseri
Malatya Ulu Mosque, Malatya
Şifaiye Madrasah, Sivas
Alay Caravanserai, Aksaray
Aleaddin Mosque, Kırşehir
Akşehir Stoneworks Museum, Konya
Ashabı Kehf Caravanserai, Kahramanmaraş
Emir Ömer Mosque, Malatya
Akçegizlemez Mosque, Konya
Stone Madrasah, Afyon
Three Mausoleums, Erzurum
Sultanhanı Caravanserai, Kayseri

We developed a parametric model by focusing on the elements and tiers explained above. Table 3 shows several variations are generated by this parametric model. Variables V1 and V4 control the size of components placed on the grid system; variables V2 and V3 provide angular changes; V5 controls the curved surfaces placed on the components. Using these variables, it is possible to create new designs in a limited way. It is seen that muqarnas compositions seen in Ottoman and Anatolian Seljuk periods have angular differences. This can be simulated with V2 and V3 variables for angular changes. Also, Uluengin²⁹ emphasizes that it is the designer’s decision to determine the height of the system without changing the muqarnas plan. With the V4 variable, this type of variation is also achieved.

Table 3.

Variations generated by the element-based modeling workflow.

Description	Parameters	Projection view	Perspective view
The original muqarnas	V1 = 0.721V2 = 0.811V3 = 0.704V4 = 15.704V5 = 0.100
Further changing the grid of points early in the model	V1 = 0.950V2 = 0.811V3 = 0.704V4 = 15.704V5 = 0.100
Changing the angle of connection between Jug and Large Biped elements	V1 = 0.721V2 = 0.550V3 = 0.704V4 = 15.704V5 = 0.100
Changing the angle of connection between Almond and Small Biped elements	V1 = 0.721V2 = 0.811V3 = 0.950V4 = 15.704V5 = 0.100
Changing the overall height of the elements	V1 = 0.721V2 = 0.811V3 = 0.704V4 = 6.000V5 = 0.100
Changing the size of the facet part of the elements	V1 = 0.721V2 = 0.811V3 = 0.704V4 = 15.704V5 = 0.900

Note: Changed variable values are indicated in bold.

Tessellation-based model

Step 1

The tessellation-based modeling starts with the analysis of the tessellation underlying the Sultanhanı muqarnas. Figure 11(a) shows the abstract geometric layout of Sultanhanı inner muqarnas. This tessellation is decomposed into two types of clusters as shown in Figure 11(b). Then, these clusters are abstracted to be reduced to basic geometric elements, or prototiles; a square and a rhombus. Finally, Figure 11(d) shows the re-composition of the prototiles. The cluster created at this phase is a non-uniform tessellation with two isosceles triangles combined into an equilateral quadrangular, and a square, which is also an approximation of the rhombic snub square tessellation (snub quadrille). The snub square tessellation is constructed by the snub operation on a square tiling. It is non-vertex-transitive and has a vertex configuration of 3.3.4.3.4. It is a semi-regular tessellation with three triangles and two squares meeting on each vertex. When the regular triangles of the Snub Square are merged into rhombi, this pattern starts to resemble the projection of the Sultanhanı muqarnas.

Figure 11.

(a) The pattern generated by the planar projection of Sultanhanı inner muqarnas, (b) de-composition of the pattern into two types of geometric relationships, (c) abstraction of the pattern of the muqarnas to the cells of an approximation of the snub square tessellation, and (d) re-composition of the components into a non-periodic tessellation.

This mathematical abstraction makes it possible to search for possible variations of the tessellation. For example, David Wells³⁴ calls this particular tessellation as an example of a hinged tessellation. He defines the solid pieces (in this case, squares) hinged at their vertices, and separated by an empty space in between. Their rotation can open out (or closed up) different variations of the tessellation without altering most of its mathematical qualities (Figure 12(a)). The combination of the prototiles generates a non-periodic tessellation. This tessellation is also called Ammann-Beenker tiling.³⁵ It was introduced by Robert Ammann and F.P.M. Beenker independently in the 1970s. The geometric relationships between the two prototiles are well-defined in this tiling. The elements of the muqarnas at Sultanhanı inner portal have congruent proportions. Figure 12 explains this by first demonstrating the prototiles defined by Amman-Beenker³⁵ (Figure 12(b)), and then generating variations of the Sultanhanı inner muqarnas configuration by utilizing David Well’s³⁴ hinged tessellation methodology.

Figure 12.

The hinged tessellation instances generated by using a parametric model.

Step 2

In this step, the elements of muqarnas are joined into three components, called A, B, and C. These components are derived from the cells of the Snub Square tessellation. The Component A is based on a rhombus prototile of the tessellation. The Component B is based on the square prototile of the tessellation. Finally, the Component C is the other rhombus prototile of the tessellation, oriented perpendicular to the Component A. Figure 13 shows the components of the Sultanhanı muqarnas in plan view. The bounding prisms of these components were used as the references for the next step.

Figure 13.

The layout of components used as the bounding volumes.

Step 3

The elevations of the components are determined by a number sequence shown at Figure 14. This numbering algorithm is different from the classical tier system. This sequence indicates the elevation level of the components directly, while the tier system uses arrows to indicate the directed continuity of the edges. This system helps to determine the location of each component while keeping the projection unchanged. The main cluster used in this operation is shown in Figure 14(a). Columns A, B, and C represent the different systems of these numbers. A is the sequence of numbers starting from 0, and then increasing evenly at every second one. The column B starts with 0 and is an integer sequence. Column C starts with 1, and is the sequence of odd numbers. When finished, this cluster is reflected two times to achieve the result shown at Figure 14(b).

Figure 14.

(a) The sequence of level numbers, creating two number series A and B, and (b) the complete numbering of Sultanhanı muqarnas.

Step 4

In order to generate the Sultanhanı muqarnas, the components A, B, and C are used as the bounding prisms while using a morph transformation. To create an abstract model of the Sultanhanı muqarnas, the Component A is morphed by the combination of the small biped, almond, and a complementary piece (Figure 15). The Component B is morphed by the combination of large biped, jug, and a complementary piece (Figure 15). Finally, the Component C is used to create the rhombus element (Figure 15).

Figure 15.

The three-dimensional components of the tessellation-based model.

The generative potentials of the tessellation-based modeling workflow was tested by a parametric model. Starting from a square grid, parametric points, and their symmetries are defined to be the base of components. To place the components, the bounding boxes are created and placed on the grid system. These bounding boxes are converted to the components A, B, and C using a morph transformation. Variable V1 controls the placements of the components A, B, and C. The modification on the V1 generates variations similar to David Well’s³⁴ hinged tessellation methodology. The horizontal size of the components is controlled by variable V2 and the heights are controlled by variable V3. Variable V4 provides a tiering system based on the logic of tessellation. Table 4 presents several design alternatives created by using this model.

Table 4.

Hypothetical variations generated by the parametric model, based on the tessellation-based modeling workflow.

Description	Parameters	Tessellation	Perspective
The original parameters of the Sultanhanı muqarnas	V1 = 0.30V2 = 40.0V3 = 40.0V4 = 4.0
The modification of the main parameter V1, creating the square prototile	V1 = 0.50V2 = 40.0V3 = 40.0V4 = 4.0
Further modification of the main parameter V1	V1 = 0.10V2 = 40.0V3 = 40.0V4 = 4.0
Height of the components (V3) is modified	V1 = 0.30V2 = 50.0V3 = 20.0V4 = 3.0

Note: Changed variable values are indicated in bold.

Block-based model

Since the Sultanhanı muqarnas was not renewed during the restorations, we do not have any information about its internal structure. For this purpose, two different sources will be used in this research. One of them is the set of drawings created in the restoration project. The second source is the Konya Akşehir Stone Madrasah muqarnas, which was reconstructed during the restoration process and has characteristics similar to the Sultanhanı muqarnas. It was built in 1250 by Anatolian Seljuk sultan II. Keykavus to be a science and culture center. During the restoration held in 2016, architects and masons found muqarnas blocks at the courtyard and reconstructed it. Although the backsides of these muqarnas blocks are not cut uniformly, the photos and drawings gave us important information about the tectonics of this muqarnas style.

Step 1

The first step of the block-based modeling of Sultanhanı muqarnas identifies the set of building blocks. Using similar muqarnas examples, survey drawings, photos and texts on the Sultanhanı inner muqarnas, we approximated the marble blocks. They are based on the combinations of the elements presented earlier. However, as Figure 16 shows, the fundamental elements are no longer the driving force of the pieces. Sultanhanı muqarnas blocks were categorized under eight main types, and seven sub-types.

Figure 16.

The blocks of the Sultanhanı muqarnas.

Step 2

The block-based model of the Sultanhanı muqarnas was completed by arranging the blocks as shown in Figure 17. The arrangement rules are based on the discontinuity of the vertical joints (for structural stability and locking the pieces together), keeping the corner angles of the stones as much as possible (preventing sharp corners), and the dimensions of the stones to be portable and affordable (avoiding a big difference between the sizes of different blocks). However, we couldn’t manage to extend this model to the full masonry system of the portal (Figure 18). A more detailed study is required to reveal how these blocks are connected to the structure of the portal.

Figure 17.

(a) The block layout, and (b) perspective view of the block layout.

Figure 18.

(Left) The digital re-construction of the Sultanhanı muqarnas by using the block-based modeling workflow; (Right) The photo of the muqarnas.

Discussion and conclusions

The three modeling experiments explained in the previous section revealed similarities and differences between them. These similarities and differences will be explained and discussed in terms of the de-composition of a whole into pieces, and the two and three dimensional placement rules of these pieces. Table 5 shows a summary of the findings.

Table 5.

The comparison of the models of the Sultanhanı muqarnas.

Models	The de-composition into pieces	The underlying lattice (2D placement rules)	Three-dimensional placement rules
Element-based model	There are five elements	An abstract tier system, using directed graphs.	The direct translation of the abstract tier system
Tessellation-based model	There are three components	The non-periodic, semi-regular tessellation	A new numbering system proposed in this paper
Block-based model	There are eight main blocks and seven sub-blocks	The bricklaying pattern of stone blocks	Basic masonry layout principles are applied

The element-based model of the Sultanhanı muqarnas was based on the standard muqarnas analysis. In this model, the three-dimensional pieces, and their two and three dimensional placement rules were well-defined. The Sultanhanı muqarnas model made with this method was used as a control model to understand the generally accepted approach to its pieces and placement rules. The modeling process resulted in five types of pieces, called elements. The forms of these elements were determined by the de-construction of the whole muqarnas into several pieces. This was a reductive process of minimizing the number of discrete types. The element-based model of the Sultanhanı muqarnas was accurate to create the visible form of the muqarnas surface. In addition, the elements obtained in this model were the members of a more general family of muqarnas pieces, making it possible to categorize the muqarnas and develop a standard language for most of them. However, this standard language was abstract, and not useful in explaining all aspects of the Sultanhanı muqarnas. For example, in the case of Sultanhanı muqarnas, the abstract and deterministic system of elements did not give correct and precise information about the actual physical construction of the object it refers to. This revealed a future research question to be addressed; “Is there another system, or common language of muqarnas, which could define both formal and structural aspects?”

The generative algorithms for element-based and tessellation-based modeling were created with the Grasshopper that enhance our exploration. We develop alternative modeling workflows based on our case study rather than a generalised muqarnas. The two and three-dimensional placement rules of the element-based approach were well-defined. The classical tier system, which utilizes a directed graph was used. Although this graph seemed to be two-dimensional, it also includes the three-dimensional placement rules. This strict determinism of this system was found efficient in re-constructing the Sultanhanı muqarnas, since it did not require any extra information to complete the three-dimensional composition. Essentially, the tier system explained how the elements were arranged in rows. Thus, this system creates a standard upon which it is possible to derive both historical and geographical connections, opening paths for more typological studies. In addition, this system was found integrated directly with the elements mentioned above. Therefore, it completes the loop of a deterministic-abstract-analytical system, leaving a very limited area to variation and customization. A future research question revealed by this result could be; “What kind of alternative placement rules could be defined, so that they do not necessarily use the standard elements, and open more possibilities for alternative designs?”

The test workflows presented in this paper offered some preliminary results which could be further advanced for an alternative understanding of muqarnas. The first test model was the tessellation-based model.

The tessellation-based modeling workflow focused on the two-dimensional geometric pattern, obtained by the projection of the muqarnas. A unique approach has been developed to define this pattern, and transform it into a three-dimensional system. The most critical feature of the pieces introduced in the tessellation-based model was the use of bounding volumes as references. In the Sultanhanı muqarnas, three distinct reference volumes were identified and named as the components. The shapes of these components were determined by the cells of a semi-regular, non-periodic tessellation. This tessellation was used as the base of the two-dimensional projection of the muqarnas. Therefore, the integration of the mathematics underneath the tessellation, and the volumetric components closes the system. This model can be seen as a partial derivation of the control model, since it has a close connection between the pieces, and the placement rules. The major differences were the introduction of the unique tessellation as a basis; and the reference volumes. Both of these features enhanced the generative potentials of the model, making it possible to create diverse alternative designs. The reason for this potential was based on the fact that the 3D morphing transformation applied to the reference volumes made it possible to use nearly any shape as the components. The referential system introduced in the tessellation-based model created an abstract reference system distinct from the geometric content of it, which enabled more diverse formal variations possible. In addition, since the first step of this workflow was a parametric model of a semi-regular tessellation; its mathematical background could be easily implemented to generate new muqarnas compositions. The major drawback of the tessellation-based system was the lack of generalization on the determination of the tessellations for a particular muqarnas. There would be a broader future study on the geometric connections between the muqarnas projections, and their corresponding tessellations. Since this model was based on a mathematical alternative to the element-based model, it also ignores the structural and tectonic features. Therefore, it is possible to state the same future research topics for this model.

Conducting a muqarnas analysis only through abstract geometric systems may ignore the structural system, which is one of the main features of the Anatolian Seljuk muqarnas. In the first two models, element-based and tessellation-based, only the visible surface of the muqarnas and the patterns under this surface were studied. In order to complete these models and fully understand their success, we needed a different perspective, a new model presented as block-based. This model was intended to capture the shape of the actual stone blocks used in the construction. In the Sultanhanı case, the block-based model had eight distinct main blocks and seven sub-blocks. The determination of these blocks were based on the photographs, and drawing of the muqarnas. The Sultanhanı muqarnas blocks revealed differences from the abstract elements of the element-based model, and the components of the tessellation-based model. The placement rules of the block-based model was based on the actual bricklaying patterns used in the construction. The block-based modeling workflow focused on the tectonic qualities of the muqarnas. In this method, instead of mathematical patterns or geometric elements, the structural features of the muqarnas were at the forefront. In particular, the stone masonry system used in the construction of the portal was integrated with the muqarnas blocks. This model was based on estimating the block shapes and their production sequences. The deterministic and survey-based construction process limited the generative potentials of this model. However, this model was found suitable for a base of the future studies on the rule-based (probably grammar-based) approaches. The shapes and the placement rules of the blocks would be further generalized in this way. Table 6 presents the summary of the general results obtained from the above findings of the case study. Figure 19 shows the visual comparison of three models of Sultanhanı muqarnas.

Table 6.

The comparison of the digital ing workflows presented in this paper.

Workflow	Analytical	Generative	Performative
Element-based	This workflow is a generally-accepted analytical solution used in muqarnas studies.	This workflow is partially generative. It can create new designs bu has a limitation created by the strict definition of the element shapes and tiers	This workflow ignores structural, and tectonic features of muqarnas.
Tessellation-based	Tessellations can open a new perspective in the analysis of muqarnas compositions	Since this workflow uses advanced geometric transformations such as morphing, it is possible to create variations from new shapes. It has the best generative performance.	This workflow ignores historical, structural, and tectonic features of muqarnas.
Block-based	The blocks of the muqarnas can open a new analytical perspective in muqarnas studies	This workflow has limited design potential. The arrangements of the blocks based on defined rules for construction.	This workflow has the best performative simulation for a defined muqarnas. It has all the muqarnas structural features.

Figure 19.

A visual comparison of the models of Sultanhanı muqarnas: (a) element-based model, (b) Tessellation-based model, and (c) block-based model.

We believe that advances in digital technologies can be used to revisit the relationships between mathematics and arts with a motivation that this would be another small step in their long-term historical developments.³⁰ Muqarnas is a deep and wide research area, containing many different instances and several different analysis approaches. Our way of understanding muqarnas systems ranges from geometry and pattern qualities to tectonic and constructive features. Although there are many studies on the geometric analysis of muqarnas, it is still an open research field, where there is no single method or solution that is capable of explaining every instance. This methodological diversity leads to a rich research field. In this study, we attempted to contribute to this diversity by explaining the similarities and differences between several modeling approaches in a limited spatio-temporal context. As an alternative to the most common element-based approach, we proposed two new modeling workflows and tested them. These are expected to help to examine the relationship between mathematics and architecture with a new and fresh look.

Footnotes

Author’s note

The research presented in this paper is partially based on Sevde Gülizar Dinçer’s master thesis; completed at Yıldız Technical University Institute for Applied Sciences Computer-aided Architectural Design Program in 2016, under the supervision of Dr. Togan Tong.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Sevde Gülizar Dinçer

Tuğrul Yazar

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