Geometric design of a masonry lattice space dome titled KARBANDI in Persian architecture

Abstract

Studies of historical buildings in Persian architecture have resulted in knowing a kind of masonry structure with a harmonic lattice geometry, which has been titled “Karbandi.” The most important and highly acclaimed feature of Karbandi is the correlation and coordination between its architectural and structural functions, which results in creating esthetic and meaningful spaces. The use of this structure is highly demanded in contemporary Persian architecture, but very little information about historical techniques survives to the present day and accordingly, there are ambiguities about its drawing and geometric design. Therefore, this research aims at discovering geometric relationships and principles of Karbandi to regulate and facilitate its design process in contemporary architecture. Toward this end, its historical samples were analyzed and their geometric features were found. As a consequence, the connection distance of dividing points on the circle is an important parameter for creating various types of Karbandi on a base. For instance, the height of a Karbandi and its elaboration are directly related to the connection distance. In addition, it was clarified that the height of a Karbandi and the size of its Shamseh are in an inverse relationship. Finally, a comprehensive classification was presented based on found geometric features.

Keywords

cross-ribbed vaulting geometric design Karbandi masonry lattice dome Persian architecture

Introduction

International Association for Shell and Spatial Structures (IASS) issued in 1984 that a space frame is a structural system, assembled of linear elements so arranged that the loads are transferred in a three-dimensional manner. In some cases, the constituent elements may be two-dimensional. Macroscopically, a space frame often takes the form of a flat or curved surface.¹ The formal definition for space structures has been given by Nooshin² in which the term “space structure” refers to a structural system that involves three dimensions. According to the above-mentioned definitions, three-dimensional action is the most important feature of space structures. Therefore, during history, other structures with different material could be assessed as a space structure. For instance, some masonry domes such as Karbandi should be considered as a space structure.³ Masonry domes and vaults are some of the main structural features of architectural heritage that must be preserved.⁴ Since these structures are able to cover a large space, they have been used more than any other covering by Iranians.⁵ Before popularity of steel elements in structural engineering, masonry structures were dominant and hence widely used since remote times, including a broad collection of structures.⁶ Besides functional and esthetic aims, this diversity reflects the local building traditions, and the historical evolution of techniques employed.⁷ Karbandi is a salient kind of these structures in Persian architecture, which is one of the most common patterns in the restoration projects and new buildings, including the Mausoleum of Omar Khayyam, Baba Tahir Hamadani, and others.⁸ This pattern is created by intersecting several rib vaults which is based on strict geometric principles. Karbandi is considered as a power of Persian architecture that elegantly integrates architecture and structure and over the centuries, it has been an element of identity in Persian architecture. The most important and highly acclaimed property of Karbandi is the correlation and coordination between its architectural and structural functions,³ which results in creating esthetic and meaningful spaces. These characteristics have attracted the attention of master builders throughout the Persian architecture history and led to the construction of many precious architectural works. Unfortunately, very little information about historical techniques survives to the present day. These techniques were a closely guarded trade secret, passed from master to apprentice and ultimately lost in history. The quest to design Karbandi is therefore an intriguing puzzle. As a guide, we have an enigmatic set of samples from the past. Therefore, we have to analyze their geometry and form to find out how the geometry of their various types can be created and accordingly, we can prescribe a comprehensive geometric design process for contemporary architects. Researches and advanced studies of Karbandi have progressed slowly in comparison with its executive progress, resulting in a lack of knowledge and experience in this field. Furthermore, most of the researches are in Farsi, which are explained as follows.

One of the basic requirements for implementing the Karbandi is to master its drawing principles. Pirnia and Bozorgmehri⁹ have presented a traditional drawing method based on dividing a circle into equal sectors and drawing intersecting equal chords between the dividing points. This resource also divides Karbandi into “plumb” and “out-of-plumb” categories. Despite the scientific significance of this reference, it has been written very briefly and various models of Karbandi have not been studied. Lorzadeh¹⁰ and Sharbaf¹¹ also introduced Karbandi in two plumb and out-of-plumb categories, but they do not refer to different types of Karbandi geometry and only depict some examples for plumb and out-of-plumb models. It should be noted that all of the above-mentioned resources are more graphic and have little explanation. Mohammadian and Faramarzi¹² presented a shape-based typology of Karbandi, but only the geometry of one type of Karbandi is studied (in this research, the geometry of Karbandi(s) has only been studied that their connection distance is equal to “a” $(d = a)$ and are on rectangular bases; refer section “If ‘d’ is equal to ‘a’ $(d = a)$ ”). Raeisi et al.¹³ have also explained a theoretical basis for Karbandi, and their research was aimed at developing a correct knowledge of Karbandi’s appearance. The research has identified three ambiguities in current models: encompass naming, recognizing geometry, and recognizing building function and their corresponding solution separately. Among English references, a resource was written by Amjad Mohammadi et al.⁸ The authors aimed at investigating a solution to broaden Karbandi’s use over irregular geometry and ultimately show that it is possible to calculate and implement a Karbandi geometry in irregular bases when some specific conditions are met. The other English reference has been conducted by Pour Ahmadi.¹⁴ In his paper, there is a new naming for different types of Karbandi. Furthermore, generalities of Karbandi geometry and definitions of its some types were presented. But the geometry of different types of Karbandi have not been analyzed due to its approach.

Based on the authors’ knowledge, none of the above-mentioned resources aimed at geometric analysis of various types of Karbandi and trying to finding out the geometric principles. Consequently, there are many ambiguities among the academic researchers in terms of geometric aspects of Karbandi, which make its design more difficult for contemporary architects so that some of who, instead of designing innovative models, copy the older examples. Understanding these facts and by considering the use of this structure in contemporary architecture, this article aims at discovering geometric relationships and principles of Karbandi types to regulate and facilitate their design process in contemporary architecture. Toward this end, this research analyzes historical samples of Karbandi in Persian architecture to figure out how the geometry and form of their different types can be generated and accordingly provides some input into the design of contemporary.

Geometry of Karbandi

The geometry of a Karbandi is the outcome of a totally strict drawing and mathematical techniques but little information about them survives to the present day. As a guide, we have an enigmatic set of samples within Persian architecture and so, analyzing these samples is necessary to achieve the research purpose. Basic studies and considering the findings of distinguished researchers such as Pirnia and Bozorgmehri,⁹ Nava’I and Haji Qassemi,¹⁵ and Besenval¹⁶ make it clear that the geometric design of all Karbandi(s) is based on two conditions:

The plan of an n-sided Karbandi is based on dividing a circle into n equal sectors and drawing intersecting equal chords between the dividing points; therefore, a circular part, called Shamseh, is usually formed inside a circumscribing circle;

The three-dimensional structure of a Karbandi is formed based on the rotation of a rib arch round the center of the circumscribing circle. Therefore, the ribs of all the arches of a Karbandi share the same dimensions and proportions (Table 1).

Table 1.

The basic geometry of all Karbandi(s): 1. Pa-barik, 2. Toranj, 3. Sousani, and 4. Shamseh (Source: authors).

Dividing the circle into 10 equal sectors for drawing the geometry of a 10-sided Karbandi	Dividing the circle into 12 equal sectors for drawing the geometry of a 12-sided Karbandi

The geometry of rib arches underlies the three-dimensional shape of Karbandi, and the ideal stress state for this shape is pure compression. The geometry of rib arches can be generated by traditional drawing manner, digital methods, and physical models based on the form that follows force principle such as hanging chains.¹⁷ By choosing each of the methods, a different form of Karbandi can be created. In this research, all of the Karbandi(s) considering the form of historical examples are generated by rotating a traditional kind of arch and consequently, all models share the similar curves.

As mentioned earlier, most resources have divided Karbandi into two “plumb” (shagooli in Farsi) and “out-of-plumb” (sar-seft in Farsi) categories,^11,18 but out-of-plumb Karbandi(s) have rarely been used in ancient Persian architecture. These two types have some characteristics that are investigated as follows.

Plumb Karbandi (shagooli in Farsi)

This category includes those types in which both parts of the rib arches that form Karbandi are placed on one plane and along each other (Figure 1). Therefore, the rib arches connect to piers and directly transfer forces.⁸ In addition, the piers of plumb Karbandi are vertical to the ground and present a better load-bearing capability in comparison with the out-of-plumb Karbandi.³ There are examples of this type of Karbandi in the Abbasian House and the men’s baths of the Ameri House in Kashan (Figure 2).

Figure 1.

Plumb Karbandi (Source: authors).

Figure 2.

Examples of plumb Karbandi: (a) Abbasian House in Kashan and (b) men’s bath of the Kashan Ameri House (Source: authors).

It should be noted that some examples have been found in Persian architecture that in their design, one or more rows of the highest components (Toranj) have been removed and led to the creation of another model of plumb Karbandi. This model is one of the most common types of plumb Karbandi that has been neglected. In order to better understand this type, geometric design process of Haj-Mohammad-Qoli Timche’s Karbandi as a successful monument is analyzed. Its design process is such that at first, a circle is drawn and then divided into 16 equal sectors (16-sided Karbandi). In the next step, the dividing points on the circle are connected to each other with a distance of 6 × 6. Thus, 16 intersecting equal chords between the dividing points are formed. Now, as shown in Table 2, 16 identical rib arches are located on each chord and then the legs of the arches are alternately cut. Furthermore, in the resulted model, two rows of the highest components (Toranj) are removed (Table 2).

Table 2.

The first stage of geometric design process of the Haj-Mohammad-Qoli Timche’s Karbandi (Source: authors).

1. A circle is divided into 16 equal sectors and the dividing points are connected to each other with a distance of 6 × 6	2. 16 identical rib arches are located on each chord	3. The legs of the arches are alternately cut

4. Full plumb Karbandi	5. Removing a row of the highest components (Toranj)	6. Removing two rows of the highest components (Toranj): Karbandi of Haj-Mohammad-Qoli Timche

In this state, two halves of rib arches are spaced apart and do not have a direct connection. Thus, it has less load-bearing capacity than a full plumb Karbandi. Therefore, several hidden rib arches are commonly used in order to increase the load-bearing capacity of the structure, which cannot be seen from the interior space (Table 3).

Table 3.

The second stage of geometric design process of the Haj-Mohammad-Qoli Timche’s Karbandi: increasing the load-bearing capacity of the Karbandi using hidden rib arches (Source: authors).

1. Full plumb Karbandi	2. Removing a row of the highest components (Toranj)	3. Removing two rows of the highest components (Toranj): Karbandi of Haj-Mohammad-Qoli Timche

Outside of building: representation of hidden rib arches (photograph by Tehrani)	Inside of building: a plumb Karbandi with removal of two rows of the highest components (Toranj)

Studies of different Karbandi examples in Persian architecture show that some of plumb Karbandi(s) were created individually and the rest were composed of two or more Karbandi(s). Furthermore, a few specific examples were found on unusual bases (the most common bases for Karbandi in Persian architecture are square, rectangle, octagon, octagon and semi-octagon, negini (a type of octagon), and kashkouli (a type of octagon), which are rectangular or made from combination of two or more identical intersecting rectangles).⁸ Therefore, the plumb Karbandi can be examined in three categories, which we titled “Simple,” “Compound,” and “Specific,” respectively, that are investigated as follows.

Simple Karbandi

As mentioned in section “Geometry of Karbandi,” the geometry of Karbandi is come from connecting the dividing points on the circumscribing circle. Therefore, the connection distance of points (e.g. 4 × 4 and 5 × 5) is an important parameter for generating various Karbandi types. We set rectangular bases as a benchmark for studying this parameter, and the other bases are considered as a combination of two or more identical intersecting rectangles (Figure 3).

Figure 3.

If $d < b$ , Shamseh will become too great to be located completely inside the base and intersects its sides (Source: authors).

Now, if the length and width of rectangle faces “a” and “b” sectors of the circumscribing circle, respectively, the number by which the dividing points on the circle will be connected to each other (e.g. 4 × 4 and 5 × 5) is “d” and the total number of sectors on the circumscribing circle is “n”; the following states will mathematically arise that need to be examined

d < b, d = b, b < d < a, d = a, a < d < \frac{n}{2}, d = \frac{n}{2}

If “d” is smaller than “b” $(d < b)$

The realization of this state is not possible on any bases, because the Shamseh becomes too great to be located completely inside the base and intersects its sides and thus the Karbandi is formed out of the base (Figure 3).

For the same reason as in the previous section, establishing the equation $(d = b)$ on the rectangular bases geometrically is impossible (Figure 4). However, it can be met on some non-rectangular bases such as squares and octagons (Table 4). In this state, Shamseh is in its maximum size and so is tangent to the sides of base. The samples of this type have eight piers and are the strongest models of Karbandi, which are usually used under the domes and bear their weight. Two salient examples of this type were used under the dome of Hakim Mosque in Isfahan and Haj-Safar-Ali Mosque in the Historic Bazaar of Tabriz (Table 4).

Figure 4.

On rectangular bases if “d” is equal to “b” $(d = b)$ , Shamseh will become too great to be located completely inside the base and intersects its sides (Source: authors).

Table 4.

Establishment of the equation $d = b$ on the non-rectangular bases (Source: authors).

Hakim Mosque in Isfahan (a 12-sided Karbandi)		Haj-Safar-Ali Mosque in Tabriz (a 28-sided Karbandi)

1. 12-sided Karbandi: dividing a circle into 12 equal sectors	2. Considering the base as a combination of two identical intersecting rectangles	1. 28-sided Karbandi: dividing a circle into 28 equal sectors	2. Considering the base as a combination of two identical intersecting rectangles

3. Connecting the points with a distance of 2 × 2	4. Erasing the lines protracted out of the base	3. Connecting the points with a distance of 6 × 6	4. Erasing the lines protracted out of the base

If “d” is between “b” and “a” $(b < d < a)$

The realization of this condition as the previous state is only possible on some non-rectangular bases such as squares and octagons. In this type, Shamseh becomes smaller than its maximum size and thus some Sousani components are formed around the Karbandi. There are two examples of this type in Imamzade-Hossein in Qazvin and Fin Garden in Kashan (Table 5 and Figure 5).

Table 5.

On the non-rectangular bases when “d” is between “b” and “a” $(b < d < a)$ (Source: authors).

Imamzade-Hossein in Qazvin (a 12-sided Karbandi; photograph by Pirnia and Bozorgmehri⁹)		Fin Garden in Kashan (a 14-sided Karbandi)

1. 12-sided Karbandi: dividing a circle into 12 equal sectors	2. Considering the base as a combination of two identical intersecting rectangles	1. 14-sided Karbandi: dividing a circle into 14 equal sectors	2. Considering the base as a combination of two identical intersecting rectangles

3. Connecting the points with a distance of 3 × 3	4. Erasing the lines protracted out of the base or in the Sousani(s)	3. Connecting the points with a distance of 5 × 5	4. Erasing the lines protracted out of the base or in the Sousani(s)5. Removing a Toranj row

Figure 5.

On the rectangular bases, if “d” is between “b” and “a” $(b < d < a)$ , Shamseh will become too great to be located completely inside the base and intersects its sides (Source: authors).

If “d” is equal to “a” $(d = a)$

This state unlike the previous ones can be met on both rectangular and non-rectangular bases. On the rectangular and square bases, Shamseh is usually in its maximum size and so is tangent to the sides of base. Therefore, on the square bases, Sousani(s) are removed and on the rectangular ones, they are formed just in two opposite sides. Moreover, in the geometry of these models, the following relations are established:

In square bases

d = \frac{n}{4}

In rectangular bases

d > \frac{n}{4}

Examples of this type are used on the peripheral portico of the Kabood Mosque in Tabriz and the Ganjali-khan baths in Kerman (Table 6).

Table 6.

Establishment of the equation $d = a$ on the rectangular and square bases (Source: authors).

Peripheral portico of the Kabood Mosque in Tabriz (a 16-sided Karbandi on the square base)	Ganjali-khan baths in Kerman (a 14-sided Karbandi on the rectangular base)

Absence of Sousani and tangency of Shamseh on all four sides of the base	Forming of Sousani(s) on just two opposite sides of Karbandi and tangency of Shamseh on two other sides

d = 4, n = 16 ⇒ $d > \frac{n}{4}$	d = 4, n = 16 ⇒ $d = \frac{n}{4}$

Another interesting equation of this type is that the four central angles formed as a result of drawing the diameters of the base are a multiple of 360 divided by the number of the sides of the Karbandi (Table 6)

\hat{a} . \hat{b}, \hat{c}, \hat{d} = (\frac{360}{n}) x, x \in N

Furthermore, the equation “ $d = a$ ” can be used in the bases that are formed by combination of two or more identical intersecting rectangles, such as the Karbandi of Sultan-Amir-Ahmad baths in Kashan (Table 7). In this state, the size of Shamseh is smaller than the previous models and thus the size of Sousani(s) became larger. The samples of this type usually have eight piers on the ground and present a good load-bearing capability.³ It should also be noted that there is always the following relation in this type

d > \frac{n}{4}

Table 7.

Establishment of the equation $d = a$ on the non-rectangular base (Source: authors).

Sultan-Amir-Ahmad baths in Kashan (a 14-sided Karbandi on the octagonal base).

1. 14-sided Karbandi: dividing a circle into 14 equal sectors	2. Considering the base as a combination of two identical intersecting rectangles	3. Connecting the points with a distance of 5 × 5 $d = 5, n = 14$ ⇒ $d > \frac{n}{4}$	4. Erasing the lines protracted out of the base or in the Sousani(s)5. Removing a row of Toranj(s)

If “d” is between “a” and “ $\frac{n}{2}$ ” $(a < d < \frac{n}{2})$

This relation can be satisfied in both rectangular and non-rectangular bases. The Shamseh component in this state is in its smallest size and therefore the Sousani components reached to their largest size. It should be noted that Sousani(s) only in this type are created in all four sides of rectangular bases. Although this type of Karbandi is geometrically perfectly logical and correct, it is not so common among historical specimens. An example of this type has been constructed in Imamzade-Hossein in Qazvin (Figure 6).

Figure 6.

If “d” is between “a” and “ $\frac{n}{2}$ ” $(a < d < \frac{n}{2})$ : Imamzade-Hossein in Qazvin (Source: authors).

If “d” is equal to “ $\frac{n}{2}$ ” $(d = \frac{n}{2})$

If “d” is equal to “ $n / 2$ ,” all the drawn chords will pass through the center of the circle and all the rib arches intersect each other at one point. Therefore, some of the components of Karbandi, including Shamseh and Toranj, are removed and resulted in a specific type of vault, which is called “Tarkin” in Farsi. This kind of vault has been extensively used in Persian architecture, such as the Malek-Timche in Historic Bazaar of Tabriz, the Jame Mosque of Isfahan, and the Jame Mosque of Ardestan (Figure 7).

Figure 7.

If “d” is equal to “ $\frac{n}{2}$ ” $(d = \frac{n}{2})$ : (a) Jameh Mosque of Ardestan (photograph by Besenval¹⁶) and (b) Malek-Timche in the Historic Bazaar of Tabriz (Source: authors).

For comparing the types of simple Karbandi and evaluating the influence of connection distance (d) on the Karbandi form, four models were drawn, which are only different in the perimeter of connection distance. By investigating the models, a number of variable form-related properties are observable. One of the most important of these features is the difference in the number of Toranj rows. As illustrated in Table 8, it is clear that the elaboration of Karbandi is directly related to the distance by which the dividing points on the circle are connected (d), so that by increasing the distance, the Karbandi becomes more complicated and contains more details.

Table 8.

The influence of connection distance (d) on the complication of Karbandi (Source: author).

$d = b$	$b < d < a$	$d = a$	$a < d < \frac{n}{2}$

Another variable quality is the height of Karbandi, which is also directly related to the connection distance of dividing points (d). The mentioned relation is due to the fact that when the connection distance is increased, the span of rib arches is also increased and thus the ribs become larger and higher (Table 9).

Table 9.

The influence of connection distance (d) on the height of Karbandi (Source: authors).

$d = b$	$b < d < a$	$d = a$	$a < d < \frac{n}{2}$

Change in the size of Shamseh and Sousani is the other point to be mentioned. As evident in the examples, when the connection distance is increased, the mid-points of the chords are closer to the center of the circle and accordingly, the size of Shamseh and Sousani becomes smaller and larger, respectively (Table 10). By collating this point with the previous one, it becomes clear that the height of a Karbandi and the size of its Shamseh are always mutually dependent. As shown in Table 8, by reducing the dimensions of the Shamseh, the height of Karbandi is increased.

Table 10.

The influence of connection distance (d) on the size of Shamseh and Sousani (Source: authors).

$d = b$	$b < d < a$	$d = a$	$a < d < \frac{n}{2}$

Compound Karbandi

Karbandi is not necessarily designed alone; sometimes two or more simple Karbandi(s) can be combined and led to the creation of more advanced models called “Compound Karbandi.” Analyzing samples of compound Karbandi within Persian architecture and considering the findings of some researchers such as Pirnia and Bozorgmehri⁹ and Mohammadian and Faramarzi¹² make it clear that the compound Karbandi can be studied in five categories, which we titled “duplicated,” “mounted,” “dual-arch,” “adjoined,” and “multiple” Karbandi.

Duplicated Karbandi

Duplicated Karbandi is composed of a full Karbandi and two identical semi-Karbandi(s). This type is usually used on the long rectangular bases, as the full Karbandi is implemented on the center of the base and due to the length of base, two semi-Karbandi(s) are also implemented on its both sides. Muzaffarieh Timche in the Historic Bazaar of Tabriz is an example of this type, which is composed of a 16-sided Karbandi and two 12-sided semi-Karbandi(s) (Figure 8).

Figure 8.

Duplicated Karbandi: Muzaffarieh Timche in the Historic Bazaar of Tabriz (Source: authors).

Mounted Karbandi

If a Karbandi is implemented on the Shamseh of another Karbandi, the resulting product will be mounted Karbandi.¹² An example of this type has been constructed in the Aqa Bozorg Mosque-School in Kashan, where a 16-sided Karbandi was implemented on another 16-sided Karbandi (Figure 9).

Figure 9.

Mounted Karbandi: Aqa Bozorg Mosque-School in Kashan (Source: authors).

Dual-arch Karbandi

As mentioned previously, Karbandi is formed based on the rotation of a rib arch round the center of the circumscribing circle; therefore, all the rib arches share the same dimensions. But some Karbandi(s) have been found that formed by the combination of two Karbandi(s) with different arches, this type is called “dual-arch Karbandi.” Two examples of this kind have been used in Sheikh-Ahmad-Jam tomb in Torbat-Jam and Nobar baths in Tabriz (Table 11). These two examples were formed by the combination of two eight-sided Karbandi(s).

Table 11.

Dual-arch Karbandi (Source: authors).

Sheikh-Ahmad-Jam tomb in Torbat-Jam	Nobar baths in Tabriz

Adjoined Karbandi

Some examples of Karbandi on uncommon bases are shown that in order to adapt Karbandi to the geometry of base, one or more Karbandi(s) have been adjoined another Karbandi. In the Aqa-Bozorg School-Mosque in Kashan, a model of this type has been used that in its design, some parts of a Karbandi have been cut and an incomplete model obtained, which has been leant against a complete Karbandi and combined with it (Figure 10).

Figure 10.

Adjoined Karbandi: Aqa-Bozorg School-Mosque in Kashan (Source: authors).

Multiple Karbandi

Whenever several Karbandi(s) are built inside the components of other Karbandi(s), the resulting product will be of multiple type. In this case, usually before completing the implementation of a Karbandi components, implementing the components of another Karbandi(s) is started and thus two or more incomplete Karbandi(s) are combined (Figure 11). Multiple Karbandi has many variations and to avoid the prolongation of the text, we refrain from examining its various types here and postpone them to another research. The examples of this type were used in Sadaqiani garden’s house in Tabriz and Mahinestan Raheb Hotel in Kashan (Figure 12).

Figure 11.

Multiple Karbandi.¹⁹

Figure 12.

Examples of multiple Karbandi: (a) Sadaqiani garden’s house in Tabriz and (b) Mahinestan Raheb Hotel in Kashan (Source: authors).

Specific Karbandi

Specific Karbandi refers to those Karbandi(s) implemented on unusual and special bases, which are not rectangular and cannot be considered as a combination of two or more identical intersecting rectangles. This type of Karbandi(s) is rarely seen in Persian architecture. Two examples of it have been constructed in Vakil baths in Shiraz and Baba-Roknadin tomb in Isfahan. In these cases, the largest side of the base faces three sectors of the circumscribing circle and the dividing points on the circle were connected to each other with a distance of 3 × 3 (Table 12).

Table 12.

Examples of specific Karbandi (Source: authors).

Vakil baths in Shiraz	Baba-Roknadin tomb in Isfahan

Out-of-plumb Karbandi (sar-seft in Farsi)

If two parts of rib arches creating a Karbandi are not in the same plane, the resulting Karbandi is “out of plumb.”¹⁸ Since two sections of rib arches are not in line with each other, the rib arches cannot act as an efficient load-bearing structure in transferring loads. Therefore, it has less load-bearing capacity than a plumb Karbandi (Figure 13). This type is used where it is required to implement the Shamseh in a form smaller than its usual standard form.

Figure 13.

Out-of-plumb Karbandi (Source: authors).

Drawing the geometry of out-of-plumb Karbandi is not different from plumb Karbandi. First, it is drawn as a plumb Karbandi and then the diameter of Shamseh is diminished to the desired size (Table 13). It should be noted that although all types of plumb Karbandi can be theoretically designed as out of plumb, it is not common among constructed examples. This type of Karbandi has been usually used on rectangular and square bases, two examples of which can be seen in the Historic Bazaar of Tabriz (Figure 14).

Table 13.

Geometry design process of an out-of-plumb Karbandi (Source: authors).

Geometry of a plumb Karbandi	Converting the geometry of a plumb Karbandi into out-of-plumb Karbandi	Geometry of an out-of-plumb Karbandi

Figure 14.

Out-of-plumb Karbandi: Historic Bazaar of Tabriz (Source: authors).

Table 14 eventually illustrates all different types of Karbandi together and provides a holistic comprehension of these structures.

Table 14.

Illustrating all different types of Karbandi (Source: authors).

		$d = b$	$b < d < a$	$d = a$	$a < d < \frac{n}{2}$	$d = \frac{n}{2}$
	Simple
		Duplicated	Mounted	Dual arch	Adjoined	Multiple
	Compound
Plumb Karbandi	Specific
Out-of-plumb	Karbandi

Conclusion

Karbandi, as one of the graceful structures in Persian architecture, has some innovative aspects such as geometric elegance, correlation, and coordination between architectural and structural functions and utilizing masonry material in the lattice dome. The using of masonry material has been a past time requirement and therefore it is not necessary to be observed in contemporary design. This research mainly focused on geometric aspects of Karbandi and figured out the geometric properties and form generation process of various types of Karbandi. Analyzing the examples of Karbandi made it evident that the distance by which the dividing points on the circumscribing circle are connected (d) to each other is an important parameter for creating various types of Karbandi. Based on this parameter, a set of Karbandi samples were drawn and geometrically investigated, then accordingly, a simple and convenient form generation method was presented. The readers need not necessarily be familiar with Karbandi to understand of what is offered in this research and they can benefit from the presented method and use it in their designs. Moreover, given this fact that the first step of structural analysis is the form generation, this research can be a prerequisite for future structural studies of Karbandi. The authors eventually came up with a comprehensive classification of Karbandi based on found geometric features. It should also be noted that in another perspective, the form of Karbandi is a kind of single-layer lamella domes, which can be formulated parametrically and created using Formex algebra. The authors are going to generate these models in the next research.

Footnotes

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 iD

Amir Amjad Mohammadi

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